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User Manual - Real Options Valuation, Inc.

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1. as a separate table and chart and one or two Uncertainty or Terminal nodes as input variables to the Probability 20 00 HUM 5 00 50 00 SESE 500 scenario table Payoff 44 50 FROM 35 00 To 50 00 STEP SIZE 2 50 Step 1 Soe one palha ia Uncertainty Nodes and Terminal Nodes Chart by Row Chart by Column Critical 131 44 v Node and ID Probabil Step 2 3750 4000 4250 45 00 4750 50 00 move r ES 5 00 cum 12393 12402 12411 12420 12428 12437 EE pU qn Completion TimeA 3000 10 00 12550 12567 12585 12602 12620 12637 12655 probability on its own or all identical Completion TimeB 70 00 150 12714 12740 12767 12793 12819 12845 12872 probability events at once Critical 1 2 20 00 12879 12914 12949 12984 13019 13054 13089 o CompletionTimeC 30 00 25 00 13044 13088 13131 13175 13219 13263 13306 9 Analyze inputs in groups licensee Time DJ 7000 30 00 13208 13261 13313 13366 13418 13471 13523 Analyze individual inputs antl 35 00 13373 13435 13496 13557 13618 13680 13741 If performing grouped analysis review the group E 40 00 13538 13608 13678 13748 13818 13888 139 58 members select any additional group members or 12 Days 45 00 13703 13782 13860 13939 14018 14097 14175 deselect any event nas 50
2. ie SK 2 a M LU DOL Joyas MTE d D 1 UC 9 X eb uud ri anite j fih AN gt X F 2 E o X i sagre Ja e gt I RISK MION D Manual i Johnathan Mun Ph D MBA MS BS CRM FRM CEC MIFC Real Options Valuation Ine REAL OPTIONS VALUATION INC sk Simulator A Risk This manual and the software described in it are furnished under license and may only be used or copied in accordance with the terms of the end user license agreement Information in this document is provided for informational purposes only is subject to change without notice and does not represent a commitment as to merchantability or fitness for a particular purpose by Real Options Valuation Inc No part of this manual may be reproduced or transmitted in any form by any means electronic or mechanical including photocopying and recording for any purpose without the express written permission of Real Options Valuation Inc Materials based on copyrighted publications by Dr Johnathan Mun Ph D MBA MS BS CRM CFC FRM MIFC Founder and CEO Real Options Valuation Inc and creator of the software Written designed and published in the United States of America Microsoft is a registered trademark of Microsoft Corporation in the U S and other countri
3. 114 Page RISK SIMULATOR 25 Percentile 75 Percentile Percentage Error Precision at 95 Confidence Figure 4 11 Simulated Results from the Stochastic Optimization Approach 115 Page Theory RISK SIMULATOR 5 RISK SIMULATOR ANALYTICAL TOOLS the applicability of each tool and through example applications complete with step by step illustrations These tools are very valuable to analysts working in the realm of tisk analysis T his chapter covers Risk Simulator s analytical tools providing detailed discussions of 5 1 Tornado and Sensitivity Tools in Simulation Tornado analysis is a powerful simulation tool that captures the static impacts of each variable on the outcome of the model That is the tool automatically perturbs each variable in the model a preset amount captures the fluctuation on the model s forecast or final result and lists the resulting perturbations ranked from the most significant to the least Figures 5 1 through 5 6 illustrate the application of a tornado analysis For instance Figure 5 1 is a sample discounted cash flow model where the input assumptions in the model are shown The question is what are the critical success drivers that affect the model s output the most That is what really drives the net present value of 96 63 or which input variable impacts this value the most The tornado chart tool can be accessed through Simulator Tools Tornado Analysis To fo
4. RISK SIMULATOR 5 13 RISK SIMULATOR 2011 2012 NEW TOOLS 5 14 Random Number Generation Monte Carlo vs Latin Hypercube and Correlation Copulas Starting with version 2011 2012 there are 6 Random Number Generators 3 Correlation Copulas and 2 Simulation Sampling Methods to choose from Figure 5 41 These preferences are set through the Simulator Options location The Random Number Generator RNG is at the heart of any simulation software Based on the random number generated different mathematical distributions can be constructed The default method is the ROV Risk Simulator proprietary methodology which provides the best and most robust random numbers As noted there are 6 supported random number generators and in general the ROV Risk Simulator default method and the Advanced Subtractive Random Shuffle method are the two approaches recommended for use Do not apply the other methods unless your model or analytics specifically calls for their use and even then we recommended testing the results against these two recommended approaches The further down the list of RNGs the simpler the algorithm and the faster it runs in comparison with the more robust results from RNGs further up the list In the Correlations section three methods are supported the Normal Copula T Copula and Quasi Normal Copula These methods rely on mathematical integration techniques and when in doubt the normal copula provides the safest
5. Figure 5 64 ROV Decision Tree EVPI MINIMAX Risk Profile 180 Page RISK SIMULATOR ROV Visual Modeler 2012 Decision Trees CAUsers File Edit Insert Properties Style Shapesand Colors Language Help gt 2 2 4 CFETTIERIEXUE Decision Tree Summary of Values Simulation Modeling Bayesian Analysis EVPI Minimax Risk Profile Sensitivity Analysis Scenario Tables Utiity Function Sensitivity analysis on the input probabilities is performed to determine its impact on the values of The sensitivity charts show the values of the decision paths under varying probability levels The decision paths First select one Decision Node to analyze below then select one probability event to numerical values are shown in the results table The location of crossover lines if any represent at test from the list If there are multiple uncertainty events with identical probabilities they can be what probabilistic events a certain decision path becomes dominant over another analyzed either independently or concurrently Step i Select one or more Decision paths to Uncertainty Nodes and Terminal Nodes Step 4 Enter the input sensitivity range Node and ID Probability Uncertainty Probabilities FROM 0 00 100 00 srEPSIZE 5 00 nature or ONE Terminal node s payoff Critical 1 1 Terminal Payoffs FROM sve SIZE m Completion Time
6. Type Two Tai 00526 02842 Certainty 90 0024 Figure 5 17 Bootstrap Simulation Results Notes The term bootstrap comes from the saying to pull oneself up by one s own bootstraps and is applicable because this method uses the distribution of statistics themselves to analyze the statistics accuracy Nonparametric simulation is simply randomly picking golf balls from a large basket with replacement where each golf ball is based on a historical data point Suppose there are 365 golf balls in the basket representing 365 historical data points Imagine that the value of each golf ball picked at random is written on a large whiteboard The results of the 365 balls picked with replacement are written in the first column of the board with 365 rows of numbers Relevant statistics e g mean median standard deviation etc are calculated on these 365 rows The process is then repeated say five thousand times The whiteboard will now be filled with 365 rows and 5 000 columns Hence 5 000 sets of statistics 1 there will be 5 000 means 5 000 medians 5 000 standard deviations etc are tabulated and their distributions shown The relevant statistics of the statistics are then tabulated where from these results one can ascertain 132 Page Theory Procedure RISK SIMULATOR how confident the simulated statistics are In other words in a simple 10 000 trial simulation say the re
7. click on the e mail Hardware ID lnk to admin realoptionsvaluation com Once we have obtained this ID a newly generated permanent license will be e mailed to you Once you obtain this license file simply save it to your hard drive if it is a zipped file first unzip its contents and save them to your hard drive Start Excel click on Risk Simulator License ot click on the License icon and click on Install License and point to this new license file Restart Excel and you are done The entire process will take less than a minute and you will be fully licensed e Once installation is complete start Microsoft Excel and if the installation was successful you should see an additional Risk Simulator item on the menu bar in Excel XP 2003 or under the new icon group in Excel 2007 2010 and a new icon bar on Excel as seen in Figure 1 1 In addition a splash screen will appear as seen in Figure 1 2 indicating that the software is functioning and loaded into Excel Figure 1 3 also shows the Risk Simulator toolbar If these items exist in Excel you are now ready to start using the software The remainder of this user manual provides step by step instructions for using the software a gt licrosoft Excel non commercial use Home Insert Page Layout Formulas Data Review View Developer Risk Simulator ZR Set Objective 2 y Rcx 2 ws COA Bove e 4 Li B D Set
8. lt analysis gt Nonparametric Wilcoxon Signed Rank Two model name Absolute Values id 114 parameter VAR77 gt Parametric One Variable T Mean model name ANOVA Randomized Block notes id 60 parameter VAR6O VAR61 VAR62 VAR63 gt Parametric One Variable Z Mean model name ANOVA Single Factor Multiple Treatments notes id 61 parameter Parametric One Variable Z Proportion WARS Parametric Two Variable F Variances model name ANOVA Two Way notes id 62 parameter tric Two Variable D dent Me VAR4O VAR41 VAR42 VAR43 VAR44 VAR45 VAR46 VAR47 VAR48 VAR49 VAR50 VAR51 1103 STEP 4 Save Optional You can save multiple analyses and notes in the profile 111 model name ARIMA 1 0 1 notes id 17 parameter VARL for future retrieval 1121 1130 Name Auto Econometrics Detailed m Notes This is a test model running AE methodology inside ROV BizStats 115 model name ARIMA 1 0 2 notes id 17 parameter VAR1 1161 p 1170 218 2 gt EDIT Parametric 2 Var T Test for Independent Unequal Variances 119 model name Auto ARIMA notes id 18 parameter VARL gt Parametric 2 Var Z Test for Independent Means 120 model name Auto Econometrics Detailed notes id 1 parameter VAR5 DE Parametric 2 Var Z Test for Indep
9. The more they drink the lower the grades no show on exams Grades and Study The more they study the higher the grades Beer and Study The more they drink the less they study drunk and partying However if you input a negative correlation between Grades and Study and assuming that the correlation coefficients have high magnitudes the correlation matrix will be nonpositive definite It would defy logic correlation requirements and matrix mathematics However smaller coefficients can sometimes still work even with the bad logic When a nonpositive or bad correlation mattix is entered Risk Simulator will automatically inform you and offers to adjust these correlations to something that is semipositive definite while still maintaining the overall structure of the correlation relationship the same signs as well as the same relative strengths 2 3 3 The Effects of Correlations in Monte Carlo Simulation Although the computations required to correlate variables in a simulation are complex the resulting effects are fairly clear Figure 2 14 shows a simple correlation model Correlation Effects Model in the example folder The calculation for revenue is simply price multiplied by quantity The same model is replicated for no correlations positive correlation 0 8 and negative correlation 0 8 between price and quantity 27 Page RISK SIMULATOR Correlation Model Without Positive Negative Correlation Correlation Correl
10. Distributional Designer allows you to create custom distributions Distributional Fitting Multiple runs multiple variables simultaneously accounts for correlations and correlation significance Distributional Fitting Single Kolmogotov Smirnov and chi square tests on continuous distributions complete with reports and distributional assumptions Hypothesis Testing tests if two forecasts are statistically similar or different Nonparametric Bootstrap simulation of the statistics to obtain the precision and accuracy of the results Overlay Charts fully customizable overlay charts of assumptions and forecasts together CDF PDF 2D 3D chatt types Principal Component Analysis tests the best predictor variables and ways to reduce the data array 8 Page RISK SIMULATOR 75 76 TT 78 19x 80 Scenatio Analysis hundreds and thousands of static two dimensional scenatios Seasonality Test tests for various seasonality lags Segmentation Clustering eroups data into statistical clusters for segmenting your data Sensitivity Analysis dynamic sensitivity simultaneous analysis Structural Break Test tests if your time series data has statistical structural breaks Tornado Analysis static perturbation of sensitivities spider and tornado analysis and scenario tables 1 4 6 Statistics and BizStats Module 81 82 83 84 Percentile Distributional Fittine using percentiles and optimiza
11. eate etse 86 EUIS Cume Forta MNT MET CE E 89 311 GARCH Volatility UD pp 90 BET ZVI URGE M enous 93 3 13 Limited Dependent Variables Logit Probit Tobit Maximum Likelihood Estimation 94 3 14 Spline Cubic Spline Interpolation and Extrapolation 97 42 OP TIMEZATICN enia ee eee dert 99 Raus QE Ran 99 4 2 Optimization with Continuous Decision V ariables etes 101 4 3 Optimization with Discrete Integer V area les iacula aeta mentia cen tton ende 106 4 4 Efficient Frontier and Advanced Optimization Settings 109 sa Se e t NP 111 5 RISK SIMULATOR ANALYTICAL 116 5 1 Tornado and Sensitivity Tools DAR RON Std dodo dnce neigt iva ute 116 SOLO ZANAN E actibus pu e ut d tdt 123 5 3 Distributional Fitting Single Variable and Multiple Variables 127 5 4 Bootstrap Simulation acte USD ai 131 3 3 esce sete ert a Na Pe Na ac a Na a erede 133 5 6 Data Extraction and Saving Simulation bie 135 RD Sun 136 5 8 Regression and Forecasting Diagnostic TOOL uias cde con nu 137 A UTI Analysis tete re ve ener 144 5 10 Exin butanal Aan era
12. Costs 15 75 16 07 V Depreci DCF Mode C24 10 10 00 10002 23 Operating Income EBITDA 873 74 837 82 8 V Amatza C25 3 10 00 10 00 24 Depreciation 10 00 10 00 V Interest DCF Mode 2 10 00 10 00 25 Amortization 3 00 3 00 E V DCF Mode 50 10 00 10 00 26 EBIT 860 74 824 82 Al v PmdB DCF Mode 35 10007 10 00 27 Interest Payments Prod DCF Mode 20 10 00 10 00 28 EBT 858 74 822 82 V DCF Made 15 15 10 00 10 00 29 Taxes 343 50 329 13 Ge 30 Net Income 515 24 493 69 p 31 Depreciation 13 00 13 00 1 Show All Variables E Use Cell Address 32 Change in Net Working Capital 0 00 0 00 B Show 12 vaiabies IF Ignore sl posable rieger values 33 Capital Expenditures 0 00 0 00 pe Es 34 Free Cash Flow 528 24 506 69 4 Ignore zero or empty values Highlight possible integer values eno 35 2 Use Global Setting 36 Investments 180000 ss 37 Analyze This Worksheet Only Analyze All Worksheets 38 39 Financial Analysis 40 Present Value of Free Cash Flow 528 24 440 60 367 26 305 91 254 62 41 Present Value of Investment Outlay 1 800 00 0 00 0 00 0 00 0 00 42 Net Cash Flows 1 271 75 505 69 485 70 455 25 445 33 Figure 5 2 Running Tornado Analysis Figure 5 3 shows the resulting tornado analysis teport which indicates that capital investment has the largest impact on net present value followed by tax
13. Optimized Variable 250 0000 Optimized Objective 450 0000 Figure 5 59 Single Variable Optimizer 172 Page Notes RISK SIMULATOR 5 26 Genetic Algorithm Optimization Genetic Algorithms belong to the larger class of evolutionary algorithms that generate solutions to optimization problems using techniques inspired by natural evolution such as inheritance mutation selection and crossover Genetic Algorithm is a search heuristic that mimics the process of natural evolution and is routinely used to generate useful solutions to optimization and search problems The genetic algorithm is available in Simulator Tools Genetic Algorithm Figure 5 60 Care should be taken in calibrating the model s inputs as the results will be fairly sensitive to the inputs the default inputs are provided as a general guide to the most common input levels and it is recommended that the Gradient Search Test option be chosen for a more robust set of results you can deselect this option to get started and then select this choice rerun the analysis and compare the results In many problems genetic algorithms may have a tendency to converge towards local optima ot even arbitrary points rather than the global optimum of the problem This means that it does not know how to sacrifice short term fitness to gain longer term fitness For specific optimization problems and problem instances other optimization algorithms may find better solu
14. 0 50 the probability of success of getting Heads Selecting the PDF 147 Page RISK SIMULATOR and setting the range of values x as from 0 to 2 with a step size of 1 this means we are requesting the values 0 1 2 for x the resulting probabilities as well as the theoretical four moments of the distribution are provided in tabular and in graphical formats As the outcomes of the coin toss ate Heads Heads Tails Tails Heads Tails and Tails Heads the probability of getting exactly no Heads is 25 one Head is 50 and two Heads is 25 Similarly we can obtain the exact probabilities of tossing the coin say 20 times as seen in Figure 5 35 Distribution Analysis This tool generates the probability density function PDF cumulative distribution function CDF and the Inverse CDF ICDF of all the distributions in Risk Simulator including theoretical moments and probability chart Distribution Binomial Trials 2 Probability 05 Figure 5 34 Distributional Analysis Tool Binomial Distribution with 2 Trials Figure 5 36 shows the same binomial distribution for 20 trials but now the CDF is computed The CDF is simply the sum of the PDF values up to the point x For instance in Figure 5 35 we see that the probabilities of 0 1 and 2 are 0 000001 0 000019 and 0 000181 whose sum is 0 000201 which is the value of the CDF at x 2 in Figure 5 36 Whereas the PDF computes the probabilities of getting exactly 2 heads
15. 122 Page Theory RISK SIMULATOR off manually or you can use the Ignore Possible Integer Values function to turn all of them off simultaneously Stock Price m Strike Price Dividend Yield 00 60 00 40 00 20 00 000 20 00 40 00 60 00 Figure 5 7 Nonlinear Spider Chart 5 2 Sensitivity Analysis While tornado analysis tornado charts and spider charts applies static perturbations before a simulation run sensitivity analysis applies dynamic perturbations created after the simulation run Tornado and spider charts are the results of static perturbations meaning that each precedent or assumption variable is perturbed a preset amount one at a time and the fluctuations in the results are tabulated In contrast sensitivity charts are the results of dynamic perturbations in the sense that multiple assumptions are perturbed simultaneously and their interactions in the model and correlations among variables are captured in the fluctuations of the results Tornado charts therefore identify which variables drive the results the most and hence suitable for simulation whereas sensitivity charts identify the impact to the results when multiple interacting variables are simulated together in the model This effect is clearly illustrated in Figure 5 8 Notice that the ranking of critical success drivers similar to the tornado chart in the previous examples However if correlations are added between the assumpt
16. Edit Profil Number of trials This is where the number of simulation trials required is entered That is running 1 000 trials means that 1 000 different iterations of outcomes based on the input assumptions will be generated You can change this number as desired but the input has to be positive integers The default number of runs is 1 000 trials You can use precision and error control later in this chapter to automatically help determine how many simulation trials to run see the section on precision and error control for details e Pause simulation on error If checked the simulation stops every time an error is encountered in the Excel model That is if your model encounters a computation error e g some input values generated in a simulation trial may yield a divide by error in one of your spreadsheet cells the simulation stops This function is important to help audit your model to make sure there are no computational errors in your Excel model However if you are sure the model works then there is no need for this preference to be checked e Turn on correlations If checked correlations between paired input assumptions will be computed Otherwise correlations will all be set to zero and a simulation is run assuming no ctoss correlations between input assumptions As an example applying correlations will yield more accurate results if indeed correlations exist and will tend to yield a lower forecast confidence
17. RISK SIMULATOR TIPS Profiles Multiple Profiles create and switch among multiple profiles in a single model This allows you to run scenatios on simulation by being able to change input parameters or distribution types in your model to see the effects on the results Profile Required Assumptions Forecasts or Decision Variables cannot be created if there is no active profile However once you have a profile you no longer have to keep creating new profiles each time In fact if you wish to run a simulation model by adding additional assumptions or forecasts you should keep the same profile Active Profile the last profile used when you save Excel will be automatically opened the next time the Excel file is opened Multiple Excel Files when switching between several opened Excel models the active profile will be from the current and active Excel model Cross Workbook Profiles be careful when you have multiple Excel files open because if only one of the Excel files has an active profile and you accidentally switch to another Excel file and set assumptions and forecasts on this file the assumptions and forecast will not run and will be invalid Deleting Profiles you can clone existing profiles and delete existing profiles but note that at least one profile must exist in the Excel file if you delete profiles Profile Location the profiles you create containing the assumptions forecasts decision variables objectives
18. ROV Decision Tree module is included in the latest version and 1s used to create and value decision tree models Additional advanced methodologies and analytics are also included o Decision Tree Models o Monte Carlo risk simulation Sensitivity Analysis O Scenario Analysis o Bayesian Joint and Posterior Probability Updating o Expected Value of Information o MINIMAX MAXIMIN o Risk Profiles Books analytical theory application and case studies are supported by 10 books Commented Cells turn cell comments on or off and decide if you wish to show cell comments on all input assumptions output forecasts and decision variables Detailed Example Models 24 example models in Risk Simulator and over 300 models in Modeling Toolkit Detailed Reports all analyses come with detailed reports Detailed User Manual step by step user manual Flexible Licensing certain functionalities can be turned on or off to allow you to customize your risk analysis experience For instance if you are only interested in the forecasting tools in Risk Simulator you may be able to obtain a special license that activates only the forecasting tools and leaves the other modules deactivated thereby saving some costs on the software Flexible Requirements works in Window 7 Vista and XP integrates with Excel 2010 2007 2003 and works in MAC operating systems running virtual machines Fully customizable colors and charts tilt 3D color chart
19. Seasonality Periods Cycle Quarters 4 Bea 05 NumberofForecastPeriods 4 Gamma 5 Periodicity v Maximum Runtime sec 20 Automatically Generate Assumption Allow Polar Parameters dx ROMA AM c Figure 3 4 Time Series Analysis This time series analysis module contains the eight time series models seen in Figure 3 3 You can choose the specific model to run based on the trend and seasonality criteria or choose the Auto Model Selection which will automatically iterate through all eight methods optimize the parameters and find the best fitting model for your data Alternatively if you choose one of the eight models you can also unselect the checkboxes and enter your own alpha beta and gamma parameters Refer to Dr Johnathan Mun s Modeling Risk Applying Monte Carlo Simulation Real Options Analysis Forecasting and Optimization Second Edition Wiley Finance 2010 for more details on the technical specifications of these parameters In addition you would need to enter the relevant seasonality periods if you choose the automatic model selection or any of the seasonal models The seasonality input has to be a positive integer e g if the data is quarterly enter 4 as the number of seasons or cycles a year or enter 12 if monthly data Next enter the number of periods to forecast This value also has to be a positive integer T
20. 143 164 stochastic 1 8 9 12 67 76 98 99 102 105 107 108 110 111 112 113 114 136 140 143 172 183 185 stochastic optimization 8 12 98 99 102 105 110 111 112 113 114 172 stock price 31 56 66 76 89 140 183 symmetric 137 t distribution 60 91 92 third moment 31 32 34 time series 1 7 9 12 65 66 67 68 69 70 76 78 80 81 85 87 96 138 140 154 156 158 168 169 183 184 time series data 9 66 67 68 78 81 85 87 96 138 140 154 156 158 168 184 title 13 14 toolbar 3 6 15 18 19 tornado 1 9 12 115 116 117 118 119 120 122 124 125 188 Tornado 9 115 116 117 118 119 120 121 122 123 125 188 189 trends 7 78 140 trials 7 11 14 18 19 29 57 38 39 40 41 42 43 50 98 99 111 113 130 147 161 186 triangular 6 11 36 58 61 Triangular 15 58 61 t statistic 94 141 types of 26 37 100 110 129 140 158 160 uniform 6 11 33 36 39 61 62 101 110 126 186 Uniform 17 39 61 upper 101 110 validity of 78 138 19 RISK SIMULATOR value 5 8 14 17 19 23 24 26 28 29 31 33 34 36 37 38 39 45 46 47 48 50 51 52 53 54 55 57 58 59 60 61 62 63 66 67 68 70 78 80 88 90 94 101 105 108 110 113 115 117 118 121 124 126 130 131 137 138 139 140 141 142 146 147 148 149 154 165 168 170 171 173 175 176 177 182 185 186 189
21. 15 93 10 00 40 00 0 6660 Asset 4 10 52 12 40 10 00 40 00 0 8480 Portfolio Total 10 7356 7 17 100 00 Return to Risk Ratio 1 4970 Figure 4 9 Asset Allocation Model Ready for Stochastic Optimization To run this model simply click on Simulator Optimization Run Optimization Alternatively and for practice you can set up the model using the following steps illustrated in Figure 4 10 1 Start a new profile Simulator New Profile For stochastic optimization set distributional assumptions on the risk and returns for each asset class That is select cell set an assumption Risk Simulator Set Input Assumption and designate your own assumption as required Repeat for cells C7 to D9 Select cell E6 and define the decision variable Simulator Optimization Set Decision ot click on the Set Decision D icon and make it a Continuous Variable Then link the decision variable s name and minimum maximum required to the relevant cells B6 F6 Then use Risk Simulator s copy on cell E6 select cells 27 to EY and use Risk Simulator s paste Risk Simulator Copy Parameter and Risk Simulator Paste Parameter or use the copy and paste icons Remember not to use Excel s regular copy and paste functions Next set up the optimization s constraints by selecting Simulator Optimization Constraints selecting ADD and selecting the cell 277 and making it equal 700 total a
22. 2 Mean Max Min 8 4 Standard Deviation SkRewness is always equal to 0 4 Excess Kurtosis ae a 5 x 6 Minimum and maximum are the distributional parameters Input requirements Maximum gt minimum either input parameter can be positive negative or zero The double log distribution looks like the Cauchy distribution where the central tendency is peaked and carries the maximum value probability density but declines faster the further it gets away from the center creating a symmetrical distribution with an extreme peak in between the minimum and maximum values Minimum and maximum are the distributional parameters The mathematical constructs for the Double Log distribution are shown below m for min lt x lt max f x 4 2b b 0 otherwise min max max min where a and b 2 2 z Es 4 forming x lt a 2 2b b d I for a lt lt 2 2b b 48 Page Erlang Distribution RISK SIMULATOR Mean 2 2 Standard Deviation CD Skewness is always equal to 0 Excess Kurtosis is a complex function and not easily represented Minimum and maximum are the distributional parameters Input requirements Maximum gt minimum either input parameter can be positive negative or zero The Erlang distribution is the same as the Gamma distribution with the requirement that the Alpha or shape parameter must be a positive integer An example applicatio
23. 2 4GB recommended e Administrative rights to install software Most new computers come with Microsoft NET Framework 2 0 3 0 already installed However if an error message pertaining to requiring Framework occurs during the installation of Risk Simulator exit the installation Then install the relevant NET Framework software included in the CD choose your own language Complete the NET installation restart the computer and then reinstall the Risk Simulator software There is a default 10 day trial license file that comes with the software To obtain a full corporate license please contact Real Options Valuation Inc at admin realoptionsvaluation com or call 1 925 271 4438 or visit our website at ww realobtionsvaluation cor Please visit this website and click on DOWNLOAD to obtain the latest software release or click on the FAQ link to obtain any updated information on licensing ot installation issues and fixes 13 Licensing If you have installed the software and have purchased a full license to use the software you will need to e mail us your Hardware ID so that we can generate a license file for you Follow the instructions below 2 Page RISK SIMULATOR e Start Excel XP 2003 2007 2010 click on the License icon or Risk Simulator License and copy down and e mail your 11 to 20 digit and alphanumeric HARDWARE ID that starts with the prefix RS you can also select the Hardware ID and do a right click
24. 5 185 06 5 411 39 5 637 71 5 864 03 6 090 35 6 316 67 6 54299 6 769 31 6 995 63 7 221 96 7 448 28 7 674 60 7 900 92 8 127 24 8 353 56 32 00 3 749 44 3 972 48 4 195 52 4 418 56 4 641 61 4 864 65 5 087 69 5 310 73 5 533 77 5 756 81 5 979 85 6 202 89 6 425 94 6 648 98 6 872 02 7 095 06 7 318 10 7 541 14 7 764 18 7 987 22 8 210 26 33 00 3 671 75 3 891 51 4 111 27 4 331 03 4 550 79 4 770 55 4 990 31 5 210 07 5 429 83 5 649 60 5 869 36 6 089 12 6 308 88 6 528 64 6 748 40 6 968 16 7 187 92 7 407 68 7 627 45 7 847 21 8 066 97 34 00 3 594 05 3 810 53 4 027 01 4243 49 4 459 97 4 676 45 4 892 94 5 109 42 5 325 90 5 542 38 5 758 86 5 975 34 6 191 82 6 408 30 6 624 79 6 841 27 7 057 75 7 274 23 7 490 71 7 707 19 7 923 67 35 00 3 516 35 3 729 55 3 942 76 4 155 96 4 369 16 4 582 36 4 795 56 5 008 76 5 221 96 5 435 16 5 648 36 5 861 57 6 074 77 6 287 97 6 501 17 6 714 37 6 927 57 7 140 77 7 353 97 7 567 17 7 780 38 36 00 3 438 66 3 648 58 3 858 50 4 068 42 4 278 34 4 488 26 4 698 18 4 908 10 5 118 03 5 327 95 5 537 87 5 747 79 5 957 71 6 167 63 6 377 55 6 587 47 6 797 39 7 007 32 7 217 24 7 427 16 7 637 08 37 00 3 360 96 3 567 60 3 774 24 3 980 88 4 187 53 4394 17 4 600 81 4 807 45 5 014 09 5 220 73 5 427 37 5 634 01 5 840 65 6 047 30 6 253 94 6 460 58 6 667 22 6 873 86 7 080 50 7 287 14 7 493 78 38 00 3 283 27 3 486 63 3 689 99 3 893
25. 50 000075 18 Days Completion Time A 10 11 20 0000 12 Days Completion Time B ID21 1 30 0000 14 Days Completion Time B ID 1 1 50 0000 18 Days Completion Time ID 1 1 20 0000 16 Days Completion Time C ID 1 2 30 0000 _ Payoff Table Probab Event Name Value SIMULATION RESULTS Mt ht 2 9 ade Bre Select the decision node to analyze EEUU OE Decision Node and Path Nu Number of Trials Mean Median Standard Deviation Variance Coefficient of Variation Maximum Minimum b od o9 E wre 248 0000 286 0000 362 0000 44 0000 48 0000 56 0000 306 0000 343 0000 435 5000 34 0000 37 0000 12 Days Completion Time A Critical 14 Days Completion Time A Critical 18 Days Completion Time A Critical 12 Days Completion Time B Critical 14 Days Completion Time B Critical 18 Days Completion Time B Critical 16 Days Completion Time C Critical 18 Days Completion Time C Critical 23 Days Completion Time C Critical 16 Days Completion Time D Critical 18 Days Completion Time D Critical EEUU OF oo 0 Rk ord ie 127 9996 129 3285130 6574 131 9864 133 3153134 6443135 91 129 118647 134 728797 Decimals 5j Copy Results Overlay Charts Figure 5 62 ROV Decision Tree Simulation Results
26. 60 61 62 63 66 91 111 124 125 126 129 175 182 192 Page RISK SIMULATOR distributions 1 6 8 9 12 16 17 26 31 33 35 36 37 38 42 46 47 52 53 56 58 62 110 126 129 130 131 132 133 146 148 153 159 160 161 175 182 183 184 e mail 2 3 186 equation 30 34 72 76 80 87 92 116 137 140 141 175 182 Erlang 6 49 52 error 1 2 14 18 21 29 56 69 72 78 80 81 94 121 130 137 138 139 155 184 errors 14 19 43 66 67 72 78 80 93 136 137 139 estimates 58 65 68 78 80 93 94 126 130 137 138 176 183 Excel 1 2 3 4 5 6 8 12 13 14 16 21 26 27 34 68 69 73 78 81 85 86 88 89 92 94 95 97 98 111 116 121 165 168 173 175 185 187 188 excess kurtosis 33 38 39 40 41 42 44 46 47 50 51 52 53 54 55 56 60 61 62 extrapolation 1 7 9 78 177 first moment 31 34 Fisher Snedecor 51 flexibility 113 fluctuations 56 122 124 136 140 154 forecast 7 8 11 14 18 19 20 21 23 25 26 28 29 30 31 36 37 65 66 67 69 70 76 78 80 81 85 86 88 89 90 92 96 98 99 111 115 123 124 130 131 132 133 137 138 140 158 168 183 184 185 187 forecast statistics 8 20 98 99 130 185 forecasting 5 7 9 11 12 36 64 65 66 67 68 70 80 85 86 88 107 136 138 140 141 154 168 169 184 Forecasting 1 7 9 12 64 65 68 69 70 73 74 76 77 78 81 85 86 8
27. Brownian Motion Mean Reversion Jump Diffusion or mixed process is the best fit It is up to the user to make this determination depending on the time series variable to be forecasted The analysis cannot determine which process if best only the user can do this e g Brownian Motion process is best for modeling stock prices but the analysis cannot determine that the historical data analyzed is from a stock or some other variable and only the user will know this Finally a good hint is that if a certain parameter is out of the normal range the process requiring this input parameter is most probably not the correct process e g if the mean reversion rate is 110 chances are mean reversion is not the correct process TIPS Distributional Analysis Charts and Probability Tables Distributional Analysis used to quickly compute the PDF CDF and ICDF of the 42 probability distributions available in Risk Simulator and to return a table of these values e Distributional Charts and Tables used to compare different parameters of the same distribution e g takes the shapes and PDF CDF ICDF values of a Weibull 184 Page RISK SIMULATOR distribution with Alpha and Beta of 2 2 3 5 and 3 5 8 and overlays them on top of one another e Overlay Charts used to compare different distributions theoretical input assumptions and empirically simulated output forecasts and overlay them on top of one another for a visual
28. For example you can use Geometric Mean to calculate average growth rate given compound interest with variable rates The Trimmed Mean calculates the arithmetic average of the data set after the extreme outliers have been trimmed As averages are prone to significant bias when outliers exist the Trimmed Mean reduces such bias in skewed distributions The Standard Error of the Mean calculates the error surrounding the sample mean The larger the sample size the smaller the error such that an infinitely large sample size the error approaches zero indicating that the population parameter has been estimated Due to sampling errors the 9596 Confidence Interval for the Mean is provided Based on an analysis ofthe sample data points the actual population mean should fall between these Lower and Upper Intervals for the Mean Median is the data point where 5096 of all data points fall above this value and 5096 below this value Among the three first moment statistics the median is least susceptible to outliers A symmetrical distribution has the Median equal to the Arithmetic Mean A skewed distribution exists when the Median is far away from the Mean The Mode measures the most frequently occurring data point Minimum is the smallest value in the data set while Maximum is the largest value Range is the difference between the Maximum and Minimum values The second moment measures a distribution s spread or width and is frequently described using measures
29. Lorentzian or Breit Wigner Distribution Chi Square Distribution RISK SIMULATOR The Cauchy distribution also called the Lorentzian or Breit Wigner distribution is a continuous distribution describing resonance behavior It also describes the distribution of horizontal distances at which a line segment tilted at a random angle cuts the x axis The mathematical constructs for the cauchy or Lorentzian distribution are as follows E y 2 eee 4 The Cauchy distribution is a special case because it does not have any theoretical moments mean standard deviation skewness and kurtosis as they are all undefined Mode location 0 and scale are the only two parameters in this distribution The location parameter specifies the peak or mode of the distribution while the scale parameter specifies the half width at half maximum of the distribution In addition the mean and variance of a Cauchy or Lorentzian distribution are undefined In addition the Cauchy distribution is the Student s T distribution with only 1 degree of freedom This distribution is also constructed by taking the ratio of two standard normal distributions normal distributions with a mean of zero and a variance of one that are independent of one another Input requirements Location Alpha can be any value Scale Beta gt 0 and can be any positive value The chi square distribution is a probability distribution used predominantly in hyp
30. Name iample Third Forecast Number of Datapoints 1000 Enabled Yes Mean 0 2861 Celt ESIA Median 0 2621 Standard Deviation 0 1593 Forecast Precision Variance 0 0254 Precision Level Average Deviation 0 1305 E Error Level Maximum 0 8358 E Minimum 0 0126 3 Range 0 8232 E Skewness 0 5797 L Kurtosis 0 2064 d 2586 Percentile 0 1590 00 75 Percentile 0 3935 gs Error Precision at 9596 0 0345 Correlation Matrix Sample First Assumption ssumption ssumption Sample First Assumption 1 00 Sample Second Assumption 0 00 1 00 Sample Third Assumption 0 00 0 00 1 00 Figure 5 21 Sample Simulation Report 136 Page Procedure RISK SIMULATOR 5 8 Regression and Forecasting Diagnostic Tool The regression and forecasting Diagnostic tool in Risk Simulator is an advanced analytical tool used to determine the econometric properties of your data The diagnostics include checking the data for heteroskedasticity nonlinearity outliers specification errors micronumerosity stationarity and stochastic properties normality and sphericity of the errors and multicollinearity Each test is described in more detail in its respective report in the model e Open the example model Examples Regression Diagnostics go to the Time Series Data worksheet and select the data including the variable names cells C5 H 55 Click on Rzse Simulator Tools Diagnostic Tool e Check the data and select from th
31. it is important to note that there is only one trial in the Bernoulli distribution and the resulting simulated value is either 0 or 1 Input requirements Probability of success gt 0 and lt 1 ie 0 0001 lt p lt 0 9999 The binomial distribution describes the number of times a particular event occurs in a fixed number of trials such as the number of heads in 10 flips of a coin or the number of defective items out of 50 items chosen Conditions The three conditions underlying the binomial distribution are e For each trial only two outcomes are possible that are mutually exclusive trials are independent what happens in the first trial does not affect the next trial 38 Page Discrete Uniform RISK SIMULATOR e probability of an event occurring remains the same from trial to trial The mathematical constructs for the binomial distribution ate as follows forn gt 0 0 1 2 0 p 1 x n x Mean np Standard Deviation Jnp 1 p Skeyness 1 2p 4np p Excess Kurtosis 1 1 p Probability of success f and the integer number of total trials 7 are the distributional parameters The number of successful trials is denoted x It is important to note that probability of success p of 0 or 1 are trivial conditions that do not require any simulations and hence are not allowed in the software Input requiremen
32. of the charts For instance if Advays On Top is selected the forecast charts will always be visible regardless of what other software are running on your computer Histogram Resolution allows you to change the number of bins of the histogram anywhere from 5 bins to 100 bins Also the Data Update feature allows you to control how fast the simulation runs versus how often the forecast chart is updated For example viewing the forecast chart updated at almost every trial will slow down the simulation as more memory is being allocated to updating the chart versus running the simulation This is merely a user preference and in no way changes the results of the simulation just the speed of completing the simulation To further increase the speed of the simulation you can minimize Excel while the simulation 1s running thereby reducing the memory required to visibly update the Excel spreadsheet and freeing up the memory to run the simulation The Char AY and Minimize All controls all the open forecast charts As shown in Figure 2 8B this forecast chart feature allows you to show all the forecast data or to filter in out values that fall within either some specified interval or some standard deviation you choose Also the precision level can be set here for this specific forecast to show the etror levels in the statistics view See the section on error and precision control later in this chapter for more details Show the following statistic on histogram i
33. using the regression equation to forecast one s retirement fund based on the firm s stocks will yield incorrect results at best In contrast suppose the outliers are caused by a single nonrecurring business condition e g merger and acquisition and such business structural changes are not forecast to recur These outliers then should be removed and the data cleansed ptior to running a regression analysis The analysis here only identifies outliers and it is up to the user to determine if they should remain or be excluded Sometimes a nonlinear relationship between the dependent and independent variables is more appropriate than a linear relationship In such cases running a linear regression will not be optimal If the linear model is not the correct form then the slope and intercept estimates and 138 Page RISK SIMULATOR the fitted values from the linear regression will be biased and the fitted slope and intercept estimates will not be meaningful Over a restricted range of independent or dependent variables nonlinear models may be well approximated by linear models this is in fact the basis of linear interpolation but for accurate prediction a model appropriate to the data should be selected A nonlinear transformation should first be applied to the data before running a regression One simple approach is to take the natural logarithm of the independent variable other approaches include taking the square root or raising the inde
34. when a statistics sampling distribution is not normally distributed or easily found these classical methods are difficult to use or are invalid In conttast bootstrapping analyzes sample statistics empirically by repeatedly sampling the data and creating distributions of the different statistics from each sampling e a simulation e Select Rusk Simulator Tools Nonparametric Bootstrap e Select only one forecast to bootstrap select the s aistc s to bootstrap enter the number of bootstrap trials and click OK Figure 5 16 MODEL A MODEL B 100 00 Cost 100 00 Nonparametric Bootstrap pratt bootstrap simulation is a distribution free technique used to estimate the reliability or a accuracy of forecast statistics i e to compute the forecast intervals of each of the statistics Select forecast to nonparametric bootstrap of the cost cells Finally define forecast outputs for thi Income A Risk Simulator Forecast Skewness V Kurtosis 25 Percentile 75 Percentile gt Certainty 100002 Figure 5 16 Nonparametric Bootstrap Simulation In essence nonparametric bootstrap simulation can be thought of as simulation based on a simulation Thus after running a simula
35. 00 50 00 13868 13955 14043 14130 14218 14305 14393 IV tuto 18 Days 20 00 letion Time B 1 1 Decision Nodes Kz 4 48 Zn xu 4999 222 14 Days 50 00 Opa 0 8 9 ren uoa 1 1809 2000 eal Completion Time C 1 2 Critical 13144 Ee M ae Saved Model ___ Name Model 1 Model 1 ADD Payoff 44 50 Figure 5 66 ROV Decision Tree Scenario Tables 181 RISK SIMULATOR 4 ROV Visual Modeler 2012 Decision Trees C Users user Desktop Screen Shots DT Modelrovdt hi A EI SD C 2 0 4 7 Decision Tree Summary of Values Simulation Modeling Bayesian Analysis EVPI Minimax Risk Profile Sensitivity Analysis Scenario Tables Utiity Function UTILITY FUNCTION GENERATION Utility functions or U x are sometimes used in place of expected values of terminal payoffs in a decision tree UU On be SPOS acie eer ofa posse outcome or an exponential extrapolation method used here They can be modeled for a decision maker who is risk averse downsides are more disastrous or painful than an equal upside potential risk neutral upsides and downsides have equal attractiveness or risk oving upside potential is more attractive Enter the minimum and maximum expected value of your terminal payoffs and the number of data points in between to compute the utility curve and table Minimum Expected Value
36. 01 0 05 and 0 10 corresponding to the 99 95 and 90 confidence levels The Coefficients with their p Values highlighted in blue indicate that they are statistically significant at the 90 confidence or 0 10 alpha level while those highlighted in red indicate that they are not statistically significant at any other alpha levels Analysis of Variance Sums or Brno F Statistic p Value Squares Squares Hypothesis Test Regression 479388 49 95877 70 4 28 0 0029 Critical F statistic 9996 confidence with df of 5 and 44 3 4651 Residual 985675 19 22401 71 Critical F statistic 95 confidence with df of 5 and 44 2 4270 Total 1465063 68 Critical F statistic 90 confidence with df of 5 and 44 1 9828 The Analysis of Variance ANOVA table provides an F test of the regression model s overall statistical significance Instead of looking at individual regressors as in the t test the F test looks at all the estimated Coefficients statistical properties The F Statistic is calculated as the ratio of the Regression s Mean of Squares to the Residual s Mean of Squares The numerator measures how much of the regression is explained while the denominator measures how much is unexplained Hence the larger the F Statistic the more significant the model The corresponding p Value is calculated to test the null hypothesis Ho where all the Coefficients are simultaneously equal to zero versus the alternate hypothesis Ha that they are all simultaneous
37. 02 Tax Rate 0 28 Price Erosion 0 11 Sales Growth Figure 5 12 Contribution to Variance Chart Notes Tornado analysis is performed before a simulation run while sensitivity analysis is performed after a simulation run Spider charts in tornado analysis can consider nonlinearities while rank correlation charts in sensitivity analysis can account for nonlinear and distributional free conditions 126 Page Theory Procedure BIZSTATS Distributional Fitting Methods RISK SIMULATOR 5 3 Distributional Fitting Single Variable and Multiple Vartables Another powerful simulation tool is distributional fitting That is determining which distribution to use for a particular input variable in a model and what the relevant distributional parameters are If no historical data exist then the analyst must make assumptions about the variables in question One approach is to use the Delphi method where a group of experts are tasked with estimating the behavior of each variable For instance a group of mechanical engineers can be tasked with evaluating the extreme possibilities of a spring coil s diameter through rigorous experimentation or guesstimates These values can be used as the variable s input parameters e g uniform distribution with extreme values between 0 5 and 1 2 When testing is not possible e g market share and revenue growth rate management can still make estimates of potential outcomes and provide the best ca
38. 14 26 27 142 correlations 8 12 14 17 26 27 28 101 122 124 142 153 182 183 186 cross sectional 65 78 data 6 7 8 9 12 19 21 26 27 28 35 36 45 55 62 64 65 66 67 68 69 70 72 73 76 78 80 81 85 86 87 89 90 93 95 96 97 103 126 129 130 131 133 134 136 137 138 139 140 142 143 152 153 154 155 156 157 158 159 160 164 165 168 169 177 183 184 185 187 decision variable 5 8 12 98 99 101 102 105 110 111 113 171 182 187 decision variables 5 12 98 99 101 102 105 110 111 113 171 182 187 decisions 98 105 113 Delphi 6 65 126 Delphi method 6 126 dependent variable 12 66 67 72 73 80 81 85 93 94 95 136 137 138 139 185 discrete 1 6 36 37 38 39 98 99 105 129 140 146 Discrete 36 38 39 105 106 107 distribution 6 9 11 12 15 16 17 20 26 28 31 32 33 35 36 37 38 39 40 41 42 43 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 65 66 76 91 92 94 99 113 124 126 129 130 131 139 146 147 148 153 159 160 161 175 182 183 186 Distribution 6 16 27 31 32 33 36 38 40 41 42 43 45 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 91 128 129 147 148 149 150 160 161 162 163 distributional 1 8 9 11 12 13 17 21 27 31 33 36 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59
39. 3 2 Running the Forecasting Tool in Risk Simulator In general to create forecasts several quick steps are required e Start Excel and enter in or open your existing historical data e Select the data and click on Simulation and select Forecasting Select the relevant sections ARIMA Multivariate Regression Nonlinear Extrapolation Stochastic Forecasting Time Series Analysis and enter the relevant inputs Figure 3 2 illustrates the Forecasting tool and the various methodologies and the following provides a quick review of the selected methodology and several quick getting started examples in using the software The example file can be found either on the start menu at 5777 Real Options Valuation Risk Simulator Examples ot accessed directly through Rak Simulator Example Models 01 ARIMA 02 Auto ARIMA 03 Auto Econometrics 04 Basic Econometrics 05 Combinatorial Fuzzy Logic 06 Cubic Spline 07 GARCH 08 J S Curves 09 Markov Chain 10 MLE LIMDEP 11 Regression Analysis 12 Neural Network 13 Nonlinear Extrapolation 14 Stochastic Processes 15 Time Series Analysis n ne 12 he n n da e e n Figure 3 2 Simulator s Forecasting Methods 16 Trendline 68 Page Theory Procedure Results Interpretation RISK SIMULATOR 3 3 Time Series Analysis Figure 3 3 lists the eight most common time series models segregated by seasonalit
40. 35 4 096 71 4 300 07 4 503 43 4 706 79 4 910 15 5 113 51 5 316 88 5 520 24 5 723 60 5 926 96 6 130 32 6 333 68 6 537 04 6 740 40 6 943 76 7 147 13 7 350 49 39 00 3 205 57 3 405 65 3 605 73 3 805 81 4 005 89 4205 97 4 406 06 4 606 14 4 806 22 5 006 30 5 206 38 5 406 46 5 606 54 5 806 62 6 006 70 6 206 79 6 406 87 6 606 95 6 807 03 7 007 11 7 207 19 40 00 3 127 87 3 324 67 3 521 48 3 718 28 3 915 08 4 111 88 4 308 68 4505 48 4 702 28 4 899 08 5 095 88 55 292 68 5 489 49 5 686 29 5 883 09 6 079 89 6 276 69 6 473 49 6 670 29 6 867 09 7 063 89 41 00 3 050 18 3 243 70 3 437 22 3 630 74 3 824 26 4 017 78 4 211 30 4 404 82 4 598 35 4 791 87 4 985 39 5 178 91 5 372 43 5 565 95 5 759 47 5 952 99 6 146 51 6 340 03 6 533 56 6 727 08 6 920 60 42 00 2 972 48 3 162 72 3 352 96 3 543 20 3 733 45 3 923 69 4 113 93 4 304 17 4 494 41 4 684 65 4874 89 5 065 13 5 255 37 5 445 61 5 635 86 5 826 10 6 016 34 6 206 58 6 396 82 6 587 06 6 777 30 43 00 2 894 79 3 081 75 3 268 71 3 455 67 3 642 63 3 829 59 4 016 55 4 203 51 4 390 47 4 577 43 4 764 40 4 951 36 5 138 32 5 325 28 5 512 24 5 699 20 5 886 16 6 073 12 6 260 08 6 447 04 6 634 01 44 00 2 817 09 3 000 77 3 184 45 3 368 13 3 551 81 3 735 49 3 919 18 4 102 86 4 286 54 4 470 22 4 653 90 4 837 58 5 021 26 5 204 94 5 388 62 5 572 30 5 755 98 5 939 67 6 123 35 6 307 03 6 490 71 45 00 2 739 39 2
41. 420 89 0 01 10 562 34 562 34 0 00 11 730 85 730 85 0 00 12 928 43 928 43 0 00 Forecast 13 1157 03 0 00 Forecast 14 1418 57 0 00 Forecast 15 1714 95 0 00 Forecast 16 2048 00 0 00 Forecast 17 2419 55 0 00 Forecast 18 2831 39 0 00 Figure 3 12 Nonlinear Extrapolation Results 79 Page Theory RISK SIMULATOR 3 7 Box Jenkins ARIMA Advanced Time Seties One very powerful advanced times series forecasting tool is the ARIMA or Auto Regressive Integrated Moving Average approach ARIMA forecasting assembles three separate tools into a comprehensive model The first tool segment is the autoregressive AR term which corresponds to the number of lagged value of the residual in the unconditional forecast model In essence the model captures the historical variation of actual data to a forecasting model and uses this variation or residual to create a better predicting model The second tool segment is the integration order I term This integration term corresponds to the number of differencing the time series to be forecasted goes through This element accounts for any nonlinear growth rates existing in the data The third tool segment is the moving average MA term which is essentially the moving average of lagged forecast errors By incorporating this lagged forecast etrors term the model in essence learns from its forecast errors or mistakes and corrects for them through a moving average calculation The ARIMA model follows the Box Jenki
42. 43 113 59 12415 109 24 105 34 10457 97 83 94 39 116 19 84 66 101 17 106 13 107 17 95 83 106 67 9242 79 64 94 15 106 00 113 45 92 63 94 54 93 05 96 19 100 85 83 34 111 82 118 12 87 17 103 66 106 93 82 45 102 74 86 82 106 68 89 61 94 56 101 34 91 32 102 02 82 51 104 46 84 72 105 05 108 40 106 59 109 43 92 49 94 52 94 00 105 92 88 13 96 41 101 45 79 93 89 68 102 91 11495 92 58 94 05 107 90 111 05 90 58 97 09 105 44 94 95 102 55 77 41 108 53 90 54 100 41 106 83 99 63 79 72 89 32 116 30 98 27 101 73 90 84 7445 102 24 103 34 96 51 114 55 93 94 106 29 102 95 142 73 98 09 108 20 105 80 106 48 102 88 104 93 103 00 99 10 108 52 101 31 88 17 90 62 96 53 106 03 109 12 104 23 90 34 95 12 102 03 100 00 118 17 99 06 81 89 104 29 92 68 114 89 102 49 119 21 106 20 88 26 92 45 105 15 103 79 100 84 95 19 85 10 97 25 87 65 97 58 111 44 99 52 89 83 97 86 90 96 97 14 Figure 5 15 Distributional Fitting Report For fitting multiple variables the process is fairly similar to fitting individual variables However the data should be arranged in columns each variable is arranged as a column and all the variables are fitted one at a time Procedure e Open a spreadsheet with existing data for fitting Select the data you wish to fit data should be in a multiple columns with multiple rows e Select Risk Simulator Tools Distributional Fitting Multi V ariable Review the data choose the relevant types of distribution you want and click OK
43. Anal 4 Hands on Exercises vd ea even Senet Set B 23 Tornado and Sensitivity Charts Nonlinear Probability Distribution Details edlen Ec bn ow did BE 24 Tools on Data Behavior T Stochastic Processes A Statistical Analysis ser Manual S Time Series Analysis Structural Break Test el S Trendline SS Tornado Analysis 33 Figure 1 1 Risk Simulator Menu and Icon Bar in Excel 2007 2010 3 Page RISK SIMULATOR eal Options luation LLC right Opti Valuation Inc 05 Al rights Figure 1 2 Risk Simulator Splash Screen xXx Hox xx gt ocr i FCO FE ROV ROV Analytical Options Help license Risk New Change Edit Setinput SetOutput Paste Remove Run RunSuper Step Reset Forecasting Run Simulator Profile Profile Profile Assumption Forecast Speed Optimization G Set Constraint Decision Tree Tools icon Menu Profile Assumptions Forecasts Editing Simulation Run Forecasting Optimization ROV BizStats ROV Decision Tree Tools Options Help license Icon BOG tc amp B CEREBRA BRK Eee S Combinatorial Cubic GARCH J S Markov MLE Neural Nonlinear Regression Stochastic Time Series Trendline Risk Set Input Set Output Run Super Reset ARIMA Auto Auto Basic Simulator Assumption Forecast Speed ARIMA Econometrics Econometrics Fuzzy Logic Spline Curves Chain LIMDEP Network Extrapolation Analysis Processes Analy
44. Excel and select Simulator Forecasting JS Curves e Select the J ot S curve type euer the required input assumptions see Figures 3 17 and 3 18 for examples and click OK to run the model and report The S curve or logistic growth cutve starts off like a J curve with exponential growth rates Over time the environment becomes saturated e g market saturation competition overcrowding the growth slows and the forecast value eventually ends up at a saturation or maximum level This model is typically used in forecasting market share or sales growth of a new product from market introduction until maturity and decline population dynamics growth of bacterial cultures and other naturally occurring variables Figure 3 18 illustrates a sample S curve J Curve Exponential Growth Curves In mathematics a quantity that grows exponentially is one whose growth rate is always proportional to its current size Such growth is said to follow an exponential law This implies that for any exponentially growing quantity the larger the quantity gets the faster it grows But it also implies that the relationship between the size of the dependent variable and its rate of growth is governed by a strict law of the simplest kind direct proportion The general principle behind exponential growth is that the larger a number gets the faster it grows Any exponentially growing number will eventually grow larger than any other number which grows at only
45. Excel model e g cell is selected in our example e Select Risk Simulator Tools Tornado Analysis e Review the precedents and rename them as needed renaming the precedents to shorter names allows a more visually pleasing tornado and spider chart and click OK Discounted Cash Flow Model Base Year 2005 Sum PV Net Benefits 1 896 63 Market Risk Adjusted Discount Rate 15 0096 Sum PV Investments 7 800 00 Private Risk Discount Rate 5 00 Net Present Value 96 63 Annualized Sales Growth Rate 2 00 internal Rate of Return 18 80 Price Erosion Rate 5 00 Return on Investment 5 37 Effective Tax Rate 40 00 2005 2006 2007 2008 2009 Product A Avg Price Unit Product B Avg Price Unit Product C Avg Price Unit Product A Sale Quantity 000s Product B Sale Quantity 000s Product C Sale Quantity 000s Total Revenues Direct Cost of Goods Sold Gross Profit Operating Expenses Sales General and Admin Costs Operating Income EBITDA Depreciation Amortization EBIT Interest Payments EBT Taxes Net Income Noncash Depreciation Amortization 13 00 13 00 13 00 13 00 13 00 Noncash Change in Net Working Capital 0 00 0 00 0 00 0 00 0 00 Noncash Capital Expenditures Free Cash Flow Investment Outlay 180 00 Financial Analysis Present Value of Free Cash Flow 528 24 440 60 367 26 305 91 254 62 Present Value of Investment Outlay 7 800 00 0 00 0 00 0 00 0 00 Net Cash Flows 1 271 76 506 69 485 70
46. Nonparametric Chi Square Goodness of Fit Nonparametric Chi Square Independence Nonparametric Chi Square Population Variance Nonparametric Friedman s Test Nonparametric Kruskal Wallis Test Nonparametric Lilliefors Test Nonparametric Runs Test Nonparametric Wilcoxon Signed Rank One Var Nonparametric Wilcoxon Signed Rank Iwo Var Parametric One Variable T Mean Parametric One Variable Z Mean Parametric One Variable Z Proportion Parametric Two Variable F Variances Parametric Two Variable T Dependent Means Parametric Two Variable T Independent Equal Variance Parametric Two Variable T Independent Unequal Variance Parametric Two Variable Z Independent Means Parametric Two Variable Z Independent Proportions Power Principal Component Analysis Rank Ascending Rank Descending Relative LN Returns Relative Returns Seasonality Segmentation Clustering Semi Standard Deviation Lower Semi Standard Deviation Upper Standard 2D Area Standard 2D Bar Standard 2D Line Standard 2D Point Standard 2D Scatter Standard 3D Area Standard 3D Bar Standard 9 Page RISK SIMULATOR 3D Line Standard 3D Point Standard 3D Scatter Standard Deviation Population Standard Deviation Sample Stepwise Regression Backward Stepwise Regression Correlation Stepwise Regression Forward Stepwise Regression Forward Backward Stochastic Processes Exponential Brownian Motion Stochastic Processes Geometric B
47. Polynomial Growth Rates 3 Select the function type and extrapolation periods are required and click OK __ Month Period _ _ x 2010 1 7 1 00 Ep 2010 2 2 6 73 Nonlinear Extrapolation is used to make statistical 2010 3 3 20 52 time series forecast projections by applying 2010 4 4 45 25 historical trends It is useful when the historical trends are nonlinear and well behaved The 2010 5 5 83 59 extrapolation is best used for short term forecasts 2000 6 6 13801 2010 7 7 210 87 2010 8 8 304 44 Automatic Selection Polynomial Function Rational Function 2010 9 9 420 89 2010 10 10 562 34 Number of Extrapolation Periods 34 2010 11 11 5730 85 2000 12 12 928 43 ok Real Options wy Valuation www realoptionsvalustion com Figure 3 11 Running a Nonlinear Extrapolation Nonlinear Extrapolation Statistical Summary Extrapolation involves making statistical projections by using historical trends that are projected for a specified period of time into the future It is only used for time series forecasts For cross sectional or mixed panel data time series with cross sectional data multivariate regression is more appropriate This methodology is useful when major changes are not expected that is causal factors are expected to remain constant or when the causal factors of a situation are not clearly understood It also helps discourage introduction of personal biases into the process Ex
48. Regression 1 Select the data area including the headers B5 G55 2 Click on Risk Simulator Forecasting Multiple Regression 3 Select the Dependent Variable in this example the variable Y and select any specific modifications as required Lag Regressors Nonlinear Regression Stepwise Regression and click OK Review the generated regression report for analytical results Se FS Multiple Regression Analysis wet vere cer De umero run res with multiple independent variables These variables can be applied through a series of lags or nonlinear transformations or re a stepwise fashion starting with the most correlated Non linear Regression Show All Steps Figure 3 7 Running a Multivariate Regression 74 Page RISK SIMULATOR Regression Analysis Report Regression Statistics R Squared Coefficient of Determination 0 3272 Adjusted R Squared 0 2508 Multiple R Multiple Correlation Coefficient 0 5720 Standard Error of the Estimates SEy 149 6720 Number of Observations 50 The R Squared or Coefficient of Determination indicates that 0 33 of the variation in the dependent variable can be explained and accounted for by the independent variables in this regression analysis However in a multiple regression the Adjusted R Squared takes into account the existence of additional independent variables or regressors and adjusts this R Squared value t
49. SOSH AHO DABIS S cap edes us oto EDDA QE oW E a UR La S c aL PA eM UA 178 5 2 5T Function Creneraton a d eee tte tiende ntl 178 6 Helpful Tips and Techniques eese 183 TIPS Assumptions Set Input Assumption User Interface sss 183 TIPSECOBY Be AS Bs cou Op Un ALAS a eA cu SU eL 183 MIPS dem eet esta aet s t a usen 184 TIPS Data Diagnostics and Statistical Analysis seen 184 TIPS Distributional Analysis Charts and Probability Tables s 184 MIPS Eficient 185 TIPS Potecast elle auci ado aH can dO MR 185 TIPS Forecast eerte eee iie 185 TIPS es petes 185 PUPS Forecasting ARIMA ere terere puppes 185 TIPS Forecasting Basic EcOmOmetrics ssscsssvssessssvssessvossssssssnssvesusossrssssossavesesssessnsnseensssscesavosesssesees 186 TIPS Forecasting Logit Probit and aede 186 TIPS Forecasting Stochastic Processes 186 TIPS Forecasting Trendliries iniia 186 SUES sc NST 71 c NN 186 TIPS Getting Started Exercises and Getting Started Videos ss 186 TIPS Hardware D TEN 187 TIPS Latin Hypercu
50. Simulator menu to set an output forecast Figure 2 5 illustrates the set forecast properties e Forecast Name Specify the name of the forecast cell This is important because when you have a large model with multiple forecast cells naming the forecast cells individually allows you to access the right results quickly Do not underestimate the importance of this simple step Good modeling practice is to use short but precise forecast names Forecast Precision Instead of relying on a guesstimate of how many trials to run in your simulation you can set up precision and error controls When an error precision combination has been achieved in the simulation the simulation will pause and inform you of the precision achieved making the required number of simulation trials an automated process rather than a guessing game Review the section on error and precision control later in this chapter for more specific details 18 Page Running the Simulation Interpreting the Forecast Results Forecast Chart RISK SIMULATOR e Show Forecast Window Allows the user to show or not show a particular forecast window The default is to always show a forecast chart lij Forecast Properties Forecast Name Income 8 Forecast Precision Precision Level V Show Forecast Window Figure 2 5 Set Output Forecast If everything looks tight simply click on Simulator Run Simulation click on the Run icon on the Risk
51. a constant rate for the same amount of time This forecast method is also called a J curve due to its shape resembling the letter J There is no maximum level of this growth curve Other growth curves include S curves and Markov Chains To generate a J curve forecast follow the instructions below 1 Click on Risk Simulator Forecasting JS Curves Real Options 2 Select Exponential J Curve and enter in the desired inputs V d l ua t Lon e g Starting Value of 100 Growth Rate of 5 percent End Period of 100 3 Click OK to run the forecast and spend some time reviewing the forecast report JSCurves The J S curves stand for J curve exponential growth and S curve logistic growth curve These curves are used in forecasting high growth rates J curve or for situations with events with initially high growth but slows down and growth matures over time as the environment becomes saturated at capacity S curve Exponential J Curve Logistic S Curve Starting Value 100 Growth Rate Saturation Level Generate forecast curve based on the following periods End Period 100 Figure 3 17 J Curve Forecast 89 Page RISK SIMULATOR Logistic S Curve A logistic function or logistic curve models the S curve of growth of some variable X The initial stage of growth is approximately exponential then as competition arises the growth slows and at maturity growth stops These functions find appl
52. a t test if the p value is less than a specified significance amount typically 0 10 0 05 or 0 01 this means that the population mean is statistically significantly greater than the hypothesized mean at 10 5 and 1 significance value or 90 95 and 99 statistical confidence Conversely if the p value is higher than 0 10 0 05 or 0 01 the population mean is statistically similar or less than the hypothesized mean Left Tailed Hypothesis Test A left tailed hypothesis tests the null hypothesis Ho such that the population mean is statistically greater than or equal to the hypothesized mean The alternative hypothesis is thatthe real population mean is statistically less than the hypothesized mean when tested using the sample dataset Using a t test if the p value is less than a specified significance amount typically 0 10 0 05 or 0 01 this means that the population mean is statistically significantly less than the hypothesized mean at 10 5 and 1 significance value or 90 95 and 99 statistical confidence Conversely if the p value is higher than 0 10 0 05 or 0 01 the population mean is statistically similar or greater than the hypothesized mean and any differences are due ti random chance Because the t test 5 more conservative and does not require a known population standard deviation as in the Z test we only use this t test Figure 5 31 Sample Statistical Analysis Tool Report Hypothesis Testing of One Variable Test f
53. and financial expectations of what the underlying data set is best represented by These parameters can be entered into a stochastic process forecast Simulation Forecasting Stochastic Processes Periodic Drit Rate 1 4896 Reversion Rate 283 8996 Jump Rate 20 419 Volatility 88 84 Long Term Value 327 72 Jump Size 237 89 Probability of stochastic model fit 46 4996 A high fit means a stochastic model is better than conventional models Runs 20 Standard Normal 1 7321 Positive 25 P Value 1 tai 0 0416 Negative 25 P l alue 2 taif 0 0833 Expected Run 26 A low p value below 0 10 0 05 0 01 means that the sequence is not random and hence suffers from stationarity problems and an ARIMA model might be more appropriate Conversely higher p values indicate randomness and stochastic process models might be appropriate Figure 5 26 Stochastic Process Parameter Estimation Multicollinearity exists when there is a linear relationship between the independent variables When this occurs the regression equation cannot be estimated at all In near collinearity situations the estimated regression equation will be biased and provide inaccurate results This situation is especially true when a stepwise regression approach is used where the statistically significant independent variables will be thrown out of the regression mix earlier than expected resulting in a regression equation that 1s neither efficient nor accurate One quick test of the prese
54. are statistically different from one another otherwise for large p values the variances are statistically identical to one another Hypothesis Test on the Means and Variances of Two Forecasts Statistical Summary Notes A hypothesis test is performed when testing the means and variances of two distributions to determine if they are statistically identical or statistically different from one another That is to see If the differences between two means and two variances that occur are based on random chance or they are in fact different from one another The two variable Hest with unequal variances the population variance of forecast 1 is expected to be different from the population variance of forecast 2 is appropriate when the forecast distributions are from different populations e g data collected from two different geographical locations two different operating business units and so forth The two variable test with equal variances the population variance of forecast 1 is expected be equal to the population variance of forecast 2 is appropriate when the forecast distributions are from similar populations e g data collected from two different engine designs with similar specifications and so forth The paired dependent two variable test is appropriate when the forecast distributions are from similar populations e g data collected from the same group of customers but on different occasions and so forth A two tailed hypot
55. are the approximate two standard error bounds If the autocorrelation is within these bounds it is not significantly different from zero at the 5 significance level Autocorrelation measures the relationship to the past of the dependent Y variable to itself Distributive lags in contrast are time lag relationships between the dependent Y variable and different independent X variables For instance the movement and direction of mortgage rates tend to follow the federal funds rate but at a time lag typically 1 to 3 months Sometimes time lags follow cycles and seasonality e g ice cream sales tend to peak during the summer months 139 Page RISK SIMULATOR and are hence related to last summer s sales 12 months in the past The distributive lag analysis Figure 5 24 shows how the dependent variable is related to each of the independent variables at various time lags when all lags are considered simultaneously to determine which time lags are statistically significant and should be considered Autocorrelation Time Lag AC PAC Lower Bound Upper Bound Q Stat Prob 1 0 0580 0 0580 0 2828 0 2828 0 1786 0 6726 2 0 1213 0 1251 0 2828 0 2828 0 9754 0 6140 3 0 0590 0 0756 0 2828 0 2828 1 1679 0 7607 4 0 2423 0 2232 0 2828 0 2828 4 4865 0 3442 5 0 0067 0 0078 0 2828 0 2828 4 4890 0 4814 6 0 2654 0 2345 0 2828 0 2828 8 6516 0 1941 7 0 0814 0 0939 0 2828 0 2828 9 0524 0 2489 8 0 0634 0 0442 0 2828 0 2828 9 3012 0 3175 9
56. buck or returns to risk ratio is maximized That is the goal is to allocate 100 of an individual s investment among several different asset classes e g different types of mutual funds or investment styles growth value aggressive growth income global index contrarian momentum etc This model is different from others in that there exists several simulation assumptions risk and return values for each asset in columns C and D as seen in Figure 4 9 A simulation is run then optimization is executed and the entire process is repeated multiple times to obtain distributions of each decision variable The entire analysis can be automated using Stochastic Optimization To run an optimization several key specifications on the model have to be identified first Objective Maximize Return to Risk Ratio C12 Decision Variables Allocation Weights E6 E9 Restrictions on Decision Variables Minimum and Maximum Required F6 G9 Constraints Portfolio Total Allocation Weights 100 E11 is set to 100 Simulation Assumptions Return and Risk Values C6 D9 The model shows the various asset classes Each asset class has its own set of annualized returns and annualized volatilities These return and tisk measures are annualized values such that they can be consistently compared across different asset classes Returns are computed using the geometric average of the relative returns while the risks are computed using the logarithmic relati
57. constraints etc are saved as an encrypted hidden worksheet This is why the profile is automatically saved when you save the Excel workbook file TIPS Right Click Shortcut and Other Shortcut Keys Right Click you can open the Risk Simulator shortcut menu by right clicking on a cell anywhere in Excel TIPS Save Saving the Excel File saves the profile settings assumptions forecasts decision variables and your Excel model including any Risk Simulator reports charts and data extracted Saving the Chart Settings saves the forecast chart settings such that the same settings can be recovered and applied to future forecast charts use the save and open icons in the forecast charts Saving and Extracting Simulated Data in Excel extracts a simulated assumptions and forecasts the Excel file itself will still have to be saved in order to save the data for retrieval later Saving Simulated Data and Charts in Risk Simulator using the Risk Simulator Data Extract and saving to a RiskSim file will allow you to reopen the dynamic and 188 Page RISK SIMULATOR live forecast chart with the same data in the future without having to rerun the simulation e Saving and Generating Reports simulation reports and other analytical reports extracted as separate worksheets in your workbook and the entire Excel file will have to be saved in order to save the data for future retrieval later TIPS Sampling and Simul
58. different operating business units The two variable t test with equal variances the population variance of forecast 1 is expected to be equal to the population variance of forecast 2 is approptiate when the forecast distributions are from similar populations e g data collected from two different engine designs with similar specifications The paired dependent two variable t test is appropriate when the forecast distributions are from the exact same population e g data collected from the same group of customers but on different occasions 134 Page Procedure RISK SIMULATOR 5 6 Data Extraction and Saving Simulation Results A simulation s raw data can be very easily extracted using Risk Simulator s Data Extraction routine Both assumptions and forecasts can be extracted but a simulation must first be run The extracted data can then be used for a variety of other analysis Open or create a model define assumptions and forecasts and ru the simulation Select Risk Simulator Tools Data Extraction Select the assumptions and or forecasts you wish to extract the data from and click OK The data can be extracted to various formats Raw data in a new worksheet where the simulated values both assumptions and forecasts can then be saved or further analyzed as required Plat text file where the data can be exported into other data analysis software Risk Simulator file where the results both assumptions and forecasts can b
59. enter in the relevant test points you wish to apply on the data e g 6 10 12 and click OK e Review the report to determine which of these test points indicate a statistically significant break point in your data and which points do not Procedure 1 Select the data to analyze e g B15 D34 and click on Risk Simulator Tools Structural Break Test and enter in the relevant test points you wish to apply on the data e g 6 10 12 and click OK 2 Review the report to determine which of these test points indicate a statistically significant break point in your data and which points do not Time Series Data B15 D34 Test Breakpoints 6 10 12 eg 15 20 23 seperate mulsipie breskports wit commas Figure 5 44 Structural Break Analysis 158 Page Procedure RISK SIMULATOR 5 18 Trendline Forecasts Trendlines can be used to determine if a set of time series data follows any appreciable trend Figure 5 45 Trends can be linear or nonlinear such as exponential logarithmic moving average power polynomial or power e Select the data you wish to analyze click on Risk Simulator Forecasting Trendline select the relevant trendlines you wish to apply on the data e g select all methods by default enter in the number of periods to forecast 6 periods and click OK Review the report to determine which of these test trendlines provide the best fit and best forecast for your data Histor
60. for the parameter of a Bernoulli distribution In this application the beta distribution is used to represent the uncertainty in the probability of occurrence of an event It is also used to describe empirical data and predict the random behavior of percentages and fractions as the range of outcomes is typically between 0 and 1 The value of the beta distribution lies in the wide variety of shapes it can assume when you vaty the two parameters alpha and beta If the parameters are equal the distribution is symmetrical If either parameter is 1 and the other parameter is greater than 1 the distribution is J shaped If alpha is less than beta the distribution is said to be positively skewed most of the values are near the minimum value If alpha is greater than beta the distribution is negatively skewed most of the values ate near the maximum value 45 Page Beta 3 and Beta 4 Distributions RISK SIMULATOR The mathematical constructs for the beta distribution are as follows AB D fy x fora gt 0 8 gt 0 x gt 0 rA 8 Mean xi a f Standard Deviation aH B 1 0 p SRewness X a jit a B 2 Bo Excess Kurtosis Xa B D aB a B 6 y 3 apla 2 3 Alpha 0 and beta are the two distributional shape parameters and is the Gamma function Conditions The two conditions underlying the beta distribution are e uncertain va
61. goals across any number of project components using minimum most likely and maximum values but it is designed to generate a distribution that more closely resembles realistic probability distributions The PERT distribution can provide a close fit to the normal or lognormal distributions Like the triangular distribution the PERT distribution emphasizes the most likely value over the minimum and maximum estimates However unlike the triangular distribution the PERT distribution constructs a smooth curve that places progressively more emphasis on values around near the most likely value in favor of values around the edges In practice this means that we trust the estimate for the most likely value and we believe that even if it is not exactly accurate as estimates seldom ate we have an expectation that the resulting value will be close to that estimate Assuming that many real world phenomena are normally distributed the appeal of the PERT distribution is that it produces a curve similar to the normal curve in shape without knowing the precise parameters of the related normal curve Minimum Most Likely and Maximum are the distributional parameters 58 Page Power Distribution RISK SIMULATOR The mathematical constructs for the PERT distribution are shown below min max x 1 2 1 B AL 2 min min 4 likely max iin ia min 4 likely max where 1 6 6 an
62. if negative correlations exist After turning on correlations here you can later set the relevant correlation coefficients on each assumption generated see the section on correlations for more details e Specify random number sequence Simulation by definition will yield slightly different results every time a simulation is run This characteristic is by virtue of the random number generation routine in Monte Carlo simulation and is a theoretical fact in all random number generators However when making 14 Page Defining Input Assumptions RISK SIMULATOR presentations sometimes you may require the same results especially when the report being presented shows one set of results and during a live presentation you would like to show the same results being generated or when you are sharing models with others and would like the same results to be obtained every time so you would then check this preference and enter in an initial seed number The seed number can be any positive integer Using the same initial seed value the same number of trials and the same input assumptions the simulation will always yield the same sequence of random numbers guaranteeing the same final set of results Note that once a new simulation profile has been created you can come back later and modify these selections To do so make sure that the current active profile is the profile you wish to modify otherwise click on Simulator Change Simulatio
63. in communication circuits The mathematical constructs for the Pareto are as follows BL LO forx L mean B 1 D B B 2 8 2 2 8 1 8 3 standard deviation skewness P 608 B 68 2 B B 3 B 4 excess Rurtosis 56 Page Pearson V Distribution RISK SIMULATOR Shape 0 and Location B are the distributional parameters Calculating Parameters There two standard parameters for the Pareto distribution location and shape The location parameter is the lower bound for the variable After you select the location parameter you can estimate the shape parameter The shape parameter is a number greater than 0 usually greater than 1 The larger the shape parameter the smaller the variance and the thicker the right tail of the distribution Input requirements Location gt 0 and can be any positive value Shape 2 0 05 The Pearson V distribution is related to the Inverse Gamma distribution where it is the reciprocal of the variable distributed according to the Gamma distribution Pearson V distribution is also used to model time delays where there is almost certainty of some minimum delay and the maximum delay is unbounded for example delay in arrival of emergency services and time to repair a machine Alpha also known as shape and Beta also known as scale are the distributional parameters The mathematical constructs for the Pearson V distribution ar
64. interface select or double click on the HWID to select its value right click to copy or click on the E mail HWID link to generate an e mail with the HWID e Troubleshooter run the Troubleshooter from the Start Programs Real Options Valuation Risk Simulator folder and run the Get HWID tool to obtain your computer s HWID TIPS Latin Hypercube Sampling LHS vs Monte Carlo Simulation MCS e Correlations when setting pairwise correlations among input assumptions we recommend using the Monte Carlo setting in the Risk Simulator Options menu Latin Hypercube Sampling is not compatible with the correlated copula method for simulation e LHS Bins a larger number of bins will slow down the simulation while providing a more uniform set of simulation results e Randomness all of the random simulation techniques in the Options menu have been tested and are all good simulators and approach the same levels of randomness when larger number of trials are run TIPS Online Resources Books Getting Started Videos Models White Papers resources available on our website www tealoptionsvaluation com download html or www tovdownloads com download html TIPS Optimization Infeasible Results if the optimization run returns infeasible results you can change the constraints from an Equal to an Inequality gt or lt and try again This also applies when you are running an efficient frontier analysis 187 Page
65. localized languages available in this module and the current language can be changed through the Language menu Insert Option nodes or Insert Terminal nodes by first selecting any existing node and then clicking on the option node icon square or terminal node icon triangle or use the functions in the Insert menu Modify individual Option Node or Terminal Node properties by double clicking on a node Sometimes when you click on a node all subsequent child nodes are also selected this allows you to move the entire tree starting from that selected node If you wish to select only that node you may have to click on the empty background and click back on that node to select it individually Also you can move individual nodes or the entire tree started from the selected node depending on the current setting right click or in the Edit menu and select Move Nodes Individually or Move Nodes Together The following are some quick descriptions of the things that can be customized and configured in the node properties user interface It is simplest to try different settings for each of the following to see its effects in the Strategy Tree o Name Name shown above the node o Value Value shown below the node o Excel Link Links the value from an Excel spreadsheet s cell Notes Notes can be inserted above or below a node o Showin Model Show any combinations of Name Value and Notes 174 Page RISK SIMULATOR Local Color ver
66. no causality and that this relationship is purely spurious Another test for multicollinearity is the use of the variance inflation factor VIF obtained by regressing each independent variable to all the other independent variables obtaining the R squared value and calculating the VIF A VIF exceeding 2 0 can be considered as severe multicollinearity A VIF exceeding 10 0 indicates destructive multicollinearity Figure 5 27 bottom 143 Page Procedure RISK SIMULATOR 5 9 Statistical Analysis Tool Another very powerful tool in Risk Simulator is the Statistical Analysis tool which determines the statistical properties of the data The diagnostics run include checking the data for various statistical properties from basic descriptive statistics to testing for and calibrating the stochastic properties of the data e Open the example model Simulator Examples Statistical Analysis go to the Data worksheet and select the data including the variable names cells C5 E 55 Click on Simulator Tools Statistical Analysis Figure 5 28 Check the data type whether the data selected are from a single variable or multiple variables arranged in rows In our example we assume that the data areas selected are from multiple variables Click OK when finished Choose the statistical tests you wish to perform The suggestion and by default is to choose all the tests Click OK when finished Figure 5 29 Spe
67. of a certain dependent variable as it depends on other independent exogenous variables Using the modeled relationship we can forecast the future values of the dependent variable The accuracy and goodness of fit for this model can also be determined Linear and nonlinear models can be fitted in the multiple regression analysis The term Neural Network is often used to refer to a network or circuit of biological neurons while modern usage of the term often refers to artificial neural networks comprising artificial neurons or nodes recreated in a software environment Such networks attempt to mimic the neurons in the human brain in ways of thinking and identifying patterns and in our situation identifying patterns for the purposes of forecasting time series data The underlying structure of the data to be forecasted is assumed to be nonlinear over time For instance a data set such as 1 4 9 16 25 1s considered to be nonlinear these data points are from a squared function The S curve or logistic growth curve starts off like a J curve with exponential growth rates Over time the environment becomes saturated e g market saturation competition overcrowding the growth slows and the forecast value eventually ends up at a saturation or maximum level This model is typically used in forecasting market share or sales growth of a new product from market introduction until maturity and decline population dynamics and other naturally occur
68. of each predicted coefficient to its standard error and are used in the typical regression hypothesis test of the significance of each estimated parameter To estimate the probability of success of belonging to a certain group eg predicting if a smoker will develop chest complications given the amount smoked per year simply compute the Estimated Y value using the MLE coefficients For example if the model is Y 1 1 0 005 Cigarettes then someone smoking 100 packs per year has an Estimated Y of 1 1 0 005 100 1 6 Next compute the inverse antilog of the odds ratio by EXP Estimated Y 1 EXP Estimated EXP 1 6 1 EXP 1 6 0 8320 So such a person has an 83 20 chance of developing some chest complications in his or her lifetime A Probit model sometimes also known as a Normit model is a popular alternative specification for a binary response model which employs a probit function estimated using maximum likelihood estimation and the approach is called probit regression The Probit and Logistic regression models tend to produce very similar predictions where the parameter estimates in a logistic regression tend to be 1 6 to 1 8 times higher than they are in a corresponding Probit model The choice of using a Probit or Logit is entirely up to convenience and the main distinction is that the logistic distribution has a higher kurtosis fatter tails to account for extreme values For example suppose that house ownership is the deci
69. only the dependent variable Y that is the Series Variable by itself or you can add in exogenous variables X7 X5 just like in a regression analysis where you have multiple independent variables You can run as many forecast periods as you wish if you use only the time series variable Y However if you add exogenous variables X note that your forecast period is limited to the number of exogenous variables data periods minus the time series variable s data periods For example you can only forecast up to 5 periods if you have time series historical data of 100 periods and only if you have exogenous variables of 105 periods 100 historical periods to match the time series variable and 5 additional future periods of independent exogenous variables to forecast the time series dependent variable In interpreting the results of an ARIMA model most of the specifications ate identical to the multivariate regression analysis see Modeling Risk Applying Monte Carlo Simulation Real Options Analysis Stochastic Forecasting and Portfolio Optimization Second Edition by Dr Johnathan Mun for more technical details about interpreting the multivariate regression analysis and ARIMA models There are however several additional sets of results specific to the ARIMA analysis as seen in Figure 3 14 The first is the addition of Akaike information criterion AIC and Schwarz criterion SC which are often used in ARIMA model selection and
70. rate average sale price quantity demanded of the product lines and so forth The report contains four distinct elements Results Interpretation A statistical summary listing the procedure performed A sensitivity table Figure 5 4 showing the starting NPV base value of 96 63 and how each input is changed e g Investment is changed from 1 800 to 1 980 on the upside with a 10 swing and from 1 800 to 1 620 on the downside with a 10 swing The resulting upside and downside values on NPV is 83 37 and 276 63 with a total change of 360 making investment the variable with the highest impact on NPV The precedent variables are ranked from the highest impact to the lowest impact A spider chart Figure 5 5 illustrating the effects graphically The y axis is the NPV target value while the x axis depicts the percentage change on each of the precedent values the central point is the base case value at 96 63 at 0 change from the base value of each precedent A positively sloped line indicates a positive relationship or effect while negatively sloped lines indicate a negative relationship e g Investment is negatively sloped which means that the higher the investment level the lower the NPV The absolute value of the slope indicates the 118 Page RISK SIMULATOR magnitude of the effect a steep line indicates a higher impact on the NPV y axis given a change in the precedent x axis A tornado chart illustrating the effe
71. rates set by senior management and market research or external data or polling and surveys data obtained from third party sources industry and sector indexes or active market research These estimates can be either single point estimates an average consensus or a set of forecast values a distribution of forecasts The latter can be entered into Risk Simulator as a custom distribution and the resulting forecasts can be simulated that is a nonparametric simulation using the estimated data points themselves as the distribution On the quantitative side of forecasting the available data or data that need to be forecasted can be divided into time series values that have a time element to them such as revenues at different years inflation rates interest rates market share failure rates cross sectional values that are time independent such as the grade point average of sophomore students across the nation in a particular year given each student s levels of SAT scores IQ and number of alcoholic beverages consumed per week or mixed panel mixture between time series and panel data e g predicting sales over the next 10 years given budgeted marketing expenses and market share projections which means that the sales data is time series but exogenous variables such as matketing expenses and market share exist to help to model the forecast predictions The Risk Simulator software provides the user several forecasting methodologies A
72. report e You can also automatically generate Models by entering a sample model and using the predefined INTEGER N variable as well as Sifting Data up down specific rows repeatedly For instance if you use the variable LLAG VZART INTEGER and you set INTEGER to be between MIN 1 and 3 then the following three models wil be run LAG VAR7 7 then LAG VAR7 2 and finally LAG V AR7 3 Also sometimes you might want to test if the time series data has structural shifts or if the behavior of the model is consistent over time by shifting the data and then running the same model For example if you have 100 months of data listed chronologically you can shift it down 3 months at a time for 10 times Le the model will be run on months 1 100 4 100 7 100 etc Using this Models section in Basic Econometrics you can run hundreds of models by simply entering a single model equation if you use these predefined integer variables and shifting methods 87 Page RISK SIMULATOR 88 Page Theory Procedure RISK SIMULATOR 3 10 J S Curve Forecasts The J curve or exponential growth curve is one where the growth of the next period depends on the current period s level and the increase is exponential This means that over time the values will increase significantly from one petiod to another This model is typically used in forecasting biological growth and chemical reactions over time Start
73. results provide the max returns or the min cost risks Uses include managing inventories financial portfolio allocation product mix project selection etc Objective Cell 52519 E Optimization Objective Maximize the value in objective cell Minimize the value in objective cell Figure 4 5 Running Discrete Integer Optimization in Risk Simulator Figure 4 6 shows a sample optimal selection of projects that maximizes the Sharpe ratio In contrast one can always maximize total revenues but as before this is a trivial process and simply involves choosing the highest returning project and going down the list until you run out of money or exceed the budget constraint Doing so will yield theoretically undesirable projects as the highest yielding projects typically hold higher risks Now if desired you can replicate the optimization using a stochastic or dynamic optimization by adding assumptions in the ENPV and or cost and or risk values For additional hands on examples of optimization in action see the case study in Chapter 11 on Integrated Risk Management in the book Real Options Analysis Tools and Techniques Second Edition Wiley Finance 2010 by Dr Johnathan Mun That case study illustrates how an efficient frontier can be generated and how forecasting simulation optimization and real options can be combined into a seamless analytical process 108 Page RISK SIMULATOR Return to Profitability R
74. selected as inputs to perturb You can set the starting and ending values to test as well as the step size or the number of steps to run between these starting and ending values The result is a scenario analysis table Figure 5 39 where the row and column headers are the two input variables and the body of the table shows the net present values E 150 P yq RISK SIMULATOR Distribution Analysis This tool generates the probability density function PDF cumulative distribution function CDF and the Inverse CDF ICDF of all the distributions in Risk Simulator including theoretical moments and probability chart Distribution Mu Sigma 0 00 Type E 2 23 0 74 0 74 Formatting Single Value Probability Range of Values Lower Bound Upper Bound Step Size Figure 5 37 Distributional Analysis Tool Normal Distribution s ICDF and Z Score 151 Page RISK SIMULATOR mi _ 1L 2 Discounted Cash Flow ROI Model 3 4 Base Year 2009 Sum PV Net Benefits 4 762 09 Discount Type Discrete End of Year Discounting 5 Start Year 2009 Sum PV Investments 1 634 22 6 Market Risk Adjusted Discount Rate 15 00 Net Present Value Model 7 Include Terminal Valuation 7 Private Risk Discount Rate 5 00 Intern
75. sensitivity charts show the values of the decision paths under varying probability levels The numerical values are shown in the results table The location of crossover lines if any represents at what probabilistic events a certain decision path becomes dominant over another 5 27 6 Scenatio Tables Scenario tables Figure 5 66 can be generated to determine the output values given some changes to the input You can choose one or more Decision paths to analyze the results of each path chosen will be represented as a separate table and chart and one or two Uncertainty or Terminal nodes as input variables to the scenario table Select or more Decision paths to analyze from the list below Select one or two Uncertainty Events or Terminal Payoffs to model e Decide if you wish to change the event s probability on its own or all identical probability events at once e Enter the input scenario range 5 27 7 Utility Function Generation Utility functions Figure 5 67 or U x are sometimes used in place of expected values of terminal payoffs in a decision tree U x can be developed two ways using tedious and detailed experimentation of every possible outcome or an exponential extrapolation method used here They can be modeled for a decision maker who is risk averse downsides more disastrous or painful than an equal upside potential risk neutral upsides and downsides have equal attractiveness or tisk loving upsid
76. series data that has seasonality and trend This methodology is found inside the ROV BizStats module in Risk Simulator at Rak Simulator ROV BizStats Combinatorial Fuzzy Logic as well as in Risk Simulator Forecasting Combinatorial Fuzzy Logic Click on Risk Simulator Forecasting Combinatorial Fuzzy Logic e Start by either manually entering data or pasting some data from the clipboard e g select and copy some data from Excel start this tool and paste the data by clicking on the Paste button 169 Page RISK SIMULATOR e Select the variable you wish to run the analysis on from the drop down list and enter in the seasonality period 4 for quarterly data 12 for monthly data etc and the desired number of Forecast Periods e g 5 Click Ryn to execute the analysis and review the computed results and charts You can also Copy the results and chart to the clipboard and paste it in another software application Note that neither neural networks nor fuzzy logic techniques have yet been established as valid and reliable methods in the business forecasting domain on either a strategic tactical or operational level Much research is still required in these advanced forecasting fields Nonetheless Risk Simulator provides the fundamentals of these two techniques for the purposes of running time series forecasts We recommend that you do not use any of these techniques in isolation but rather in combination with t
77. set of Trials 20 Probability of success 0 5 and Random X or Number of Successful Trials 10 where the Probability of Success is allowed to change from 0 0 25 0 50 and is shown as the row variable and the Number of Successful Trials is also allowed to change from 0 1 2 8 and is shown as the column variable PDF is chosen and hence the results in the table show the probability that the given events occur For instance the probability of getting exactly 2 successes when 20 trials are run where each trial has a 25 chance of success is 0 0669 or 6 69 162 Page RISK SIMULATOR ROV PROBABILITY DISTRIBUTIONS Distributi Charts and Tables Arcsine Minimum Maximum Random X Percentile PDF CDF ICDF Mean Stdev Skew Kurtosis Binomial Trials Probability Random X Percentile PDF CDF ICDF Mean Stdev Skew Decimals 4 Bernoulli Probability Random X Percentile PDF CDF ICDF Mean Stdev Skew Kurtosis 0 5000 0 5000 1 0000 0 5000 0 0000 2 0000 Cauchy Alpha Beta Random X Percentile PDF CDF ICDF Random X Percentile Kurtosis This tool lists all the probability distributions available in Real Options Valuation Inc s suite of products 06 05 PDF CDF ICDF Mean Stdev Skew Chi Square OF 0 4608 0 9590 0 2644 0 2857 0 1597 0 5963 0 1200 Location Random X Percentile PDF CDF ICDF Mean Stdev Skew Kurtosis 10 2
78. should never show a Tornado chart with only the key variables without showing some less critical variables as a contrast to their effects on the output e Default Values the default testing points can be increased from the 10 value to some larger value to test for nonlinearities the Spider chart will show nonlinear lines and Tornado charts will be skewed to one side if the precedent effects are nonlinear e Zero Values and Integers inputs with zero or integer values only should be deselected in the Tornado analysis before it is run Otherwise the percentage perturbation may invalidate your model e g if your model uses a lookup table where Jan 1 Feb 2 Mar 3 etc perturbing the value 1 at a 10 value yields 0 9 and 1 1 which makes no sense to the model e Chart Options try various chart options to find the best options to turn on or off for yout model TIPS Troubleshooter ROV Troubleshooter tun this troubleshooter to obtain your computer s HWID for licensing purposes to view your computer settings and prerequisites and to re enable Risk Simulator if it has been accidentally disabled 190 Page RISK SIMULATOR INDEX acquisition 137 156 allocation 98 99 101 102 103 104 105 110 111 alpha 45 52 69 70 126 137 analysis 1 5 9 11 12 31 33 36 51 54 67 68 69 70 72 73 74 76 78 80 81 93 98 99 110 113 115 117 119 120 121 122 124 125 133 134 137 139 1
79. shows another spider chart where nonlinearities are fairly evident the lines on the graph are not straight but cutved The model used is Tornado and Sensitivity Charts Nonlinear which uses the Black Scholes option pricing model as an example Such nonlinearities cannot be ascertained from a tornado chart and may be important information in the model or provide decision makers with important insight into the model s dynamics Figure 5 2 shows the Tornado analysis tool s user interface Notice that there are a few new enhancements starting in Risk Simulator version 4 and beyond Here are some tips on running Tornado analysis and details on the new enhancements e Tornado analysis should never be run just once It is meant as a model diagnostic tool which means that it should ideally be run several times on the same model For g 121 Page RISK SIMULATOR instance in a large model Tornado can be run the first time using all of the default settings and all precedents should be shown select Show All Variables The result may be a large report and long and potentially unsightly Tornado charts Nonetheless this analysis provides a great starting point to determine how many of the precedents are considered critical success factors For example the Tornado chart may show that the first 5 variables have high impact on the output while the remaining 200 variables have little to no impact in which case a second Tornado analysis is run s
80. significant The same approach can be applied to the F Statistic bv comparing the calculated F Statistic with the critical F values at various significance levels 83 RISK SIMULATOR Autocorrelation Time Lag AC PAC Lower Bound Upper Bound Q Stat Prob 1 0 9921 0 9921 0 0958 0 0958 431 1216 2 0 9841 0 0105 0 0958 0 0958 856 3037 09760 0 0109 0 0958 0 0958 12754818 4 09678 0 0142 0 0958 0 0958 16885499 5 0 9594 0 0098 0 0958 0 0958 20954625 6 0 9509 0 0113 0 0958 0 0958 24961572 7 0 9423 0 0124 0 0958 0 0958 28905594 8 0 9336 0 0147 0 0958 0 0958 327856689 9 0 9247 0 0121 0 0958 0 0958 3 660 1152 10 0 9156 0 0139 0 0958 0 0958 40351192 11 0 9066 0 0049 0 0958 0 0958 44035117 12 0 8975 0 0068 0 0958 0 0958 4 765 6032 13 0 8883 0 0097 0 0958 0 0958 5121 0697 14 0 8791 0 0087 0 0958 0 0958 5 470 0032 15 0 8698 0 0064 0 0958 0 0958 5 812 4256 16 0 8605 0 0056 0 0958 0 0958 5 148 3694 17 08512 0 0062 0 0958 0 0958 64778620 18 0 8419 0 0038 0 0958 0 0958 68009622 19 0 8326 0 0003 0 0958 00958 741177708 20 0 8235 0 0002 0 0958 0 0958 7 428 3952 If autocorrelation AC 1 is nonzero it means that the series is first order serially correlated If dies off more or less geometrically with increasing lag it implies that the series follows a low order autoregressive process If AC K drops to zero a
81. simply a waste of time and resources That is in optimization the model is put through a rigorous set of algorithms where multiple iterations ranging from several to thousands of iterations are required to obtain the optimal results Hence generating one iteration at a time is a waste of time and resources The same portfolio can be solved using Risk Simulator in under a minute as compared to multiple hours using such a backward approach Also such a simulation optimization approach will typically yield bad results and it is not a stochastic optimization approach Be extremely careful of such methodologies when applying optimization to your models The next two sections provide examples of optimization problems One uses continuous decision variables while the other uses discrete integer decision variables In either model you can apply discrete optimization dynamic optimization stochastic optimization or even the efficient frontiers with shadow pricing Any of these approaches can be used for these two examples Therefore for simplicity only the model setup is illustrated and it is up to the user to decide which optimization process to run Also the continuous model uses the nonlinear 100 RISK SIMULATOR optimization approach because the portfolio risk computed is a nonlinear function and the objective is a nonlinear function of portfolio returns divided by portfolio risks and integer optimization is an example of a li
82. such as Standard Deviations Variances Quartiles and Inter Quartile Ranges Standard Deviation indicates the average deviation of all data points from their mean Itis a popular measure as is associated with risk higher standard deviations mean a wider distribution higher risk or wider dispersion of data points around the mean and its units are identical to original data set s The Sample Standard Deviation differs from the Population Standard Deviation in that the former uses a degree of freedom correction to account for small sample sizes Also Lower and Upper Confidence Intervals are provided for the Standard Deviation and the true population standard deviation falls within this interval If your data set covers every element of the population use the Population Standard Deviation instead The two Variance measures are simply the squared values ofthe standard deviations The Coefficient of Variability is the standard deviation of the sample divided by the sample mean proving a unit free measure of dispersion that can be compared across different distributions you can now compare distributions of values denominated in millions of dollars with one in billions of dollars or meters and kilograms etc The First Quartile measures the 25th percentile of the data points when arranged from its smallest to largest value The Third Quartile is the value of the 75th percentile data point Sometimes quartiles are used as the upper and lower ranges of a distrib
83. the estimate of residual variance lowering the chance of rejecting the null hypothesis that is creating higher prediction errors They may be due to recording errors which may be correctable or they may be due to the dependent variable values not all being sampled from the same population Apparent outliers may also be due to the dependent variable values being from the same but non normal population However a point may be an unusual value in either an independent or dependent variable without necessarily being an outlier in the scatter plot In regression analysis the fitted line can be highly sensitive to outliers In other words least squares regression is not resistant to outliers thus neither is the fitted slope estimate A point vertically removed from the other points can cause the fitted line to pass close to it instead of following the general linear trend of the rest of the data especially if the point is relatively far horizontally from the center of the data However great care should be taken when deciding if the outliers should be removed Although in most cases when outliers are removed the regression results look better a priori justification must first exist For instance if one is regressing the performance of a particular firm s stock returns outliers caused by downturns in the stock market should be included these are not truly outliers as they are inevitabilities in the business cycle Forgoing these outliers and
84. to target a long term value making it useful for forecasting time series variables that have a long term rate such as interest rates and inflation rates these are long term target rates by regulatory authorities or the market The Jump Diffusion process is useful for forecasting time series data when the variable can occasionally exhibit random jumps such as oil prices or price of electricity discrete exogenous event shocks can make prices jump up or down Finally these three stochastic processes can be mixed and matched as required The results on the right indicate the mean and standard deviation of all the iterations generated at each time step If the Show All Iterations option is selected each iteration pathway will be shown in a separate worksheet The graph generated below shows a sample set of the iteration pathways Stochastic Process Brownian Motion Random Walk with Drift Start Value 100 Steps 50 00 Jump Rate N A Drift Rate 5 0096 Iterations 10 00 Jump Size N A Volatility 25 00 Reversion Rate N A Random Seed 1720050445 Horizon 5 Long Term Value N A Figure 3 10 Stochastic Forecast Result 77 Page Time 0 0000 0 1000 0 2000 0 3000 0 4000 0 5000 0 6000 0 7000 0 8000 0 9000 1 0000 1 1000 1 2000 1 3000 1 4000 1 5000 1 6000 1 7000 1 8000 1 9000 2 0000 2 1000 2 2000 2 3000 2 4000 2 5000 2 6000 2 7000 2 8000 2 9000 3 0000 3 1000 3 2000 3 3000 3 4000 3 5000 3 6000 3 7000 3 8000 3 9000 4
85. type and much more Hands on Exercises detailed step by step guide to running Risk Simulator including guides on interpreting the results Multiple Cell Copy and Paste allows assumptions decision variables and forecasts to be copied and pasted 5 Page RISK SIMULATOR 13 14 15 16 17 18 19 20 Profiling Aallows multiple profiles to be created in a single model different scenarios of simulation models can be created duplicated edited and run in a single model Revised Icons in Excel 2007 2010 a completely reworked icon toolbar that is more intuitive and user friendly There are four sets of icons that fit most screen resolutions 1280 x 760 and above Right Click Shortcuts access all of Risk Simulator s tools and menus using a mouse right click ROV Software Integration works well with other ROV software including Real Options SLS Modeling Toolkit Basel Toolkit ROV Compiler ROV Extractor and Evaluator ROV Modeler ROV Valuator ROV Optimizer ROV Dashboard ESO Valuation Toolkit and others RS Functions in Excel insert RS functions for setting assumptions and forecasts and right click support in Excel Troubleshooter allows you to re enable the software check for your system requirements obtain the Hardware ID and others Turbo Speed Analysis runs forecasts and other analyses tools at blazingly fast speeds enhanced in version 5 2 The analyses and results remain th
86. values 11 14 15 19 20 21 23 24 25 26 27 28 33 36 37 45 46 48 51 54 56 58 61 65 66 67 68 69 76 78 80 81 88 93 94 96 97 98 99 100 101 102 105 107 110 115 117 121 126 133 134 137 138 139 142 146 147 149 153 154 157 161 168 177 182 183 184 185 189 vatiance 31 32 47 51 54 57 60 74 92 113 133 136 137 142 volatility 7 12 66 76 89 90 140 183 Weibull 6 62 63 124 161 182 183 Yes No 38 200 RISK SIMULATOR Copyright 2005 2012 Dr Johnathan Mun All rights reserved Real Options Valuation Inc 4101F Dublin Blvd Ste 425 Dublin California 94568 U S A Phone 925 271 4438 Fax 925 369 0450 admin realoptionsvaluation com www tisksimulator com www tealoptionsvaluation com 201 Page
87. 0 0204 0 0673 0 2828 0 2828 9 3276 0 4076 10 0 0190 0 0865 0 2828 0 2828 9 3512 0 4991 11 0 1035 0 0790 0 2828 0 2828 10 0648 0 5246 12 0 1658 0 0878 0 2828 0 2828 11 9466 0 4500 13 0 0524 0 0430 0 2828 0 2828 12 1394 0 5162 14 0 2050 0 2523 0 2828 0 2828 15 1738 0 3664 15 0 1782 0 2089 0 2828 0 2828 17 5315 0 2881 16 0 1022 0 2591 0 2828 0 2828 18 3296 0 3050 17 0 0861 0 0808 0 2828 0 2828 18 9141 0 3335 18 0 0418 0 1987 0 2828 0 2828 19 0559 0 3884 19 0 0869 0 0821 0 2828 0 2828 19 6894 0 4135 20 0 0091 0 0269 0 2828 0 2828 19 6966 0 4770 Distributive Lags P Values of Distributive Lag Periods of Each Independent Variable Variable 1 2 3 4 5 6 7 8 9 10 11 12 1 0 8467 0 2045 0 3336 0 8105 0 9757 0 1020 0 9205 0 1267 0 5431 0 9110 0 7495 0 4016 x2 0 6077 0 9900 0 8422 0 2851 0 0638 0 0032 0 8007 0 1551 0 4823 0 1126 0 0519 0 4383 x3 0 7394 0 2396 0 2741 0 8372 0 9808 0 0464 0 8355 0 0545 0 6828 0 7354 0 5093 0 3500 X4 0 0061 0 6739 0 7932 0 7719 0 6748 0 8627 0 5586 0 9046 0 5726 0 6304 0 4812 0 5707 x5 0 1591 0 2032 0 4123 0 5599 0 6416 0 3447 0 9190 0 9740 0 5185 0 2856 0 1489 0 7794 Figure 5 24 Autocorrelation and Distributive Lag Results Another requirement in running a regression model is the assumption of normality and sphericity of the error term If the assumption of normality is violated or outliers are present then the linear regression goodness of fit test may not be the most powerful or informat
88. 00 010 0 000000 000 0 000000 000 Variables variable Initial Lower Upper No Name Status value Bound Bound Optimal values have been found Do you wish to replace the existing decision variables with the optimized values or revert to the original inputs Figure 4 6 Optimal Selection of Projects That Maximizes the Sharpe Ratio 4 4 Efficient Frontier and Advanced Optimization Settings The middle graphic in Figure 4 5 shows the constraints set for the example optimization Within this function if you click on the Efficient Frontier button after you have set some constraints you can make the constraints changing That is each of the constraints can be created to step through between some maximum and minimum value As an example the constraint in cell 77 lt 6 can be set to run between 4 and 8 Figure 4 7 Thus five optimizations will be run each with the following constraints J17 lt 4 J17 lt 5 J17 lt 6 J17 lt 7 and J17 lt 8 The optimal results will then be plotted as an efficient frontier and the report will be generated Figure 4 8 Specifically here are the steps required to create a changing constraint e n an optimization model Le model with Objective Decision Variables and Constraints already set up click on Rask Simulator Optimization Constraints and click on Efficient Frontier e Select the constraint you want to change or step e g J17 enter in the parameters
89. 0000 4 1000 4 2000 4 3000 4 4000 4 5000 4 6000 4 7000 4 8000 4 9000 5 0000 Mean 100 00 106 32 105 92 105 23 109 84 107 57 108 63 107 85 109 61 109 57 110 74 111 53 111 07 107 52 108 26 106 36 112 42 110 08 109 64 110 18 112 23 114 32 111 14 111 03 112 04 112 98 115 74 115 11 114 87 113 28 115 72 120 05 116 69 118 31 116 35 115 71 118 69 121 66 121 40 125 19 129 65 129 61 125 86 125 70 126 72 129 52 132 28 138 47 139 69 140 85 143 61 Stdev 0 00 4 05 4 70 8 23 11 18 14 67 19 79 24 18 24 46 27 99 30 81 35 05 34 10 32 85 37 38 32 19 32 16 31 24 31 87 36 43 37 63 33 10 38 42 37 69 37 23 40 84 43 69 43 64 43 70 42 25 43 43 50 48 42 61 45 57 40 82 40 33 41 45 45 34 45 03 48 19 55 44 53 82 49 68 53 79 49 70 50 28 49 70 56 77 66 32 65 95 68 65 Theory Procedure Results Interpretation Notes RISK SIMULATOR 3 6 Nonlinear Extrapolation Extrapolation involves making statistical projections by using historical trends that are projected for a specified period of time into the future It is only used for time series forecasts For cross sectional or mixed panel data time series with cross sectional data multivariate regression is more appropriate Extrapolation is useful when major changes are not expected that is causal factors are expected to remain constant or when the causal factors of a situation are not clearly understood It also hel
90. 1 Quartile 7 3rd Quartile Show Decimals FE as Histogram Resolution Chart X Axis 4 4 j Statistics 4 Faster i Display Control 7 Always Show Window On Top Data Update Interval Semitransparent When Inactive Faster gt Update PValue 0 2782 Figure 2 9 Forecast Chart Global View 2 3 4 Using Forecast Charts and Confidence Intervals In forecast charts you can determine the probability of occurrence called confidence intervals That is given two values what ate the chances that the outcome will fall between these two values Figure 2 10 illustrates that there is a 90 probability that the final outcome in this case the level of income will be between 0 2653 and 1 3230 The two tailed confidence interval can be obtained by first selecting T 0 Tai as the type entering the desired certainty value e g 90 and hitting on the keyboard The two computed values corresponding to the certainty value will then be displayed In this example there is a 5 probability that income will be below 0 2653 and another 5 probability that income will be above 1 3230 That is the two tailed confidence interval is a symmetrical interval centered on the median or 50th percentile value Thus both tails will have the same probability Alternatively a one tail probability can be computed Figure 2
91. 1050 ___5208 Prod Avg Price Prod A Quantity Prod B Quantity Sensitivity analysis creates dynamic perturbations i e Prod C Quantity Total Cost of G Gross Operatin SG amp A Cc Opera Deprecia Amortizal EBIT Interest F EBT Taxes Net In Deprecia Change Capital E Free InvestMems multiple assumptions are perturbed simultaneously to identify the impact to the results It is used to identify critical success factors of the forecast Select the forecast s on which to run dynamic sensitivity analysis Forecast Name Worksheet Cell Net Present Value Financial Analysis Present Value of Free Cash Flow 528 24 440 60 gt 62 Present Value of Investment Outlay 1 800 00 0 00 0 00 0 00 0 00 Net Cash Flows 1 271 76 506 69 485 70 465 25 445 33 Figure 5 10 Running Sensitivity Analysis The results of the sensitivity analysis comprise a report and two key charts The first is a nonlinear rank correlation chart Figure 5 11 that ranks from highest to lowest the assumption forecast correlation pairs These correlations are nonlinear and nonparametric making them free of any distributional requirements 1 an assumption with a Weibull distribution can be compared to another with a beta distribution The results from this chart are fairly similar to that of the tornado analysis seen previously of course without the capital investment value wh
92. 11 shows a left tail selection at 95 confidence i e choose Left Tai as the type enter 25 as the certainty level and hit TAB on the keyboard This means that there is a 95 probability that the income will be below 1 3230 or a 5 probability that income will be above 1 3230 corresponding perfectly with the results seen in Figure 2 10 25 Page RISK SIMULATOR Income Risk Simulator Forecast Type lt Figure 2 11 Forecast Chart One Tail Confidence Interval In addition to evaluating what the confidence interval is Le given a probability level and finding the relevant income values you can determine the probability of a given income value For instance what is the probability that income will be less than or equal to 1 To obtain the answer select the L Tai probability type enter 7 into the value input box and hit TAB The corresponding certainty will then be computed in this case as shown in Figure 2 12 there is a 67 70 probability income will be at or below 1 For the sake of completeness you can select the Rzo 12 gt probability type and enter the value 7 in the value input box and hit TAB The resulting probability indicates the right tail probability past the value 1 that is the probability of income exceeding 1 in this case as shown in Figure 2 13 we see that there is a 32 30 probability of income exceeding 1 The sum o
93. 14 0 1038 0 0362 Conclusion The sample data is normally distributed at 127 00 0 02 0 16 0 1180 0 0420 the 175 alpha level 153 00 0 02 0 18 0 1504 0 0296 177 00 0 02 0 20 0 1851 0 0149 186 00 0 02 0 22 0 1994 0 0206 188 00 0 02 0 24 0 2026 0 0374 198 00 0 02 0 26 0 2193 0 0407 222 00 0 02 0 28 0 2625 0 0175 231 00 0 02 0 30 0 2797 0 0203 240 00 0 02 0 32 0 2975 0 0225 246 00 0 02 0 34 0 3096 0 0304 251 00 0 02 0 36 0 3199 0 0401 265 00 0 02 0 38 0 3494 0 0306 280 00 0 02 0 40 0 3820 0 0180 285 00 0 02 0 42 0 3931 0 0269 286 00 0 04 0 46 0 3953 0 0647 291 00 0 02 0 48 0 4065 0 0735 303 00 0 02 0 50 0 4336 0 0664 311 00 0 02 0 52 0 4519 0 0681 Figure 5 32 Sample Statistical Analysis Tool Report Normality Test 146 Page RISK SIMULATOR Stochastic Process Parameter Estimations Statistical Summary stochastic process is a sequence of events or paths generated by probabilistic laws That is random events can occur over time but are governed by specific statistical and probabilistic rules The main stochastic processes include Random Walk or Brownian Motion Mean Reversion and Jump Diffusion These processes can be used to forecast a multitude of variables that seemingly follow random trends but yet are restricted by probabilistic laws The process generating equation is known in advance but the actual results generated is unknown The Random Walk Brownian Motion process can be used to forecast stock prices prices of commodities
94. 179 Page RISK SIMULATOR ROV Visual Modeler 2012 Decision Trees C Users user Desktop Screen Shots DT Model rovdt File Edit Insert Properties Style Colors Language Help 5 0 9 0 4 7 P A Decision Tree Summary of Values Simulation Modeling Bayesian Analysis EVPI Minimax Risk Profile Sensitivity Analysis Scenario Tables Utility Function This Bayesian analysis tool can be performed on any two uncertainty events that are linked along a path For instance in the example on amy 62 074 the right uncertainties A and B are linked where event A occurs first in the timeline and event B occurs second First Event A is Market a Research with 2 outcomes Favorable Unfavorable Second Event B is Market conditions also with 2 outcomes Strong and Weak This 45 AND 55 37 93 ee les or reliability je an yds on ond ample te cate cor emu he ca Nd shown in the grid on the right as well as which results are used as inputs in the decision tree in the figure Uncertainty u 22 86 re 56 506 a ana teia 40 Unfavorabie given Strong e iaa Compute Bayesian updated Posterior Probabilities given Prior and Reliability Joint Probabilities more Bayesian Analysis Results Compute Reliabilty Joint Probabilties g
95. 2 73 74 78 80 81 85 86 93 94 136 137 138 139 140 141 Regression 7 9 65 67 68 72 73 74 75 80 86 136 regression analysis 66 67 72 81 85 86 93 94 136 137 relative returns 66 89 100 110 report 8 14 69 74 76 78 81 86 87 88 89 92 95 96 97 108 111 117 121 124 126 135 136 138 140 142 155 156 157 158 164 165 185 188 return 31 67 92 93 100 101 103 104 110 121 161 172 183 returns 31 32 33 66 76 85 99 100 101 102 103 104 105 110 111 137 153 186 tisk 5 8 11 12 28 31 32 33 34 36 52 67 100 101 102 105 104 105 107 110 111 115 119 172 173 175 177 Risk Simulator 1 2 3 4 5 6 12 13 14 15 16 17 18 19 25 26 27 29 30 31 33 37 65 66 67 68 69 70 73 76 78 81 85 86 87 88 89 90 91 92 94 95 97 98 99 100 101 102 103 105 107 108 111 115 116 120 123 126 129 130 gt T 197 Page RISK SIMULATOR 132 134 135 136 138 143 146 148 149 152 153 154 155 156 157 158 160 161 164 168 169 170 171 172 173 182 183 185 186 187 188 189 running 5 6 14 19 21 28 73 87 98 108 119 120 130 137 139 153 164 169 175 185 186 sales 41 42 61 65 67 68 72 88 138 154 sample 11 29 34 35 41 60 69 74 76 87 88 100 107 110 115 126 130 131 132 137 139 143 149 152 153 161 164 176 save 3 14 134 164 187 saving 5 134 18
96. 212 33 223 35 228 86 234 38 2 46 1 140 70 171 94 1 203 18 1 234 41 1 265 65 1 296 89 1 328 13 Optimization gt Create Forecast Statistics Table 157 50 157 50 157 50 157 50 1575 15 75 15 75 15 75 15 75 15 75 Data Deseasonalization amp Detrending 967 45 998 69 7 029 93 1 061 16 1 092 40 1 123 64 Options 10 00 510 00 ___510 00 ___510 00 __ 51000 510 00 Languages gt Data EMTACHOTNEXDOHE 3 00 3 00 3 00 3 00 3 00 3 00 3 00 Data Open Import 954 45 985 69 1 016 93 1 048 16 1 079 40 1 110 64 1 141 88 License x 7 00 About Risk Simulator Distributional Analysis eee e Check for Updates Distributional Charts amp Tables Distributional Designer 0 00 0 00 WWWWRRKE User Manual Help Distributional Fitting Single Variable 0 00 0 00 35 7 Distributional Fitting Multi Variable 36 Investment Outlay dit Correlations 200 37 0 Hypothesis Testing 38 Net Free Cash Flow WB 5 73 _ 5584 47 603 21 621 36 5657 64 5675 78 55 444 64 39 d 40 Financial Analysis Overlay Charts 41 Present Value of Free principal component Analysis 77 5384 30 344 89 5308 92 276 47 5247 23 220 91 1 547 71 42 Present Value of Investmt 22 50 00 0 00 0 00 0 00 0 00 0 00 0 00 43 Discount
97. 3 0 4513 24 140 699997 141 3839 0 6939 25 141100006 141 0731 0 0270 26 141500006 141 8311 0 2311 27 141 839994 142 2065 0 3065 28 142100006 142 4709 0 3709 29 142 699997 142 6402 0 0598 30 142 899994 143 4561 0 5561 31 142899994 143 3532 0 4532 32 143 5 143 4040 0 0960 33 143 800003 144 2784 0 4784 34 144 100006 144 2966 0 1966 35 144 800003 144 7374 0 0626 36 145 199997 145 5692 0 3692 37 145199997 145 7582 0 5582 38 145599997 145 6649 0 0351 38 146 146 4605 0 4605 40 146 399994 146 5176 0 1176 41 146 800003 147 0891 0 2891 42 146 600006 147 4066 0 8066 Figure 3 14 Box Jenkins ARIMA Forecast Report 84 Page Theory Procedure Notes RISK SIMULATOR 38 AUTO ARIMA Box Jenkins ARIMA Advanced Time Series While the analyses are identical AUTO ARIMA differs from ARIMA in automating some of the traditional ARIMA modeling It automatically tests multiple permutations of model specifications and returns the best fitting model Running the Auto ARIMA is similar to regular ARIMA forecasting with the difference being that the P D Q inputs are no longer required and different combinations of these inputs ate automatically run and compared e Start Excel and enter your data or open an existing worksheet with historical data to forecast the illustration shown in Figure 3 15 uses the example file Advanced Forecasting Models in the Examples menu of Risk Simulator e Inthe Auto ARIMA worksheet
98. 3 00 4 00 5 00 6 00 7 00 28 EBT 858 74 889 98 921 21 952 45 983 69 1 013 93 1 044 16 1 074 40 1 104 64 1 134 88 29 Taxes 343 50 355 99 368 49 380 98 393 48 405 57 417 67 429 76 441 86 453 95 30 Net Income 515 24 533 99 552 73 571 47 590 21 608 36 626 50 644 64 662 78 680 93 31 Noncash Depreciation Amortization 13 00 13 00 13 00 13 00 13 00 13 00 13 00 13 00 13 00 13 00 32 Noncash Change in Net Working Capital 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 33 Noncash Capital Expenditures 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 34 Free Cash Flow 528 24 546 99 565 73 584 47 603 21 621 36 639 50 657 64 675 78 5 444 64 35 36 Investment Outlay 500 00 1 500 00 Figure 5 38 Scenatio Analysis Tool SCENARIO ANALYSIS TABLE Output Variable 5656 Initial Base Case Value 3 127 87 Column Variable 12 Mii 10 Max 30 Steps Initial Base Case Value 10 00 Row Variable 5059 Min 0 3 Max 0 5 Steps 0 01 Initial Base Case Value 40 00 10 00 11 00 12 00 13 00 14 00 15 00 16 00 17 00 18 00 19 00 20 00 21 00 22 00 23 00 24 00 25 00 26 00 27 00 28 00 29 00 30 00 30 00 3 904 83 4 134 43 4 364 04 4 593 64 4 823 24 5 052 84 5 282 44 5 512 04 5 741 64 5 971 24 6 200 85 6 430 45 6 660 05 6 889 65 7 119 25 7 348 85 7 578 45 7 808 05 8 037 65 8 267 26 8 496 86 31 004 3 827 14 4 053 46 4 279 78 4 506 10 4 732 42 4 958 74
99. 30 00 e Completion TimeB 70 00 change all identical probabiliies payoffs im Critical 1 2 5 00 10 00 15 00 20 00 E ME a iti 12220 12418 12616 12814 13012 Analyze individual probability payoff line If performing a grouped analysis review the group Time At members select any additional group members or 12 Days 30 00 PEREGO mut T 14 Days 50 00 AE T a ds T1 Time zar 2095 zx Node and ID Ims jJ r9 20 5 6 e r 18 Days i Critical wre Completion Time 2 Saved Model Name Model 1 Model 1 Probabilities Figure 5 65 ROV Decision Tree Sensitivity Analysis 4 ROV Visual Modeler 2012 Decision Trees CAU Shots DT Modelrovdt i Sk E File Edit Insert Properties Style Shapesand Colors Language Help 2 0 9 4 2 2 0 amp 4ze Decision Tree I Summary of Values Simulation Modeling Bayesian Analysis I EVPI Minimax Risk Profile Sensitivity Analysis Scenario Tables Utility Function Scenario tables can be generated to determine the output values given some changes to the input You Step 4 Enter the input scenario range can choose one or more Decision paths to analyze the results of each path chosen will be represented
100. 31 437 597 197 General Number of Trials 1000 Stop Simulation on Error No Random Seed 123456 Enable Correlations Yes Assumptions Name aple First Assumption Name e Second Assumption Name iple Third Assumption Enabled Yes Enabled Yes Enabled Yes Celt SESS Celt SESS E 10 Dynamic Simulation No Dynamic Simulation No Dynamic Simulation No Range Range Range Minimum Infinity Minimum Infinity Minimum Infinity Maximum Infinity Maximum Infinity Maximum Infinity Distribution Normal Distribution Triangular Distribution Beta Mean 100 Minimum 10 Aipha 2 Standard Deviation 10 Most Likely 0 Beta 5 Maximum 10 Forecasts Name Sample First Forecast Number of Datapoints 1000 Enabled Yes Mean 100 0400 12 Median 99 8427 Standard Deviation 9 8331 Forecast Precision Variance 96 6903 Precision Level Average Deviation 7 8397 E Error Level Maximum 134 5452 E Minimum 66 9132 E Range 67 6320 E Skewness 0 1121 E Kurtosis 0 1401 E 2596 Percentile 93 3563 7596 Percentile 106 3153 149 12992 Error Precision at 9596 0 0061 nple Second Forecast Number of Datapoints 1000 Enabled Yes Mean 0 0806 Cell 5313 Median 0 0755 Standard Deviation 41171 Forecast Precision Variance 16 9506 Precision Level Average Deviation 3 3389 Enor Level Maximum 9 3923 Minimum 9 7671 3 Range 19 1594 E Skewness 0 0494 B Kurtosis 0 5394 25 Percentile 2 8924 75 Percentile 2 8015 Error Precision at 95 3 1644
101. 34 00 Min expected value payoff to generate the start of the U x curve EV TerminalPayoffs U1 x Maximum Expected Value 435 50 expected value to generate the end of the U x curve 34 0000 0 1446 248 0000 0 6798 42 1939 0 1762 286 0000 0 7311 U x Points to Compute 50 Number of steps to take between the min and max U x 50 3878 0 2066 362 0000 0 8103 If you had 50 50 gamble where you either earn X lose 5 2 versus not playing and getting a 0 payoff what would es 0 2359 Se 9 12020 this X be For example if you are indifferent between bet where you win 100 or lose 50 with equal probability 66 7755 0 2641 48 0000 0 1978 compared to not playing at all then your X is 100 Enter the X in the Positive Earnings box below Note that the larger X is 74 9694 0 2913 56 0000 0 2268 the less risk averse you are whereas a smaller X indicates that you are more risk averse 83 1633 0 3175 306 0000 0 7547 Ee E UU obtain the results You can also apply the computed U x values to the m 108 88 decision tree to re run it or revert the tree back to using expected values of 107 7449 0 3903 34 0000 0 1446 the payoffs 115 9388 0 4128 37 0000 0 1563 o sana a anar rann asnan ld 669345 l B Risk Averse Utity Function Positive Ubity Values Orly 0 09 20 0 9 0 ew OF 20 5 09 qi s 8 19 U3 Risk Averse Utility Function Calibrated between 0 and 1 U4 Risk Averse Uti
102. 40 143 149 156 168 169 172 176 177 183 186 188 189 annualized 90 100 101 110 approach 11 13 26 36 64 65 66 67 72 78 80 92 93 94 98 99 100 104 110 113 119 126 138 141 153 186 ARIMA 1 7 9 12 65 66 68 80 81 82 84 85 86 88 89 92 95 97 138 184 asset 31 99 100 101 103 104 110 111 asset classes 100 101 110 assumption 11 14 15 16 17 19 36 37 80 92 98 99 111 113 122 124 126 137 139 175 182 183 185 assumptions 5 6 8 11 12 13 14 15 16 17 26 27 28 37 65 67 69 70 76 88 89 91 92 98 101 102 105 107 110 111 113 115 122 123 125 126 134 135 137 139 158 161 171 175 182 183 184 186 187 autocorrelation 8 66 73 81 138 140 141 behavior 45 47 66 78 86 87 126 140 Beta 45 46 47 49 51 53 54 56 57 58 62 63 161 183 binomial 6 38 39 40 41 42 43 93 146 147 161 Binomial 38 41 147 148 149 bootstrap 1 12 130 131 Bootstrap 8 130 131 Box Jenkins 1 12 66 80 82 84 85 Brownian Motion 10 140 183 causality 142 191 Page RISK SIMULATOR center of 31 118 137 coefficient of determination 136 confidence interval 23 24 29 60 130 132 148 constraints 12 98 99 100 102 105 108 111 171 186 187 Continuous 36 45 98 100 103 104 111 correlation 7 8 11 14 26 27 28 37 80 81 124 138 141 142 160 182 183 correlation coefficient
103. 44 00 40 00 3 43 189 83 186 40 9 00 11 00 10 00 16 71 176 55 159 84 11 03 13 48 12 25 23 18 170 07 146 90 45 00 55 00 50 00 30 53 162 72 132 19 31 50 38 50 35 00 40 15 153 11 13 64 16 67 15 15 48 05 145 20 18 00 22 00 20 00 138 24 57 03 13 50 16 50 15 00 116 80 76 64 4 50 5 50 5 00 90 59 102 69 1 80 2 20 2 00 95 08 98 17 9 00 11 00 10 00 97 09 96 16 1 80 2 20 2 00 96 16 97 09 2 70 3 30 3 00 96 63 96 63 E 0 00 0 00 0 00 96 63 96 63 0 00 0 00 0 00 Sales Growth 0 022 Depreciation a Interest 22 18 Amortization 27 33 Capex 0 Net Capital 0 119 Page Notes RISK SIMULATOR Figure 5 3 Tornado Analysis Report Remember that tornado analysis is a static sensitivity analysis applied on each input variable in the model that is each variable is perturbed individually and the resulting effects are tabulated This approach makes tornado analysis a key component to execute before running a simulation One of the very first steps in risk analysis is capturing and identifying the most important impact drivers in the model The next step is to identify which of these important impact drivers are uncertain These uncertain impact drivers are the critical success drivers of a project where the results of the model depend on these critical success drivers These variables are the ones that should be simulated Do not waste time si
104. 465 25 445 33 117 Page RISK SIMULATOR Figure 5 1 Sample Model B E rcm H J K L 2 Discounted Cash Flow Model 3 4 Base Year 2005 Sum PV Net Benefits 1 896 63 5 Market Risk Adjusted Discount Rate 15 0096 Sum PV investments 1 900 00 5 Private Risk Discount Rate 5 00 Net Present Value 96 63 1 E Annualized Sales Growth Rate 2 00 Interna Rate of Return 18 80 8 Price Erosion Rate 5 00 Return on Investment 5 37 3 Effective Tax Rate 40 00 10 Tornado Analysis 11 2005 2006 12 Prod A Avg Price 10 00 9 50 Tomado a creates static perturbations i e each precedent 4 is perturbed one at a time to identify the impact to the results It is 13 Prod B Avg Price 12 25 11 64 used to identify critical success factors of a model before running 14 Prod C Avg Price 15 15 14 38 simulations 15 Prod A Quantity 50 00 51 00 16 Prod B Quantity 35 00 35 70 Review the precedents below and make any necessary changes 17 Prod C Quantity 20 00 20 40 Selection Name Worksheet Cell Base Value Upside Downside Test Points 18 Total Revenues 1231 75 1 193 57 V Market DCFMode C5 015 10 00 10 002 19 Cast of Goods Sold 184 76 179 03 V Investm DCF Mode C36 1800 10 00 10 00 20 Gross Profit 1 046 99 1 014 53 9 DCF Mode 0 10 00 10 007 21 Operating Expenses 157 50 150 65 Changei DCF Mode C32 0 10 00 10 00 22
105. 5 05 Cosine 10 20 155 05 Minimum Maximum Random X Percentile PDF CDF ICDF Mean 23730 0 4661 10 2644 10 2857 0 1597 0 5963 0 1200 0 1551 0 5782 15 0000 15 0000 21762 0 0000 0 5938 DF Numerator 10 DF Denominator 20 Pop Success 50 Alpha Location Factor Random X Percentile PDF CDF ICDF Mean Stdev Skew Kurtosis 1 5552 0 7667 10 5289 10 5714 0 3194 0 5963 0 1200 Discrete Uniform Minimum Maximum Random X Percentile PDF ICDF Mean Stdev Skew Kurtosis Run ROV PROBABILITY DISTRIBUTIONS Distributions Charts and Tables Overlay Chart tool Distribution Arcsine A Charts and Tables Minimum Maximum Random X 0 0 Chart Table 10 20 12 PDF B Result Parameter Change First Parameter Change Second Parameter This tool generates a probability table and comparative charts for a chosen distribution as well as the different shapes based on different input parameters To view multiple distributions use Risk Simulator s Chart Theoretical Distribution a a Simulated Distribution Trials 10000 Sed 123 Ge l kE 45669355 11 2669 12 3338 13 4007 14 4675 15 5344 16 6013 17 6682 18 7351 Chart Type 2D Area Gridii
106. 5 188 models 2 5 6 7 9 12 14 28 58 66 67 69 70 80 81 86 87 89 90 92 94 99 105 138 154 155 164 165 169 173 184 185 187 Monte Carlo 1 5 6 11 12 13 14 17 19 26 27 29 35 36 37 65 67 70 73 74 76 81 98 153 173 175 186 multicollinearity 8 12 73 136 141 142 Multinomial SLS 2 195 Page RISK SIMULATOR 1 2 6 8 9 11 12 13 14 18 30 37 66 67 72 73 76 81 85 87 99 110 113 121 122 126 129 141 143 149 156 160 161 164 165 177 182 183 184 185 186 187 multiple multiple regression 1 67 141 multiple variables 8 87 129 143 156 160 164 165 multivariate 8 12 72 73 74 78 80 81 93 153 Mum 0 i 1 65 70 73 74 76 81 89 90 107 negative binomial 6 41 42 43 148 nonlinear 1 7 9 12 66 67 68 73 78 80 96 98 99 115 121 124 125 137 142 158 189 normal 6 11 17 26 30 33 36 39 47 53 54 55 58 60 76 91 92 93 94 126 130 137 148 153 175 183 184 Normal 6 21 55 91 94 150 153 175 182 184 null hypothesis 81 126 133 137 138 139 157 objective 12 98 99 100 101 102 105 111 optimal 98 99 103 104 107 108 113 137 176 optimal decision 99 113 176 optimization 1 7 8 9 12 66 89 93 98 99 100 101 102 103 104 105 107 108 110 111 113 129 171 172 186 option 2 17 69 70 120 121 126 134 161 165 172 173 174 175 182 out
107. 7 seasonality 9 12 68 69 70 138 154 155 168 169 second moment 31 33 34 159 sensitivity 1 9 12 117 119 122 124 125 177 Sensitivity 5 9 115 119 120 122 123 124 173 175 177 180 significance 8 60 81 94 130 133 137 138 139 142 157 simulation 1 5 6 7 8 11 12 13 14 15 17 18 19 21 25 26 27 28 29 35 36 37 65 67 69 76 98 99 101 102 105 107 110 111 113 115 119 122 125 124 125 126 130 131 132 134 135 153 158 171 173 175 182 185 186 187 188 Simulation 1 6 7 8 11 12 13 14 15 17 18 19 25 27 35 36 65 68 70 73 74 76 81 98 110 113 115 130 131 134 135 153 175 178 186 188 single 6 8 14 19 21 65 67 69 70 72 87 99 113 116 126 137 143 153 160 164 170 171 186 188 Single Asset SLS 2 skew 31 32 34 Skew 32 33 34 skewness 32 33 38 39 40 41 42 44 46 47 50 51 52 53 54 55 56 60 61 62 130 SLS 1 2 6 gt gt gt Spearman 26 27 142 gt 9 specification errors 136 spider 1 9 12 116 117 120 122 spread 28 31 198 Page RISK SIMULATOR standard deviation 11 17 21 28 30 31 32 33 34 36 39 41 44 46 47 50 51 52 54 55 60 61 76 94 98 99 113 126 131 133 139 142 148 175 182 static 9 12 98 101 102 105 108 115 119 122 140 182 statistics 1 8 9 12 19 20 21 26 28 29 31 34 81 98 99 113 126 130 131
108. 7 88 89 92 95 97 136 138 158 168 184 185 forecasts 5 6 8 12 13 15 17 18 26 30 65 66 68 69 70 78 80 90 98 123 132 133 134 135 138 158 161 169 182 184 187 fourth moment 31 33 34 Frequency 35 36 functions 1 6 7 11 15 16 18 34 36 78 87 98 105 111 121 138 153 173 177 182 185 188 193 Page RISK SIMULATOR functions of 138 gallery 16 17 gamma 6 47 52 60 69 70 148 Gamma 46 49 52 57 58 62 geometric 6 40 41 55 76 100 110 Geometric 10 38 40 geometric average 100 110 goodness of fit 138 139 goodness of fit tests 138 growth 53 65 67 80 88 110 126 140 growth rate 65 67 80 88 126 140 heteroskedasticity 7 8 12 66 73 89 136 137 138 139 Histogram 21 35 36 Holt Winter 10 69 71 hypergeometric 6 40 41 Hypergeometric 40 hypothesis 1 12 47 51 60 74 81 94 129 130 132 133 139 142 157 icon 3 6 15 16 18 19 85 89 97 101 105 111 173 174 184 icons 6 16 85 111 161 165 182 187 independent variable 66 67 72 73 81 85 86 87 93 94 137 139 141 142 156 185 inflation 55 65 138 140 142 inputs 13 15 17 27 45 46 50 53 55 61 62 66 67 68 76 85 92 98 105 111 113 115 121 149 153 160 161 172 176 177 184 185 189 installation 2 3 integer 1 12 14 39 41 43 47 49 52 60 67 70 87 98 99 105 121 189 interest 65 66 96 13
109. 8 140 149 152 interest rate 65 66 96 138 140 investment 99 105 110 117 118 119 124 jump diffusion 2 12 67 76 140 183 Kolmogorov Smirnov test 129 kurtosis 33 47 56 94 194 Page RISK SIMULATOR lags 9 73 80 81 138 least squares 73 93 94 137 least squares regression 93 94 137 linear 7 12 26 66 68 72 80 93 96 98 100 137 139 141 142 158 Ljung Box Q statistics 81 138 logistic 6 7 12 48 53 54 67 88 93 94 Lognormal 54 55 lower 14 27 32 40 48 54 57 60 101 110 117 124 126 management 58 65 113 126 152 market 7 26 33 65 67 88 92 126 137 140 142 152 176 matrix 26 27 141 160 182 183 mean 2 11 12 17 20 21 28 29 31 32 33 36 39 46 47 53 54 55 60 66 67 76 89 92 94 99 113 126 131 137 139 140 148 155 156 175 182 183 185 Mean 9 34 39 41 47 50 51 52 53 54 55 60 61 91 140 183 mean reversion 12 67 76 140 183 185 mix 141 model 6 7 8 11 13 14 15 17 18 19 27 28 29 36 37 57 62 65 66 67 69 70 73 76 80 81 85 86 87 88 89 92 93 94 95 97 98 99 100 101 102 105 108 110 111 113 115 116 119 120 121 122 123 124 126 134 135 136 137 138 139 140 143 149 153 158 164 165 168 170 171 172 174 175 177 183 184 185 186 187 188 189 Model 8 9 13 15 18 27 28 70 89 94 95 97 100 106 111 116 149 158 159 173 18
110. 90 141 00 140 50 140 40 140 00 140 00 139 90 139 80 139 60 139 60 139 60 140 20 141 30 141 20 140 90 140 90 140 70 141 10 141 60 141 90 142 10 142 70 142 90 142 90 143 50 143 80 144 10 144 80 145 20 145 20 145 70 Series Data M2 286 70 287 80 289 10 290 10 292 30 293 90 295 30 296 40 296 50 296 60 297 20 297 80 298 30 298 50 299 20 300 10 301 00 302 20 304 20 306 80 308 20 309 60 311 00 312 30 314 20 316 60 318 10 319 90 322 30 324 10 325 70 327 60 329 30 331 20 333 50 335 50 337 60 340 20 M3 289 00 290 10 291 30 292 30 294 50 296 10 297 40 298 50 298 50 298 60 299 20 299 80 300 30 300 50 301 30 302 20 303 00 304 30 306 40 309 20 310 70 312 20 313 80 315 30 317 30 320 00 321 70 323 80 326 50 328 70 330 60 332 60 334 50 336 60 339 00 341 00 343 20 346 20 Box Jenkins ARIMA Forecasts Autoregressive Integrated Moving Average ARIMA forecasts apply advanced econometric modeling tecniques to forecast time series data by first back fitting to historical data and then forecasting the future Advanced knowledge of econometrics is required to properly model ARIMA Please see the ARIMA example Excel model for more details However to get started quickly following the instructions below 1 Risk Simulator Forecasting ARIMA 2 Click on the Time Series Variable link icon and select the area B5 B440 3 Try different P D Q values and select a Forecas
111. 919 79 3 100 20 3 280 60 3 461 00 3 641 40 3 821 80 4 002 20 4 182 60 4 363 00 4543 40 4 723 80 4 904 20 5 084 61 5 265 01 5 445 41 5 625 81 5 806 21 5 986 61 6 167 01 6 347 41 46 00 2 661 70 2 838 82 3 015 94 3 193 06 3 370 18 3 547 30 3 724 42 3 901 54 4 078 66 4 255 79 4 432 91 4 610 03 4 787 15 4 964 27 5 141 39 5 318 51 5 495 63 5 672 75 5 849 87 6 027 00 6 204 12 47 00 2 584 00 2 757 84 2 931 68 3 105 52 3 279 37 3 453 21 3 627 05 3 800 89 3 974 73 4 148 57 4 322 41 4 496 25 4 670 09 4 843 93 5 017 77 5 191 62 5 365 46 5 539 30 5 713 14 5 886 98 6 060 82 48 00 2 506 31 2 676 87 2 847 43 3 017 99 3 188 55 3 359 11 3 529 67 3 700 23 3 870 79 4 041 35 4 211 91 4 382 48 4 553 04 4 723 60 4 894 16 5 064 72 5 235 28 5 405 84 5 576 40 5 746 96 5 917 52 49 00 2 428 61 2 595 89 2 763 17 2 930 45 3 097 73 3 265 01 3 432 29 3 599 58 3 766 86 3 934 14 4 101 42 4 268 70 4 435 98 4 603 26 4 770 54 4 937 82 5 105 10 5 272 38 5 439 67 5 606 95 5 774 23 50 00 2 350 91 2 514 91 2 678 92 2 842 92 3 006 92 3 170 92 3 334 92 3 498 92 3 662 92 3 826 92 3 990 92 4 154 92 4 318 92 4 482 92 4 646 93 4 810 93 4 974 93 5 138 93 5 302 93 5 466 93 5 630 93 Figure 5 39 Scenario Analysis Table 152 Page RISK SIMULATOR 5 12 Segmentation Clustering Tool A final analytical technique of interest is that of segmentation clustering Figure 5 40 illustrat
112. Bayes Analysis ROV Visual Modeler 2012 Decision Trees C Users user Desktop Screen Shots DT Model rovdt sf File Edit Insert Properties Style Shapes and Colors Language Help SO 2 a o s ao xebaeGan 2 P A Decision Tree Summary of Values Simulation Modeling Bayesian Analysis EVPI Minimax Risk Profile sensitivity Analysis Scenario Tables Utility Function Expected Value of Perfect Information Minimax and Maximin Analysis Risk Profiles and Value of Imperfect Information This tool computes the Expected Value of Perfect Information EVPI Minimax and Maximin Analysis as well as the Risk Profile and Value of Imperfect Information To get started enter the number of decision branches or strategies under consideration e g build a large medium small facility and the number of uncertain events or states of nature outcomes e g good market bad market and enter the expected payoffs under each scenario Input Assumptions Minimax and Maximin Analysis Risk Profile Decision Branches Minimax minimizing the maximum regret Maximin Strategy 1 Risk Profile maximizing the minimum payoff are two alternate gt approaches to finding the optimal decision path Uncertainty Events or States These two approaches not used often but still provide added insight into the decision making process Enter the number of decision branches or
113. Decision d New Change Edit SetInput SetOutput Copy Paste Remove Run RunSuper Step Reset Forecasting Run ROV ROV Analytical Options Help license Next Profile Profile Profile Assumption Forecast Speed M Optimization Set Constraint pizstats Decision Tree Tools T d icon l New Simulation Profile kssumptions Forecasts Editing Simulation Run Forecasting Optimization ROV BizStats ROV Decision Tree Tools Options Help License Icon Edit Simulation Profile 5 83 Change Simulation Profile D E F G H 1 J K L M N R U RSet Input Assumption Set Output Optimization BrampteModels 2 Copy Parameter Run Optimization i 80 01 Advanced Forecasting Models Paste Parameter 3 Set Objective Bi 02 Basic Simulation Model chet Mine Ve Remove Parameter D Set Decision Cece Mode BE Correlated Simulation Create Forecast Statistics Table Close All Charts Constraints Bj 04 Correlation Risk Effects Model English Create Report Minimize All Charts Genetic Algorithm gt BE 05 Cost Estimation Model Chinese Simplified Gs Data Deseasonalization amp Detrendin BE Run Simulation Goal Seek BH 06 Data Fitting Chinese Traditional Si gBrh 57 v Data Extraction Export M Run Super Speed Simulation Single Variable Optimizer Mil Dis openimpoit 89 07 DCF ROI and Volatility French Francais Ib step Simulation x Bj Hyp
114. Drift Rate 5 86 Reversion Rate N A Jump Rate 16 33 Volatility 7 0496 Long Term Value N A Jump Size 21 33 Probability of stochastic model fit 4 6396 Figure 5 33 Sample Statistical Analysis Tool Report Stochastic Parameter Estimation 5 10 Distributional Analysis Tool The Distributional Analysis tool is a statistical probability tool in Risk Simulator that is useful in a vatiety of settings It can be used to compute the probability density function PDF which is also called the probability mass function PMF for discrete distributions these terms are used interchangeably where given some distribution and its parameters we can determine the probability of occurrence given some outcome x In addition the cumulative distribution function CDF can be computed which is the sum of the PDF values up to this x value Finally the inverse cumulative distribution function ICDF is used to compute the value x given the cumulative probability of occurrence This tool is accessible via Rusk Simulator Tools Distributional Analysis As an example of its use Figure 5 34 shows the computation of a binomial distribution Le a distribution with two outcomes such as the tossing of a coin where the outcome is either Head or Tail with some prescribed probability of heads and tails Suppose we toss a coin two times Setting the outcome Head as a success we use the binomial distribution with Trials 2 tossing the coin twice and Probability
115. E use as the universal error measure on multiple forecast models for direct comparisons of the accuracy of each model TIPS Forecasting ARIMA e Forecast Periods the number of exogenous data rows has to exceed the time series data rows by at least the desired forecast periods e g if you wish to forecast 5 periods into the future and have 100 time series data points you will need to have 185 Page RISK SIMULATOR at least 105 or more data points on the exogenous variable Otherwise just run ARIMA without the exogenous variable to forecast as many periods as you wish without any limitations TIPS Forecasting Basic Economettics e Variable Separation with Semicolons separate independent variables using semicolon TIPS Forecasting Logit Probit and Tobit Data Requitements the dependent variables for running logit and probit models must be binary only 0 and 1 whereas the Tobit model can take binary and other numerical decimal values The independent variables for all three models can take any numerical value TIPS Forecasting Stochastic Processes e Default Sample Inputs when in doubt use the default inputs as a starting point to develop your own model e Statistical Analysis Tool for Parameter Estimation use this tool to calibrate the input parameters into the stochastic process models by estimating them from your raw data e Stochastic Process Model sometimes if the stochastic process user inte
116. I ASSET ALLOCATION OPTIMIZATION MODEL eo 11 12 13 14 15 16 17 18 20 21 22 24 25 26 27 28 29 30 31 32 Asset Class 1 Asset Class 2 Asset Class 3 Asset Class 4 Asset Class 5 Asset Class 6 Asset Class 7 Asset Class 8 Asset Class 9 Required Required Returns Risk Return to Allocation ean oe Minimum Maximum Ranking Ranking Risk Ranking Ranking Allocation Allocation Hi Lo Lo Hi Hi Lo Hi Lo 10 54 12 36 5 00 35 00 0 8524 g 2 1 11 25 16 23 5 00 35 00 0 6929 7 8 10 1 11 84 15 64 5 00 35 00 0 7570 6 f g 1 10 64 12 35 5 00 35 00 0 8615 8 1 5 1 13 25 13 28 5 00 35 00 0 9977 5 4 2 1 14 21 14 39 5 00 35 00 0 9875 3 6 3 1 15 5396 14 2596 5 0096 35 00 1 0898 1 5 1 1 14 95 16 44 5 00 35 00 0 9094 2 9 4 1 14 16 16 50 5 00 35 00 0 8584 4 10 6 1 10 06 12 50 5 00 35 00 0 8045 10 3 8 1 Asset Class 10 Portfolio Total Return to Risk Ratio 12 6419 4 58 100 00 2 7596 Specifications of the optimization model Objective Decision Variables Restrictions on Decision Variables Constraints Maximize Return to Risk Ratio C18 Allocation Weights E6 E15 Minimum and Maximum Required F6 G15 Portfolio Total Allocation Weights 100 E17 is set to 100 Additional specifications 1 One can alw
117. ISK SIMULATOR Conditions The three conditions underlying the negative binomial distribution are e The number of trials is not fixed e trials continue until the 7th success e The probability of success is the same from trial to trial The mathematical constructs for the Pascal distribution are shown below 1 f x 8 05 1 0 otherwise for all x gt s k x 1 T F x Pie 1 gt s 0 otherwise S p Standard Deviation Js 1 Skewness e yr p p 6p 6 Excess Kurtosis 1 p Successes Required and Probability are the distributional parameters Input requirements Successes required gt 0 and is an integer 0 lt Probability lt 1 The Poisson distribution describes the number of times an event occurs in a given interval such as the number of telephone calls per minute or the number of errors per page in a document Conditions The three conditions underlying the Poisson distribution are e The number of possible occurrences in any interval is unlimited occurrences are independent The number of occurrences in one interval does not affect the number of occurrences in other intervals The average number of occurrences must remain the same from interval to interval 43 Page RISK SIMULATOR The mathematical constructs for the Poisson are as follows P x 4 for xand gt 0 Mean A Standard D
118. Notes Notice that the statistical ranking methods used in the distributional fitting routines are the chi square test and Kolmogorov Smirnov test The former is used to test discrete distributions and the latter continuous distributions Briefly a hypothesis test coupled with an internal optimization routine is used to find the best fitting parameters on each distribution tested and the results are ranked from the best fit to the worst fit 130 Page Theory Procedure Results Interpretation Revenue 20000 Revenue 200 00 Cost Income 100 00 Income 10000 To replicate this model start by creating a Simulation Proff Simulation New Profile then set the random seed to be revenue cells and provide them a Normal distribution with deviation of 20 select one of the revenue cell and click on select Normal and enter the relevant parameters Then each the simulation RISK SIMULATOR 5 4 Bootstrap Simulation Bootstrap simulation is a simple technique that estimates the reliability or accuracy of forecast statistics or other sample raw data Essentially bootstrap simulation is used in hypothesis testing Classical methods used in the past relied on mathematical formulas to describe the accuracy of sample statistics These methods assume that the distribution of a sample statistic approaches a normal distribution making the calculation of the statistic s standard etror or confidence interval relatively easy However
119. PA otras 147 SA1 Scenaro THAD STS a RENE NOURRITURE E 150 3 12 Sennoniaton CABRIO Toila aniei iet uie ener perennis 153 5 13 RISK SIMULATOR 2011 2012 NEW TOOLS sioe ae pla dies 154 5 14 Random Number Generation Monte Carlo vs Latin Hypercube and Correlation Copulas 154 5 15 Deseasonalizing and Detrending 155 5 16 Prnapal Component Anah Ts e RES SU RR E Nea EINER ADI SUR GERI 157 DAF Aut Dra Analis cts 157 N tabi pra une 159 Dy RDU Oe Checkin Tool ARO eR Te rer sss 159 5 20 Percentile Distributional Obi a ostia tis 160 5 21 Distribution Charts and Tables Probability Distribution Tool s 162 5 23 Neural Network and Combinatorial Fuzzy Logic Forecasting Metbodologies 169 524 Optimizer Goal iat tara aia a notano iaaiaee 171 172 5 26 Genetic 173 5 27 ROV Denson Tree Module sino ebbe a AAS 174 5 271 Decision Treeni aa a 174 5 27 2 S Modelirip pendit 176 5373 Baye _ 177 5 27 4 Expected Value of Perfect Information MINIMAX MAXIMIN Analysis Risk Profiles and Value of Imperfect Information seen 177 5 215 a cb odis ub ocupa NEEDED LEA LANE 178 nnb
120. PTIMIZATION MODEL m Required Required Returns Risk Returnto Allocation sisi ir oer ird possa Minimum Maximum Ranking Ranking Risk Ranking Ranking Allocation Allocation Hi Lo Lo Hi Hi Lo Hi Lo Asset Class 1 10 5496 12 3696 5 0096 35 00 0 8524 9 2 7 4 Asset Class 2 11 25 16 23 5 00 35 00 0 6929 7 8 10 10 Asset Class 3 11 84 15 64 5 00 35 00 0 7570 6 ri 9 9 Asset Class 4 10 64 12 35 5 00 35 00 0 8615 8 1 5 3 Asset Class 5 13 25 13 28 5 00 35 00 0 9977 5 4 2 2 Asset Class 6 14 21 14 39 5 00 35 00 0 9875 3 6 3 5 Asset Class 7 15 53 14 25 5 00 35 00 1 0898 1 1 1 Asset Class 8 14 9596 16 44 5 00 35 0096 0 9084 2 8 4 7 Asset Class 9 14 16 16 50 5 00 35 00 0 8584 4 10 6 8 Asset Class 10 10 06 12 50 5 0096 35 00 0 8045 10 3 8 6 Portfolio Total 12 6920 4 52 Return to Risk Ratio 2 8091 Figure 4 3 Continuous Optimization Results 105 Page Procedure RISK SIMULATOR 4 3 Optimization with Discrete Integer Variables Sometimes the decision variables are not continuous but are discrete integers e g 0 and 1 We can use optimization with discrete integer variables as on off switches or go no go decisions Figure 4 4 illustrates a project selection model with 12 projects listed The example here uses the Discrete Optimization file found either on the start menu at 57277 Real Options Valuation Risk Simulator Examples or accessed directly through Risk Simulator Example Mod
121. RIMA Autoregressive Integrated Moving Average Auto ARIMA Auto Econometrics Basic Econometrics Combinatorial Fuzzy Logic Cubic Spline Curves Custom Distributions GARCH Generalized Autoregressive Conditional Heteroskedasticity J Curve Markov Chain Maximum Likelihood Logit Probit Tobit Multivariate Regression Neural Network Forecasts Nonlinear Extrapolation Ce nno wnbwIrc PP RP RP RP gt S Curve nN Stochastic Processes N Time Series Analysis and Decomposition 18 Trendlines The analytical details of each forecasting method fall outside the purview of this user manual For more details please review Modeling Risk Applying Monte Carlo Simulation Real Options Analysis Stochastic Forecasting and Portfolio Optimization Second Edition by Dr Johnathan Mun Wiley Finance 2010 who is also the creator of the Risk Simulator software Nonetheless the following illustrates some of the more common approaches and several quick getting started 65 Page ARIMA Auto ARIMA Basic Econometrics Auto Econometrics Combinatorial Fuzzy Logic Cubic Spline Curves Custom Distributions GARCH RISK SIMULATOR examples in using the software More detailed descriptions and example models of each of these techniques are found throughout this chapter and the next All other forecasting approaches are fairly easy to apply within Risk Simulator Autoregressive integrated m
122. Roe m EE paths that exist e g building a large medium or small facility as well as the uncertainty events 80 states of nature under each path e g good economy vs bad economy Then complete the payoff table for the various scenarios and Compute the Minimax and Maximin results You car cick on m Load Example to see a sample calculation Sum of Probabilities Expected Value 14 Strateqy 2 Risk Profile 20 Payoff 20 00 306 00 Expected Value of Perfect Information The Expected Value of Perfect Information EVPI i e assuming you had perfect foresight and knew exactly what to do through market research or other means to better discern the probabilistic outcomes EVPI computes if there is added value in such information i e if market research will add value as compared to more naive estimates of the probabilistic states of nature To get started enter the number of decision branches or strategies under consideration e g build a large medium small facility and the number of uncertain events or states of nature outcomes e g good market bad market and enter the expected payoffs under each scenario Expected Value of Perfect Information of States of Nature Expected Value of Without Perfect Information of States of Nature Path 2 is optimal Expected Value of Perfect Information Path 1 is optimal
123. Simulator toolbar and the simulation will proceed You may also reset a simulation after it has run to rerun it Risk Simulator Reset Simulation ot the reset simulation icon on the toolbar or to pause it during a run Also the step function Rusk Simulator Step Simulation ot the szep simulation ion on the toolbar allows you to simulate a single trial one at a time useful for educating others on simulation 1 you can show that at each trial all the values in the assumption cells are being replaced and the entire model is recalculated each time You can also access the run simulation menu by right clicking anywhere in the model and selecting Run Simulation Risk Simulator also allows you to run the simulation at extremely fast speed called Super Speed To do this click on Risk Simulator Run Super Speed Simulation use the run super speed icon Notice how much faster the super speed simulation runs In fact for practice Reset Simulation and then Edit Simulation Profile and change the Number of Trials to 100 000 and Run Super Speed It should only take a few seconds to run However please be aware that super speed simulation will not run if the model has errors VBA visual basic for applications or links to external data sources or applications In such situations you will be notified and the regular speed simulation will be run instead Regular speed simulations are always able to run even with errors VBA or external links
124. Super Speed Simulation Then in the run optimization user interface select Stochastic Optimization on the Method tab and set it to run 500 trials and 20 optimization runs and click OK This approach will integrate the super speed simulation with optimization Notice how much faster the stochastic optimization runs You can now quickly rerun the optimization with a higher number of simulation trials Notice that if there are input simulation assumptions in the optimization model i e these input assumptions required in order to run the dynamic or stochastic optimization routines the Statistics tab is now populated in the Run Optimization user interface You can select from the drop down list the statistics you want such as average standard deviation coefficient of variation conditional mean conditional variance a specific percentile and so forth This means that if you run a stochastic optimization a simulation of thousands of trials will first run then the selected statistic will be computed and this value will be temporarily placed in the simulation assumption cell then an optimization will be run based on this statistic and then the entire process is repeated multiple times This method is important and useful for banking applications in computing conditional Value at Risk or conditional VaR Statistics Preferences Options Controls Global View Type Two Tal 10 303500 03104 Certainty 90 008
125. The estimated parameters for GARCH M with the t distribution are those five parameters in the mean and conditional variance equations plus another parameter the degrees of freedom for the t distribution In contrast for the GJR models the mean equations are the same in the six variations and the differences are that the conditional variance equations and the assumption on 2 can be either a normal distribution or t distribution The estimated parameters for EGARCH and GJR GARCH with normal distribution are those four parameters in the conditional vatiance equation The estimated parameters for GARCH EARCH and GJR GARCH with t distribution are those parameters in the conditional variance equation plus the degrees of freedom for the t distribution More technical details of GARCH methodologies fall outside of the scope of this book 3 12 Markov Chains A Markov chain exists when the probability of a future state depends on a previous state and when linked together form a chain that reverts to a long run steady state level This approach is typically used to forecast the market share of two competitors The required inputs are the starting probability of a customer in the first store the first state will return to the same store in the next period versus the probability of switching to a competitor s store in the next state e Start Excel and select Simulator Forecasting Markov Chain Enter in the required input assumptions
126. The final step in Monte Carlo simulation is to interpret the resulting forecast charts Figures 2 6 through 2 13 show the forecast chart and the corresponding statistics generated after running the simulation Typically the following elements are important in interpreting the results of a simulation The forecast chart shown in Figure 2 6 is a probability histogram that shows the frequency counts of values occurring in the total number of trials simulated The vertical bars show the frequency of a particular x value occutring out of the total number of trials while the 19 Page Forecast Statistics RISK SIMULATOR cumulative frequency smooth line shows the total probabilities of all values at and below x occurring in the forecast The forecast statistics shown in Figure 2 7 summarize the distribution of the forecast values in terms of the four moments of a distribution See the Understanding the Forecast Statistics section later in this chapter for more details on what some of these statistics mean You can rotate between the histogram and statistics tabs by depressing the space bar Income Risk Simulator Forecast Percentage Error Precision at 95 Confidence Figure 2 7 Forecast Statistics 20 Page Preferences Options Controls Global View versus Normal View RISK SIMULATOR 2 2 3 Forecast Chart Tabs The preferences tab in the forecast chart Figure 2 8A allows you to change the look and feel
127. a points within a time series of data e g yield curve interest rates macroeconomic variables like inflation rates and commodity prices or market returns and also used to extrapolate outside of the given or known range making it useful for forecasting Spline Interpolation and Extrapolation Results B 4 AN Te Real Options 15 4 21 Interpolate Valuation 20 4 13 Interpolate 25 413 Interpolate These are the known value 3 0 4 16 Interpolate inputs in the Cublic Spline 3 5 4 19 Interpolate Interpolation and Extrapolation 40 42296 interpolate E 4 5 4 24 Interpolate 5 0 4 26 Interpolate Observation KnownX Known Y 55 4 29 Interpolate 1 0 0833 4 55 6 0 4 32 Interpolate 2 0 2500 447 6 5 4 35 Interpolate 3 0 5000 452 7 0 4 38 Interpolate 4 1 0000 4 39 441 Interpolate 5 2 0000 4 13 8 0 444 Interpolate 6 3 0000 4 16 8 5 447 Interpolate 7 5 0000 4 26 9 0 450 Interpolate 8 7 0000 438 95 4 53 Interpolate 9 10 0000 456 10 0 4 56 Interpolate 10 20 0000 4 88 10 5 4 59 Interpolate 11 30 0000 4 84 Figure 3 23 Spline Forecast Results 98 Page Static Optimization Dynamic Optimization RISK SIMULATOR 4 OPTIMIZATION his chapter looks at the optimization process and methodologies in more detail in connection with using Risk Simulator These methodologies include the use of continuous versus discrete integer optimization as well as static versus dynamic and s
128. ables have different impacts on different subgroups of the population such as to test if a new marketing campaign activity major event acquisition divestiture and so forth have an impact on the time series data Suppose for example a data set has 100 time series data points You can set various breakpoints to test for instance data points 10 30 and 51 This means that three structural break tests will be performed data points 1 9 compared with 10 100 data points 1 29 compated with 30 100 and 1 50 compared with 51 100 to see if there is a break in the 157 Page Procedure RISK SIMULATOR underlying structure at the start of data points 10 30 and 51 A one tailed hypothesis test is performed on the null hypothesis HO such that the two data subsets are statistically similar to one another that is there is no statistically significant structural break The alternative hypothesis Ha is that the two data subsets are statistically different from one another indicating a possible structural break If the calculated p values are less than or equal to 0 01 0 05 or 0 10 then the hypothesis is rejected which implies that the two data subsets are statistically significantly different at the 1 5 and 10 significance levels High p values indicate that there is no statistically significant structural break e Select the data you wish to analyze e g B15 D34 click on Risk Simulator Tools Structural Break Test
129. aco shell boxes would you then need to sample or trials run to obtain this level of precision Here the 2 taco shells is the error level while the 90 is the level of precision If sufficient numbers of trials ate run then the 90 confidence interval will be identical to the 90 precision level where a more precise measure of the average is obtained such that 90 of the time the error and hence the confidence will be 2 taco shells As an example say the average is 20 units then the 90 29 Page RISK SIMULATOR confidence interval will be between 18 and 22 units with this interval being precise 90 of the time where in opening all 1 million boxes 900 000 of them will have between 18 and 22 broken taco shells The number of trials required to hit this precision is based on the sampling 5 where Z is the error of 2 taco shells X is the sample vn average Z is the standard normal Z score obtained from the 90 precision level 5 is the sample standard deviation and n is the number of trials required to hit this level of error with the specified precision Figures 2 17 and 2 18 illustrate how precision control can be performed on multiple simulated forecasts in Risk Simulator This feature prevents the user from having to decide how many trials to run in a simulation and eliminates all possibilities of guesswork Figure 2 17 illustrates the forecast chart with a 95 precision level set This value can be changed and will be reflect
130. ady state level This approach is typically used to forecast the market share of two competitors The required inputs are the starting probability of a customer in the first store the first state will return to the same store in the next period versus the probability of switching to a competitor s store in the next state Maximum likelihood estimation MLE is used to forecast the probability of something occurring given some independent variables For instance MLE is used to predict if a credit line or debt will default given the obligor s characteristics 30 years old single salary of 100 000 per year and having a total credit card debt of 10 000 or the probability a patient will have lung cancer if the person is a male between the ages of 50 and 60 smokes 5 packs of cigarettes per month and so forth In these circumstances the dependent variable is limited i e limited to being binary 1 and 0 for default die and no default live or limited to integer values like 1 2 3 etc and the desired outcome of the model is to predict the probability of an event occurring Traditional regression analysis will not work in these situations the predicted probability is usually less than zero or greater than one and many of the required regression assumptions are violated such as independence and normality of the errors and the errors will be fairly large Multivariate regression is used to model the relationship structure and characteristics
131. al Lognormal distribution but adds a Location or Shift parameter The Lognormal distribution starts from a minimum value of 0 whereas this Lognormal 3 or Shifted Lognormal distribution shifts the starting location to any other value Mean Standard Deviation and Location Shift are the distributional parameters Input requirements Mean gt 0 Standard Deviation gt 0 Location can be any positive or negative value including zero The normal distribution is the most important distribution in probability theory because it describes many natural phenomena such as people s IQs or heights Decision makers can use the normal distribution to describe uncertain variables such as the inflation rate or the future price of gasoline Conditions The three conditions underlying the normal distribution are Some value of the uncertain variable is the most likely the mean of the distribution e uncertain variable could as likely be above the mean as it could be below the mean symmetrical about the mean e The uncertain variable is more likely to be in the vicinity of the mean than further away The mathematical constructs for the normal distribution are as follows _ x 4 2 forallvaluesof x 2 and u while o gt 0 Mean u Standard Deviation SRewness 0 this applies to all inputs of mean and standard deviation Excess Kurtosis 0 this applies to all inputs of mean and standard
132. al Rate of Return 55 68 8 Terminal Period Growth Rate 2 00 Retum on Investment 191 40 9 Effective Tax Rate 40 0096 Profitability Index 2 91 d R Scenario Analysi 1 2009 2010 2011 Serio Analysis P 7 2018 12 Product A Avg Price Unit 10 00 10 50 11 00 Start by entering the cell addresses for the output and input test variables e g A1 14 50 13 Product B Avg Price Unit 12 25 12 50 12 75 Location of Output Variable G6 8 14 50 14 Product C Avg Price Unit 15 15 15 30 15 45 eg zn 16 50 15 Product A Sale Quantity 000s 50 50 pacers olen 50 16 Product B Sale Quantity 0005 35 35 S5 Next enter the starting value ending value and number of steps or the step size to test 35 E e 1 231 3 1 268 A 1 305 1 562 a d E 2232 1309 7 Stating Value Stating Value 19 Direct Cost of Goods Sold 5184 76 190 28 195 79 d 10 234 38 20 Gross Profit 1 046 99 1 078 23 1 109 46 05 Eod Ye 1 328 13 21 Operating Expenses 157 50 157 50 157 50 C Steps 20 157 50 22 Sales General and Admin Costs 15 75 1575 1575 Step Size 001 C Step Size 15 75 23 Operating Income EBITDA 873 74 904 98 936 21 4 1 154 88 24 Depreciation 10 00 10 00 10 00 Cancel 10 00 25 Amortization 53 00 53 00 3 00 53 00 26 EBIT 860 74 891 98 923 21 1 141 88 27 Interest Payments 2 00 2 00 2 00 2 00 2 00
133. al statistics for stochastic optimization including percentiles as well as conditional means which are critical in computing conditional value at risk measures Search Algorithm simple fast and efficient search algorithms for basic single decision variable and goal seek applications Super Speed Simulation in Dynamic and Stochastic Optimization runs simulation at super speed while integrated with optimization 1 4 5 Analytical Tools Module 59 60 61 62 63 64 65 60 67 68 69 70 71 72 73 74 Check Model tests for the most common mistakes in your model Correlation Editor allows latge correlation matrices to be directly entered and edited Create Report automates report generation of assumptions and forecasts in a model Create Statistics Report generates comparative report of all forecast statistics Data Diagnostics runs tests on heteroskedas city micronumerosity outliers nonlinearity autocorrelation normality sphericity nonstationarity multicollinearity and correlations Data Extraction and Export extracts data to Excel or flat text files and Risk Sim files runs statistical reports and forecast result reports Data Open and Import rettieves previous simulation run results Deseasonalization and Detrending deseasonalizes and detrends your data Distributional Analysis computes exact PDF CDF and ICDF of all 42 distributions and generates probability tables
134. amp PROBIT SAMPLE DATA MLE LIMDEF iive probabilities or values exceeding 100 Only these LIMDEP models are when dependent variables are limited 1 3 0 1 0 1 0 1 1 2 0 2 0 1 0 1 1 1 4 fan La Logit Probit Tobit Figure 3 21 Maximum Likelihood Module 96 Page Theory RISK SIMULATOR 3 14 Spline Cubic Spline Interpolation and Extrapolation Sometimes there are missing values in a time series data set For instance interest rates for years 1 to 3 may exist followed by years 5 to 8 and then year 10 Spline curves can be used to interpolate the missing years interest rate values based on the data that exist Spline curves can also be used to forecast or extrapolate values of future time periods beyond the time period of available data The data can be linear or nonlinear Figure 3 22 illustrates how a cubic spline is run and Figure 3 23 shows the resulting forecast report from this module The Known X values represent the values on the x axis of a chart in our example this is Years of the known interest rates and usually the x axis values ate those that are known in advance such as time or years and the Known Y values represent the values on the y axis in our case the known Interest Rates The y axis variable is typically the variable you wish to interpolate missing values from or extrapolate the values into the future 3 Cubic Spline Interpolation and Extrapolation 4 H The cu
135. an replace any one or both of these parameters with your own percentiles and this tool will perform a fitting to obtain the relevant distributional parameters Step 1 Select the distribution and parameter estimation type Step 2 Enter the relevant inputs Triangular Minimum MostLikely Percentile Parameter Triangular Percentile Percentile Maximum Triangular Percentile MostLikely Percentile Triangular Minimum Percentile Percentile Percentile 10 Percentile 4 45 Triangular Percentile Percentile Percentile Percentile Triangular Mean Stdev Percentile Uniform Uniform Minimum Percentile Step 3 Run curve fit and review the empirical versus theoretical distributions Uniform Percentile Maximum Uniform Percentile Percentie Fitted Square G Uniform Mean Stdev Alpha 0 7113 Weibull Beta 0 2935 Weibull Alpha Percentile Location 25176 Weibull Percentile Beta Weibull Percentile Percentile Theoretical Weibull Mean Stdev 2 5300 Weibull 3 2 6600 Weibull 3 Percentile Beta Location 3 8900 Weibull 3 Alpha Percentile Location 2 8836 Weibull 3 Alpha Beta Percentile Weibull 3 Percentile Percentile Location Weibull 3 Percentile Beta Percentile 3 4054 Weibull 3 Alpha Percentile Percentile ji 19 3606 Weibull 3 Percentile Percentile Percentile Weibull 3 Mean Stdev Percentile 0 5255 Figure 5 47 Percentile D
136. ances extreme values may have a larger smaller or unbalanced impact e g nonlinearities may occur where increasing or decreasing economies of scale and scope creep occurs for larger or smaller values of a variable and only a wider range will capture this nonlinear impact A tornado chart lists all the inputs that drive the model starting from the input variable that has the most effect on the results The chart is obtained by perturbing each precedent input at some consistent range e g 10 from the base case one at a time and comparing their results to the base case A spider chart looks like a spider with a central body and its many legs protruding The positively sloped lines indicate a positive relationship while a negatively sloped line indicates a negative relationship Further spider charts can be used to visualize linear and nonlinear relationships The tornado and spider charts help identify the critical success factors of an output cell in order to identify the inputs to simulate The identified critical variables that are uncertain are the ones that should be simulated Do not waste time simulating variables that are neither uncertain nor have little impact on the results Result 1 1 1 1 2 0 Base Value 96 6261638553219 Input Changes Ouiput Output Effective Input Input Base Case Precedent Cell Downside Upside Range Downside Upside Value 276 63 83 37 360 00 1 620 00 1 980 00 1 800 00 219 73 26 47 246 20 36 00
137. and most conservative results The t copula provides for extreme values in the tails of the simulated distributions whereas the quasi normal copula returns results that are between the values derived by the other two methods In the Simulation methods section Monte Carlo Simulation MCS and Latin Hypercube Sampling LHS methods are supported Note that Copulas and other multivariate functions are not compatible with LHS because LHS can be applied to a single random variable but not over a joint distribution In reality LHS has very limited impact on the model output s accuracy the more distributions there are in a model since LHS only applies to distributions individually The benefit of LHS is also eroded if one does not complete the number of samples nominated at the beginning that is if one halts the simulation run in mid simulation LHS also applies a heavy burden on a simulation model with a large number of inputs because it needs to generate and organize samples from each distribution prior to running the first sample from a distribution This can cause a long delay in running a large model without providing much more additional accuracy Finally LHS is best applied when the distributions are well behaved and symmetrical and without any correlations Nonetheless LHS is a powerful approach that yields a uniformly sampled distribution where MCS can sometimes generate lumpy distributions sampled data can sometimes be more heavily concentrated i
138. and other stochastic time series data given drift or growth rate and a volatility around the drift path The Mean Reversion process can be used to reduce the fluctuations of the Random Walk process by allowing the path to target a long term value making it useful for forecasting time series variables that have a long term rate such as interest rates and inflation rates these are long term target rates by regulatory authorities or the market The Jump Diffusion process is useful for forecasting time series data when the variable can occasionally exhibit random jumps such as oil prices or price of electricity discrete exogenous event shocks can make prices jump up or down Finally these three stochastic processes can be mixed and matched as required Statistical Summary The following are the estimated parameters for a stochastic process given the data provided It is up to you to determine ifthe probability of fit similar to a goodness of fit computation is sufficient to warrant the use of a stochastic process forecast and if so whether itis a random walk mean reversion ora jump diffusion model or combinations thereof In choosing the right stochastic process model you will have to rely on past experiences and a priori economic and financial expectations of what the underlying data set is best represented by These parameters can be entered into a stochastic process forecast Simulation Forecasting Stochastic Processes Annualized
139. apacity S curve gt 3 000 Exponential J Curve 9 Logistic S Curve 2 000 Growth Phase Starting Value 200 Growth Rate 10 1 000 400 Saturation Level 6000 0 Generate forecast curve based on the following periods 0 20 30 40 60 80 100 EndPeriod 100 Period Figure 3 18 S Curve Forecast 311 GARCH Volatility Forecasts Theory The generalized autoregressive conditional heteroskedasticity GARCH model is used to model historical and forecast future volatility levels of a marketable security e g stock prices commodity prices oil prices etc The data set has to be a time series of raw price levels GARCH will first convert the prices into relative returns and then run an internal optimization to fit the historical data to a mean reverting volatility term structure while assuming that the volatility is heteroskedastic in natute changes over time according to some economettic characteristics The theoretical specifics of a GARCH model are outside the purview of this user manual For more details on GARCH models please refer to Advanced Analytical Models by Dr Johnathan Mun Wiley Finance 2008 Procedure e Start Excel and open the example file Advanced Forecasting Model go to the GARCH worksheet select the data and click on Rak Simulator Forecasting GARCH e Click on the link icon select the Data Location enter the required input assumptions see Figure 3 19 and click OK to run the m
140. arameter you define the Weibull distribution can be used to model the exponential and Rayleigh distributions among others The Weibull distribution is very flexible When the Weibull shape parameter is equal to 1 0 the Weibull distribution is identical to the exponential distribution The Weibull location parameter lets you set up an exponential distribution to start at a location other than 0 0 When the shape parameter is less than 1 0 the Weibull distribution becomes a steeply declining curve A manufacturer might find this effect useful in describing part failures during a burn in period The mathematical constructs for the Weibull distribution ate as follows a l x a x d alz PLE Mean BT a Standard Deviation 2 7 234 87 3r 8 267 1438 Ira 257 r a SRewness Excess Kurtosis 6 1 8 1217 1 BP 28 3F 0 28 4r 8 38 F0 487 Ira 2 r a ef Shape a and central location scale are the distributional parameters and is the Gamma function Input requirements Shape Alpha 0 05 Scale Beta gt 0 and can be any positive value 62 Weibull 3 Distribution RISK SIMULATOR The Weibull 3 distribution uses the same constructs as the original Weibull distribution but adds a Location or Shift parameter The Weibull distribution starts from a minimum value of 0 whereas this Weibull 3 or Shifted Wei
141. ard the tail of the distribution while the median remains constant Another way of seeing this relationship is that the mean moves but the standard deviation variance or width may still remain constant If the third moment is not considered then looking only at the expected returns e g median or mean and tisk standard deviation a positively skewed project might be incorrectly chosen For example if the horizontal axis represents the net revenues of a project then clearly a left or negatively skewed distribution might be preferred because there is a higher probability of greater returns Figure 2 22 as compared to a higher probability for lower level returns Figure 32 Page Measuring the Catastrophic Tail Events in a Distribution the Fourth Moment RISK SIMULATOR 2 23 Thus in a skewed distribution the median is a better measure of returns as the medians for both Figures 2 22 and 2 23 are identical risks are identical and hence a project with a negatively skewed distribution of net profits is a better choice Failure to account for a project s distributional skewness may mean that the incorrect project could be chosen e g two projects may have identical first and second moments that is they both have identical returns and risk profiles but their distributional skews may be very different O1 O2 Skew 0 KurtosisXS 0 f H2 Figure 2 22 Third Moment Left Skew O1 O2 Skew g
142. ast 100 00 Cost 100 00 100 00 Income Hypothesis Testing Satsicn Preferences Options Controls Figure 5 18 Hypothesis Testing 133 Hypothesis testing is used to determine if two or more forecast distributions have the same mean and variance i e if they are statistically different from one another their differences are due to random chance Simulation Model Simulation Model kA Please select the two forecasts on which to run a hypothesis test Results Interpretation RISK SIMULATOR A two tailed hypothesis test is performed on the null hypothesis such that the two variables population means are statistically identical to one another The alternative hypothesis Ha is such that the population means ate statistically different from one another If the calculated p values are less than or equal to 0 01 0 05 or 0 10 this means that the null hypothesis is rejected which implies that the forecast means statistically significantly different at the 1 5 and 10 significance levels If the null hypothesis is not rejected when the p values are high the means of the two forecast distributions are statistically similar to one another The same analysis is performed on variances of two forecasts at a time using the pairwise F test If the p values are small then the variances and standard deviations
143. at Goal Seek works only with one variable input value If you want to accept more than one input value use Risk Simulator s advanced Optimization routines Figure 5 58 shows how Goal Seek is applied to a simple model 171 Page RISK SIMULATOR One Variable Target Seek Set cell Result To value 300 By changing cell 1 E _Cancel Figure 5 58 Goal Seek 5 25 Single Variable Optimizer The Single Variable Optimizer tool is a search algorithm used to find the solution of a single variable within a model just like the goal seek routine discussed previously If you want the maximum or minimum possible result from a model but ate not sure what input value the formula needs to get that result use the Rusk Simulator Tools Single Variable Optimizer feature Figure 5 59 Note that this tool runs very quickly but is only applicable to finding one variable input If you want to accept more than one input value use Risk Simulator s advanced Optimization routines Note that this tool is included in Risk Simulator because if you require a quick optimization computation for a single decision variable this tool provides that capability without having to set up an optimization model with profiles simulation assumptions decision variables objectives and constraints Objective Cell 8 Maximize Minimize Variable Cell 1 Min 50 250 Tolerance 0 000000001 Max Iterations 100
144. ation Price 2 00 2 00 2 00 Quantity 1 00 1 00 1 00 Revenue Figure 2 14 Simple Correlation Model The resulting statistics are shown in Figure 2 15 Notice that the standard deviation of the model without correlations is 0 1450 compared to 0 1886 for the positive correlation and 0 0717 for the negative correlation That is for simple models negative correlations tend to reduce the average spread of the distribution and create a tight and more concentrated forecast distribution as compated to positive correlations with larger average spreads However the mean remains relatively stable This implies that correlations do little to change the expected value of projects but can reduce or increase a project s tisk A si Revenue Positive Correlation Risk Simulator Forecast 53 Revenue Negative Correlation Risk Simulator 53 Histogram Statistic Preferences Options Histogram Statistic Preferences Options Percentage Error Precision at 95 Confidence Figure 2 15 Correlation Results Figure 2 16 illustrates the results after running a simulation extracting the raw data of the assumptions and computing the correlations between the variables The figure shows that the input assumptions are recovered in the simulation That is you enter 0 8 and 0 8 correla
145. ation function font and so forth Hitting Escape before pasting allows you to maintain the target cell s values and computations and pastes only the Risk Simulator parameters e Copy and Paste on Multiple Cells select multiple cells for copy and paste with contiguous and noncontiguous assumptions TIPS Correlations e Set Assumption set pairwise correlations using the set input assumption dialog ideal for entering only several correlations e Edit Correlations set up a correlation matrix by manually entering or pasting from Windows clipboard ideal for large correlation matrices and multiple correlations e Multiple Distributional Fitting automatically computes and enters pairwise correlations ideal when performing multiple variable fitting to automatically compute the correlations for deciding what constitutes a statistically significant correlation TIPS Data Diagnostics and Statistical Analysis e Stochastic Parameter Estimation in the Statistical Analysis and Data Diagnostic reports there is a tab on stochastic parameter estimations that estimates the volatility drift mean reversion rate and jump diffusion rates based on historical data Be aware that these parameter results are based solely on historical data used and the parameters may change over time and depending on the amount of fitted historical data Further the analysis results show all parameters and do not imply which stochastic process model e g
146. ation Techniques e Random Number Generator there are six supported random number generators see the user manual for details and in general the ROV Risk Simulator default method and the Advanced Subtractive Random Shuffle method are the two recommended approaches to use Do not apply the other methods unless your model or analytics specifically calls for their uses and even then we recommended testing the results against these two recommended approaches TIPS Software Development Kit SDK and DLL Libraries e SDK DLL and OEM all of the analytics in Risk Simulator can be called outside of this software and integrated in any user proprietary software Contact admin realoptionsvaluation com for details on using our Software Development Kit to access the Dynamic Link Library DLL analytics files TIPS Starting Risk Simulator with Excel ROV Troubleshooter tun this troubleshooter to obtain your computer s HWID for licensing purposes to view your computer settings and prerequisites and to re enable Risk Simulator if it has been accidentally disabled e Start Risk Simulator when Excel Starts you can let Risk Simulator start automatically when Excel starts each time or start it manually from the Start Programs Real Options Valuation Risk Simulator shortcut location This preference can be set in the Risk Simulator Options menu TIPS Super Speed Simulation e Model Development if you wish to run super speed in your mo
147. ation tests nonlinearities etc before a proper model can be constructed See Modeling Risk Apphing Monte Carlo Simulation Real Options Analysis Forecasting and Optimization Second Edition Wiley Finance 2010 by Dr Johnathan Mun for more detailed analysis and discussion of multivariate regression as well as how to identify these regression pitfalls e Start Excel and open yout historical data if required the illustration below uses the file Multiple Regression in the examples folder e Check to make sure that the data is arranged in columns select the entire data area including the variable name and select Rak Simulator Forecasting Multiple Regression e Select the dependent variable and check the relevant options lags stepwise regression nonlinear regression etc and click OK 73 Page RISK SIMULATOR Results Figure 3 8 illustrates a sample multivariate regression result report The report comes complete Interpretation with all the regression results analysis of variance results fitted chart and hypothesis test results The technical details of interpreting these results are beyond the scope of this user manual See Modeling Risk Apphing Monte Carlo Simulation Real Options Analysis Forecasting and Optimization Second Edition Wiley Finance 2010 by Dr Johnathan Mun for more detailed analysis and discussion of multivariate regression as well as the interpretation of regression ts Multivariate
148. ays maximize portfolio total returns or minimize the portfolio total risk 2 Incorporate Monte Carlo simulation in the model by simulating the returns and volatility of each asset class and apply Simulation Optimization techniques 3 The portfolio can be optimized as is without simulation using Static Optimization techniques Figure 4 1 Continuous Optimization Model 11 Procedure RISK SIMULATOR Referring to Figure 4 1 column E Allocation Weights holds the decision variables which are the variables that need to be tweaked and tested such that the total weight is constrained at 100 cell E17 Typically to start the optimization we set these cells to a uniform value where in this case cells E6 to E15 are set at 10 each In addition each decision variable may have specific restrictions in its allowed range In this example the lower and upper allocations allowed are 5 and 35 as seen in columns F and G This means that each asset class may have its own allocation boundaries Next column H shows the return to risk ratio which is simply the return percentage divided by the risk percentage where the higher this value the higher the bang for the buck Columns I through L show the individual asset class rankings by returns risk return to risk ratio and allocation In other words these rankings show at a glance which asset class has the lowest risk or the highest return and so forth The portfolio s total r
149. babilistic rules They are useful for forecasting random events e g stock prices interest rates price of electricity Methods Starting Value Brownian Motion Random Walk with Drift Growth or Drift Rate Exponential Brownian Motion Random Walk with Drift Mean Reversion Process with Drift Annualized Volatility Jump Diffusion Process with Drift Forecast Horizon Years Jump Diffusion Process with Drift and Mean Reversion Reversion Rate Long Term Value Jump Rate Jump Size Figure 3 9 Stochastic Process Forecasting Stochastic Process Forecasting Statistical Summary A stochastic process is a sequence of events or paths generated by probabilistic laws That is random events can occur over time but are governed by specific statistical and probabilistic rules The main stochastic processes include Random Walk or Brownian Motion Mean Reversion and Jump Diffusion These processes can be used to forecast a multitude of variables that seemingly follow random trends but yet are restricted by probabilistic laws The Random Walk Brownian Motion process can be used to forecast stock prices prices of commodities and other stochastic time series data given a drift or growth rate and a volatility around the drift path The Mean Reversion process can be used to reduce the fluctuations of the Random Walk process by allowing the path
150. be Sampling LHS vs Monte Carlo Simulation MCS 187 TIPS Online RESOULCES xe EE He ee idee e erected e erede 187 TIRS OPU ZANO de e iac a ed at 187 TIPS estimer ccu 188 TIPS Right Click Shortcut and Other Shortcut Keys eee 188 IPS SAVE sena dua tua ons notet aee me 188 TIPS Sampling and Simulation Techniques e 189 TIPS Software Development Kit SDK and DLL Libraries ee 189 TIPS State Risk Simulator with c Rue de 189 TIPS Super Speed quests m ee ER EEG Ete ELA REL ERBEN ELLE Reb 189 TIPS Tornado Analysls c uentos 189 EIPS trc ee a IR eR e led 190 RISK SIMULATOR 1 INTRODUCTION TO RISK SIMULATOR 1 1 Welcome to Risk Simulator T he Risk Simulator is a Monte Carlo simulation Forecasting and Optimization software The software is written in Microsoft and functions together with Excel as an add in This software is also compatible and often used with the Real Options Super Lattice Solver SLS software and Employee Stock Options Valuation Toolkit ESOV software also developed by Real Options Valuation Inc Note that although we attempt to be thorough in this user manual the manual is absolutely not a substitute for the Training DVD live t
151. be used to both interpolate missing data points within a time series of data e g yield 238 curve interest rates macroeconomic variables like inflation rates and commodity 39 prices or market returns and is also used to extrapolate outside of the given or known 10 range making it useful for forecasting 41 Known X Values 15225 2 Known Y Values 01505 44 Generate 2 spline curve based on the following X values 45 Starting 1 Ending 50 Step Size 05 46 4T 48 Figure 3 22 Cubic Spline Module 97 Page RISK SIMULATOR Procedure e Start Excel and open the example file Advanced Forecasting Model go to the Cubic Spline worksheet sect the set excluding the headers and click on Simulator Forecasting Cubic Spline e The data location is automatically inserted into the user interface if you first select the data or you can also manually click on the link icon and link the Known X values and Known Y values see Figure 3 22 for an example then enter in the required Starting and Ending values to extrapolate and interpolate as well as the required Step Size between these starting and ending values Click OK to run the model and report see Figure 3 23 Cubic Spline Forecasts The cubic spline polynomial interpolation and extrapolation model is used to fill in the gaps of missing values and for forecasting time series data whereby the model can be used to both interpolate missing dat
152. bic spline polynomial interpolation and extrapolation model is used 6 to fill in the gaps of missing spot yields and term structure of interest rates 7 whereby the model be used to both interpolate missing data points within 8 a time series of interest rates as well as other macroeconomic variables such 9 as inflation rates and commodity prices or market returns and also used to 10 extrapolate outside of the given or known range useful for forecasting purposes 11 12 43 14 Years Yields 15 0 0833 4 55 These the yields 16 0 2500 4 47 that are known and Al 0 5000 4 52 are used as inputs in 18 1 0000 4 39 the Cubic Spline 19 2 0000 4 13 Interpolation and 20 3 0000 4 16 Extrapolation model 21 5 0000 4 26 22 7 0000 4 38 23 10 0000 4 56 24 20 0000 4 88 25 30 0000 4 84 26 Edd 28 To run the Cubic Spline forecast click Risk Simulator Forecasting 29 Cubic Spline and then click on the link icon and select C15 C25 as the Known 30 X values values on the x axis of a time series chart and D15 D25 as the Known 31 Y values make sure the length of Known X and Y values the same Enter 32 the desired forecast periods e g Starting 1 Ending 50 Step Size 0 5 Click 233 and review the generated forecasts and chart Cubic Spline 36 The cublic spline polynomial interpolation and extrapolation model is used to fil in the 37 gaps eed LLL the model can 1
153. bjective r tz R A Lj gt D Set Decision w New Change Edit SetInput SetOutput Paste Remove Run RunSuper Step Reset Forecasting Run Analytical Profile Profile Profile Assumption Forecast Speed M Optimization Set Constraint Tools B New Simulation Profile Assumptions Forecasts Editing Simulation Run Forecasting Optimization Tools X Edit Simulation Profile Discounted Cash Flow ROI Model Change Simulation Profile To T e Tr Te TuHT TT T T ae Discounted Cash Flow ROI Model tie Set Output Forecast I Copy Parameter 2009 Sum PV Net Benefits 4 762 09 Discount Type Discrete End of Year Discounting 2009 Sum PV Investments 1 634 22 lunt Rate 15 00 Net Present Value 3 127 87 Model Include Terminal Valuation F gt Remove Parameter 5 00 Internal Rate of Retum 55 68 2 0096 Retum on Investment 191 40 Charts 40 00 Profitability Index 2 91 Minimize All Charts 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 B Run Simulation 511 50 512 00 13 50 59 Run Super Speed Simulation 51225 3 13 00 1325 14 00 smis 5 sisas siseo siszs sisso s1eos s1620 s1635 Reset Simulation 4 20 j 1 342 00 1 378 75 1 415 50 1 452 25 1 489 00 1 525 75 1 562 50 Forecasting E T 201 30 206 81
154. bull distribution shifts the starting location to any other value Alpha Beta and Location or Shift are the distributional parameters Input requirements Alpha Shape 2 0 05 Beta Central Location Scale gt 0 and can be any positive value Location can be any positive or negative value including zero 63 Page RISK SIMULATOR 3 FORECASTING orecasting is the act of predicting the future It can be based on historical data or speculation about the future when no history exists When historical data exist a quantitative or statistical approach is best but if no historical data exist then potentially a qualitative or judgmental approach is usually the only recourse Figure 3 1 lists the most common methodologies for forecasting QUALITATIVE QUANTI Use Risk Simulator to run Monte Carlo Simulations use distributional fitting or nonparametric custom distributions Figure 3 1 Forecasting Methods 64 Page RISK SIMULATOR 3 1 Different Types of Forecasting Techniques Generally forecasting can be divided into quantitative and qualitative approaches Qualitative forecasting is used when little to no reliable historical contemporaneous or comparable data ate available Several qualitative methods exist such as the Delphi or expert opinion approach a consensus building forecast by field experts marketing experts or internal staff members management assumptions target growth
155. cal significance In other words a p value for a correlation pair that is less than a given significance value is statistically significantly different from zero indicating that there is significant a linear relationship between the two variables The Peatson s R between two variables x and y is related to the covariance cov measure COV x y where R The benefit of dividing the covariance by the product of the two 45 variables standard deviation s is that the resulting correlation coefficient is bounded between 1 0 and 1 0 inclusive This makes the correlation a good relative measure to compate among different variables particularly with different units and magnitude The Spearman rank based nonparametric correlation is also included in the report The Spearman s R is related to the Pearson s R in that the data is first ranked and then correlated The rank correlation provides a better estimate of the relationship between two variables when one or both of them is nonlinear It must be stressed that a significant correlation does not imply causation Associations between variables in no way imply that the change of one variable causes another variable to change When two variables that are moving independently of each other but in a related path they may be correlated but their relationship might be spurious e g a correlation between sunspots and the stock market might be strong but one can surmise that there is
156. cally compute and enter all pairwise correlations or after setting some assumptions use the edit correlation tool to enter your correlation matrix e Equations in an Assumption Cell only empty cells or cells with static values can be set as assumptions however there might be times when a function or equation is required in an assumption cell and this can be done by first entering the input assumption in the cell and then typing in the equation or function when the simulation is being run the simulated values will replace the function and after the simulation completes the function or equation is again shown TIPS Copy and Paste e Copy and Paste using Escape when you select cell and use the Risk Simulator Copy function it copies everything into Windows clipboard including the cell s value equation function color font and size as well as Risk Simulator assumptions forecasts or decision variables Then as you apply the Risk Simulator Paste function you have two options The first option is to apply the Risk Simulator Paste directly and all cell values color font equation functions and parameters will be pasted into the new cell The second option is to first click 183 Page RISK SIMULATOR Escape on the keyboard and then apply the Risk Simulator Paste Escape tells Risk Simulator that you wish to paste only the Risk Simulator assumption forecast or decision variable and not the cell s values color equ
157. cally compute option is set as default and when you click RUN on a completed Decision Tree model the decision nodes will be updated with the results o Uncertainty Nodes Event Names Probabilities and Set Simulation Assumptions You can add probability event names probabilities and simulation assumptions only after the uncertainty branches are created o Terminal Nodes Manual Input Excel Link and Set Simulation Assumptions The terminal event payoffs can be entered manually or linked to an Excel cell e g if you have a large Excel model that computes the payoff you can link the model to this Excel model s output cell or set probability distributional assumptions for running simulations e View Node Properties Window is available from the Edit menu and the selected node s properties will update when a node is selected Decision Tree module also comes with the following advanced analytics o Monte Carlo Simulation Modeling on Decision Trees Bayes Analysis for obtaining posterior probabilities o Expected Value of Perfect Information MINIMAX and MAXIMIN Analysis Risk Profiles and Value of Imperfect Information o Sensitivity Analysis Scenario Analysis o Utility Function Analysis 5 27 2 Simulation Modeling This tool runs Monte Carlo risk simulation on the decision tree Figure 5 62 It allows you to set probability distributions as input assumptions for running simulations You can either set an assumption
158. ch of all linked precedents across multiple worksheets in the same workbook e Selecting Use Global Setting is useful when you have a large model and wish to test all the precedents at say 50 instead of the default 10 Instead of having to change each precedent s test values one at a time you can select this option change one setting and click somewhere else in the user interface to change the entire list of the precedents Deselecting this option will allow you the control to change test points one precedent at a time Zero or Empty Values is an option turned on by default where precedent cells with zero or empty values will not be run in the Tornado analysis This is the typical setting e 200 002 Possible Integer Values is an option that quickly identifies all possible precedent cells that currently have integer inputs This function is sometimes important if your model uses switches e g functions such as IF a cell is 1 then something happens and IF a cell has a 0 value something else happens or integers such as 1 2 3 etc which you do not wish to test For instance 10 of a flag switch value of 1 will return a test value of 0 9 and 1 1 both of which are irrelevant and incorrect input values in the model and Excel may interpret the function as an error This option when selected will quickly highlight potential problem areas for Tornado analysis and then you can determine which precedents to turn on or
159. chasing Product A will continue to purchase Product in the next period switch to a competitive brand B Real Options V Valuation Probability of Staying at State 1 if Starting at State 1 9 90 wwwresloptionsvalustion com Probability of Staying at State 2 if Starting at State 2 2 80 Figure 3 20 Markov Chains Switching Regimes 313 Limited Dependent Variables Logit Probit Tobit Maximum Likelihood Estimation Theory The term Limited Dependent Variables describes the situation where the dependent variable contains data that are limited in scope and range such as binary responses 0 or 7 or truncated ordered or censored data For instance given a set of independent variables e g age income education level of credit card or mortgage loan holders we can model the probability of default using maximum likelihood estimation MLE The response or dependent variable is binary That is it can have only two possible outcomes that we denote as 7 and 0 e g Y may represent presence absence of a certain condition defaulted not defaulted on previous loans success failure of some device answer yes no on a survey etc We also have a vector of independent variable regressors X which are assumed to influence the outcome Y A typical ordinary least squares regression approach is invalid because the regression errors are heteroskedastic and non normal and the resulting estimated probability estimates wil
160. comparison TIPS Efficient Frontier e Frontier Variables to access the frontier variables first set the model s Constraints before setting efficient frontier variables TIPS Forecast Cells e Forecast Cells with No Equations you can set output forecasts on cells without any equations or values simply ignore the warning message but be aware that the resulting forecast chart will be empty Output forecasts are typically set on empty cells when there are macros that are being computed and the cell will be continually updated TIPS Forecast Charts versus Spacebar hit TAB on the keyboard to update the forecast chart and to obtain the percentile and confidence values after you enter some inputs and hit the Spacebar to rotate among the various tabs in the forecast chart e Normal versus Global View click on these views to rotate between a tabbed interface and a global interface where all elements of the forecast charts are visible at once Copy copies the forecast chart or the entire global view depending on whether you are in the normal or global view TIPS Forecasting e Cell Link Address if you first select the data in the spreadsheet and then run a forecasting tool the cell address of the selected data will be automatically entered into the user interface Otherwise you will have to manually enter in the cell address or use the link icon to link to the relevant data location Forecast RMS
161. confidence or 0 10 alpha level while those highlighted in red indicate that they are not statistically significant at any other alpha levels Analysis of Variance Sums of Mean of Squares Squares F Statistic p Value Hypothesis Test Regression 38415447527 19207723 7638 3171851 1034 0 0000 Critical F statistic 9996 confidence with af of 2 and 432 4 6546 Residual 2616 0549 6 0557 Critical F statistic 95 confidence with af of 2 and 432 3 0166 Total 38418063 5026 19207729 8195 Critica F statistic 8096 confidence with af of 2 and 432 2 3149 The Analysis of Variance ANOVA table provides an F test of the regression model s overall statistical significance Instead of looking at individual regressors as in the Hest the F test looks at ali the estimated Coefficients statistical properties The F Statistic is calculated as the ratio of the Regression s Mean of Squares to the Residual s Mean of Squares The numerator measures how much of the regression is explained while the denominator measures how much is unexplained Hence the larger the F Statistic the more significant the model The corresponding p Value is calculated to test the nuli hypothesis Ho where all the Coefficients are simultaneously equal to zero versus the alternate hypothesis Ha that they are ail simultaneously different from zero indicating significant overall regression model If the p Value is smaller than the 0 01 0 05 or 0 10 alpha significance then the regression is
162. cts in another graphical manner where the highest impacting precedent is listed first The x axis is the NPV value with the center of the chart being the base case condition Green bars in the chart indicate a positive effect while red bars indicate a negative effect Therefore for investments the red bar on the right side indicates a negative effect of investment on higher NPV in other words capital investment and NPV are negatively correlated The opposite is true for price and quantity of products A to C their green bars are on the right side of the chart Tornado and Spider Charts Statistical Summary One of the powerful simulation tools is the tornado chart it captures the static impacts of each variable on the outcome of the model That is the tool automatically perturbs each precedent variable in the model a user specified preset amount captures the fluctuation on the model s forecast or final result and lists the resulting perturbations ranked from the most significant to the least Precedents are all the input and intermediate variables that affect the outcome of the model For instance if the model consists of A B C where C D E then B D and E are the precedents for A C is not a precedent as it is only an intermediate calculated value The range and number of values perturbed is user specified and can be set to test extreme values rather than smaller perturbations around the expected values In certain circumst
163. d A2 6 6 max min max min and B is the Beta function 4Mode Max Mean 6 Standard Deviation D H Skew 7 eee u Min Max 4 Input requirements Minimum lt Most Likely lt Maximum and can be positive negative or zero The Power distribution is related to the exponential distribution in that the probability of small outcomes is large but exponential decreases as the outcome value increases Alpha also known as shape is the only distributional parameter The mathematical constructs for the Power distribution are shown below ax F x x Mean ze 1 Standard Deviation 2 2 2 Skew E a at3 Excess Kurtosis is a complex function and cannot be readily computed Input requirements Alpha gt 0 59 Page Power 3 Distribution Student s t Distribution RISK SIMULATOR The Power 3 distribution uses the same constructs as the original Power distribution but adds a Location or Shift parameter and a multiplicative Factor parameter The Power distribution starts from a minimum value of 0 whereas this Power 3 or Shifted Multiplicative Power distribution shifts the starting location to any other value Alpha Location or Shift and Factor are the distributional parameters Input requirements Alpha gt 0 05 Location or Shift can be any positive or negative value including zero Factor gt 0 The Student s t dis
164. d below and click on Load Example to see the sample inputs corresponding to the selected analysis and the results shown in the grid on the right as well as which results are used as inputs in the decision tree in the figure e STEP 1 Enter the names for the first and second uncertainty events and choose how many probability events states of nature or outcomes each event has STEP 2 Enter the names of each probability event or outcome e STEP 3 Enter the second event s prior probabilities and the conditional probabilities for each event or outcome The probabilities must sum to 100 5 27 4 Expected Value of Perfect Information MINIMAX and MAXIMIN Analysis Risk Profiles and Value of Impertect Information This tool Figure 5 64 computes the Expected Value of Perfect Information EVP MINIMAX and MAXIMIN Analysis as well as the Risk Profile and the Value of Imperfect Information To get started enter the number of decision branches or strategies under consideration e g build a large medium or small facility the number of uncertain events or states of nature outcomes e g good market bad market and the expected payoffs under each scenario The Expected Value of Perfect Information that is assuming you had perfect foresight and knew exactly what to do through market research or other means to better discern the probabilistic outcomes computes if there is added value in such information i e if market res
165. d when the sample size is less than 30 but is also appropriate and in fact provides more conservative results with larger data sets This t test can be applied to three types of hypothesis tests a two tailed test a right tailed test and a left tailed test All three tests and their respective results are listed below for your reference Two Tailed Hypothesis Test A two tailed hypothesis tests the null hypothesis Ho such that the population mean is statistically identical to the hypothesized mean The alternative hypothesis is that the real population mean is statistically different from the hypothesized mean when tested using the sample dataset Using a t test if the computed p value is less than a specified significance amount typically 0 10 0 05 or 0 01 this means that the population mean is statistically significantly different than the hypothesized mean at 10 5 and 1 significance value or at the 90 95 and 99 statistical confidence Conversely if the p value is higher than 0 10 0 05 or 0 01 the population mean is statistically identical to the hypothesized mean and any differences are due to random chance Right Tailed Hypothesis Test Arighttailed hypothesis tests the null hypothesis Ho such that the population mean is statistically less than or equal to the hypothesized mean The alternative hypothesis is that the real population mean is statistically greater than the hypothesized mean when tested using the sample dataset Using
166. del test run a few super speed simulations while the model is being constructed to make sure that the final product will run the super speed simulation Do not wait until the final model is complete before testing super speed to avoid having to backtrack to identify where any broken links or incompatible functions exist e Regular Speed when in doubt regular speed simulation always works TIPS Tornado Analysis e Tornado Analysis the tornado analysis should never be run just once It is meant as a model diagnostic tool which means that it should ideally be run several times on 189 Page RISK SIMULATOR the same model For instance in a large model Tornado can be run the first time using all of the default settings and all precedents should be shown select Show All Variables This single analysis may result in a large report and long and potentially unsightly Tornado charts Nonetheless it provides a great starting point to determine how many of the precedents are considered critical success factors For example the Tornado chart may show that the first 5 variables have high impact on the output while the remaining 200 variables have little to no impact in which case a second tornado analysis is run showing fewer variables For the second run select Show Top 10 Variables if the first 5 are critical thereby creating a nice report and a Tornado chart that shows a contrast between the key factors and less critical factors You
167. deviation Mean and standard deviation are the distributional parameters Input requirements Standard deviation gt 0 and can be any positive value Mean can take on any value 55 Page Parabolic Distribution Pareto Distribution RISK SIMULATOR The parabolic distribution is a special case of the beta distribution when Shape Scale 2 Values close to the minimum and maximum have low probabilities of occurrence whereas values between these two extremes have higher probabilities or occurrence Minimum and maximum are the distributional parameters The mathematical constructs for the Parabolic distribution are shown below x Ur x 0 0 0 f Bana ora gt 0 B 0 x p Where the functional form above is for a Beta distribution and for a Parabolic function we set Alpha Beta 2 and a shift of location in Minimum with a multiplicative factor of Maximum Minimum Mens Min 2 A Standard Deviation Skewness 0 Excess Kurtosis 0 8571 Minimum and Maximum are the distributional parameters Input requirements Maximum gt minimum either input parameter can be positive negative or zero The Pareto distribution is widely used for the investigation of distributions associated with such empirical phenomena as city population sizes the occurrence of natural resources the size of companies personal incomes stock price fluctuations and error clustering
168. dologies forward backward correlation forward backwatd 43 Neural Network Forecasts linear nonlinear logistic hyperbolic tangent and cosine 44 Nonlinear Extrapolation nonlinear time series forecasting 45 S Curve logistic S curves 46 Time Series Analysis 8 time series decomposition models for predicting levels trends and seasonalities 47 Trendlines forecasting and fitting using linear nonlinear polynomial power logarithmic exponential and moving averages with goodness of fit 1 4 4 Optimization Module 48 49 50 51 52 Linear Optimization multiphasic optimization and general linear optimization Nonlinear Optimization detailed results including Hessian matrices LaGrange functions and more Static Optimization quick runs for continuous integers and binary optimizations Dynamic Optimization simulation with optimization Stochastic Optimization quadratic tangential central forward and convergence criteria 7 Page RISK SIMULATOR 53 54 55 56 57 58 Efficient Frontier combinations of stochastic and dynamic optimizations on multivariate efficient frontiers Genetic Algorithms used for a variety of optimization problems Multiphasic Optimization testing for local versus global optimum allowing better control over how the optimization is run and increases the accuracy and dependency of the results Percentiles and Conditional Means addition
169. double click to run a model S and go to the Command console V You can teplicate the model or create your own and click Run Command X when ready Each line in the console represents a model and its relevant parameters e entire bizstats profile where data and multiple models are created and saved can be edited directly in XML Z by opening the XML Editor from the File menu Changes to the profile can be programmatically made here and takes effect once the file is saved e Click on the data grid s column header s to select the entire column s or variable s and once selected you can right click on the header to Auto Fit the column Cut 165 RISK SIMULATOR Copy Delete or Paste data You can also click on and select multiple column headers to select multiple variables and right click and select Visualize to chart the data e Ifa cell has a large value that is not completely displayed click on and hover your mouse over that cell and you will see a popup comment showing the entire value or simply resize the variable column drag the column to make it wider double click on the column s edge to auto fit the column or right click on the column header and select auto fit e Use the up down left right keys to move around the grid or use the Home and End keys on the keyboard to move to the far left and far right of a row You can also use combination keys such as to jump to the top left cell Ct
170. e e The number of possible occurrences in any unit of measurement is not limited to a fixed number e The occurrences ate independent The number of occurrences in one unit of measurement does not affect the number of occurrences in other units The average number of occurrences must remain the same from unit to unit The mathematical constructs for the gamma distribution are as follows a l _ 6 Ne I a B o Standard Deviation with any valueof gt 0 and 8 gt 0 f x SRewness 2 a Excess Kurtosis 6 a Shape parameter alpha 0 and scale parameter beta 8 are the distributional parameters and J is the Gamma function When the alpha parameter is a positive integer the gamma distribution is called the Erlang distribution used to predict waiting times in queuing systems where the Erlang distribution is the sum of independent and identically distributed random variables each having a memoryless exponential distribution Setting 7 as the number of these random variables the mathematical construct of the Erlang distribution is f x lt for all x gt 0 and all positive integers of 7 1 Input requirements Scale beta gt 0 and can be any positive value Shape alpha 2 0 05 and any positive value Location can be any value 52 Page Laplace Distribution Logistic Distribution RISK SIMULATOR The Laplace distribution is also sometimes called the double exponential di
171. e Maximize Return to Risk Ratio C18 Decision Variables Allocation Weights E6 E15 Restrictions on Decision Variables Minimum and Maximum Required F6 G15 Constraints Total Allocation Weights Sum to 100 E17 Open the example file and start a new profile by clicking on Risk Simulator New Profile and provide it a name first step in optimization is to set the decision variables Select cell E6 set the first decision variable Stimulator Optimization Set Decision and click on the link icon to select the name cell B6 as well as the lower bound and upper bound values at cells and G6 Then using Risk Szulator s copy copy this cell E6 decision variable and paste it to the remaining cells in E7 o E15 102 Page RISK SIMULATOR e The second step in optimization is to set the constraint There is only one constraint here that is the total allocation in the portfolio must sum to 100 So click on Risk Simulator Optimization Constraints and select ADD to add a new constraint Then select the cell E77 and make it egual to 100 Click OK when done final step in optimization is to set the objective function and start the optimization by selecting the objective cell C78 and Simulator Optimization Run Optimization and then selecting the optimization of choice Static Optimization Dynamic Optimization Stochastic Optimization To get started select Static Optimizatio
172. e e g B11 K30 click on Risk Simulator Tools Principal Component Analysis and click OK 2 Review the generated report for the computed results Principal Component Analysis Principal Component Analysis is a way of identifying patterns in data and recasting the data in such as way as to highlight their similarities and differences Patterns of data are very difficult to find in high dimensions when multiple variables exist and higher dimensional graphs are very difficult to represent and interpret Once the patterns in the data are found they can be compressed and the number of dimensions is now reduced This reduction of data dimensions does not mean much reduction in loss of information Instead similar levels of information can now be obtained by less number of variables Data Location B11 K30 Figure 5 43 Principal Component Analysis 5 17 Structural Break Analysis A structural break tests whether the coefficients in different data sets are equal and this test is most commonly used in time series analysis to test for the presence of a structural break Figure 5 44 A time series data set can be divided into two subsets Structural break analysis is used to test each subset individually and on one another and on the entire data set to statistically determine if indeed there is a break starting at a particular time period The structural break test is often used to determine whether the independent vari
173. e highest returning asset and the remaining 25 to the second best returns asset Optimization is not required However when allocating the portfolio this way the risk is a lot higher as compared to when maximizing the returns to risk ratio although the portfolio returns by themselves are higher e In contrast one can minimize the total portfolio risk but the returns will now be less Table 4 1 illustrates the results from the three different objectives being optimized and shows that the best approach is to maximize the returns to risk ratio that is for the same amount of tisk this allocation provides the highest amount of return Conversely for the same amount of return this allocation provides the lowest amount of risk possible This approach of bang for the buck or returns to risk ratio is the cornerstone of the Markowitz efficient frontier in modern portfolio theory That is if we constrained the total portfolio risk level and successively increased it over time we will obtain several efficient portfolio allocations for different risk characteristics Thus different efficient portfolio allocations can be obtained for different individuals with different risk preferences Portfolio Portfolio Returns Risk ta Objective Risk Ratio Maximize Returns to Risk Ratio 12 69 4 52 2 8091 Maximize Returns 13 97 6 77 2 0636 Minimize Risk 12 38 4 46 2 7754 Table 4 1 Optimization Results ASSET ALLOCATION O
174. e Click on the Charts and Tables tab Figure 5 49 select a distribution A e g Arcsine choose if you wish to run the CDF ICDF or PDF B enter the relevant inputs and click Run Chart or Run Table C You can switch between the Charts and Table tab to view the results as well as try out some of the chart icons E to see the effects on the chart e You can also change two parameters H to generate multiple charts and distribution tables by entering the From To Step input or using the Custom inputs and then hitting Run For example as illustrated in Figure 5 50 run the Beta distribution and select PDF G select Alpha and Beta to change H using custom I inputs and enter the relevant input parameters 2 5 5 for Alpha and 5 3 5 for Beta J and click Run Chart This will generate three Beta distributions K Beta 2 5 Beta 5 3 and Beta 5 5 L Explore various chart types gridlines language and decimal settings M and try rerunning the distribution using theoretical versus empirically simulated values N e Figure 5 51 illustrates the probability tables generated for a binomial distribution where the probability of success and number of successful trials random variable X are selected to vary O using the From To Step option Try to replicate the calculation as shown and click on the Table tab P to view the created probability density function results This example uses a binomial distribution with a starting input
175. e Dependent Variable Y drop down menu Click OK when finished Figure 5 22 Multiple Regression Analysis Data Set Diagnostic Tool This tool is used to diagnose forecasting problems in a set of multiple variables Dependent Variable Dependent Variable Y Figure 5 22 Running the Data Diagnostic Tool A common violation in forecasting and regression analysis is heteroskedasticity that is the variance of the errors increases over time see Figure 5 23 for test results using the Diagnostic tool Visually the width of the vertical data fluctuations increases or fans out over time and typically the coefficient of determination R squared coefficient drops significantly when heteroskedasticity exists If the variance of the dependent variable is not constant then the 137 Page RISK SIMULATOR error s variance will not be constant Unless the heteroskedasticity of the dependent variable is pronounced its effect will not be severe The least squares estimates will still be unbiased and the estimates of the slope and intercept will either be normally distributed if the errors are normally distributed or at least normally distributed asymptotically as the number of data points becomes large if the errors are not normally distributed The estimate for the variance of the slope and overall variance will be inaccurate but the inaccuracy is not likely to be substantial if the independent variable values are symmetric about
176. e but are governed by specific statistical and probabilistic rules The main stochastic processes include random walk or Brownian motion mean reversion and jump diffusion These processes can be used to forecast a multitude of variables that seemingly follow random trends but restricted by probabilistic laws The process generating equation is known in advance but the actual results generated ate unknown Figure 5 26 The Random Walk Brownian Motion process can be used to forecast stock prices prices of commodities and other stochastic time series data given a drift or growth rate and volatility around the drift path The Mean Reversion process can be used to reduce the fluctuations of the Random Walk process by allowing the path to target a long term value making it useful for forecasting time seties variables that have a long term rate such as interest rates and inflation rates these are long term target rates by regulatory authorities or the market The Jump Diffusion process is useful for forecasting time series data when the variable can occasionally exhibit random jumps such as oil prices or price of electricity discrete exogenous event shocks can make prices jump up or down These processes can also be mixed and matched as required A note of caution is required here The stochastic parameters calibration shows all the parameters for all processes and does not distinguish which process is better and which is worse or which process i
177. e potential is more attractive Enter the minimum and maximum expected value of your terminal payoffs and the number of data points in between to compute the utility curve and table If you had a 50 50 gamble where you either earn X or lose X 2 versus not playing and getting a 0 payoff what would this X be For example if you are indifferent between a bet where you can win 100 or lose 50 with equal probability compared to not playing at all then your X is 100 Enter the X in the Positive Earnings box below Note that the larger X is the less risk averse you are whereas a smaller X indicates that you are more risk averse Enter the required inputs select the U x type and click Compute Utility to obtain the results You can also apply the computed U x values to the decision tree to re run it or revert the tree back to using expected values of the payoffs 178 Page RISK SIMULATOR TE RON Vinal Modele SS _ EE ae mj Ta tet Te Dupard Umpep 9E 2 5 0 4 SOs SSH 9 435 OS Deer Tret mary of valor acaiym EVP Mea Profe Arai maro Fats ity fuesen d Urcartenty 3 Ka Jam wa ma v ae 20800 1500 m5 insert Above label insert abet l Comment Gard Event Name Wero were row nos The nn has been competes Ui Shen Name hem voe how Notes E 204 00 000
178. e retrieved at a later time by selecting Risk Simulator Tools Data Open Import The third option is the most popular selection that is to save the simulated results as a risksim file where the results can be retrieved later and a simulation does not have to be rerun each time Figure 5 20 shows the dialog box for extracting or exporting and saving the simulation results Hi Data Extraction t3 Data Extraction is used to obtain the raw data generated in a simulation The data can be extracted from both assumptions and forecasts The raw data can then be manipulated and additional analysis can be performed as desired Select the parameter s to extract Extract Name Worksheet Sample Second 3 Sheet1 E13 v Sample Second Sheetl _ 9_ Sample Third Sheet E10 New Excel Worksheet New Excel Worksheet Risk Simulator Data risksim Text File txt Extraction Format Figure 5 20 Sample Simulation Report 135 Page i RISK SIMULATOR 5 7 Create Report After a simulation is run you can generate a report of the assumptions and forecasts used in the simulation run as well as the results obtained during the simulation run Procedure e Open or create a model define assumptions and forecasts and run the simulation e Select Risk Stimulator Create Report Figure 5 21 Simulation Example Profile 77RA 9255 1 7 44 122
179. e same but are now computed vety quickly reports are generated very quickly as well Web Resources Case Studies and Videos download free models getting started videos case studies whitepapers and other materials from our website 1 4 2 Simulation Module 21 22 223 24 25 26 27 28 29 6 random number generators ROV Advanced Subtractive Generator Subtractive Random Shuffle Generator Long Period Shuffle Generator Portable Random Shuffle Generator Quick IEEE Hex Generator and Basic Minimal Portable Generator 2 sampling methods Monte Carlo and Latin Hypercube 3 Correlation Copulas applying Normal Copula T Copula and Quasi Normal Copula for correlated simulations 42 probability distributions arcsine Bernoulli beta beta 3 beta 4 binomial Cauchy chi square cosine custom discrete uniform double log Erlang exponential exponential 2 F distribution gamma geometric Gumbel max Gumbel min hypergeometric Laplace logistic lognormal arithmetic and lognormal log lognormal 3 arithmetic and lognormal 3 log negative binomial normal parabolic Pareto Pascal Pearson V Pearson VI PERT Poisson power power 3 Rayleigh t and 12 triangular uniform Weibull Weibull 3 Alternate Parameters using percentiles as an alternate way of inputting parameters Custom Nonparametric Distribution make your own distributions for running historical simulations and applying the Del
180. e same from trial to trial The mathematical constructs for the geometric distribution are as follows P x p l p for0 lt p lt landx 1 2 n ee p 1 2 Standard Deviation 2 p 1 SRewness 2 Excess Kurtosis 6p 6 1 Probability of success is the only distributional parameter The number of successful trials simulated is denoted x which can only take on positive integers Input requirements Probability of success gt 0 and 1 Le 0 0001 p X 0 9999 It is important to note that probability of success p of 0 or 1 are trivial conditions that do not require any simulations and hence are not allowed in the software The hypergeometric distribution 1s similar to the binomial distribution in that both describe the number of times a particular event occuts in a fixed number of trials The difference is that binomial distribution trials are independent whereas hypergeometric distribution trials change the probability for each subsequent trial and are called trials without replacement For example suppose a box of manufactured parts is known to contain some defective parts You choose a part from the box find it is defective and remove the part from the box If you choose another part from the box the probability that it is defective is somewhat lower than for the first part because you have already removed a defective part If you had replaced the defective part the probabil
181. e shown below y Del B Ta ipa ee 202 a l p Standard Deviation a 1 2 Skew a Kaa Excess Kurtosis 3 4 Input requirements Alpha Shape gt 0 Beta Scale gt 0 57 Page Pearson VI Distribution PERT Distribution RISK SIMULATOR The Pearson VI distribution is related to the Gamma distribution where it is the rational function of two variables distributed according to two Gamma distributions Alpha 1 also known as shape 1 Alpha 2 also known as shape 2 and Beta also known as scale are the distributional parameters The mathematical constructs for the Pearson VI distribution are shown below Gu py B a 1 x By X I Jo Mean Ba 05 1 2 Standard Deviation ala 051 a 71 2 Skew 2 a 2 20 1 a a 1 a 3 Excess Kurtosis 3 2 2 a p 3 a 4 a 1 3 Input requirements Alpha 1 Shape 1 gt 0 Alpha 2 Shape 2 gt 0 Beta Scale gt 0 The PERT distribution is widely used in project and program management to define the worst case nominal case and best case scenarios of project completion time It is related to the Beta and Triangular distributions PERT distribution can be used to identify risks in project and cost models based on the likelihood of meeting targets and
182. earch will add value as compared to more naive estimates of the probabilistic states of nature To get started enter the number of decision branches or strategies under consideration e g build a large medium or small facility and the number of uncertain events or states of nature outcomes e g good market bad market and enter the expected payoffs under each scenario MINIMAX minimizing the maximum regret and MAXIMIN maximizing the minimum payoff are two alternate approaches to finding the optimal decision path These two approaches ate not used often but still provide added insight into the decision making process Enter the number of decision branches or paths that exist e g building a large medium or small facility as well as the uncertainty events or states of nature under each path e g good economy vs bad economy Then complete the payoff table for the various scenatios and Compute the MINIMAX and MAXIMIN results You can also click on Load Example to see a sample calculation 177 Page Procedure RISK SIMULATOR 5 27 5 Sensitivity Sensitivity analysis Figure 5 65 on the input probabilities is performed to determine the impact of inputs on the values of decision paths First select one Decision Node to analyze below and then select one probability event to test from the list If there are multiple uncertainty events with identical probabilities they can be analyzed either independently or concurrently The
183. ed Payback gt S asonality Test 44 Segmentation Clustering 45 Risk Analysis Sensitivity A 46 Base Case PV at Time 0 TT 5384 30 5344 89 299 60 260 27 226 00 196 41 1 238 69 4T PV of Cash Flow at Time Scenario Analysis 94 5441 94 396 62 5355 26 317 94 284 32 254 05 1 779 86 48 Intermediate X Variable Statistical Analysis S ET infomation Model Structural Break Test H Figure 5 46 Model Checking Tool 5 20 Percentile Disttibutional Fitting Tool The Percentile Distributional Fitting tool Figure 5 47 is another alternate way of fitting probability distributions There are several related tools and each has its own uses and advantages e Distributional Fitting Percentiles using an alternate method of entry percentiles and first second moment combinations to find the best fitting parameters of a specified distribution without the need for having raw data This 160 Page Procedure RISK SIMULATOR method is suitable for use when there are insufficient data only when percentiles and moments ate available or as a means to recover the entire distribution with only two or three data points but the distribution type needs to be assumed or known e Distributional Fitting Single Variable using statistical methods to fit your raw data to all 42 distributions to find the best fitting distribution and its input parameters Multiple data points are required for a good f
184. ed in the Statistics tab as shown in Figure 2 18 error equation of x tZ R Income Risk Simulator Forecast Histogram Statistics Preferences Options Controls Global View Data Filter Show all data Show only data between nfinty _ Show only data within _____6 standard deviation s Statistic Precision level used to calculate the error 9534 Show the following statistic s on the histogram Mean Median 1st Quartile 3rd Quartile Show Decimals ChartX Axis 4 Confidence 4 Statistics 4 and Infinity Percentage Error Precision at 95 Confidence Figure 2 18 Computing the Error 30 Page Measuring the Center of the Distribution the First Moment Measuring the Spread of the Distribution the Second Moment Skew 0 KurtosisXS 0 RISK SIMULATOR 2 3 5 Understanding the Forecast Statistics Most distributions can be defined up to four moments The first moment describes a distribution s location or central tendency expected returns the second moment describes its width or spread risks the third moment its directional skew most probable events and the fourth moment its peakedness or thickness in the tails catastrophic losses or gains All four moments should be calculated in practice and interpreted to provide a more comprehensive view of the project under analysis Risk Simula
185. een Show only data within 6 2 standard deviation s Statistic Precision level used to calculate the error 95 Show the following statistic s on the histogram Mean Median 1st Quartile 3rd Quartile Show Decimals Chart X Axis 4 Confidence 4 Statistics 4 Figure 2 8B Forecast Chart Options d 0 s Char Type Chan Overlay Continuous v Min Max Au Tite Income 1000 Trials X Axis vj Y Axis V ChanXAxis 4 4 Decimal Distribution Fitting Done Actual Theoretical Continuous Logistic Mean 086 085 Discrete Stdev 0 19 0 23 Fit Stats 0 03 2 Decimals Skew 0 12 0 00 P Value 0 2782 Kut 945 120 Figure 2 8C Forecast Chart Controls 22 Page RISK SIMULATOR Income Risk Simulator Forecast She CCEELEDD DDD ORR B OTD us Percentage Eror Precision at 95 Confidence Char Overlay Continuous Data Filter Min Max Show all data xas Te Show only data between Infinity and Infinity Curva 7 4 Show only data within standard deviation s Distribution Fitting Done Statistic ns Theoretical 8 Continuous Precision level used to calculate the error 95 4 c Show the following statistic s on the histogram Fit Stats 0 03 023 0 00 Mean Median 15
186. egy Tree to be encrypted with up to a 256 bit password encryption Be careful when a file is being encrypted because if the passwotd is lost the file can no longer be opened Capturing the Screen or printing the existing model can be done through the File menu The captured screen can then be pasted into other software applications e Add Duplicate Rename and Delete a Strategy Tree can be performed through right clicking the Strategy Tree tab or the Edit menu e You can also Insert File Link and Insert Comment on any option or terminal node or Insert Text or Insert Picture anywhere in the background or canvas area e You can Change Existing Styles or Manage and Create Custom Styles of your Strategy Tree this includes size shape color schemes and font size color specifications of the entire Strategy Tree e Insert Decision Insert Uncertainty or Insert Terminal nodes by selecting any existing node and then clicking on the decision node icon square uncertainty node icon circle or terminal node icon triangle or use the functionalities in the Insert menu 175 Page RISK SIMULATOR Modify individual Decision Uncertainty or Terminal nodes properties by double clicking on a node The following are some additional unique items in the Decision Tree module that can be customized and configured in the node properties user interface o Decision Nodes Custom Override or Auto Compute the value on a node The automati
187. el assumes that there is a latent unobservable variable Y This variable is linearly dependent on the X variables via a vector of f coefficients that determine their interrelationships In addition there is a normally distributed error term U to capture random influences on this relationship The observable variable Y is defined to be equal to the latent variables whenever the latent variables are above zero and is assumed to be zero otherwise That is Y Y if Y gt 0 and Y 0 if Y 0 If the relationship parameter fj is estimated by using ordinary least squares regression of the observed Y on X the resulting regression estimators are inconsistent and yield downward biased slope coefficients and an upward biased intercept Only MLE would be consistent for a Tobit model In the Tobit model there is an ancillary statistic called sigma which is equivalent to the standard error of estimate in a standard ordinary least squares regression and the estimated coefficients are used the same way as a tegression analysis 95 Page RISK SIMULATOR Procedure e Start Excel and open the example file Advanced Forecasting Model go to the MLE worksheet sect the data set including the headers and click on Risk Simulator Forecasting Maximum Likelihood e Select the dependent variable from the drop down list see Figure 3 21 and click OK to run the model and report Binary Logistic Maximum Likelihood Forecast Logit Probit Tobit LOGIT
188. els Each project has its own returns ENPV and NPV for expanded net present value and net present value the ENPV is simply the NPV plus any strategic real options values costs of implementation risks and so forth If required this model can be modified to include required full time equivalences FTE and other resources of various functions and additional constraints can be set on these additional resources The inputs into this model are typically linked from other spreadsheet models For instance each project will have its own discounted cash flow or returns on investment model The application here is to maximize the portfolio s Sharpe ratio subject to some budget allocation Many other versions of this model can be created for instance maximizing the portfolio returns or minimizing the risks or adding constraints where the total number of projects chosen cannot exceed 6 and so forth and so on All of these items can be run using this existing model Open the example file and start a new profile by clicking on Simulator New Profile and provide it a name first step in optimization is to set up the decision variables Set the first decision variable by selecting cell J4 select Simulator Optimization Set Decision click on the link icon to select the name cell 84 and select the Bzvary variable Then using Risk Simulator s copy copy this cell J4 decision variable and paste the decision variable to the
189. endent Proportions A 121 VAR6 VAR VARB Power 1220 1 1230 gt Relative LN Returns g notes _id 2 parameter VARS Relative Returns aa basic tags ox caret Segmentation Clustering Sa T Devon toner Seer erates UE Figure 5 55 ROV BizStats XML Editor 168 Page Procedure Procedure RISK SIMULATOR 5 23 Neural Network and Combinatorial Fuzzy Logic Forecasting Methodologies The term Neural Network is often used to refer to a network or circuit of biological neurons while modern usage of the term often refers to artificial neural networks comprising artificial neurons or nodes recreated in a software environment Such networks attempt to mimic the neurons in the human brain in ways of thinking and identifying patterns and in our situation identifying patterns for the purposes of forecasting time series data In Risk Simulator the methodology is found inside the ROV BizStats module located at Rise Simulator ROW BixStats Neural Network as well as in Simulator Forecasting Neural Network Figure 5 56 shows the Neural Network forecast methodology Click on Simulator Forecasting Neural Network e Start by either manually entering data or pasting some data from the clipboard e g select and copy some data from Excel start this tool and paste the data by clicking on the Paste button e Select if you wish t
190. entent 12 2 2 2 Running a Monte Carlo Simulation 13 2 2 3 Forecast Chart Tabs ee retten pta d cago caede e edet 21 2 3 4 Using Forecast Charts and Confidence Intervals eee 23 2 3 Correlations and Precision Control soe e qute petam rice eA 26 231 The Basies of Correlations erede a 26 2 3 2 Applying Correlations in Risk Simulator 27 2 3 3 The Effects of Correlations in Monte Carlo Simulation sees 27 24 Precision and Error Control seen ep p RR RR RET RE n RD ORAS 29 2 3 5 Understanding the Forecast Statistics seen 31 2 3 6 Understanding Probability Distributions for Monte Carlo Simulation 35 24 Discrete 1212 28 BoP CORDE EAD tendono 45 FORECASTING cite DEED A RE SI AS 64 3 1 Diferent Types of Forecasting TOD TIES doce 65 3 2 Running the Forecasting Tool in Risk Simulatot 68 Sb 69 ed RERO SS isa 72 2o OUR ORI So 76 toa E Paid Ah Fable feu on fub Pa Def eye 78 3 7 Box Jenkins ARIMA Advanced Time Series uisus et dated Fabr Fan laps 80 3 8 AUTO ARIMA Box Jenkins ARIMA Advanced Tittte Sertes essere 82 TOO
191. enues and many other time series data are typically autocorrelated where the value in the current period is related to the value in a previous period and so forth clearly the inflation rate in March is related to Februaty s level which in turn is related to January s level etc Ignoting such blatant relationships will yield biased and less accurate forecasts In such events an autocortelated regression model or an ARIMA model may be better suited Risk Simulator Forecasting ARIMA Finally the autocorrelation functions of a series that is nonstationary tend to decay slowly see the nonstationary report in the model If autocorrelation AC 1 is nonzero it means that the series is first order serially correlated If AC E dies off more or less geometrically with increasing lag it implies that the series follows a low order autoregressive process If AC k drops to zero after a small number of lags it implies that the series follows a low order moving average process Partial correlation PAC k measures the correlation of values that are k periods apart after removing the correlation from the intervening lags If the pattern of autocorrelation can be captured by an autoregression of order less than k then the partial autocorrelation at lag k will be close to zero Ljung Box Q statistics and their p values at lag k have the null hypothesis that there is no autocorrelation up to order k The dotted lines in the plots of the autocorrelations
192. eracting lagged and mixed variables to be automatically run on your data to determine the best fitting econometric model that describes the behavior of the dependent variable It is useful for modeling the effects of the variables and for forecasting future outcomes while not requiring the analyst to be an expert econometrician In contrast the term fuzzy logic is derived from fuzzy set theory to deal with reasoning that is approximate rather than accurate as opposed to crisp logic where binary sets have binary logic fuzzy logic variables may have a truth value that ranges between 0 and 1 and is not constrained to the two truth values of classic propositional logic This fuzzy weighting schema is used together with a combinatorial method to yield time series forecast results Sometimes there are missing values in a time series data set For instance interest rates for years 1 to 3 may exist followed by years 5 to 8 and then year 10 Spline curves can be used to interpolate the missing years interest rate values based on the data that exist Spline curves can also be used to forecast or extrapolate values of future time periods beyond the time period of available data The data can be linear or nonlinear Using Risk Simulator expert opinions can be collected and a customized distribution can be generated This forecasting technique comes in handy when the data set is small or the goodness of fit is bad when applied to a distributional fitti
193. es Other product names mentioned herein may be trademarks and or registered trademarks of the respective holders Copyright 2005 2012 Dr Johnathan Mun All rights reserved Real Options Valuation Inc 4101F Dublin Blvd Ste 425 Dublin California 94568 U S A Phone 925 271 4438 Fax 925 369 0450 admin realoptionsvaluation com www tisksimulator com www realoptionsvaluation com Real Options Valuation Table of Contents 1 INTRODUCTION TO RISK SIMULATOR 1 For Welcome to Risk ose asin ape 1 1 2 Installation Requirements and ts te tis te t tet dan den 2 LAT SU tans A dote dp SUE 2 14 Whats Newin 2 TAT General Capabilities eterno teens eee Ee Eee eek 5 14 2 Simulatioti Module d 6 1 4 3 Forecasung Module e n tete it te Rie ORE 7 14 4 Optimization Module nante RR NI d deba tie 7 1 4 5 Analytical Tools Module nisse e Radio qi AS SARI NN INN 8 1 4 6 Statistics and BizStats Module eee tentent tentent 9 2 MONTE CARLO RISK SIMULATION eee e eee eere nnn 11 VAI Deak I PRI 018177517 TERM MR 11 2 2 Gui Stared wih Rsk SHULL 12 2 2 1 A High Level Overview of the Software esee t
194. es a sample dataset You can select the data and run the tool through Simulator Tools Segmentation Clustering Figure 5 40 shows a sample segmentation of two groups That is taking the original data set we run some internal algorithms a combination or k means hierarchical clustering and other method of moments in order to find the best fitting groups or natural statistical clusters to statistically divide or segment the original data set into two groups You can see the two group memberships in Figure 5 40 Clearly you can segment this data set into as many groups as you wish This technique is valuable in a variety of settings including marketing market segmentation of customers into various customer relationship management groups etc physical sciences engineering and others Cluster and Segmentation Analysis Clustering and segmentation analysis is used to mathematically separate a set of data into different segment groups or clusters Selected Data 176 Options Showall 2 segmentation clusters Showcluster number 2 Show cluster numbership for value SEGMENTATION AND CLUSTER ANALYSIS RESULT Groups Sample Ordered Data 2 1 1 00 1 2 1 00 1 3 2 00 1 4 3 00 1 5 2 00 1 6 4 00 1 7 15 00 1 8 16 00 1 9 14 00 1 10 15 00 1 11 125 00 2 12 126 00 2 13 128 00 2 14 129 00 2 15 130 00 2 16 175 00 2 17 179 00 2 18 174 00 2 Figure 5 40 Segmentation Clustering Tool and Results 153
195. es of the true population b values in the following regression equation 60 b1X1 b2X2 bnXn The Standard Error measures how accurate the predicted Coefficients are and the t Statistics are the ratios of each predicted Coefficient to its Standard Error The t Statistic is used in hypothesis testing where we set the null hypothesis Ho such that the real mean of the Coefficient 0 and the alternate hypothesis Ha such that the real mean of the Coefficient is not equal to 0 A t test is is performed and the calculated t Statistic is compared to the critical values at the relevant Degrees of Freedom for Residual The t test is very important as it calculates if each of the coefficients is statistically significant in the presence of the other regressors This means that the t test statistically verifies whether a regressor or independent variable should remain in the regression or it should be dropped The Coefficient is statistically significant if its calculated t Statistic exceeds the Critical t Statistic at the relevant degrees of freedom df The three main confidence levels used to test for significance are 90 95 and 99 If a Coefficient s t Statistic exceeds the Critical level it is considered statistically significant Alternatively the p Value calculates each t Statistic s probability of occurrence which means that the smaller the p Value the more significant the Coefficient The usual significant levels for the p Value are 0
196. eturns in cell C17 is SUMPRODUCT C6 C15 E6 E15 that is the sum of the allocation weights multiplied by the annualized returns for each asset class In other words we have Rp R R R Rp where Rp is the return on the portfolio ate the individual returns on the projects and are the respective weights or capital allocation across each project In addition the portfolio s diversified risk in cell D17 is computed by taking m 1 E 273 Op 2 06 gt gt 20 0 0 0 1 i l j l Here ate the respective cross correlations between the asset classes hence if the cross correlations are negative there are risk diversification effects and the portfolio risk decreases However to simplify the computations here we assume zero correlations among the asset classes through this portfolio risk computation but assume the correlations when applying simulation on the returns as will be seen later Therefore instead of applying static correlations among these different asset returns we apply the correlations in the simulation assumptions themselves creating a more dynamic relationship among the simulated return values Finally the return to risk ratio or Sharpe ratio is computed for the portfolio This value is seen in cell C18 and represents the objective to be maximized in this optimization exercise To summarize we have the following specifications in this example model Objectiv
197. eviation NE Skewness d 1 Excess Kurtosis 9 Rate or Lambda A is the only distributional parameter Input requirements Rate gt 0 and lt 1000 Le 0 0001 lt rate lt 1000 44 Page Arcsine Distribution Beta Distribution RISK SIMULATOR 2 5 Continuous Distributions The arcsine distribution is U shaped and is a special case of the beta distribution when both shape and scale are equal to 0 5 Values close to the minimum and maximum have high probabilities of occurrence whereas values between these two extremes have very small probabilities of occurrence Minimum and maximum are the distributional parameters The mathematical constructs for the Arcsine distribution are shown below The probability density function PDF is denoted f x and the cumulative distribution function CDF is denoted 1 for0 lt xxl f x 24 ayx x 0 otherwise 0 lt 0 2 an x for0 lt x lt 1 1 x 1 Min Max 2 E Standard Deviation ae SRewness 0 for all inputs Mean Excess Kurtosis 1 5 for all inputs Minimum and maximum are the distributional parameters Input requirements Maximum gt minimum either input parameter can be positive negative or zero The beta distribution is very flexible and is commonly used to represent variability over a fixed range One of the more important applications of the beta distribution is its use as a conjugate distribution
198. exactly Plotting data is one guide to selecting a probability distribution The following steps provide another process for selecting probability distributions that best describe the uncertain variables in your spreadsheets e Look at the variable in question List everything you know about the conditions surrounding this variable You might be able to gather valuable information about the uncertain variable from historical data If historical data are not available use your own judgment based on experience listing everything you know about the uncertain variable Review the descriptions of the probability distributions e Select the distribution that characterizes this variable A distribution characterizes a variable when the conditions of the distribution match those of the variable Monte Carlo simulation in its simplest form is a random number generator that is useful for forecasting estimation and risk analysis A simulation calculates numerous scenarios of a model by repeatedly picking values from a user predefined probability distribution for the uncertain variables and using those values for the model As all those scenarios produce associated results in a model each scenario can have a forecast Forecasts are events usually with formulas or functions that you define as important outputs of the model These usually are events such as totals net profit or gross expenses Simplistically think of the Monte Carlo simulation a
199. f 67 70 and 32 30 is of course 100 the total probability under the curve 24 Page RISK SIMULATOR F Income Risk Simulator Forecast l Figure 2 13 Forecast Chart Probability Evaluation TIPS e The forecast window is resizable by clicking on and dragging the bottom right corner of the forecast window tis also advisable that the current simulation be reset Rak Simulator Reset Simulation before rerunning a simulation Remember that you will need to hit TAB on the keyboard to update the chart and results when you type in the certainty values or right and left tail values 25 Page RISK SIMULATOR e You can also hit the spacebar on the keyboard repeatedly to cycle among the histogram to statistics preferences options and control tabs e In addition if you click on Simulator Options you can access several different options for Risk Simulator including allowing Risk Simulator to start each time Excel starts or to only statt when you want it to by going to Szart Programs Real Options Valuation Risk Simulator Risk Simulator changing the cell colors of assumptions and forecasts and turning cell comments on and off cell comments will allow you to see which cells are input assumptions and which are output forecasts as well as their respective input parameters and names Do spend some time playing around with the forecast chart outputs and various bell
200. ferent from zero at approximately the 5 significance level Finding the right ARIMA model takes practice and experience These AC PAC SC and AIC diagnostic tools are highly useful in helping to identify the correct model specification 81 Page RISK SIMULATOR ARIMA is an advanced modeling technique fj used to model and forecast time series data data that have a time component to it e g interest rates inflation sales revenues gross domestic product Time Series Variable ssa g Exogenous Variable Autoregressive Order AR p 118 Differencing Order I d o Moving Average Order MA q 01 Maximum Iterations 100 Forecast Periods 54 Backcast CX Figure 3 13 Box Jenkins ARIMA Forecast Tool 82 Page RISK SIMULATOR ARIMA Autoregressive Integrated Moving Average Regression Statistics R Squared Coefficient of Determination 0 9999 Akaike Information Criterion AIC 46213 Adjusted R Squared 0 9999 Schwarz Criterion SC 4 6632 Muitipie R Multiple Correlation Coefficient 1 0000 Log Likelihood 1005 1340 Standard Error of the Estimates SEV 297 5246 Durbin Watson DW Statistic 1 8588 Number of Observations 435 Number of Iterations 5 Autoregressive Integrated Moving Average or ARIMA p d q models are the extension of the AR mode that use three components for modeling the seria correlation in the time series data The first component is the autoregre
201. fetime of a given object has the same distribution regardless of the time it existed In other words time has no effect on future outcomes Conditions The condition underlying the exponential distribution is e exponential distribution describes the amount of time between occurrences The mathematical constructs for the exponential distribution are as follows forx gt 0 2 gt 0 Mean L Sandi Deaton lt gt Skewness 2 this value applies to all success rate inputs Excess Kurtosis 6 this value applies to all success rate inputs Success rate 4 is the only distributional parameter The number of successful trials is denoted x Input requirements Rate gt 0 The Exponential 2 distribution uses the same constructs as the original Exponential distribution but adds a Location or Shift parameter The Exponential distribution starts from a minimum value of 0 whereas this Exponential 2 or Shifted Exponential distribution shifts the starting location to any other value Rate or Lambda and Location or Shift are the distributional parameters Input requirements Rate Lambda gt 0 Location can be any positive or negative value including zero The extreme value distribution Type 1 is commonly used to describe the largest value of a response over a period of time for example in flood flows rainfall and earthquakes Other applications include the breaking strengths of materials con
202. for Min Max and Step Size Figure 4 7 click ADD and then click OK and OK again You should deselect D17 lt 5000 constraint before running e Run Optimization as usual Rak Simulator Optimization Run Optimization You can choose static dynamic or stochastic e results will be shown as a user interface Figure 4 8 Click on Create Report to generate a report worksheet with all the details of the optimization runs 109 Page RISK SIMULATOR m Parameters MIN j MAX e STEP SIZE fi Changing Constraints J 17 lt MIN 4 MAX 8 STEP 1 Figure 4 7 Generating Changing Constraints in an Efficient Frontier Efficient Frontier F Optimization Complete Problem Parameters Number of variables 12 Number of functions 3 Objective function willbe Maximized 1 017 lt 5000 J17 lt 4 Functions Starting Values Function Lower Upper Function No Name Status Type Initial Value Bound Bound No Name Initial Value Final Value 1 G OBJ 2 45726 1 G 245726 3 46137 2 G ere RNGE 3197 43710 1 10 0 2 319743710 1472 56292 S5 z 3 G ses RNGE 800000 1 10 3 G 8 00000 0 00000 Variables Efficient Frontier Analysis Step 1 Constraints are Starting Values FinalResuts 20227 lt 5000 Problem Parameters Number of variables is 12 Number of functions is 3 Variable Initial Lower Upper Variable Object
203. for the selected node or set a new assumption and use this new assumption or use previously created assumptions in a numerical equation or formula For example you can set a new assumption called Normal e g normal distribution with a mean of 100 and standard deviation of 10 and run a simulation in the decision tree or use this assumption in an equation such as 100 Normal 15 25 Create your own model in the numerical expression box You can use basic computations or add existing variables into your equation by double clicking on the list of existing variables New vatiables can be added to the list as required either as numerical expressions or assumptions 176 Page Procedure RISK SIMULATOR 5 27 3 Bayes Analysis This Bayesian analysis tool Figure 5 63 can be used on any two uncertainty events that are linked along a path For instance in the example on the right Figure 5 63 uncertainties A and B ate linked where event A occurs first in the timeline and event B occurs second First Event A is Market Research with 2 outcomes Favorable or Unfavorable Second Event B is Market Conditions also with 2 outcomes Strong and Weak This tool is used to compute joint marginal and Bayesian posterior updated probabilities by entering the prior probabilities and reliability conditional probabilities or reliability probabilities can be computed when you have posterior updated conditional probabilities Select the relevant analysis desire
204. fter smali number of Jags it implies that the series follows a low order moving average process correlation PAC K measures the correlation of values that are k periods apart after removing the correlation from the intervening lags If the pattern of autocorrelation can be captured by an autoregression of order less than k then the partial autocorrelation at lag k will be close to zero Ljung Box Q statistics and their p values at lag k has the nuli hypothesis that there is no autocorrelation up to order The dotted lines in the plots of the autocorrelations are the approximate two standard error bounds If the autocorrelation is within these bounds it is not significantly different from zero at approximately the 596 significance level Forecasting Period Actual Y Forecast F Error E 2 139 399994 139 6056 0 2056 3 139599997 140 0069 0 3069 4 139 699997 140 2586 0 5586 5 140599997 140 1343 0 5657 6 141 199997 141 6948 0 4948 7 141 699997 141 6741 0 0259 8 141 899994 142 4339 0 5339 8 141 142 3587 1 3587 10 140 5 141 0456 0 5466 11 140 399994 140 9447 0 5447 12 140 140 8451 0 8451 13 140 140 2846 0 2946 14 139899994 140 5663 0 6663 15 139 800003 140 2823 0 4823 16 139 600006 140 2726 0 6726 17 139 600006 139 9775 0 3775 18 139 600006 140 1232 0 5231 19 140199997 140 0513 0 1487 20 141 300003 140 8862 0 3138 21 141199997 14241738 0 9738 22 140 899994 1414377 0 5377 23 140 899994 141 351
205. ge Triangular Distribution Uniform Distribution RISK SIMULATOR The triangular distribution describes a situation where you know the minimum maximum and most likely values to occur For example you could describe the number of cars sold per week when past sales show the minimum maximum and usual number of cars sold Conditions The three conditions underlying the triangular distribution are e The minimum number of items is fixed e The maximum number of items is fixed The most likely number of items falls between the minimum and maximum values forming a triangular shaped distribution which shows that values near the minimum and maximum are less likely to occur than those near the most likely value The mathematical constructs for the triangular distribution are as follows 20200 for Min lt lt Likely Min Likely min 2 Max x for Likely x Max Max Min Max Likely Mean gin Likely Max Standard Deviation 3 Min Likely Max Min Max Min Likely Max Likely ipiis d J2 Min Max 2Likely 2Min Max Likely Min 2Max Likely 5 Min Likely MinMax MinLikely MaxLikely Excess Kurtosis 0 6 this applies to all inputs of Max and Likely Minimum value most likely value 1263 and maximum value Max are the distributional parameters Input requirements M
206. he maximum runtime is set at 300 seconds Typically no changes ate required However when forecasting with a significant amount of historical data the analysis might take slightly longer and if the processing time exceeds this runtime the process will be terminated You can also elect to have the forecast automatically generate assumptions That is instead of single point estimates the forecasts will be assumptions Finally the polar parameters option allows you to optimize the alpha beta and gamma parameters to include zeto and one Certain forecasting software allows these polar parameters while others do not Risk Simulator allows you to choose which to use Typically there is no need to use polar parameters 70 Page RISK SIMULATOR Holt Winter s Multiplicative Summary Statistics Alpha Beta Gamma RMSE Alpha Beta Gamma RMSE 0 00 0 00 0 00 914 824 0 00 0 00 0 00 914 824 0 10 0 10 0 10 415 322 0 10 0 10 0 10 415 322 0 20 0 20 0 20 187 202 0 20 0 20 0 20 187 202 0 30 0 30 0 30 118 795 0 30 0 30 0 30 118 795 0 40 0 40 0 40 101 794 0 40 0 40 0 40 101 794 0 50 0 50 0 50 102 143 The analysis was run with alpha 0 2429 beta 1 0000 gamma 0 7797 and seasonality 4 Time Series Analysis Summary When both seasonality and trend exist more advanced models are required to decompose the data into their base elements a base case level L weighted by the alpha parameter a trend component b
207. he other Risk Simulator forecasting methodologies to build more robust models Neural Network Forecast STEP 1 Data Manually enter your data paste from another application or load an example dataset with analysis C Un gt WNP gt mj STEP 2 Choose analysis type variable and forecast period to run Cosine with Hyperbolic Tangent Hyperbolic Tangent Layers Testing Periods Forecast Periods Apply Multiphased Optimization indicates negative values Period Actual Y Forecast F 581 5000 613 3528 613 5197 613 6203 613 7188 613 8520 614 0608 614 2046 614 3029 614 4223 614 5671 614 7154 614 8963 614 9954 615 0992 615 2115 Figure 5 56 Neural Network Forecast 170 Page RISK SIMULATOR Manually enter your data paste from another application or load an example dataset with analysis VAR9 VARIO Results RMSE 707 039492 Auto ARIMA RMSE 249 495091 Forecast F Error E 802 4484 863 9179 971 7020 1NR ANIR DN Ov un b C NJ Figure 5 57 Fuzzy Logic Time Series Forecast 5 24 Optimizer Goal Seek The Goal Seek tool is a search algorithm applied to find the solution of a single variable within a model If you know the result that you want from a formula or a model but are not sure what input value the formula needs to get that result use the Rak Simulator Tools Goal Seek feature Note th
208. hematical constructs for the logistic distribution are as follows a x nd for any valueof dee Mean a Standard Deviation 8 SkRewness this applies to all mean and scale inputs Excess Kurtosis 1 2 this applies to all mean and scale inputs Mean 0 and scale 8 ate the distributional parameters 53 Page Lognormal Distribution RISK SIMULATOR Calculating Parameters There are two standard parameters for the logistic distribution mean and scale The mean parameter is the average value which for this distribution is the same as the mode because this is a symmetrical distribution After you select the mean parameter you can estimate the scale parameter The scale parameter is a number greater than 0 The larger the scale parameter the greater the variance Input requirements Scale Beta gt 0 and can be any positive value Mean Alpha can be any value The lognormal distribution is widely used in situations where values are positively skewed for example in financial analysis for security valuation or in real estate for property valuation and where values cannot fall below zero Stock prices are usually positively skewed rather than normally symmetrically distributed Stock prices exhibit this trend because they cannot fall below the lower limit of zero but might increase to any price without limit Similarly real estate prices illustrate positive skewness as propert
209. hesis test is performed on the null hypothesis Ho such that the two variables population means are statistically identical to one another The alternative hypothesis is that the population means are statistically different from one another if the calculated p values are less than or equal to 0 01 0 05 or 0 10 this means that the hypothesis is rejected which implies that the forecast means are statistically significantly different at the 196 596 and 1096 significance levels If the nuli hypothesis is not rejected when the p values are high the means of the two forecast distributions are statistically similar to one another The same analysis is performed on variances of two forecasts at a time using the pairwise F Test If the p values are smali then the variances and standard deviations are statistically different from one another otherwise for large p values the variances are statistically identical to one another Result Hypothesis Test Assumption Unequal Variances Computed tstatistic 0 32947 P value for t statistic 0 74181 Computed F statistic 1 026723 P value for F statistic 0 351212 Figure 5 19 Hypothesis Testing Results The two variable t test with unequal variances the population variance of forecast 1 is expected to be different from the population variance of forecast 2 is appropriate when the forecast distributions are from different populations e g data collected from two different geographical locations or two
210. howing fewer variables For example select the Show Top 10 Variables if the first 5 are critical thereby creating a nice report and Tornado chart that shows a contrast between the key factors and less critical factors You should never show Tornado chart with only the key variables You need to show some less critical variables as a contrast to their effects on the output Finally the default testing points can be increased from the 10 of the parameter to some larger value to test for nonlinearities the Spider chart will show nonlinear lines and Tornado charts will be skewed to one side if the precedent effects are nonlinear e Selecting Use Cell Address is always a good idea if your model is large as it allows you to identify the location worksheet name and cell address of a precedent cell If this option is not selected the software will apply its own fuzzy logic in an attempt to determine the name of each precedent variable in a large model the names might sometimes end up being confusing with repeated variables or the names that are too long possibly making the Tornado chart unsightly The Analyze This Worksheet and Analyze All Worksheets options allow you to control whether the precedents should only be part of the current worksheet or include all worksheets in the same workbook This option comes in handy when you are only attempting to analyze an output based on values in the current sheet versus performing a global sear
211. ibution Gamma Distribution Gumbel Maximum Distribution 52 25 128 85 49 08 49 01 166 19 52 81 50 51 197 52 50 74 49 72 279 06 47 98 Figure 5 13 Single Variable Distributional Fitting 128 Page RISK SIMULATOR F 03 0 03 0 03 0 03 0 03 0 04 0 05 0 05 0 05 0 07 0 08 0 10 0 12 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 Standard Deviation 10 40 Kolmogorov Smirnov Test Statistic J Test Statistic 0 02 Automatically Generate Assumption Figure 5 14 Distributional Fitting Result 129 Page RISK SIMULATOR Single Variable Distributional Fitting Statistical Summary Fitted Assumption 99 44 Fitted Distribution Normal Distribution Mu 99 28 Sigma 10 17 Koimogorov Smirnov Statistic 0 03 for Test Statistic 0 9727 Actual Theoretical Mean 99 14 99 28 Standard Deviation 10 20 10 17 Skewness 0 12 0 00 Excess Kurtosis 0 10 0 00 Original Fitted Data 93 75 99 66 86 95 111 86 99 55 95 55 97 32 87 25 90 68 85 86 98 74 88 76 97 70 99 75 90 05 106 63 103 21 66 48 104 38 123 26 103 65 92 85 84 18 109 85 86 04 102 26 105 36 97 64 109 15 110 98 108 09 95 38 93 24 83 86 100 17 110 17 103 72 120 52 95 09 115 18 83 64 90 23 92 44 92 37 92 70 110 81 72 67 104 23 96 47 121 15 94 92 77 26 103 45 96 75 93 91 101 91 124 14 90 95 107 13 92 02 96 43 96 35 88 30 108 48 11350 101 40 104 72 102
212. ical Sales Revenues Year Quarter Sales 2006 1 1 2006 2 2 2006 3 3 2006 4 4 Trendline 2007 1 5 2007 2 6 Selected Trendlines 2007 3 7 Linear Exponential 2007 4 8 n 2008 1 9 Logarithmic Polynomial Order 2 2008 2 10 Power Moving Average Order 2 2008 3 11 2008 4 12 Generate forecasts 6 periods 2009 1 13 i 2009 2 14 Cancel 2009 3 15 2009 4 16 2010 1 17 2010 2 18 2010 3 19 2010 4 20 Figure 5 45 Trendline Forecasts 5 19 Model Checking Tool After a model is created and after assumptions and forecasts have been set you can run the simulation as usual or run the Check Model tool Figure 5 46 to test if the model has been set up correctly Alternatively if the model does not run and you suspect that some settings may be incorrect run this tool from Rak Simulator Tools Check Model to identify where there might be problems with your model Note that while this tool checks for the most common model problems as well as for problems in Risk Simulator assumptions and forecasts it is in no way comprehensive enough to test for all types of problems It is still up to the model developer to make sure the model works propertly 159 Page RISK SIMULATOR x id 95 06 17 ROI and Vol xls Compatibility Mode Microsoft Excel non Insert Page Layout Formulas Data Review View Developer Risk Simulator f tiii iC Al 2 m Set O
213. ications in a range of fields from biology to economics For example in the development of an embryo a fertilized ovum splits and the cell count grows 1 2 4 8 16 32 64 etc This is exponential growth But the fetus can grow only as large as the uterus can hold thus other factors start slowing down the increase in the cell count and the rate of growth slows but the baby is still growing of course After a suitable time the child is born and keeps growing Ultimately the cell count is stable the person s height is constant the growth has stopped at maturity The same principles can be applied to population growth of animals or humans and the market penetration and revenues of a product with an initial growth spurt in market penetration but over time the growth slows due to competition and eventually the market declines and matures 1 Click on Risk Simulator Forecasting JS Curves Real Options 2 Enter in the required inputs see below for an example V Valua tion 3 Click OK and review the forecast report www realoptions valuation com 7 000 JSCurves 8000 RS The J S curves stand for J curve exponential growth and S curve Maturity and logistic Mei curve These curves are used in forecasting high Saturation Phase growth rates J curve or for situations with events with initially high growth but slows down and growth matures over time as the 5 000 e 4000 environment becomes saturated at c
214. ich we decided was a known value and hence was not simulated with one special exception Tax rate was relegated to a much lower position in the sensitivity analysis chart Figure 5 11 as compared to the tornado chart Figure 5 6 This is because by itself tax rate will have a significant impact but once the other variables are interacting in the model it appeats that tax rate has less of a dominant effect because tax rate has a smaller distribution as historical tax rates tend not to fluctuate too much and also because tax rate is a straight percentage value of the income before taxes where other precedent variables have a larger effect on This example proves that performing sensitivity analysis after a simulation run is important to ascertain if there are any interactions in the model and if the effects of certain variables still hold The second chart Figure 5 12 illustrates the percent variation explained That is of the fluctuations in the forecast how much of the variation can be explained by each 125 Page RISK SIMULATOR of the assumptions after accounting for all the interactions among variables Notice that the sum of all variations explained is usually close to 100 there are sometimes other elements that impact the model but that cannot be captured here directly and if correlations exist the sum may sometimes exceed 100 due to the interaction effects that are cumulative Figure 5 11 Rank Correlation Chart 3
215. ics are the ratios of each predicted Coefficient to its Standard Error The t Statistic is used in hypothesis testing where we set the nuli hypothesis Ho such that the real mean of the Coefficient 0 and the alternate hypothesis Ha such that the real mean of the Coefficient is not equal to 0 A Hest is is performed and the calculated t Statistic is compared to the critical values at the relevant Degrees of Freedom for Residual The ttest is very important as it calculates if each of the coefficients is statistically significant in the presence of the other regressors This means that the Hest statistically verifies whether a regressor or independent variable should remain in the regression or it should be dropped The Coefficient is statistically significant if its calculated t Statistic exceeds the Critical t Statistic at the relevant degrees of freedom df The three main confidence levels used test for significance are 9096 9596 and 999 if a Coefficient s t Statistic exceeds the Critical level it is considered statistically significant Alternatively the p Value calculates each t Statistic s probability of occurrence which means that the smaller the p Value the more significant the Coefficient The usual significant levels for the p Value are 0 01 0 05 and 0 10 corresponding to the 9996 9596 and 9996 confidence levels The Coefficients with their p Values highlighted in blue indicate that they are statistically significant at the 9096
216. identification That is AIC and SC are used to determine if a particular model with a specific set of p d and q parameters is a good statistical fit SC imposes a greater penalty for additional coefficients than the AIC but generally the model with the lowest the AIC and SC values should be chosen Finally an additional set of results called the autocorrelation AC and partial autocorrelation PAC statistics are provided in the ARIMA report For instance if autocorrelation AC 1 is nonzero it means that the series is first order serially correlated If AC dies off more or less geometrically with increasing lags it implies that the series follows a low order autoregressive process If AC drops to zero after a small number of lags it implies that the series follows a low order moving average process In contrast PAC measutes the correlation of values that are amp periods apart after removing the correlation from the intervening lags If the pattern of autocorrelation can be captured by an autoregression of order less than amp then the partial autocorrelation at lag amp will be close to zero The Ljung Box Q statistics and their p values at lag amp are also provided where the null hypothesis being tested is such that there is no autocorrelation up to order amp The dotted lines in the plots of the autocortelations the approximate two standard error bounds If the autocorrelation is within these bounds it is not significantly dif
217. in Most Likely lt Max and can take any value However Min lt Max and can take any value With the uniform distribution all values fall between the minimum and maximum and occur with equal likelihood Conditions The three conditions underlying the uniform distribution are e The minimum value is fixed The maximum value is fixed e All values between the minimum and maximum occur with equal likelihood 61 Page Weibull Distribution Rayleigh Distribution RISK SIMULATOR The mathematical constructs for the uniform distribution are as follows 1 for all valuessuch that Min lt Max Max Min 2 Y Standard Deviation Hao Skewness this applies to all inputs of Min and Max Excess Kurtosis 1 2 this applies to all inputs of Min and Maximum value Max and minimum value Mz are the distributional parameters Input requirements Min lt Max and can take any value The Weibull distribution describes data resulting from life and fatigue tests It is commonly used to describe failure time in reliability studies as well as the breaking strengths of materials in reliability and quality control tests Weibull distributions are also used to represent various physical quantities such as wind speed The Weibull distribution is a family of distributions that can assume the properties of several other distributions For example depending on the shape p
218. ion e Select the extrapolation type automatic selection polynomial function or rational function and enter the number of forecast period desired Figure 3 11 and click OK The results report shown in Figure 3 12 shows the extrapolated forecast values the error measurements and the graphical representation of the extrapolation results The error measurements should be used to check the validity of the forecast and are especially important when used to compate the forecast quality and accuracy of extrapolation versus time series analysis When the historical data is smooth and follows some nonlinear patterns and curves extrapolation is better than time series analysis However when the data patterns follow seasonal cycles and a trend time series analysis will provide better results 78 Page RISK SIMULATOR Note that Nonlinear Extrapolation involves making statistical projections by using historical trends that are projected for a specified period of time into the future It is only used for time series forecasts Extrapolation is fairly reliable relatively simple and inexpensive However extrapolation which assumes that recent and historical trends will continue produces large forecast errors if discontinuities occur within the projected time period 1 Enter the historical data and select the data area E13 E24 Historical Sales Revenues 2 Click on Risk Simulator Forecasting Nonlinear Extrapolation
219. ion named for the famous gambling capital of Monaco is a very potent methodology For the practitioner simulation opens the door for solving difficult and complex but practical problems with great ease Monte Carlo creates artificial futures by generating thousands and even millions of sample paths of outcomes and looks at their prevalent characteristics For analysts in a company taking graduate level advanced math coutses is just not logical or practical A brilliant analyst would use all available tools at his or her disposal to obtain the same answer the easiest and most practical way possible And in all cases when modeled correctly Monte Carlo simulation provides similar answets to the more mathematically elegant methods So what is Monte Carlo simulation and how does it work 2 1 What Is Monte Carlo Simulation Monte Carlo simulation in its simplest form is a random number generator that is useful for forecasting estimation and risk analysis simulation calculates numerous scenatios of a model by repeatedly picking values from a user predefined probability distribution for the uncertain variables and using those values for the model As all those scenarios produce associated results in a model each scenario can have a forecast Forecasts ate events usually with formulas or functions that you define as important outputs of the model These usually are events such as totals net profit or gross expenses Simplistically think of
220. ion adequately describes a set of data it is one of the most powerful statistical tools for detecting departures from normality and is powerful for testing normal tails However in non normal distributions this test lacks power compared to others e Distributional Fitting Kolmogorov Smirnov KS A nonparametric test for the equality of continuous probability distributions that can be used to compare a sample with a reference probability distribution making it useful for testing abnormally shaped distributions and non normal distributions e Distributional Fitting Kuiper s Statistic Related to the KS test making it as sensitive in the tails as at the median and also makes it invariant under cyclic 127 Page RISK SIMULATOR transformations of the independent variable making it invaluable when testing for cyclic variations over time The AD provides equal sensitivity at the tails as the median but it does not provide the cyclic invariance e Distributional Fitting Schwarz Bayes Information Criterion The SC BIC introduces a penalty term for the number of parameters in the model with a larger penalty than AIC Results Interpretation The null hypothesis being tested is such that the fitted distribution is the same distribution as the population from which the sample data to be fitted comes Thus if the computed p value is lower than a critical alpha level typically 0 10 or 0 05 then the distribution is the wr
221. ions a very different picture results as shown in Figure 5 9 Notice for instance that price erosion had little impact on NPV but when some of the input assumptions are correlated the interaction that exists between these correlated variables makes price erosion have more impact 123 Page Procedure RISK SIMULATOR 0 22 C Price 0 17 Tax Rate 0 05 Price Erosion 0 03 Sales Growth 0 0 0 1 02 0 3 0 4 0 5 0 6 Figure 5 8 Sensitivity Chart Without Correlations 0 21 Price Erosion 0 18 Tax Rate 0 03 Sales Growth 0 0 0 1 0 2 0 3 0 4 0 5 0 6 Figure 5 9 Sensitivity Chart With Correlations Open or create a model define assumptions and forecasts and 7 7 the simulation the example here uses the Tornado and Sensitivity Charts Linear file Select Risk Simulator Tools Sensitivity Analysis Select the forecast of choice to analyze and click OK Figure 5 10 124 Page Results Interpretation RISK SIMULATOR Discounted Cash Flow Model Base Year 2005 Sum PV Net Benefits 1 896 63 Market Risk Adjusted Discount Rate 15 00 Sum PV Investments 1 800 00 Private Risk Discount Rate 5 00 Net Present Value Annualized Sales Growth Rate 2 00 Intemal Rate of Return 18 80 Price Erosion Rate 5 00 Return on Investment 5 37 Effective Tax Rate 40 00 2005 2006 2007 2008 2009 Prod A Avg Price s1000 5950 __ 59031 5857 8 15 Prod B Avg Price __51225 _ si10 amp _ 5
222. is used for optimizing multiple decision variables subject to constraints to maximize or minimize an objective and can be run either as a static optimization dynamic and stochastic optimization under uncertainty together with Monte Carlo simulation or as a stochastic optimization with super speed simulations The software can handle linear and nonlinear optimizations with binary integer and continuous variables as well as generate Markowitz efficient frontiers e The Analytical Tools Module allows you to run segmentation clustering hypothesis testing statistical tests of raw data data diagnostics of technical forecasting assumptions e g heteroskedasticity multicollinearity and the like sensitivity and scenario analyses overlay chart analysis spider charts tornado charts and many other powerful tools ROV BizStats over 130 business statistics and analytical models e ROV Decision Tree decision tree models Monte Carlo risk simulation on decision trees sensitivity analysis scenario analysis Bayesian joint and posterior probability updating expected value of information MINIMAX MAXIMIN risk profiles Real Options Super Lattice Solver is a software that complements Risk Simulator used for solving simple to complex real options problems 12 Page Starting a New Simulation Profile RISK SIMULATOR The following sections walk you through the basics of the Simulation Module in Risk Simulator while futu
223. ish to set the assumption on and using the mouse right click access the shortcut Risk Simulator menu to set an input assumption In addition for expert users you can set input assumptions using the Risk Simulator RS Functions select the cell of choice click on Excel s Insert Function select the All Category and scroll down to the RS functions list we do not recommend using RS functions unless you are an expert user For the examples going forward we suggest following the basic instructions in accessing menus and icons As shown in Figure Z4 there are several key areas in the Assumption Properties worthy of mention e Assumption Name This is an optional area to allow you to enter in unique names for the assumptions to help track what each of the assumptions represents Good modeling practice is to use short but precise assumption names e Distribution Gallery This area to the left shows all of the different distributions available in the software To change the views right click anywhere in the gallery 16 Page RISK SIMULATOR and select large icons small icons or list There are over two dozen distributions available e Input Parameters Depending on the distribution selected the required relevant parameters are shown You may either enter the parameters directly or link them to specific cells in your worksheet Hard coding or typing the parameters is useful when the assumption parameters are assumed not to change Linking
224. isk Ratio Index ENPV Cost Risk Risk Selection Optimization Complete Project 4 458 00 1 732 44 54 96 12 00 833 1 26 Project 2 1 954 00 859 00 1 914 92 98 0096 1 02 3 27 Project 3 1 599 00 1 845 00 1 551 03 97 0096 1 03 1 87 Project 4 2 251 00 1 645 00 51 01295 45 0096 222 237 Project 5 849 00 458 00 925 41 109 00 0 92 285 Project 6 758 00 52 00 560 92 74 00 1 35 15 58 Project 7 2 845 00 758 00 5 633 10 198 00 051 475 Project 8 1 235 00 115 00 926 25 75 00 1 33 1174 Project 9 1 945 00 125 00 2 100 60 108 0096 0 93 16 56 Project 10 52 250 00 458 00 1 912 50 85 0096 1 18 591 Project 11 549 00 4500 263 52 48 00 208 1320 Project 12 525 00 105 00 309 75 59 00 1 69 6 00 Total 5 776 00 3 694 44 1 539 26 64 Sanna Eus 12345678 9 1011 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Humber of Iterations ENPV is the expected NPV of each credit line or project while Cost can be the total cost of administration as well as required capital holdings to cover the credit line and Risk is the Number of variables is 12 Coefficient of Variation of the credit line s ENPV Number of functions is 3 Objective function will be MAXimized Starting values Functions Function Initial Lower Upper Name Status value Bound Bound 1 2 4573 2 eree 3197 4371 1 000000E 010 NGE 6 0000 1 0000
225. istribution Zo Normal Distribution Zu T Distribution SESRCEEM lt Ao 2 lt Ao 2 Variance in E Mean Equation amp GARCH M Standard Deviation in Mean Equation GARCH M Log Variance in Mean Equation c Ao 6 amp 0 4 0 2 y c Aln o7 0 4 2_ 2 2 0 c Ao 6 0 4 2 _ 2 2 0 y c Aln o 0 4 2 2 a aE fo 0 2 E S0 6 0 06 EGARCH y E y 6 O Z zi O Z B In o In o7 2 v B In o7 E E a Eds r a Z Eds t 1 O 1 1 0 1 Ele E Behe NG v DF v 2Nz GJR GARCH y y mi 0 4 0 4 2 2 r d 1 if e 0 0 otherwise d um 1 1 0 amp 2 2 re d ife lt 0 d 0 otherwise 92 Page Theory Procedure Notes RISK SIMULATOR For the GARCH M models the conditional variance equations ate the same in the six variations but the mean questions are different and assumption on Z can be either normal distribution or t distribution The estimated parameters for GARCH M with normal distribution are those five parameters in the mean and conditional variance equations
226. istributional Fitting Tool 161 Page Distributional Analysis Distributional Charts and Tables Overlay Charts Procedure RISK SIMULATOR 5 21 Distribution Charts and Tables Probability Distribution Tool Distributional Charts and Tables is a new Probability Distribution tool that is a very powerful and fast module used for generating distribution charts and tables Figures 5 48 through 5 51 Note that there are three similar tools in Risk Simulator but each does very different things Used to quickly compute the PDF CDF and ICDF of the 42 probability distributions available in Risk Simulator and to return a probability table of these values The Probability Distribution tool described here used to compare different parameters of the same distribution e g the shapes and PDF CDF ICDF values of a Weibull distribution with Alpha and Beta of 2 2 3 5 and 3 5 8 and overlays them on top of one anothet Used to compare different distributions theoretical input assumptions and empirically simulated output forecasts and to overlay them on top of one another for a visual comparison Run ROV BizStats at Simulator Distributional Charts and Tables click on the Apply Global Inputs button to load a sample set of input parameters or enter your own inputs and click Run to compute the results The resulting four moments and CDF ICDF PDF are computed for each of the 45 probability distributions Figure 5 48
227. it and the distribution type may or may not be known ahead of time e Distributional Fitting Multiple Variables using statistical methods to fit your raw data on multiple variables at the same time This method uses the same algorithms as the single variable fitting but incorporates a pairwise correlation matrix between the variables Multiple data points are required for a good fit and the distribution type may or may not be known ahead of time e Custom Distribution Set Assumption using nonparametric resampling techniques to generate a custom distribution with the existing raw data and to simulate the distribution based on this empirical distribution Fewer data points are required and the distribution type is not known ahead of time Click on Simulator Tools Distributional Fitting Percentiles choose the probability distribution and types of inputs you wish to use enter the parameters and click Run to obtain the results Review the fitted R square results and compare the empirical versus theoretical fitting results to determine if your distribution is a good fit Data Fitting Subject Matter Expert Curve Fit This data fitting method allows you to enter custom percentiles in lieu of one or more regular input parameters to determine the theoretical distribution and is useful when soliciting subject matter expert opinions For instance instead of entering Mean and Standard Deviation for a Normal distribution you c
228. it any sets of historical data Simply compute the expected returns and the volatility of the historical data or estimate them using comparable external data or make assumptions about these values See Modeling Risk Applying Monte Carlo Simulation Real Options Analysis Forecasting and Optimization Second Edition Wiley Finance 2010 by Dr Johnathan Mun for more details on how each of the inputs are computed e g mean reversion rate jump probabilities volatility etc e Start the module by selecting Simulator Forecasting Stochastic Processes e Select the desired process enter the required inputs click on Update Chart a few times to make sure the process is behaving the way you expect it to and click OK Figure 3 9 Figure 3 10 shows the results of a sample stochastic process The chart shows a sample set of the iterations while the report explains the basics of stochastic processes In addition the forecast values mean and standard deviation for each time period are provided Using these values you can decide which time period is relevant to your analysis and set assumptions based on these mean and standard deviation values using the normal distribution These assumptions can then be simulated in your own custom model 76 Page RISK SIMULATOR Stochastic Processes are sequences of events or paths generated by probabilistic laws where random events can occur over time but are governed by specific statistical and pro
229. ities would have remained the same and the process would have satisfied the conditions for a binomial distribution 40 Page Negative Binomial Distribution RISK SIMULATOR Conditions The three conditions underlying the hypergeometric distribution are total number of items or elements the population size is a fixed number a finite population The population size must be less than or equal to 1 750 e sample size the number of trials represents a portion of the population known initial probability of success in the population changes after each trial The mathematical constructs for the hypergeometric distribution are as follows N N Lx pij o DN Bc ET NIN E E n N n Mean N Standard Deviation N nN n N N 1 SRewness N 1 N N JN n N n Excess Kurtosis complex function The number of items in the population or Population Size N trials sampled or Sample Size and number of items in the population that have the successful trait or Population Successes N are the distributional parameters The number of successful trials is denoted x Input requirements Population Size 2 and integer Sample Size gt 0 and integer Population Successes gt 0 and integer Population Size gt Population Successes Sample Size lt Population Successes Population Size lt 1750 The negative binomial distributi
230. ive function will be maximized No Name Status Value Bound Bound No Name Initial Value Final Value 1 x UL 100000 0 1 1 x 100000 1 00000 SORN Vaives 2 X UL 1 00000 0 1 2 x 1 00000 000000 Function Initial Lower upper 3 x UL 100000 0 1 3 x 1 00000 0 00000 mes ame VUE ci Bold 4 x UL 1 00000 0 1 4 x 10000 1 00000 3 6 2 4578 EU Sa 5 x UL 4 00000 0 1 5 x 4 00000 0 00000 R G RNGE 3197 4371 1 000000E 010 0 000000 000 a 6 x UL 100000 0 1 6 x 100000 000000 7 x UL 100000 0 1 7 x 100000 0 00000 Optimal values have been found Do you wish to replace the existing decision variables with the optimized values 8 x UL 1 00000 0 1 8 x 100000 000000 everttothe original inputs 9 x UL 1 00000 0 1 9 x 100000 000000 10 x UL 100000 0 1 10 X 100000 0 00000 1 x UL 100000 0 1 11 100000 1 00000 12 x UL 1 00000 0 1 12 x 1 00000 1 00000 Objective Binding Super Infeas Norm of Hessian Step Degen No Function Constrs Basics Constr Red Grad Size Step 1 320543710 0 12 2 0 57590 1 0 2 355285 0 11 1 0 28146 1 1 288211 0 10 1 0 34697 1 0061 Figure 4 8 Efficient Frontier Results 10 Page RISK SIMULATOR 4 5 Stochastic Optimization This example illustrates the application of stochastic optimization using a sample model with four asset classes each with different risk and return characteristics The idea here is to find the best portfolio allocation such that the portfolio s bang for the
231. ive test available and this could mean the difference between detecting a linear fit or not If the errors are not independent and not normally distributed it may indicate that the data might be autocorrelated or suffer from nonlinearities or other more destructive errors Independence of the errors can also be detected in the heteroskedasticity tests Figure 5 25 The Normality test on the errors performed is a nonparametric test which makes no assumptions about the specific shape of the population from which the samples are drawn allowing for smaller sample data sets to be analyzed This test evaluates the null hypothesis of whether the sample errors were drawn from a normally distributed population versus an alternate hypothesis that the data sample is not normally distributed If the calculated D statistic is greater than or equal to the D critical values at various significance values then reject the null hypothesis and accept the alternate hypothesis the errors are not normally distributed Otherwise if the D statistic is less than the D critical value do not reject the null hypothesis the errors are normally distributed The Normality test relies on two cumulative frequencies one derived from the sample data set and the second from a theoretical distribution based on the mean and standard deviation of the sample data 140 Page RISK SIMULATOR Test Result Errors Hune Observed Expected O E Regression Error Average 0 00 Frequenc
232. iven Prior and Posterior Probabilities less common inhale oe Sl Shi oe ee Se cco ee EIS Prob Strong 400 each event has Prob Weak 55 00 First Event Name Market Research Probability Events or States Prob Favorable Strong 60 00 Prob Favorable Weak 30 00 Prob Unfavorable Strong 40 00 Prob Unfavorable Weak 70 00 States Market Research Market Conditions Joint and Marginal Probabilities Second Event Name Market Conditions Probability Events or States STEP 2 Enter the names of each probability event or outcome Favorable Strong Prob Favorable 43 50 Unfavorable Weak Prob Unfavorable 56 50 Prob Strong Favorable 27 00 Prob Weak Favorable 16 50 Prob Strong Unfavorable 18 00 STEP 3 Enter the second event s prior probabilities and the conditional probabilities for each event or outcome The probabilities must sum to 100 Prob Weak n Unfavorable 38 50 Conditional Probabities Reiabilties Saved Model Posterior or Updated Probabilities Events Prior P x Favorable Unfavorable SUM Nane Prob Strong Favorable 62 07 Prob Weak Favorable 37 93 Strong 45 00 60 00 40 00 10000 4 Prob Strong Unfavorable 31 86 5500 30 00 70 00 10000 Prob Weak Unfavorable 68 14 100 00 Figure 5 63 ROV Decision Tree
233. l based models generate and extract simulation forecasts distributions of results perform distributional fitting automatically finding the best fitting statistical distribution compute correlations maintain relationships among simulated random variables identify sensitivities creating tornado and sensitivity charts test statistical hypotheses finding statistical differences between pairs of forecasts run bootstrap simulation testing the robustness of result statistics and run custom and nonparametric simulations simulations using historical data without specifying any distributions or their parameters for forecasting without data ot applying expert opinion forecasts The Forecasting Module can be used to generate automatic time series forecasts with and without seasonality and trend multivariate modeling relationships among variables nonlinear extrapolations curve fitting stochastic processes random walks mean reversions jump diffusion and mixed processes Box Jenkins ARIMA econometric forecasts Auto ARIMA basic econometrics and auto econometrics modeling relationships and generating forecasts exponential J curves logistic S curves GARCH models and their multiple variations modeling and forecasting volatility maximum likelihood models for limited dependent variables logit tobit and probit models Markov chains trendlines spline curves and others The Optimization Module
234. l return nonsensical values of above 7 or below 0 MLE analysis handles these problems using an iterative optimization routine to maximize a log likelihood function when the dependent vatiables are limited A Logit or Logistic regression is used for predicting the probability of occurrence of an event by fitting data to a logistic curve It is a generalized linear model used for binomial regression and like many forms of regression analysis it makes use of several predictor variables that may be either numerical or categorical MLE applied in a binary multivariate logistic analysis is used to model dependent variables to determine the expected probability of success of belonging to certain group The estimated coefficients for the Logit model are the logarithmic odds ratios and cannot be interpreted directly as probabilities A quick computation is first required and the approach is simple Specifically the Logit model is specified as Estimated Y LN P 1 P or conversely EXP Estimated Y 1 EXP Estimated and the coefficients f are the log odds ratios So 94 Page RISK SIMULATOR taking the antilog or EEXP Bj we obtain the odds ratio of 7 This means that with an increase in a unit of fj the log odds ratio increases by this amount Finally the rate of change is the probability dP dX The standard error measures how accurate the predicted coefficients are and the t statistics are the ratios
235. ld happen to the portfolio s outcome and optimal decisions if the constraint were now 1 5 million or 2 million and so forth This is the concept of the Markowitz efficient frontiers in investment finance whereby one can determine what additional returns the portfolio will generate if the portfolio standard deviation is allowed to increase slightly This process is similar to the dynamic optimization process with the exception that one of the constraints is allowed to change and with each change the simulation and optimization process is run This process is best applied manually using Risk Simulator That is run a dynamic or stochastic optimization then rerun another optimization with a constraint and repeat that procedure several times This manual process is important because by changing the constraint the analyst can determine if the results are similar or different and hence whether it is worthy of any additional analysis or the analyst can determine how far a marginal increase in the constraint should be to obtain a significant change in the objective and decision variables One item is worthy of consideration There exist other software products that supposedly perform stochastic optimization but in fact they do not For instance after a simulation is run then one iteration of the optimization process is generated and then another simulation is run then the second optimization iteration is generated and so forth This approach is
236. liers 8 136 137 138 139 parameter 16 17 38 40 44 45 46 47 48 49 50 51 52 53 54 55 56 57 59 60 62 63 69 92 94 121 141 164 168 183 Parameter 55 111 141 146 183 185 pareto 56 Pareto 6 56 57 pause 18 19 Pearson 6 26 27 57 58 142 point estimate 65 69 70 99 113 Poisson 6 43 44 50 52 population 29 34 41 53 56 60 67 88 126 132 133 137 139 156 196 Page RISK SIMULATOR portfolio 1 98 99 100 101 102 103 104 105 110 113 precision 1 8 14 18 21 29 prediction 80 137 138 price 27 31 54 55 66 76 89 117 118 119 122 140 141 149 probability 1 6 8 9 11 19 23 24 32 35 36 37 38 39 40 41 42 43 45 47 48 51 53 55 58 59 60 67 92 93 94 146 147 148 159 160 161 175 176 177 183 Probability 5 9 17 25 35 36 38 39 40 41 42 43 146 161 162 163 173 183 profile 13 14 15 27 69 101 105 111 126 164 165 186 187 p value 81 126 133 138 142 157 random 6 11 12 14 26 35 36 37 45 46 47 50 52 53 67 76 80 94 130 131 132 140 141 153 161 186 188 random number 6 11 14 36 153 188 range 17 31 45 52 93 99 101 110 113 115 138 147 177 183 rank correlation 26 27 124 125 142 rate 7 31 44 49 50 55 66 76 94 96 117 119 124 138 140 149 155 183 185 ratio 47 51 78 94 99 101 103 104 105 107 110 regression 7 66 67 7
237. lify the user interface we allow users to enter the more common Pearson s product moment correlation e g computed using Excel s CORREL function while in the mathematical codes we convert these simple correlations into Spearman s rank based correlations for distributional simulations 2 3 2 Applying Correlations in Risk Simulator Correlations can be applied in Risk Simulator in several ways When defining assumptions Rak Simulator Ser Input Assumption simply enter the correlations into the correlation matrix grid in the Distribution Gallery e With existing data run the Mult Fit tool Rak Simulator Tools Distributional Fitting Multiple Variables to perform distributional fitting and to obtain the correlation matrix between pairwise variables If a simulation profile exists the assumptions fitted will automatically contain the relevant correlation values e With existing assumptions you can click on Risk Simulator Tools Edit Correlations to enter the pairwise correlations of all the assumptions directly in one user interface Note that the correlation matrix must be positive definite That is the correlation must be mathematically valid For instance suppose you are trying to correlate three variables grades of graduate students in a particular year the number of beers they consume a week and the number of hours they study a week One would assume that the following correlation relationships exist Grades and Beer
238. list the models alphabetically categorically and by data input requirements note that in certain Unicode languages e g Chinese Japanese and Korean there is no alphabetical arrangement and therefore the first option will be unavailable The software can handle different regional decimal and numerical settings e g one thousand dollars and fifty cents can be written as 1 000 50 or 1 000 50 or 1000 50 and so forth The decimal settings be set in ROV BizStats menu Data Decimal Settings However when in doubt please change the computer s regional settings to English USA and keep the default North America 1 000 50 in ROV BizStats this setting is guaranteed to work with ROV BizStats and the default examples 166 Page RISK SIMULATOR a a File Data Language Help STEP 1 Data Manually enter your data paste from another application A STEP 2 Analysis Choose an analysis and enter the B or load an example dataset with analysis E rey se keen View Alphabetical Absolute Values ABS VARS 13 ANOVA Randomized Blocks Multiple Treatme VARGVARZNARS ANOVA Single Factor Multiple Treatments 3 0 ANOVA Two Way Analysis ARIMA F Auto ARIMA Auto Econometrics Detailed 3294 Auto Econometrics Quick Autocorrelation and Partial Autocorrelation m AI d dii Basic Econometrics So Control Chart C Control Chart NP Runs the current analysis in Step 2 or selected saved a
239. lity Function Calibrated between 0 and 100 Ring Uy Fanon beteen 1 is Function Calibrated between 0 and 100 0 50 100 150 200 250 300 350 400 450 EV Figure 5 67 ROV Decision Tree Utility Functions 182 Page RISK SIMULATOR 6 Helpful Tips and Techniques The following are some quick helpful tips and shortcut techniques for advanced users of Risk Simulator For details on using specific tools refer to the relevant sections in this user manual TIPS Assumptions Set Input Assumption User Interface Quick Jump select any distribution and type in any letter and it will jump to the first distribution starting with that letter e g click on Normal and type in W and it will take you to the Weibull distribution e Right Click Views select any distribution right click and select the different views of the distributions large icons small icons list e Tab to Update Charts after entering some new input parameters e g you type in a new mean or standard deviation value hit TAB on the keyboard or click anywhere on the user interface away from the input box to see the distributional chart automatically update e Enter Correlations enter pairwise correlations directly here the columns are resizable as needed use the multiple distributional fitting tool to automati
240. lity in the right tail and 2 50 in the left tail leaving 95 in the center or confidence interval area which is equivalent to a 97 50 area for one tail The result is the familiar Z score of 1 96 Therefore using this Distributional Analysis tool the standardized scores for other distributions and the exact and cumulative probabilities of other distributions can all be obtained quickly and easily 149 Page RISK SIMULATOR Distribution Analysis This tool generates the probability density function PDF cumulative distribution function CDF and the Inverse CDF ICDF of all the distributions in Risk Simulator including theoretical moments and probability chart Distribution Trials Probability 0 411901 0 588099 0 748278 0 868412 0 942341 0 979305 0 994091 0 998712 0 999799 0 999980 0 999999 1 000000 Figure 5 36 Distributional Analysis Tool Binomial Distribution s CDF with 20 Trials 5 11 Scenatio Analysis Tool The Scenario Analysis tool in Risk Simulator allows you to run multiple scenarios quickly and effortlessly by changing one or two input parameters to determine the output of a variable Figure 5 38 illustrates how this tool works on the discounted cash flow sample model Model 7 in Risk Simulator s Example Models folder In this example cell G6 net present value is selected as the output of interest whereas cells C9 effective tax rate and C12 product price are
241. llocation and do not forget the sign Select cell C72 the objective to be maximized and make it the objective Simulator Optimization Set Objective or click on the icon Run the optimization by going to Simulator Optimization Run Optimization Review the different tabs to make sure that all the required inputs in steps 2 and 3 are correct Select Stochastic Optimization and let it run for 500 trials repeated 20 times Click OK when the simulation completes and a detailed stochastic optimization report will be generated along with forecast charts of the decision variables 112 Page RISK SIMULATOR Decision Variable Properties Decision Name Asset 1 Decision Type Continuous 1 15 2 35 10 55 Lower Bound 0 1 1 Upper Bound 04 Integer e g 1 2 3 Lower Bound Bound Binary 0 or 1 v SES11 100 Optimization Summary Optimization is used to allocate resources where the results provide the max returns or the min cost risks Uses include managing inventories financial portfolio allocation product mix project selection etc Run on static model without simulations Usually run to determine the intial optimal portfolio before more advanced optimizations are applied Dynamic Optimization simulation is first run the results of the simulation are applied in the model and then an optimization is applied to the simulated values Number
242. llow along the first example open the Tornado and Sensitivity Charts Linear file in the examples folder Figure 5 2 shows this sample model where cell G6 containing the net present value is chosen as the target result to be analyzed The target cell s precedents in the model are used in creating the tornado chart Precedents are all the input and intermediate variables that affect the outcome of the model For instance if the model consists of A B C and where C D E then B D and E are the precedents for A C is not a precedent as it is only an intermediate calculated value Figure 5 2 also shows the testing range of each precedent variable used to estimate the target result If the precedent variables are simple inputs then the testing range will be a simple perturbation based on the range chosen e g the default is 10 Each precedent variable can be perturbed at different percentages if required A wider range is important as it is better able to test extreme values rather than smaller perturbations around the expected values In certain circumstances extreme values may have a larger smaller or unbalanced impact e g nonlinearities may occur where increasing or decreasing economies of scale and scope creep in for larger or smaller values of a variable and only a wider range will capture this nonlinear impact 116 Page Procedure RISK SIMULATOR e Select the single output cell i e a cell with a function or equation in an
243. ly different from zero indicating a significant overall regression model If the p Value is smaller than the 0 01 0 05 or 0 10 alpha significance then the regression is significant The same approach can be applied to the F Statistic by comparing the calculated F Statistic with the critical F values at various significance levels Forecasting Period Actual Y Forecast Error E RMSE 140 4048 a 521 0000 299 5124 221 4876 2 367 0000 487 1243 120 1243 3 443 0000 353 2789 89 7211 4 365 0000 276 3296 88 6704 5 614 0000 776 1336 162 1336 6 385 0000 298 9993 86 0007 7 286 0000 354 8718 68 8718 8 397 0000 312 6155 84 3845 9 764 0000 529 7550 234 2450 10 427 0000 347 7034 79 2966 11 153 0000 266 2526 113 2526 12 231 0000 264 6375 33 6375 13 524 0000 406 8009 117 1991 14 328 0000 272 2226 55 7774 15 240 0000 231 7882 8 2118 16 286 0000 257 8862 28 1138 17 285 0000 314 9521 29 9521 18 569 0000 335 3140 233 6860 19 96 0000 282 0356 186 0356 20 498 0000 370 2062 127 7938 21 481 0000 340 8742 140 1258 22 468 0000 427 5118 40 4882 23 177 0000 274 5298 97 5298 24 198 0000 294 7795 96 7795 25 458 0000 295 2180 162 7820 Figure 3 8 Multivariate Regression Results 75 Page Theory Procedure Results Interpretation RISK SIMULATOR 3 5 Stochastic Forecasting A stochastic process is nothing but a mathematically defined equation that can create a series of outcomes over time o
244. maa 288 18 00 om L a 2 d Eme 1 mam mn Completion Tee a L 0030 00000 Vo ru 6148400 00000 V ve 3 21908 toss 1 96 a a now beet hors roe insert Notes delon node be ts um to 100 Label ronde shape romt iabe Override nama wth labe value wh abel Overman eit abel ranch Event hae Pease rote tat f are matr e of eventa fe probabiliter ed NUN e era rome Phim mane Cemeitenl 200 Completion 07000 T barge Ort Cos Lm Figure 5 61 ROV Decision Tree Decision Tree 4 ROV Visual Modeler 2012 Decision Trees C Users user Desktop Screen Shots DT Model rovdt A a File Edit Insert Properties Style Colors Language Help Oo A gt Decision Tree summary of Values Simulation Modeling ANALYSIS RESULTS Input Probabilities Event Name Completion Time A Critical ID 1 1 1 30 0000 Completion Time B Critical 10 1 1 2 70 0000 Completion Time C Critical ID 1 2 1 30 000076 Completion Time D Critical ID 1 2 2 70 0000 12 Days Completion Time 10 11 30 0000 14 Days Completion Time A 10 11
245. minator degree of freedom 7 is related to the chi square distribution in that 2 In 4 A 1 E yalm m Mean m 2 2 Standard Deviation 2m m n 2 jor all m gt 4 n m 2 m 4 2 m 2n 2 2 m 4 m 6 n m n 2 _ 12 16 20m 8m m 44n 2 5m n 22n 5mn n m 6 m 8 n m 2 Excess Kurtosis The numerator degree of freedom 7 and denominator degree of freedom mw are the only distributional parameters Input requirements Degrees of freedom numerator amp degrees of freedom denominator must both be integers gt 0 51 Page Gamma Distribution Erlang Distribution RISK SIMULATOR The gamma distribution applies to a wide range of physical quantities and is related to other distributions lognormal exponential Pascal Erlang Poisson and chi square It is used in meteorological processes to represent pollutant concentrations and precipitation quantities The gamma distribution is also used to measure the time between the occurrence of events when the event process is not completely random Other applications of the gamma distribution include inventory control economic theory and insurance risk theory Conditions The gamma distribution is most often used as the distribution of the amount of time until the rth occurrence of an event in a Poisson process When used in this fashion the three conditions underlying the gamma distribution ar
246. mulating variables that are neither uncertain nor have little impact on the results Tornado charts assist in identifying these critical success drivers quickly and easily Following this example it might be that price and quantity should be simulated assuming that the required investment and effective tax rate are both known in advance and unchanging Base Value 96 6261638553219 input Changes Output Output Effective Input Input Base Case Downside Upside Range Downside Upside Value 276 63 83 37 360 00 1 620 00 1 950 00 1 800 00 219 73 26 47 3 36 00 44 00 343 189 03 9 00 11 00 1671 176 99 11 03 13 48 2318 17007 45 00 55 00 3053 162 72 31 50 38 50 4015 153 11 A 1364 16 67 48 05 145 20 18 00 22 00 138 24 57 03 13 50 15 50 116 80 76 64 4 50 5 5096 9059 102 69 1 8096 2 2096 9508 98 47 i 9 00 11 00 97 09 96 16 1 80 2 20 9616 97 09 2 70 3 30 96 63 96 63 0 00 0 00 96 63 96 63 0 00 0 00 120 Pa E 09 Notes RISK SIMULATOR Figure 5 5 Spider Chart Price Erosion Sales Growth Depreciation Interest Amortization Capex Net Capital r T T T I T I I 7 150 100 50 100 Figure 5 6 Tornado Chart Although the tornado chart is easier to read the spider chart is important for determining if there are any nonlineatities in the model For instance Figure 5 7
247. n Check to make sure the objective cell is set for C78 and select Maximize You can now review the decision variables and constraints if required ot click OK to run the static optimization Once the optimization is complete you may select Revert to revert back to the original values of the decision variables as well as the objective or select Replace to apply the optimized decision variables Typically Replace is chosen after the optimization is done Figure 4 2 shows the screen shots of these procedural steps You can add simulation assumptions on the model s returns and risk columns C and D and apply the dynamic optimization and stochastic optimization for additional practice Decision Variable Properties Decision Name AssetClesst E Decision Type Continuous e g 1 15 2 35 10 55 Lower Bound 005 Upper Bound 035 Integer e g 1 2 3 Lower Bound Upper Bound Binary 0 or 1 Constraint 100 103 RISK SIMULATOR Constraints v SES17 100 Optimization Summary Optimization is used to allocate resources where the results provide the max returns or the min cost risks Uses include managing inventories financial portfolio allocation product mix project selection etc Static Optimization Run on static model without simulations Usually run to determine the intial optimal portfolio before more advanced optimizations are applied Dynamic O
248. n Profile select the profile you wish to change and click OK Figure 2 2 shows an example where there are multiple profiles and how to activate a selected profile Then click on Rak Simulator Edit Simulation Profile and make the required changes You can also duplicate or rename an existing profile When creating multiple profiles in the same Excel model make sure to provide each profile a unique name so you can tell them apart later on Also these profiles are stored inside hidden sectors of the Excel xls file and you do not have to save any additional files The profiles and their contents assumptions forecasts etc are automatically saved when you save the Excel file Finally the last profile that is active when you exit and save the Excel file will be the one that is opened the next time the Excel file is accessed Change Active Simulation 2010 10 14 Second Profile Book 1 2010 10 14 N A 1 2010 10 14 N A V View simulation profiles in all workbooks Delete Duplicate Figure 2 2 Change Active Simulation The next step is to set input assumptions in your model Note that assumptions can only be assigned to cells without any equations or functions typed in numerical values that are inputs in a model whereas output forecasts can only be assigned to cells with equations and functions outputs of a model Recall that assumptions and forecasts cannot be set unless a simulation profile already exi
249. n of the Erlang distribution is the calibration of the rate of transition of elements through a system of compartments Such systems are widely used in biology and ecology e g in epidemiology an individual may progress at an exponential rate from being healthy to becoming a disease cartier and continue exponentially from being a carrier to being infectious Alpha also known as shape and Beta also known as scale are the distributional parameters The mathematical constructs for the Erlang distribution are shown below i JAB f x BaD for x 0 0 otherwise a l py 1 gt 0 F x e Dig for x 0 otherwise Mean af Standard Deviation ap Skew E 6 Excess 3 Alpha and Beta are the distributional parameters Input requirements Alpha Shape gt 0 and is an Integer Beta Scale gt 0 49 Page Exponential Distribution Exponential 2 Distribution Extreme Value Distribution or Gumbel Distribution RISK SIMULATOR The exponential distribution is widely used to describe events recurring at random points in time such as the time between failures of electronic equipment or the time between arrivals at a setvice booth It is related to the Poisson distribution which describes the number of occurrences of an event in a given interval of time An important characteristic of the exponential distribution is the memoryless property which means that the future li
250. n one area of the distribution as compared to a more uniformly sampled distribution every part of the distribution will be sampled when LHS is applied 154 Page RISK SIMULATOR Options Random Number Generator Minimize Excel and All Charts When Running ROV Risk Simulator Default Start Risk Simulator with Excel Always Show Forecast Windows on Top Se eee p e orecasts ision Variables Shuffle Advanced Subtractive Random Shuffle Correlation Normal Default gt T Copula DF 30 5 Quasi Normal DF 30 Simulation Monte Carlo Simulation Default Quick IEEE Hex Basic Minimal Portable Parameters Color Scheme Assumption Decision Forecast e Latin Hypercube Sampling LHS st Language Engish pois jp ome LHS is not recommended when ther correlated assumptions Figure 5 41 Risk Simulator Options 5 15 Deseasonalizing and Detrending Data The data deseasonalization and detrending tool removes any seasonal and trending components in your original data Figure 5 42 In forecasting models the process usually includes removing the effects of accumulating data sets from seasonality and trend to show only the absolute changes in values and to allow potential cyclical patterns to be identified after removing the general drift tendency twists bends and effects of seasonal cycles of a set of time
251. nalysis in Step 4 view the results charts Control Chart R and statistics copy the results and charts to Control Chart U dipboard or generate reports Control Chart X lt STEP 4 Save Optional Tou con save pa fikle and notes in the pofle 139 69 290 10 292 29 892 3 140 69 292 29 2945 8854 141 19 293 89 296 10 677 58388 lt o8 avERBEESdSB U Number of Dependent Variables Tested 3 Number of Econometric Models Tested 61 Auto Econometrics Detailed L VAR LNARZILN VAR3 This is a test model running AE methodology inside ROV BizStats LN VAR2 LN VAR3 LN VAR2 4 N VAR LN VAR 1 Absolute Values LN VAR1 4LN VAR3 VAR2 ANOVA Randomized Block LN VAR 2 LN VAR3 ANOVA Single Factor Multiple Treatments He TAN ANOVA Two Way LN VAR 2 VAR2 LN VAR 1 4LN VAR2 ARIMA 1 0 1 LN VAR2 4LN VAR3 LN VAR 1 LN VAR2 LN VAR3 LN VAR3 LN VAR2 VAR3 LN VAR1 LN VAR3 VAR2 VAR3 VAR 1 LN VAR2 File Data Language Help STEP 1 Data Manually enter your data paste from another application Example P or load an example dataset with analysis Command R Bime 7j Dataset Md tb66992555t6p 5 0 6 8 Volatility GARCH M Volatility GJR GARCH Volatilit
252. nce Although they sound similar the concepts are significantly different from one another A simple illustration is in order Suppose you ate a taco shell manufacturer and are interested in finding out how many broken taco shells there are on average in a box of 100 shells One way to do this is to collect a sample of prepackaged boxes of 100 taco shells open them and count how many of them are actually broken You manufacture 1 million boxes a day this is your population but you randomly open only 10 boxes this is your sample size also known as your number of trials in a simulation The number of broken shells in each box is as follows 24 22 4 15 33 32 4 1 45 and 2 The calculated average number of broken shells is 18 2 Based on these 10 samples or trials the average is 18 2 units while based on the sample the 80 confidence interval is between 2 and 33 units that is 80 of the time the number of broken shells is between 2 and 33 based on this sample size or number of trials run However how sure are you that 18 2 is the correct average Are 10 trials sufficient to establish this The confidence interval between 2 and 33 is too wide and too variable Suppose you require more accurate average value where the error is 2 taco shells 90 of the time this means that if you open all 1 million boxes manufactured in a day 900 000 of these boxes will have broken taco shells on average at some mean unit 2 taco shells How many more t
253. nce of multicollinearity in a multiple regression equation is that the R squared value is relatively high while the t statistics are relatively low Another quick test is to create a correlation matrix between the independent A high cross correlation indicates a potential for autocorrelation The rule of thumb is that a correlation with an absolute value greater than 0 75 is indicative of severe multicollinearity 142 Page RISK SIMULATOR Correlation Matrix CORRELATION X2 x3 x4 x5 x1 0 333 0 959 0 242 0 237 2 1000 0 349 0 319 0 120 1000 0 196 0 227 4 000 0 290 Variance Inflation Factor VIF 2 x3 x4 KS x1 1 12 12 46 1 06 1 06 2 WA 1 14 1 11 1 01 x3 MA 1 04 1 05 x4 MWA 1 09 Figure 5 27 Multicollinearity Errors The Correlation Matrix lists the Pearson s Product Moment Correlations commonly referred to as the Pearson s R between variable pairs The correlation coefficient ranges between 1 0 and 1 0 inclusive The sign indicates the direction of association between the variables while the coefficient indicates the magnitude or strength of association The Pearson s R only measures a linear relationship and is less effective in measuring nonlinear relationships To test whether the correlations are significant a two tailed hypothesis test is performed and the resulting p value s is listed In Figure 5 27 top P values less than 0 10 0 05 and 0 01 are highlighted in blue to indicate statisti
254. nd some time going through the reports generated to get a better understanding of the statistical tests performed sample reports are shown in Figures 5 30 through 5 33 Data Set This tool is used to describe and find statistical relationships in a set of raw data Selected Data Variable X1 Variable X2 Variable X3 18308 185 1148 18068 7729 100484 16728 14630 4008 38927 22322 3711 3136 197 Datais from a single variable Data comprises multiple variables in columns Cancel Figure 5 28 Running the Statistical Analysis Tool 144 Page RISK SIMULATOR Statistical Analyses Figure 5 29 Statistical Tests Descriptive Statistics Analysis of Statistics Almost all distributions can be described within 4 moments some distributions require one moment while others require two moments and so forth Descriptive statistics quantitatively capture these moments The first moment describes the location of a distribution mean median and mode and is interpreted as the expected value expected returns or the average value of occurrences The Arithmetic Mean calculates the average of all occurrences by summing up all of the data points and dividing them by the number of points The Geometric Mean is calculated by taking the power root of the products of all the data points and requires them to all be positive The Geometric Mean is more accurate for percentages or rates that fluctuate significantly
255. nditional heteroskedasticity models are used in forecasting the volatility of financial instruments using the prices themselves The GARCH P Q model allows for different positive P and Q integer lag parameters for the mean news and variance equations Note than only positive data values can be used in a GARCH volatility forecast Periodicity is the number of periods per year e g 12 for monthly data 252 for daily trading data 365 for daily data to annualize the volatility or keep as 1 for periodic volatility Base is the predictive base periods this means how many periods back you would like to use as forecast base to predict future volatility e g enter in 12 if using the past 12 periods Variance Targeting means if you wish the volatility forecast to revert to an imputed long run mean over time Mko rid to your raw price data in chronological order past to present in a single column with multiple rows Data Location 8 2428 8 Generate a GARCH P Q model for Q 1 Periodicity 252 Base 1 Forecast Periods 10 Apply Variance Targeting GARCH GARCH M D TGARCH M EGARCH D GJR GARCH GJR TGARCH Figure 3 19 GARCH Volatility Forecast 91 3 11 1 GARCH Equations RISK SIMULATOR The accompanying table lists some of the GARCH specifications used in Risk Simulator with two underlying distributional assumptions one for normal distribution and the other for the t d
256. near optimization model its objective and all of its constraints are linear Therefore these two examples encapsulate all of the procedures aforementioned 4 2 Optimization with Continuous Decision Variables Figure 4 1 illustrates the sample continuous optimization model The example here uses the Continuous Optimization file found either on the start menu at Start Real Options Valuation Risk Simulator Examples or accessed directly through Risk Simulator Example Models In this example there are 10 distinct asset classes e g different types of mutual funds stocks or assets where the idea is to most efficiently and effectively allocate the portfolio holdings such that the best bang for the buck is obtained that is to generate the best portfolio returns possible given the risks inherent in each asset class To truly understand the concept of optimization we will have to delve deeply into this sample model to see how the optimization process can best be applied As mentioned the model shows the 10 asset classes each with its own set of annualized returns and annualized volatilities These return and risk measures are annualized values such that they can be consistently compared across different asset classes Returns are computed using the geometric average of the relative returns while the risks are computed using the logarithmic relative stock returns approach sb E F H l J K L mm N
257. nes Figure 5 49 ROV Probability Distribution PDF and CDF Charts 163 ge RISK SIMULATOR ROV PROBABILITY DISTRIBUTIONS 00000 Fr Chart n ical Distributi Simulated Distribution Trials N 1000 Sed 12 Distributions Charts and Tables e cd genere endorse chem for aa wel aniio cen hapaa based on parodie hm Overay tool Distribution Beta Charts and Tables c Change First Parameter Second Parameter Apa 2 PDF eR 5 From OCA From To Series To ws EX don Custom Step 01 01 o 0 460800 255 i5 3 5 e g Choose Gamma distribution set Alpha and Beta as parameters to change and enter 2 3 and 5 9 the two custom input copy Chat Table boxes for generating Gamma 2 5 and Gamma 3 9 charts SES m l 699491 Cz 0 dX Index dms 0 0083 0 2483 0 4883 0 7284 0 9684 Decimals D Language English M Chat Type 20 Line Gridlines Run Close Charts and Tables This tool generates a probability table and comparative charts for a chosen distribution as well as the different shapes based on different input parameters To
258. ng routine The generalized autoregressive conditional heteroskedasticity GARCH model is used to model historical and forecast future volatility levels of a marketable security e g stock prices commodity prices and oil prices The data set has to be a time series of raw price levels GARCH will first convert the prices into relative returns and then run an internal optimization to fit the historical data to a mean reverting volatility term structure while assuming that the volatility is heteroskedastic in nature changes over time according to some econometric characteristics Several variations of this methodology are available in Risk Simulator including EGARCH EGARCH T GARCH M GJR GARCH GJR GARCH T IGARCH and T GARCH 66 Page J Curve Markov Chain Maximum Likelihood on Logit Probit and Tobit Multivariate Regression Neural Network Forecast Nonlinear Extrapolation S Curve Stochastic Processes RISK SIMULATOR The J curve or exponential growth curve is where the growth of the next period depends on the current period s level and the increase is exponential This means that over time the values will increase significantly from one period to another This model is typically used in forecasting biological growth and chemical reactions over time A Markov chain exists when the probability of a future state depends on a previous state and when linked together form a chain that reverts to a long run ste
259. ns methodology with each term representing steps taken in the model construction until only random noise remains Also ARIMA modeling uses correlation techniques in generating forecasts ARIMA can be used to model patterns that may not be visible in plotted data In addition ARIMA models can be mixed with exogenous variables but make sure that the exogenous variables have enough data points to cover the additional number of periods to forecast Finally be aware that due to the complexity of the models this module may take longer to run There are many reasons why an ARIMA model is superior to common time series analysis and multivariate regressions The common finding in time series analysis and multivariate regression is that the error residuals are correlated with their own lagged values This serial correlation violates the standard assumption of regression theory that disturbances are not correlated with other disturbances The primary problems associated with serial correlation are e Regression analysis and basic time series analysis are no longer efficient among the different linear estimators However as the error residuals can help to predict current error residuals we can take advantage of this information to form a better prediction of the dependent variable using ARIMA e Standard errors computed using the regression and time series formula are not correct and are generally understated and if there are lagged dependent variables se
260. nstructions on how you would like to run a simulation That is all the assumptions forecasts run preferences and so forth Having profiles facilitates creating multiple scenarios of simulations That is using the same exact model several profiles can be created each with its own specific simulation properties and requirements The same person can create different test scenarios using different distributional assumptions and inputs or multiple persons can test their own assumptions and inputs on the same model e Start Excel and create a new model or open an existing one you can use the Basic Simulation Model example to follow along Click on Simulator New Simulation Profile e Specify a title for your simulation as well as all other pertinent information Figure 2 1 13 Page RISK SIMULATOR Simulation Properties Profile Name New Simulation Profile Simulation Settings Number of trials 1 0005 7 Pause simulation on error V Turn on correlations Specify random number sequence Seed 999 Figure 2 1 New Simulation Profile e Title Specifying a simulation title allows you to create multiple simulation profiles in a single Excel model Thus you can now save different simulation scenario profiles within the same model without having to delete existing assumptions and changing them each time a new simulation scenario is required You can always change the profile s name later Rak Simulator
261. o a more accurate view of the regression s explanatory power Hence only 0 25 of the variation in the dependent variable can be explained by the regressors The Multiple Correlation Coefficient Multiple R measures the correlation between the actual dependent variable Y and the estimated or fitted Y based on the regression equation This is also the square root of the Coefficient of Determination R Squared The Standard Error of the Estimates SEy describes the dispersion of data points above and below the regression line or plane This value is used as part of the calculation to obtain the confidence interval of the estimates later Regression Results Intercept xi x2 x3 X4 X5 Coefficients 57 9555 0 0035 0 4644 25 2377 0 0086 16 5579 Standard Error 108 7901 0 0035 0 2535 14 1172 0 1016 14 7996 t Statistic 0 5327 1 0066 1 8316 1 7877 0 0843 1 1188 p Value 0 5969 0 3197 0 0738 0 0807 0 9332 0 2693 Lower 5 161 2966 0 0106 0 0466 3 2137 0 2132 13 2687 Upper 95 277 2076 0 0036 0 9753 53 6891 0 1961 46 3845 Degrees of Freedom Hypothesis Test Degrees of Freedom for Regression 5 Critical t Statistic 99 confidence with df of 44 2 6923 Degrees of Freedom for Residual 44 Critical t Statistic 95 confidence with df of 44 2 0154 Total Degrees of Freedom 49 Critical t Statistic 90 confidence with df of 44 1 6802 The Coefficients provide the estimated regression intercept and slopes For instance the coefficients are estimat
262. o run a Linear or Nonlinear Neural Network model enter in the desired number of Forecast Periods e g 5 the number of hidden Layers in the Neural Network e g 3 and number of Testing Periods e g 5 Click Ryn to execute the analysis and review the computed results and charts You can also Capy the results and chart to the clipboard and paste it in another software application Note that the number of hidden layers in the network is an input parameter and will need to be calibrated with your data Typically the more complicated the data pattern the higher the number of hidden layers you would need and the longer it would take to compute It is recommended that you start at 3 layers The testing period is simply the number of data points used in the final calibration of the Neural Network model and we recommend using at least the same number of periods you wish to forecast as the testing period In contrast the term fuzzy logic is derived from fuzzy set theory to deal with reasoning that is approximate rather than accurate as opposed to crisp logic where binary sets have binary logic fuzzy logic variables may have a truth value that ranges between 0 and 1 and is not constrained to the two truth values of classic propositional logic This fuzzy weighting schema is used together with a combinatorial method to yield time series forecast results in Risk Simulator as illustrated in Figure 5 57 and is most applicable when applied to time
263. ocess Geometric Brownian Data VAR63 Stochasti Diffusi gt Varl Var2 Var3 5 001 AutoEconometricsDetailed VAR5 VARG VAR7 VAR8 0 1 0 AE res Dump are 6 010 PrincipalComponentAnalysis VAR6 VAR7 VAR8 VAR9 VAR10 emp x 7 Stochastic Process Mean Reversion Structural Break w sm 3 Time Series Analysis Auto k Time Series Analysis Double Exponential Sm STEP 3 Run Runs the current analysis in Step 2 or selected IWe Serieo Analysis Double Moving Average saved analysis in Step 4 view the results charts Time Series Analysis Holt Winter s Additive and statistics copy the results and charts to Time Series Analysis Holt Winter s Multiplica dipboard or generate reports Time Series Analysis 5 Additive STEP 4 Save Optional You can save multiple analyses and notes in the profile for future retrieval 21685 9352 92 8151 5 4049 232 7934 1 5207 Name Auto Econometrics Detailed Reduced Data Matrix 70 0219 0 1338 0 0235 0 0909 0 5140 70 0322 0 0459 70 1152 0 1 Notes This is a test model running AE methodology inside ROV BizStats 0 1169 0 5155 0 1681 0 1824 0 2595 0 0431 0 1284 0 1010 0 1 0 0241 0 1154 70 0339 0 0683 0 6488 0 0436 0 0438 0 1124 0 2 ADD 0 0526 0 1003 0 0813 0 0735 0 0150 0 0816 0 3112 0 1061 0 0 0 0776 0 1985 0 0619 0 0869 0 1055 0 1270 0 0311 0 0154 0 0 EDIT Parametric 2 Var T Test for Independent Unequal Variance
264. odel and report 90 Page Note RISK SIMULATOR The typical volatility forecast situation requires P 1 Q 1 Periodicity number of periods year 12 for monthly data 52 for weekly data 252 or 365 for daily data Base minimum of 1 and up to the periodicity value and Forecast Periods number of annualized volatility forecasts you wish to obtain There are several GARCH models available in Risk Simulator including EGARCH EGARCH T GARCH M GJR GARCH GJR GARCH T IGARCH and T GARCH See the chapter in Modeling Risk Second Edition by Dr Johnathan Mun Wiley Finance 2010 on GARCH modeling for more details on what each specification is for IK Real Options V a ua tion Historical Data Days Generalized Autoregressive Conditional Heteroskedasticity GARCH To run a GARCH model enter in the relevant time series data then click on Risk Simulator Forecasting GARCH and click on the data location ink icon select the historical data area e g C8 C2428 Enter in the required inputs e g P 1 Q 1 Daily Trading Periodicity 252 Predictive Base 1 Forecast Periods 10 and click OK Review the generated forecast report For practice run each of the GARCH variations and compare the results Refer to the user manual for the functional form and specifications for each model variation GARCH GARCH M TGARCH TGARCH M EGARCH EGARCH T GJR GARCH GJR TGARCH GARCH GARCH or generalized autoregressive co
265. of Determination R Squared The Standard Error of the Estimates SEY describes the dispersion of data points above and below the regression line or plane This value is used as part of the calculation to obtain the confidence interval of the estimates later The AIC and SC are often used in model selection SC imposes a greater penalty for additional coefficients Generally the user should select model with the lowest value of the AIC and SC The Durbin Watson statistic measures the seria correlation in the residuals Generally DW less than 2 implies positive seria correlation Regression Results Intercept AR 1 MA 1 Coefficients 0 0626 1 0055 0 4936 Standard Error 0 3408 0 0006 0 0420 Statistic 0 2013 1691 1373 11 7633 pValue 0 8406 0 0000 0 0000 Lower 596 0 4498 1 0065 0 5628 Upper 95 0 5749 1 0046 0 4244 Degrees of Freedom Hypothesis Test Degrees of Freedom for Regression 2 Critical tStatistic 9996 confidence with af of 432 25873 Degrees of Freedom for Residual 432 Critical t Statistic 95 confidence with af of 432 1 9655 Tota Degrees of Freedom 434 Critical t Statistic 90 confidence with df of 432 1 6484 The Coefficients provide the estimated regression intercept and slopes For instance the coefficients are estimates of the true population b values in the foliowing regression equation Y Bo BX 8 X The Standard Error measures how accurate the predicted Coefficients are and the t Statist
266. of Periods Per S I Cycle m o 3 Review the two reports generated for more details on the methodology application and resulting Detrend Data charts and deseasonalized detrended data v Li v i Linear Exponential nality Test Logarithmic Polynomial Order 6 Iv Power Moving Average Order 3 Time Series Data 89 828 M Static Mean M Difference Order 1 Maximum Seasonality Period to Test e Static Median Rate Order 1 E ea Doa 29 30 Figure 5 42 Deseasonalization and Detrending Data 156 Page RISK SIMULATOR 5 16 Principal Component Analysis Principal Component Analysis is a way of identifying patterns in data and recasting the data in such a way as to highlight their similarities and differences Figure 5 43 Patterns of data are very difficult to find in high dimensions when multiple variables exist and higher dimensional graphs are very difficult to represent and interpret Once the patterns in the data are found they can be compressed and the number of dimensions is now reduced This reduction of data dimensions does not mean much reduction in loss of information Instead similar levels of information can now be obtained with a smaller number of variables Procedure e Select the data to analyze e g B11 K30 click on Rak Simulator Tools Principal Component Analysis and click OK Review the generated report for the computed results Procedure 1 Select the data to analyz
267. of Simulation Trials 50 Stochastic Optimization Similar to dynamic optimization but the process is repeated several times The final decision variables will each have its own forecast chart indicating its optimal range Number of Simulation Trials 500 Number of Optimization Runs 204 Figure 4 10 Setting Up the Stochastic Optimization Problem 113 Page Results Interpretation Super Speed Simulation with Optimization Simulation Statistics for Stochastic and Dynamic Optimization RISK SIMULATOR Stochastic optimization is performed when a simulation is run first and then the optimization is run Then the whole analysis is repeated multiple times As shown in Figure 4 11 for the example optimization the result is a distribution of each decision variable rather than a single point estimate This means that instead of saying you should invest 30 69 in Asset 1 the results show that the optimal decision is to invest between 30 35 and 31 04 as long as the total portfolio sums to 100 This way the results provide management or decision makers a range of flexibility in the optimal decisions while accounting for the risks and uncertainties in the inputs You can also run stochastic optimization with super speed simulation To do this first reset the optimization by resetting all four decision variables back to 25 Next Run Optimization click on the Advanced button Figure 4 10 and select the checkbox for Run
268. on for additional practice 106 Page RISK SIMULATOR gt 3S I s e eh emi o Seno Return to Profitability Risk Ratio index Project 1 54 96 8 33 1 26 Project 2 1 914 92 1 02 3 27 Project 3 1 551 03 1 03 1 87 Project 4 1 012 95 2 22 2 37 Project 5 925 41 0 92 2 85 Project 6 560 92 1 35 15 58 Project 7 5 633 40 051 4 75 Project 8 926 25 1 33 11 74 Project 9 2 100 60 0 93 16 56 Project 10 1 912 50 1 18 591 Project 11 263 52 2 08 13 20 Project 12 309 75 1 69 6 00 ENPV Cost Risk Risk 9 Selection Total 17 218 00 8 197 44 7 007 40 70 12 00 Goat MAX 5000 lt 6 Sharpe Ratio 2 4573 ENPY is the expected NPV of each credit line or project while Cost can be the total cost of administration as well as required capital holdings to cover the credit line and Risk is the Coefficient of Variation of the credit line s ENPV Figure 4 4 Discrete Integer Optimization Model Decision Variable Properties Decision Name Decision Type Continuous e g 1 15 2 35 10 55 Lower Bound Upper Bound Bj Integer e g 1 2 3 Lower Bound Upper Bound Ej Binary 0 or 1 Constraints Current Constraints v 50517 lt 5000 5 517 6 Efficient Frontier 107 Page Results Interpretation RISK SIMULATOR Optimization Summary Optimization is used to allocate resources where the
269. on is useful for modeling the distribution of the number of additional trials required in addition to the number of successful occurrences required R For instance in order to close a total of 10 sales opportunities how many extra sales calls would you need to make above 10 calls given some probability of success in each call The x axis shows the number of additional calls required or the number of failed calls The number of trials is not fixed the trials continue until the Rth success and the probability of success is the same from trial to trial Probability of success and number of successes required are the distributional parameters It is essentially superdistribution of the geometric and binomial 41 Page Pascal Distribution RISK SIMULATOR distributions This distribution shows the probabilities of each number of trials in excess of R to produce the required success R Conditions The three conditions underlying the negative binomial distribution are The number of trials is not fixed trials continue until the 7th success e probability of success is the same from trial to trial The mathematical constructs for the negative binomial distribution are as follows p P x PART p p forx r r l and0 lt p l r 1 x Mean r p Standard Deviation Skewness 22 2 Excess Kurtosis p 6p 6 r 1 p Probability of success p and required successes R a
270. ong distribution Conversely the higher the p value the better the distribution fits the data Roughly you can think of p value as a percentage explained that is if the p value is 0 9727 Figure 5 14 then setting a normal distribution with a mean of 99 28 and a standard deviation of 10 17 explains about 97 27 of the variation in the data indicating an especially good fit Both the results Figure 5 14 and the report Figure 5 15 show the test statistic p value theoretical statistics based on the selected distribution empirical statistics based on the raw data the original data to maintain a record of the data used and the assumption complete with the relevant distributional parameters Le if you selected the option to automatically generate assumption and if a simulation profile already exists The results also rank all the selected distributions and how well they fit the data Student s T Triangular Uniform 47 56 185 86 53 30 49 71 204 77 53 09 50 24 145 61 52 09 50 36 219 85 45 81 R Sinele Fit Distribution fitting takes existing raw data and statistically finds the best fitting distribution i e by optimizing the parameters of each distribution and performing statistical hypotheses tests Distribution Type Fit to Continuous Distributions C Fit to Discrete Distributions Select Distributions to Fit Cauchy Distribution ChiSquare Distribution Exponential Distribution F Distr
271. or Normality The Normality test is a form of nonparametric test which makes no assumptions about the specific shape ofthe population from which the sample is drawn allowing for smaller sample data sets to be analyzed This test evaluates the null hypothesis of whetherthe data sample was drawn from a normally distributed population versus an alternate hypothesis that the data sample is not normally distributed If the calculated p value is less than or equal to the alpha significance value then reject the null hypothesis and acceptthe alternate hypothesis Otherwise if the p value is higher than the alpha significance value do not reject the null hypothesis This test relies on two cumulative frequencies one derived from the sample data set the second from a theoretical distribution based on the mean and standard deviation of the sample data An alternative to this test is the Chi Square test for normality The Chi Square test requires more data points to run compared to the Normality test used here Test Result Relative Data Average 331 92 PEO COE Standard Deviation 172 91 47 00 0 02 0 02 0 0497 0 0297 D Statistic 0 0859 68 00 0 02 0 04 0 0635 0 0235 D Critical at 196 0 1150 87 00 0 02 0 06 0 0783 0 0183 D Critical at 596 0 1237 96 00 0 02 0 08 0 0862 0 0062 D Critical at 1096 0 1473 102 00 0 02 0 10 0 0918 0 0082 Null Hypothesis The data is normally distributed 108 00 0 02 0 12 0 0977 0 0223 114 00 0 02 0
272. or instance before running a stochastic optimization problem a discrete optimization is first run to determine if there exist solutions to the optimization problem before a more protracted analysis is performed Next Dynamic Optimization is applied when Monte Carlo simulation is used together with optimization Another name for such a procedure is Simulation Optimization That is a simulation is first run then the results of the simulation are then applied in the Excel model and then an optimization is applied to the simulated values In other words a simulation is run for N trials and then an optimization process is run for M iterations until the optimal results are obtained or an infeasible set is found That is using Risk Simulator s optimization module you can choose which forecast and assumption statistics to use and replace in the model after the simulation is run Then these forecast statistics can be applied in the optimization process This approach is useful when you have a large model with many interacting assumptions and forecasts and when some of the forecast statistics are required in the optimization For example if the standard deviation of an assumption or forecast is required in the optimization 99 Page Stochastic Optimization Efficient Frontier RISK SIMULATOR model e g computing the Sharpe ratio in asset allocation and optimization problems where we have mean divided by standard deviation of the portfolio
273. othesis testing and is related to the gamma and standard normal distributions For instance the sum of independent normal distributions is distributed as a chi square Q with degrees of freedom d Zi 22 22 02 The mathematical constructs for the chi square distribution are as follows 0 5 52 x7 for all x gt 0 I k 2 Mean k Standard Deviation 42k Skewness 2 E k Excess Kurtosis 12 k T is the gamma function Degrees of freedom is the only distributional parameter The chi square distribution can also be modeled using a gamma distribution by setting the Shape parameter equal to and the scale equal to 25 where S is the scale Input requirements Degrees of freedom gt 1 and must be an integer lt 300 47 Page Cosine Distribution Double Log Distribution RISK SIMULATOR The cosine distribution looks like a logistic distribution where the median value between the minimum and maximum have the highest peak or mode carrying the maximum probability of occutrence while the extreme tails close to the minimum and maximum values have lower probabilities Minimum and maximum are the distributional parameters The mathematical constructs for the Cosine distribution are shown below E for min lt x lt max 242b b 0 otherwise where imt max p _ max min 2 1 1 sin for min lt x lt max F x 5 2 1 for x gt max Min Max
274. othesis Testing and Bootstrap Simulation German Deutsch Be Mi DisanosticToo B Multiple Regression talian Italiano leset Simulation h I Distributional Analysis aal Mode BP 10 Nonlinear Extrapolation Japanese 12538 ample Models E Distributional Charts amp Tables ET 11 Optimization Continuous 209 Foregsng 0 Distributional Designe een ee Bj 12 Optimization Discrete Portuguese Portugu s amp Distributional Fi Single Variabl Optimization f Auto ARIMA eens FERMO Elo Vale Bj 13 Optimization Stochastic Russian Fi Multi Auto Econometrics 5 Distributional Fitting Multi Variable 5 charts Osa z ROV BizStat 7 Basic Econometrics Distributional Fitting Percentiles Bj 15Queuing Models izStats 3 i ROV Decision Tree Ub Ct Cometations 58 16 Regression Diagnostics Hypothesis Testi amp options sil esti 35 Hypothesis Testing Bj 17 Retirement Funding with VBA Macros B Languages p B VB aa Bj 18 Statistical Analysis 1 License 2S comes eo Bj 19 stochastic Processes i ds About Risk Simulator ee oee ee Bj 20 Time Series ARIMA 4 Seasonality Test Check for Updates a paren td Bj 21 Time Series Forecasting 2 4s tation Clust Resources p 6 Multiple Regression Analysis d ina Bj 22 Tornado and Sensitivity Charts Linear x tivity
275. ou can enter in econometric Speier o Pa pesa eset ice min ac ie de rai e at more independent vari HETES CN VART and are d abl ie two ums virgen D a om inthe ia derer valen E LAVAR LNVARA VARS RS 3 RESIDUAL VAR3 VAR4 Independent Variables _ I VAR3 VAR4 LAG VARS 1 DIFF VARE ae R Squared Coefficient of Determination 0 5231 Adjusted R Squared 0 4663 Multiple R Multiple Correlation Coefficient 0 7233 Standard Error of the Estimates SEy 0 4666 INTEGER1 Min ANOVA F Statistic 9 2137 ANOVA p Value INTEGER2 Min INTEGER3 Min Intercept Figure 3 16 Basic Econometrics Module Notes To run an econometric model simply select the data B5 G55 including headers and click on R k Simulator Forecasting Basic Econometrics You can then type in the variables and their modifications for the dependent and independent variables Figure 3 16 Note that only one variable is allowed as the Dependent Variable Y whereas multiple variables are allowed in the Independent Variables X section sepatated by a semicolon and that basic mathematical functions can be used e g LN LOG LAG TIME RESIDUAL DIFF Click on Show Results to preview the computed model and click OK to generate the econometric model
276. outcomes of the more risky stock are relatively more unknown than the less risky stock The vertical axis in Figure 2 21 measures the stock prices thus the mote risky stock has a wider range of potential outcomes This range is translated into a distribution s width the horizontal axis in Figure 2 20 where the wider distribution represents the riskier asset Hence width or spread of a distribution measures a variable s risks Notice that in Figure 2 20 both distributions have identical first moments or central tendencies but the distributions are clearly very different This difference in the distributional width is measurable Mathematically and statistically the width or risk of a variable can be measured through several different statistics including the range standard deviation 0 variance coefficient of variation and percentiles Measuring the Skew of the Distribution the Third Moment RISK SIMULATOR Skew 0 KurtosisXS 0 Figure 2 20 Second Moment Stock prices Figure 2 21 Stock Price Fluctuations The third moment measures a distribution s skewness that is how the distribution is pulled to one side or the other Figure 2 22 illustrates a negative skew or left skew where the tail of the distribution points to the left Figure 2 23 illustrates a positive skew or right skew where the tail of the distribution points to the right The mean is always skewed tow
277. oving average ARIMA also known as Box Jenkins ARIMA is an advanced econometric modeling technique ARIMA looks at historical time series data and performs backfitting optimization routines to account for historical autocorrelation the relationship of one value versus another in time and the stability of the data to correct for the nonstationary characteristics of the data and this predictive model learns over time by correcting its forecasting errors Advanced knowledge in econometrics is typically required to build good predictive models using this approach The Auto ARIMA module automates some of the traditional ARIMA modeling by automatically testing multiple permutations of model specifications and returns the best fitting model Running the Auto ARIMA is similar to regular ARIMA forecasts The difference being that the P D Q inputs are no longer required and different combinations of these inputs are automatically run and compared Econometrics refers to a branch of business analytics modeling and forecasting techniques for modeling the behavior of or forecasting certain business economic finance physics manufacturing operations and any other variables Running the Basic Econometrics models are similar to regular regression analysis except that the dependent and independent variables are allowed to be modified before a regression is run Similar to basic econometrics but Auto Econometrics allows thousands of linear nonlinear int
278. ows you the number of employees in each wage group as a fraction of all employees you can estimate the likelihood or probability that an employee drawn at random from the whole group earns a wage within a given interval For example assuming the same conditions exist at the time the sample was taken the probability is 0 33 a one in three chance that an employee drawn at random from the whole group earns between 8 00 and 8 50 an hour 35 Page Selecting the Right Probability Distribution Monte Carlo Simulation RISK SIMULATOR 0 33 Probability 7 00 7 50 8 00 8 50 9 00 Hourly Wage Ranges in Dollars Figure 2 26 Frequency Histogram II Probability distributions are either discrete or continuous Discrete probability distributions describe distinct values usually integers with no intermediate values and are shown as a series of vertical bars A discrete distribution for example might describe the number of heads in fout flips of a coin as 0 1 2 3 or 4 Continuous distributions are actually mathematical abstractions because they assume the existence of every possible intermediate value between two numbers That is a continuous distribution assumes there is an infinite number of values between any two points in the distribution However in many situations you can effectively use a continuous distribution to approximate a discrete distribution even though the continuous model does not necessarily describe the situation
279. pendent variable to the second or third power and run a regression or forecast using the nonlinearly transformed data Diagnostic Results Heteroskedasticity Micronumerosity Outliers Nonlinearity W Test Hypothesis Test Approximation Natural Natural Number of Nonlinear Test Hypothesis Test Variable p value result result Lower Bound Upper Bound Potential Outliers p value result Y no problems 7 86 671 70 2 Variable X1 0 2543 Homoskedastic no problems 21377 95 64713 03 3 0 2458 linear Variable X2 0 3371 Homoskedastic no problems 77 47 445 93 2 0 0335 nonlinear Variable X3 0 3649 Homoskedastic no problems 5 77 15 69 3 0 0305 nonlinear Variable X4 0 3066 Homoskedastic no problems 295 96 628 21 4 0 9298 linear Variable x5 0 2495 Homoskedastic no problems 3 35 9 38 3 0 2727 linear Figure 5 23 Results from Tests of Outliers Heteroskedasticity Micronumerosity and Nonlinearity Another typical issue when forecasting time series data is whether the independent variable values are truly independent of each other or are actually dependent Dependent variable values collected over a time series may be autocorrelated For serially correlated dependent variable values the estimates of the slope and intercept will be unbiased but the estimates of their forecast and variances will not be reliable and hence the validity of certain statistical goodness of fit tests will be flawed For instance interest rates inflation rates sales rev
280. phi method Distribution Truncation enabling data boundaries Excel Functions set assumptions and forecasts using functions inside Excel Multidimensional Simulation simulation of uncertain input parameters 6 Page RISK SIMULATOR 30 Precision Control determines if the number of simulation trials run is sufficient 31 Super Speed Simulation runs 100 000 trials in a few seconds 1 4 3 Forecasting Module 32 ARIMA autoregressive integrated moving average models ARIMA P D Q 33 Auto ARIMA tuns the most common combinations of ARIMA to find the best fitting model 34 Auto Econometrics runs thousands of model combinations and permutations to obtain the best fitting model for existing data linear nonlinear interacting lag leads rate difference 35 Basic Econometrics econometric and linear nonlinear and interacting regression models 36 Combinatorial Fuzzy Logic Forecasts time series forecast methods 37 Cubic Spline nonlinear interpolation and extrapolation 38 GARCH volatility projections using generalized autoregressive conditional heteroskedasticity models GARCH GARCH M TGARCH TGARCH M EGARCH EGARCH T GJR GARCH and GJR TGARCH 39 J Curve exponential J curves 40 Limited Dependent Variables Logit Probit and Tobit 41 Markov Chains two competing elements over time and market share predictions 42 Multiple Regression regular linear and nonlinear regression with stepwise metho
281. pproach as repeatedly picking golf balls out of a large basket with replacement The size and shape of the basket depend on the distributional input assumption e g a normal distribution with a mean of 100 and a standard deviation of 10 versus a uniform distribution or a triangular distribution where some baskets 36 Page RISK SIMULATOR are deeper or more symmetrical than others allowing certain balls to be pulled out more frequently than others The number of balls pulled repeatedly depends on the number of trials simulated For a large model with multiple related assumptions imagine a very large basket wherein many smaller baskets reside Each small basket has its own set of golf balls that are bouncing around Sometimes these small baskets are linked with each other if there is a correlation between the variables and the golf balls are bouncing in tandem while other times the balls are bouncing independent of one another The balls that are picked each time from these interactions within the model the large central basket are tabulated and recorded providing a forecast output result of the simulation With Monte Carlo simulation Risk Simulator generates random values for each assumption s probability distribution that are totally independent In other words the random value selected for one trial has no effect on the next random value generated Use Monte Carlo sampling when you want to simulate real world what if scenarios for your s
282. preadsheet model The two following sections provide a detailed listing of the different types of discrete and continuous probability distributions that can be used in Monte Catlo simulation 37 Page Bernoulli or Yes No Distribution Binomial Distribution RISK SIMULATOR 2 4 Discrete Distributions The Bernoulli distribution is a discrete distribution with two outcomes e g head or tails success or failure 0 or 1 It is the binomial distribution with one trial and can be used to simulate Yes No or Success Failure conditions This distribution is the fundamental building block of other more complex distributions For instance e Binomial distribution a Bernoulli distribution with higher number of 7 total trials that computes the probability of x successes within this total number of trials Geometric distribution a Bernoulli distribution with higher number of trials that computes the number of failures required before the first success occurs Negative binomial distribution a Bernoulli distribution with higher number of trials that computes the number of failures before the Xth success occurs The mathematical constructs for the Bernoulli distribution are as follows 1 for x 0 p forx p for x 1 or P n p 1 Mean p Standard Deviation 4 p l p Skewness _1 2P 1 Excess Kurtosis 6 1 1 Probability of success f is the only distributional parameter Also
283. ps discourage introduction of personal biases into the process Extrapolation is fairly reliable relatively simple and inexpensive However extrapolation which assumes that recent and historical trends will continue produces large forecast errors if discontinuities occur within the projected time period That is pure extrapolation of time series assumes that all we need to know is contained in the historical values of the seties that is being forecasted If we assume that past behavior is a good predictor of future behavior extrapolation is appealing This makes it a useful approach when all that is needed are many short term forecasts This methodology estimates the function for any arbitrary x value by interpolating a smooth nonlinear curve through all the x values and using this smooth curve extrapolates future x values beyond the historical data set The methodology employs either the polynomial functional form or the rational functional form a ratio of two polynomials Typically a polynomial functional form is sufficient for well behaved data however rational functional forms ate sometimes more accurate especially with polar functions ie functions with denominators approaching zero e Start Excel and open your historical data if required the illustration shown next uses the file Nonlinear Extrapolation from the examples folder e Select the time series data and select Risk Simulator Forecasting Nonlinear Extrapolat
284. ptimization A simulation is first run the results of the simulation are applied in the model and then an optimization is applied to the simulated values Number of Simulation Trials 1000 Stochastic Optimization Similar to dynamic optimization but the process is repeated several times The final decision variables will each have its own forecast chart indicating its optimal range Number of Simulation Trials 1000 Number of Optimization Runs 204 Figure 4 2 Running Continuous Optimization in Risk Simulator Results The optimization s final results are shown in Figure 4 3 where the optimal allocation of assets Interpretation for the portfolio is seen in cells E6 E15 That is given the restrictions of each asset fluctuating between 5 and 35 and where the sum of the allocation must equal 100 the allocation that maximizes the return to risk ratio can be identified from the data provided in Figure 4 3 A few important things have to be noted when reviewing the results and optimization procedures performed thus far e The correct way to run the optimization is to maximize the bang for the buck or returns to risk Sharpe ratio as we have done 104 Page RISK SIMULATOR e If instead we maximized the total portfolio returns the optimal allocation result is trivial and does not require optimization to obtain That is simply allocate 5 the minimum allowed to the lowest eight assets 35 the maximum allowed to th
285. r Distribution The triangular distribution describes a Minimum Maximum Regular Input Percentile Input 7 Enable Data Boundary In finity E Infinity Enable Dynamic Simulations Minimum M maximum and most likely values to occur deber For example you could describe the number of cars sold per week when past You can enter pairwise correlations here when there are other assumptions available A short description of the selected distribution is provided Figure 2 4 Assumption Properties Defining Output Forecasts Click on the Link icons to link the input to any Excel cell location Enter the required input parameters Optional view of alternate percentile inputs Optional data boundaries Optional multidimensional simulation The next step is to define output forecasts in the model Forecasts can only be defined on output cells with equations or functions The following describes the set forecast process e Select the cell you wish to set a forecast e g cell G70 in the Basic Simulation Model example e Click on Risk Simulator Set Output Forecast ot click on the set output forecast icon on the Risk Simulator icon toolbar Figure 1 3 e Entet the relevant information and click OK Note that you can also set output forecasts by selecting the cell you wish to set the forecast on and using the mouse right click access the shortcut Risk
286. r different parameters e Click Run I to compute the results J You can view any relevant analytical results charts or statistics from the various tabs in Step 3 e If required you can provide model name to save into the profile in Step 4 L Multiple models can be saved in the same profile Existing models can be edited deleted M and rearranged in order of appearance and all the changes can be saved O into a single profile with the file name extension bizstats e The data grid size can be set in the menu where the grid can accommodate up to 1 000 variable columns with 1 million rows of data per vatiable The menu also allows you to change the language settings and decimal settings for your data get started it is always a good idea to load the example file A that comes complete with some data and precreated models S You can double click on any of these models to run them and the results ate shown in the report area J which sometimes can be a chart or model statistics T U Using this example file you can now see how the input parameters H are entered based on the model description G and you can proceed to create your own custom models e Click on the variable headers D to select one multiple variables at once and then right click to add delete copy paste or visualize P the variables selected e Models can also be entered using a Command console V W X To see how this wotks
287. raining courses and books written by the software s creator e g Dr Johnathan Mun s Real Options Analysis 2nd Edition Wiley Finance 2005 Modeling Risk Applying Monte Carlo Simulation Real Options Analysis Forecasting and Optimization 2nd Edition Wiley Finance 2010 and Valuing Employee Stock Options 2004 FAS 123R Wiley Finance 2004 Please visit our website at www realoptionsvaluation com for more information about these items The Risk Simulator software has the following modules e Monte Carlo Simulation runs parametric and nonparametric simulation of 42 probability distributions with different simulation profiles truncated and correlated simulations customizable distributions precision and error controlled simulations and many other algorithms Forecasting runs Box Jenkins ARIMA multiple regression nonlinear extrapolation stochastic processes and time series analysis e Optimization Under Uncertainty runs optimizations using discrete integer and continuous vatiables for portfolio and project optimization with and without simulation e Modeling and Analytical Tools runs tornado spider and sensitivity analysis as well as bootstrap simulation hypothesis testing distributional fitting etc ROV BizStats over 130 business statistics and analytical models e ROV Decision Tree decision tree models Monte Carlo risk simulation on decision trees sensitivity analysis scenario analysis Bayesian join
288. re chapters provide more details about the applications of other modules To follow along make sure you have Risk Simulator installed on your computer to proceed In fact it is highly recommended that you first watch the getting started videos on the web www tealoptionsvaluation com risksimulator html or attempt the step by step exercises at the end of this chapter before coming back and reviewing the text in this chapter This approach is recommended because the videos will get you started immediately as will the exercises whereas the text in this chapter focuses more on the theory and detailed explanations of the properties of simulation 2 2 2 Running a Monte Carlo Simulation Typically to run a simulation in your existing Excel model the following steps have to be performed 1 Start a new simulation profile or open an existing profile Define input assumptions in the relevant cells Define output forecasts in the relevant cells N Run simulation 5 Interpret the results If desired and for practice open the example file called Basic Simulation Model and follow along with the examples below on creating a simulation The example file can be found either on the start menu at Szart Real Options Valuation Risk Simulator Examples ot accessed directly through Risk Simulator Example Models To start a new simulation you will first need to create a simulation profile A simulation profile contains a complete set of i
289. re higher e g the Value at Risk of a project might be significant 33 Page The Functions of Moments RISK SIMULATOR 01 02 Skew 0 Kurtosis gt 0 Figure 2 24 Fourth Moment Ever wonder why these risk statistics are called moments In mathematical vernacular moment means raised to the power of some value In other words the third moment implies that in an equation three is most probably the highest power In fact the equations below illustrate the mathematical functions and applications of some moments for a sample statistic For example notice that the highest power for the first moment average is one the second moment standard deviation is two the third moment skew is three and the highest power for the fourth moment is fout First Moment Arithmetic Average or Simple Mean Sample xa The Excel equivalent function is AVERAGE The Excel equivalent function is STDEV for a sample standard deviation The Excel equivalent function is STDEVP for a population standard deviation Third Moment Skew Sample n x x n 1 n 2 5 5 skew The Excel equivalent function is SKEW Fourth Moment Kurtosis Sample kurtosis 1 3 n 1 1 2 3 5 2 3 The Excel equivalent function is KURT 34 Page RISK SIMULATOR 2 3 6 Understanding Probability Distributions for Monte Carlo Simulation This section demonstrates the power of Monte Carlo
290. re the distributional parameters Input requirements Successes required must be positive integers gt 0 and 8000 Probability of success gt 0 lt 1 that is 0 0001 lt lt 0 9999 It is important to note that probability of success p of 0 or 1 are trivial conditions that do not require any simulations and hence are not allowed in the software The Pascal distribution is useful for modeling the distribution of the number of total trials required to obtain the number of successful occurrences required For instance to close a total of 10 sales opportunities how many total sales calls would you need to make given some probability of success in each call The x axis shows the total number of calls required which includes successful and failed calls The numbet of trials is not fixed the trials continue until the Rth success and the probability of success is the same from trial to trial Pascal distribution is related to the negative binomial distribution Negative binomial distribution computes the number of events required in addition to the number of successes required given some probability in other words the total failures whereas the Pascal distribution computes the total number of events required in other words the sum of failures and successes to achieve the successes required given some probability Successes required and probability are the two distributional parameters 42 Page Poisson Distribution R
291. remaining cells in 5 75 This is the best method if you have only several decision variables and you can name each decision variable with a unique name for identification later e The second step in optimization is to set the constraint There are two constraints here the total budget allocation in the portfolio must be less than 5 000 and the total number of projects must not exceed 6 So click on Simulator Optimization Constraints and select ADD to add a new constraint Then select the cell 1277 and make it less than or equal to lt 5000 Repeat by setting cell 77 lt 6 e final step in optimization is to set the objective function and start the optimization by selecting cell C79 and Simulator Optimization Set Objective Then run the optimization using Simulator Optimization Run Optimization and selecting the optimization of choice Static Optimization Dynamic Optimization or Stochastic Optimization To get started select Static Optimization Check to make sure that the objective cell is either the Sharpe ratio ot portfolio returns to tisk ratio and select Maximize You can now review the decision variables and constraints if required or click OK to run the static optimization Figure 4 5 shows the screen shots of these procedural steps You can add simulation assumptions on the model s ENPV and risk columns C and E and apply the dynamic optimization and stochastic optimizati
292. rface hangs for a long time chances are your inputs are incorrect and the model is not correctly specified if the mean reversion rate is 110 mean reversion is probably not the correct process Try with different inputs or use a different model TIPS Forecasting Trendlines e Forecast Results scroll to the bottom of the report to see the forecasted values TIPS Function Calls e RS Functions there are functions that you can use inside your Excel spreadsheet to set input assumption and get forecast statistics To use these functions you need to first install RS Functions which include Start Programs Real Options Valuation Risk Simulator Tools and Install Functions and then run a simulation before setting the RS functions inside Excel Refer to the example model 24 for examples on how to use these functions TIPS Getting Started Exercises and Getting Started Videos e Getting Started Exercises there are multiple step by step hands on examples and results interpretation exercises available in the Start Programs Real Options 186 Page RISK SIMULATOR Valuation Risk Simulator shortcut location These exercises are meant to quickly get you up to speed with the use of the software e Getting Started Videos these are all available for free on our website www tealoptionsvaluation com download html or www tovdownloads com download html TIPS Hardware ID e Right Click HWID Copy in the Install License user
293. riable is a random value between 0 and a positive value e The shape of the distribution can be specified using two positive values Input requirements Alpha and beta both gt 0 and can be any positive value The original Beta distribution only takes two inputs Alpha and Beta shape parameters However the output of the simulated value is between 0 and 1 In the Beta 3 distribution we add an extra parameter called Location or Shift where we are not free to move away from this 0 to 1 output limitation therefore the Beta 3 distribution is also known as a Shifted Beta distribution Similarly the Beta 4 distribution adds two input parameters Location or Shift and Factor The original beta distribution is multiplied by the factor and shifted by the location and therefore the Beta 4 is also known as the Multiplicative Shifted Beta distribution The mathematical constructs for the Beta 3 and Beta 4 distributions are based on those in the Beta distribution with the relevant shifts and factorial multiplication e g the PDF and CDF will be adjusted by the shift and factor and some of the moments such as the mean will similarly be affected the standard deviation in contrast is only affected by the factorial multiplication whereas the remaining moments are not affected at all Input requirements Location gt lt 0 location can take on any positive or negative value including zero Factor gt 0 46 Page Cauchy Distribution or
294. ring phenomenon Sometimes variables cannot be readily predicted using traditional means and these variables are said to be stochastic Nonetheless most financial economic and naturally occurring phenomena e g motion of molecules through the air follow a known mathematical law or relationship Although the resulting values are uncertain the underlying mathematical structure is known and can be simulated using Monte Carlo risk simulation The processes supported in Risk Simulator include Brownian motion random walk mean reversion jump diffusion and mixed processes useful for forecasting nonstationary time series variables 67 Page Time Series Analysis and Decomposition Trendlines RISK SIMULATOR In well behaved time series data typical examples include sales revenues and cost structures of large corporations the values tend to have up to three elements a base value trend and seasonality Time series analysis uses these historical data and decomposes them into these three elements and recomposes them into future forecasts In other words this forecasting method like some of the others described first performs a back fitting backcast of historical data before it provides estimates of future values forecasts Trendlines can be used to determine if a set of time series data follows any appreciable trend Trends can be linear or nonlinear such as exponential logarithmic moving average power polynomial or power
295. rl End to the bottom right cell Shift Up Down to select a specific area and so forth e You can enter short notes for each variable on the Notes row Remember to make yout notes short and simple Try out the various chart icons on the Visualize tab to change the look and feel of the charts e g rotate shift zoom change colors add legend and so forth e The Copy button is used to copy the Results Charts and Statistics tabs in Step 3 after a model is run If no models are run then the copy function will only copy a blank page e The Report button will only run if there are saved models in Step 4 or if there is data in the grid else the report generated will be empty You will also need Microsoft Excel to be installed to run the data extraction and results reports and Microsoft PowerPoint available to run the chart reports e When in doubt about how to run a specific model or statistical method start the Example profile and review how the data is setup in Step 1 or how the input parameters are entered in Step 2 You can use these as getting started guides and templates for your own data and models language can be changed in the Language menu Note that currently there 10 languages available in the software with more to be added later However sometimes certain limited results will still be shown in English e You can change how the list of models in Step 2 is shown by changing the View drop list You can
296. ror MAPE is a relative error statistic measured as an average percent error of the historical data points and is most appropriate when the cost of the forecast error is more closely related to the percentage error than the numerical size of the error Finally an associated measure is the Theil s U statistic which measures the naivety of the model s forecast That is if the Theil s U statistic is less than 1 0 then the forecast method used provides an estimate that is statistically better than guessing Period Actual Forecast Fit Error Measurements 1 684 20 RMSE 71 8132 2 584 10 MSE 5157 1348 3 765 40 MAD 53 4071 4 892 30 MAPE 4 50 5 885 40 684 20 Theil s U 0 3054 6 677 00 667 55 7 1006 60 935 45 8 1122 10 1198 09 9 1163 40 1112 48 10 993 20 887 95 11 1312 50 1348 38 12 1545 30 1546 53 13 1596 20 1572 44 14 1260 40 1299 20 15 1735 20 1704 77 16 2029 70 1976 23 17 2107 80 2026 01 18 1650 30 1637 28 19 2304 40 2245 93 20 2639 40 2643 09 Forecast 21 2713 69 Forecast 22 2114 79 Forecast 23 2900 42 Forecast 24 3293 81 Figure 3 5 Example Holt Winter s Forecast Report 71 Page Theory RISK SIMULATOR 3 4 Multivariate Regression It is assumed that the user is sufficiently knowledgeable about the fundamentals of regression analysis The general bivariate linear regression equation takes the form of Y 8 8 X e where is the intercept A is the slope and 215 the error term It is bivariate as there are only two
297. rownian Motion Stochastic Processes Jump Diffusion Stochastic Processes Mean Reversion with Jump Diffusion Stochastic Processes Mean Reversion Structural Break Sum Time Series Analysis Auto Time Series Analysis Double Exponential Smoothing Time Series Analysis Double Moving Average Time Series Analysis Holt Winter s Additive Time Series Analysis Holt Winter s Multiplicative Time Series Analysis Seasonal Additive Time Series Analysis Seasonal Multiplicative Time Series Analysis Single Exponential Smoothing Time Series Analysis Single Moving Average Trend Line Difference Detrended Trend Line Exponential Detrended Trend Line Exponential Trend Line Linear Detrended Trend Line Linear Trend Line Logarithmic Detrended Trend Line Logarithmic Trend Line Moving Average Detrended Trend Line Moving Average Trend Line Polynomial Detrended Trend Line Polynomial Trend Line Power Detrended Trend Line Power Trend Line Rate Detrended Trend Line Static Mean Detrended Trend Line Static Median Detrended Variance Population Variance Sample Volatility EGARCH Volatility EGARCH T Volatility GARCH Volatility GARCH M Volatility GJR GARCH Volatility GJR TGARCH Volatility Log Returns Approach Volatility TGARCH Volatility TGARCH M Yield Curve Bliss and Yield Curve Nelson Siegel 10 Page RISK SIMULATOR 2 MONTE CARLO RISK SIMULATION onte Carlo risk simulat
298. rrelations Pearson s correlation coefficient is the most common correlation measure and is usually referred to simply as the correlation coefficient However Pearson s correlation is a parametric measure which means that it requires both correlated variables to have an underlying normal distribution and that the relationship between the variables is linear When these conditions are violated which is often the case in Monte Carlo simulation the nonparametric counterparts become more important Spearman s rank correlation and Kendall s tau are the two alternatives The Spearman correlation is most commonly used and is most appropriate when applied in the context of Monte Carlo simulation there is no dependence on normal distributions or linearity meaning that correlations between different variables with different 26 Page RISK SIMULATOR distribution can be applied To compute the Spearman correlation first rank all the x and y variable values and then apply the Pearson s correlation computation In the case of Risk Simulator the correlation used is the more robust nonparametric Spearman s rank correlation However to simplify the simulation process and to be consistent with Excel s correlation function the correlation inputs required are the Pearson s correlation coefficient Risk Simulator will then apply its own algorithms to convert them into Spearman s rank correlation thereby simplifying the process However to simp
299. s Correlation Matrix Parametric 2 Var Z Test for Independent Means 1 0000 0 3333 0 9590 0 2422 0 2374 DB Parametric 2 Var Z Test for Independent Proportions 0 3333 1 0000 0 3494 0 3187 0 1200 0 9590 0 3494 1 0000 0 1964 0 2271 0 2422 0 3187 0 1964 1 0000 0 2905 0 2374 0 1200 0 2271 0 2905 1 0000 470279784 3284 670889 8820 112410 0992 1222792 7730 7829 8444 670889 8820 8614 6500 175 2712 6886 4692 16 9438 11241n naa 175 2712 213n 247 1123 1 RAAR D STEP 2 Analysis Choose an analysis and enter the parameters required see example parameter inputs below from another application North America 1 000 50 Europe and Latin America 1 000 50 c D UR Multiple Regression Linear VARG VAR7 VARS VAR9 VA Multiple Regression Nonlinear Nonlinear Regression XML Edi I itor Nonparametric Chi Square Goodness of Fit 7 D Var name VAR95 notes data gt Nonparametric Chi Square Independence aE abes gt Nonparametric Chi Square Population Varia Data var name VAR97 notes gt Nonparametric Friedman s Test var name VAR98 notes data gt Nonparametric Kruskal Wallis Test var name VAR99 notes data gt Nonparametric Lilliefors Test var name VARL00 notes data gt Nonparametric Runs Test lt data gt Nonparametric Wilcoxon Signed Rank One
300. s a user preference for whether the mean median first quartile and fourth quartile lines 25th and 75th percentiles should be displayed on the forecast chart As shown in Figure 2 8C this tab has all the functionalities in allowing you to change the type color size zoom tilt 3D and other things in the forecast chart as well as to generate overlay charts PDF CDF and run distributional fitting on your forecast data see the Data Fitting sections for more details on this methodology Figures 2 8A to 2 8C show the forecast chart s Normal View where the forecast chart user interface is divided into tabs making it small and compact In contrast Figure 2 9 shows the Global View where all elements are located a single interface The results are identical in both views and selecting which view is a matter of personal preference You can switch between these two views by clicking on the link located at the top right corner called Global View and Local View Income Risk Simulator Forecast Global View Histogram Statistics Preferences Options Controls Display Control 7 Always Show Window On Top 7 Semitransparent When Inactive Copy Chart Histogram Resolution Faster y Simulation Higher Resolution Data Update Interval Faster y Update Figure 2 8A Forecast Chart Preferences 21 Page RISK SIMULATOR Data Filter Show all data Show only data betw
301. s and whistles especially the Controls tab 2 3 Correlations and Precision Control 2 3 1 The Basics of Correlations The correlation coefficient is a measure of the strength and direction of the relationship between two variables and it can take on any value between 1 0 and 1 0 That is the correlation coefficient can be decomposed into its sign positive or negative relationship between two variables and the magnitude or strength of the relationship the higher the absolute value of the correlation coefficient the stronger the relationship The correlation coefficient can be computed in several ways The first approach is to manually compute the correlation r of two variables x and y using Te n xy yi The second approach is to use Excel s CORREL function For instance if the 10 data points for x and y are listed in cells A1 B10 then the Excel function to use is CORREL A1 A10 B1 B10 The third approach is to run Risk Simulator s Multi Fit Tool and the resulting correlation matrix will be computed and displayed It is important to note that correlation does not imply causation Two completely unrelated random variables might display some correlation but this does not imply any causation between the two e g sunspot activity and events in the stock market are correlated but there is no causation between the two There are two general types of correlations parametric and nonparametric co
302. s more appropriate to use It is up to the user to make this determination For instance if we see 283 reversion rate chances are mean teversion process is inappropriate or a very high jump rate of say 100 most probably means that a jump diffusion process is probably not appropriate and so forth Further the analysis cannot determine what the variable is and what the data source is For instance is the raw data from historical stock prices or is it the historical prices of electricity or inflation rates or the molecular motion of subatomic particles and so forth Only the user would know about the raw data and hence using a priori knowledge and theory be able to pick the correct process to 141 Page RISK SIMULATOR use e g stock prices tend to follow a Brownian motion random walk whereas inflation rates follow a mean revetsion process or a jump diffusion process is more appropriate should you be forecasting the price of electricity Statistical Summary The following are the estimated parameters for a stochastic process given the data provided It is up to you to determine if the probability of fit similar to a goodness of fit computation is sufficient to warrant the use of a stochastic process forecast and if so whether it is a random walk mean reversion or a jump diffusion model or combinations thereof In choosing the right stochastic process model you will have to rely on past experiences and a priori economic
303. s option allows the user to perform a quick due diligence test of the input assumption For instance if setting a normal distribution with some mean and standard deviation inputs you can click on the percentile input to see what the corresponding 10th and 90th percentiles are e Enable Dynamic Simulation This option is unchecked by default but if you wish to run a multidimensional simulation i e if you link the input parameters of the assumption to another cell that is itself an assumption you are simulating the inputs or simulating the simulation then remember to check this option Dynamic simulation will not work unless the inputs are linked to other changing input assumptions Note If you are following along with the example continue by setting another assumption on cell G9 This time use the Uniform distribution with a minimum value of 0 9 and a maximum value of 1 1 Then proceed to defining the output forecasts in the next step 17 Page RISK SIMULATOR Right mouse click in the distribution gallery to change how you would like to view the list of 42 distributions Enter the assumption name here Assumption Properties Assumption Name Revenue Aj Custom Bernoulli eeta s Binomial i l Chi Sauare y Most Likely Discrete Uniform 2 Erang 8 Exponential 2 Gumbel Maximum Hypergeometric Logistic IK Lognormai 3 Wl Parabolic 2 Pascal Triangula
304. se most likely case and worst case scenarios However if reliable historical data are available distributional fitting can be accomplished Assuming that historical patterns hold and that history tends to repeat itself then historical data can be used to find the best fitting distribution with their relevant parameters to better define the variables to be simulated Figures 5 13 5 14 and 5 15 illustrate a distributional fitting example This illustration uses the Data Fitting file in the examples folder e Opena spreadsheet with existing data for fitting e Select the data you wish to fit data should be in a single column with multiple rows e Select Risk Simulator Tools Distributional Fitting Single V ariable e Select the specific distributions you wish to fit to or keep the default where all distributions are selected and click OK Figure 5 13 e Review the results of the fit choose the relevant distribution you want and click OK Figure 5 14 Please note that ROV BizStats has some additional methods and algorithms for data fitting including the following methods e Distributional Fitting Akaike Information Criterion AIC Rewards goodness of fit but also includes a penalty that is an increasing function of the number of estimated parameters although AIC penalizes the number of parameters less strongly than other methods e Distributional Fitting Anderson Darling AD When applied to testing if a normal distribut
305. see Figure 3 20 for an example and click OK to run the model and report Set both probabilities to 10 and rerun the Markov chain and you will see the effects of switching behaviors very clearly in the resulting chart 93 Page RISK SIMULATOR Markov Chain Forecast The Markov Process is useful for studying the evolution of systems over multiple and repeated trials in successive time periods The system s state at a particular time is unknown and we are interested in knowing the probability that a particular state exists For instance Markov Chains are used to compute the probability that a particular machine or equipment will continue to function in the next time period or whether a consumer purchasing Product A will continue to purchase Product A in the next period or switch to a competitive brand B To generate a Markov process follow the instructions below Makov Chain 1 Click on Risk Simulator Forecasting Markov Chain 2 Enter in the relevant state probabilities e g 90 and 80 Markov chains are very powerful analytical tools used to model the rcents and click OK switching behavior between one state of nature versus another and pe eventually settling on a long term steady state equilibrium market 3 Review the forecast report generated share For instance Markov Chains are used to compute the probability that a particular machine or equipment will continue to function in the next time period or if a consumer pur
306. select the data and click on Rask Simulator Forecasting AUTO ARIMA You can also access this method through the forecasting icons ribbon or right clicking anywhere in the model and selecting the forecasting shortcut menu e Click on the link icon and link to the existing time series data enter the number of forecast periods desired and click OK For ARIMA and Auto ARIMA you can model and forecast future periods by either using only the dependent variable Y that is the Time Series Variable by itself or you can add in exogenous variables X7 X2 just like in a regression analysis where you have multiple independent variables You can run as many forecast periods as you wish if you use only the time series variable Y However if you add exogenous variables X note that your forecast period is limited to the number of exogenous variables data periods minus the time series variable s data periods For example you can only forecast up to 5 periods if you have time series historical data of 100 periods and only if you have exogenous variables of 105 periods 100 historical periods to match the time series variable and 5 additional future periods of independent exogenous variables to forecast the time series dependent variable 85 Page RISK SIMULATOR al RSPAS eanan wN Procedure gt B Cc D Sample Historical Time M1 138 90 139 40 139 70 139 70 140 70 141 20 141 70 141
307. series data For example a detrended data set may be necessaty to see a more accurate account of a company s sales in a given year mote clearly by shifting the entire data set from a slope to a flat surface to better expose the underlying cycles and fluctuations Many time series data exhibit seasonality where certain events repeat themselves after some time period or seasonality period e g ski resorts revenues are higher in winter than in summer and this predictable cycle will repeat itself every winter Seasonality periods represent how many periods would have to pass before the cycle repeats itself e g 24 hours in a day 12 months in a year 4 quarters in a year 60 minutes in an hour etc For deseasonalized and detrended data a seasonal index greater than 1 indicates a high period or peak within the seasonal cycle and a value below 1 indicates a dip in the cycle 155 Page RISK SIMULATOR Procedure e Select the data you wish to analyze e g 9 28 and click on Simulator Tools Deseasonalization Data Deseasonalization and Detrending and Dewendind Select Deseasonalize Data and or Detrend Data select any detrending models you wish to run enter in the relevant orders e g polynomial order moving average order difference order and rate order and click OK e Review the two reports generated for more details on the methodology application and resulting charts and deseasonalized detrended data Proced
308. simulation but to get started with simulation one first needs to understand the concept of probability distributions begin to understand probability consider this example You want to look at the distribution of nonexempt wages within one department of a large company First you gather raw data in this case the wages of each nonexempt employee in the department Second you organize the data into a meaningful format and plot the data as a frequency distribution on a chart To create a frequency distribution you divide the wages into group intervals and list these intervals on the chart s horizontal axis Then you list the number or frequency of employees in each interval on the chart s vertical axis Now you can easily see the distribution of nonexempt wages within the department A glance at the chart illustrated in Figure 2 25 reveals that most of the employees approximately 60 out of a total of 180 earn from 7 00 to 9 00 per hour Number of Employees 7 00 7 50 8 00 8 50 9 00 Hourly Wage Ranges in Dollars Figure 2 25 Frequency Histogram I You can chart this data as a probability distribution A probability distribution shows the number of employees in each interval as a fraction of the total number of employees To create a probability distribution you divide the number of employees in each interval by the total number of employees and list the results on the chart s vertical axis The chart in Figure 2 26 sh
309. sion to be modeled and this response variable is binary home purchase or no home purchase and depends on a series of independent variables X such as income age and so forth such that I Bo t fX where the larger the value of I the higher the probability of home ownership For each family a critical I threshold exists where if exceeded the house is purchased otherwise no home is purchased and the outcome probability D is assumed to be normally distributed such that P CDF I using a standard normal cumulative distribution function CDF Therefore using the estimated coefficients exactly like those of a regression model and using the Estimated Y value apply a standard normal distribution you can use Excel s NORMSDIST function or Risk Simulatot s Distributional Analysis tool by selecting Normal distribution and setting the mean to be 0 and standard deviation to be 7 Finally to obtain a Probit or probability unit measure set I 5 because whenever the probability P 0 5 the estimated I is negative due to the fact that the normal distribution is symmetrical around a mean of zero The Tobit Model Censored Tobit is an econometric and biometric modeling method used to desctibe the relationship between a non negative dependent variable Y and one or more independent variables X The dependent variable in a Tobit econometric model is censored it is censored because values below zero are not observed The Tobit mod
310. sis icon Menu Assumptions Forecasts Simulation Run Forecasting Icon RA M z E m i E A z um U Ls Ls op o Diagnostic Distribution Edit Fitting Hypothesis Nonparametric Overlay Seasonal Segment Sensitivity Scenario Statistical Structural Tornado Next Analysis Correlations e Testing Bootstrap Charts Test amp Cluster Analysis Analysis Analysis Break Test Analysis icon Analytical Tools d d 949 x Risk ROV ROV Check Create Data Trend Data Simulator BizStats Decision Tree Model Report Seasons Extraction Tool Menu ROV BizStats ROV Decision Tree Icon e HOES x Be Risk New Change SetInput Set Output Copy Paste Remove Run Run Super Reset Run Set Set Set Example Help User Simulator Profile Profile Assumption Forecast Speed Optimization Objective Decision Constraint Models Manual xd Menu Profile Assumptions Forecasts Editing Simulation Run Optimization Help Icon Figure 1 3 Risk Simulator Icon Toolbars in Excel 2007 2010 4 Page RISK SIMULATOR 1 4 What s New in Version 2012 The following lists the main capabilities of Risk Simulator where the highlighted items indicate the latest additions to version 2011 2012 1 4 1 General Capabilities 1 10 11 12 Available in 11 languages English French German Italian Japanese Korean Portuguese Russian Spanish Simplified Chinese and Traditional Chinese
311. ssive AR term The AR p model uses the lags of the time series in the equation An AR p mode has the form v a t y t 1 a p v t p e t The second component is the integration order term Each integration order corresponds to differencing the time series I 1 means differencing the data once I d means differencing the data d times The third component is the moving average term The MA q model uses the q lags of the forecast errors to improve the forecast An MA q mode has the form v e 0 b 1 e t 1 b q e t q Finally an ARMA p q model has the combined form y a 1 Wit 1 a p v e b 1 e t 12 b g e q The R Squared or Coefficient of Determination indicates the percent variation in the dependent variable that can be explained and accounted for by the independent variables in this regression analysis However in a multiple regression the Adjusted R Squared takes into account the existence of additional independent variables or regressors and adjusts this R Squared value to a more accurate view the regression s explanatory power However under some ARIMA modeling circumstances e g with nonconvergence models the R Squared tends to be unreliable The Multiple Correlation Coefficient Multiple R measures the correlation between the actua dependent variable Y and the estimated or fitted Y based on the regression equation This correlation is also the square root of the Coefficient
312. stribution because it can be constructed with two exponential distributions with an additional location parameter spliced together back to back creating an unusual peak in the middle The probability density function of the Laplace distribution is reminiscent of the normal distribution However whereas the normal distribution is expressed in terms of the squared difference from the mean the Laplace density is expressed in terms of the absolute difference from the mean making the Laplace distribution s tails fatter than those of the normal distribution When the location parameter is set to zero the Laplace distribution s random variable is exponentially distributed with an inverse of the scale parameter Alpha also known as location and Beta also known as scale are the distributional parameters The mathematical constructs for the Laplace distribution are shown below fo ool 25 B 1 5 exp when x lt amp Mean a Standard Deviation 1 4142 8 SkRewness is abvays equal to 0 as it is a symmetrical distribution Excess Kurtosis is always equal to 3 Input requirements Alpha Location can take on any positive or negative value including zero Beta Scale gt 0 The logistic distribution is commonly used to describe growth that is the size of a population expressed as a function of a time variable It also can be used to describe chemical reactions and the course of growth for a population or individual The mat
313. struction design and aircraft loads and tolerances The extreme value distribution is also known as the Gumbel distribution The mathematical constructs for the extreme value distribution ate as follows 6 where z e for 8 gt 0 and any value of x and 50 Page F Distribution or Fisher Snedecor Distribution RISK SIMULATOR 0 5772158 Standard Deviation 1 6 Skewness 12V 6 1 2020569 _ 1 13955 this applies for all values of mode and scale Exess Kurtosis 5 4 this applies for all values of mode and scale Mode a and scale 8 are the distributional parameters Calculating Parameters There are two standard parameters for the extreme value distribution mode and scale The mode parameter is the most likely value for the variable the highest point on the probability distribution After you select the mode parameter you can estimate the scale parameter The scale parameter is a number greater than 0 The larger the scale parameter the greater the variance Input requirements Mode Alpha can be any value Scale Beta gt 0 The F distribution also known as the Fisher Snedecor distribution is another continuous distribution used most frequently for hypothesis testing Specifically it is used to test the statistical difference between two variances in analysis of variance tests and likelihood ratio tests The F distribution with the numerator degree of freedom 7 and deno
314. sts Do the following to set new input assumptions in your model sure a Simulation Profile exists open an existing profile or start a new profile Risk Simulator New Simulation Profile 15 Page RISK SIMULATOR e Select the cell you wish to set an assumption on e g cell in the Basic Simulation Model example Click on Rak Simulator Set Input Assumption ot click on the set input assumption icon in the Risk Simulator icon toolbar Select the relevant distribution you want enter the relevant distribution parameters e g Triangular distribution with 1 2 2 5 as the minimum most likely and maximum values and hit OK to insert the input assumption into your model Figure 2 3 Assumption Properties Assumption Name Revenue 7 Minimum Triangular 1 E w Most Likely X 2 Maximum i a i T Regular Input Arcsine Bernoulli Percentile Input 7 Enable Data Boundary Triangular Distribution The triangular distribution describes a Minimum E situation where you know the minimum maximum and most likely values to occur uix For example you could describe the number of cars sold per week when past sales show the minimum maximum and _ toa t uM prospect olt Figure 2 3 Setting an Input Assumption Enable Dynamic Simulations Note that you can also set assumptions by selecting the cell you w
315. sulting forecast average is found to be 5 00 How certain is the analyst of the results Bootstrapping allows the user to ascertain the confidence interval of the calculated mean statistic indicating the distribution of the statistics Finally bootstrap results are important because according to the Law of Large Numbers and the Central Limit Theorem in statistics the mean of the sample means is an unbiased estimator and approaches the true population mean when the sample size increases 5 5 Hypothesis Testing A hypothesis test is performed when testing the means and variances of two distributions to determine if they are statistically identical or statistically different from one another that is whether the differences are based on random chance or if they are in fact statistically significant e Run a simulation e Select Risk Simulator Tools Hypothesis Testing e Select only o forecasts to test at a time select the type of hypothesis test you wish to run and click OK Figure 5 18 MODEL A MODEL B Revenue 200 00 Revenue 200 00 Cost Income To replicate this model start by creating a Simulati Simulation New Profile then set the random seg revenue cells and provide them a Normal distributio deviation of 20 select one of the revenue cell and c select Normal and enter the relevant parameters 1 each of the cost cells Finally define forecast outp the simulation F Income A Risk Simulator Forec
316. sus Global Color Node colors can be changed locally to a node or globally o Label Inside Shape Text can be placed inside the node you may need to make the node wider to accommodate longer text o Branch Event Name Text can be placed on the branch leading to the node to indicate the event leading to this node o Select Real Options A specific real option type can be assigned to the current node Assigning real options to nodes allows the tool to generate a list of required input variables Global Elements are all customizable including elements of the Strategy Tree s Background Connection Lines Option Nodes Terminal Nodes and Text Boxes For instance the following settings can be changed for each of the elements o Font settings on Name Value Notes Label Event names o Node Size minimum and maximum height and width o Borders line styles width and color o Shadow colors and whether to apply a shadow or not o Global Color o Global Shape e The Edit menu s View Data Requirements Window command opens a docked window on the right of the Strategy Tree such that when an option node or terminal node is selected the properties of that node will be displayed and can be updated directly This feature provides an alternative to double clicking on a node each time e Example Files are available in the File menu to help you get started on building Strategy Trees Protect File from the File menu allows the Strat
317. t 0 KurtosisXS 0 pa uo Ha Figure 2 23 Third Moment Right Skew The fourth moment or kurtosis measures the peakedness of a distribution Figure 2 24 illustrates this effect The background denoted by the dotted line is a normal distribution with a kurtosis of 3 0 or an excess kurtosis KurtosisXS of 0 0 Risk Simulator s results show the KurtosisXS value using 0 as the normal level of kurtosis which means that a negative KurtosisXS indicates flatter tails platykurtic distributions like the uniform distribution while positive values indicate fatter tails leptokurtic distributions like the student s t or lognormal distributions The distribution depicted by the bold line has a higher excess kurtosis thus the area under the curve is thicker at the tails with less area in the central body This condition has major impacts on risk analysis As shown for the two distributions in Figure 2 24 the first three moments mean standard deviation and skewness can be identical but the fourth moment kurtosis is different This condition means that although the returns and risks are identical the probabilities of extreme and catastrophic events potential large losses or large gains occurting are higher for a high kurtosis distribution e g stock market returns are leptokuttic or have high kurtosis Ignoring a project s kurtosis may be detrimental Typically a higher excess kurtosis value indicates that the downside risks a
318. t Period of choice e g 1 0 0 for PDQ and 5 for Forecast 4 Click OK to run ARIMA and review the ARIMA report for details of the results Auto ARIMA Auto ARIMA runs the most common low order PDQ combinations and finds the best fit using Adjusted R Squared Akaike and Schwarz Criterion and ranks them from best to worst Time Series Variable B5 B440 5 Exogenous Variable Maximum Iterations Forecast Periods Backcast ARIMA is an advanced modeling technique used to model and forecast time series data data that have a time component to it e g interest rates inflation sales revenues gross domestic product Time Series Variable B5 B440 Exogenous Variable Autoregressive Order Differencing Order I d Moving Average Order MA a Maximum Iterations lisp P E e8 es s Forecast Periods Backcast AUTO ARIMA Models Proper ARIMA modeling requires testing of the autoregressive and moving average of the errors on the time series data in order to calibrate the correct PDQ inputs Nonetheless you can use the AUTO ARIMA forecasts to automatically test all possible combinations of the most frequently occurring PDQ values to find the best fitting ARIMA model To do so following these steps 1 Risk Simulator Forecasting AUTO ARIMA 2 Click on the Time Series Variable link icon and select the area B5 B440 3 Click OK to run ARIMA and review the ARIMA repor
319. t and posterior probability updating expected value of information MINIMAX MAXIMIN tisk profiles 1 Page RISK SIMULATOR Real Options SLS software is used for computing simple and complex options and includes the ability to create customizable option models This software has the following modules e Single Asset SLS for solving abandonment chooser contraction deferment and expansion options as well as for solving customized options e Multiple Asset and Multiple Phase SLS for solving multiphase sequential options options with multiple underlying assets and phases combination of multiphase sequential with abandonment chooser contraction deferment expansion and switching options it can also be used to solve customized options e Multinomial SLS for solving trinomial mean reverting options quadranomial jump diffusion options and pentanomial rainbow options e Excel Add In Functions for solving all the above options plus closed form models and customized options in an Excel based environment 1 2 Installation Requirements and Procedures To install the software follow the on screen instructions The minimum requirements for this software are Pentium IV processor or later dual core recommended e Windows XP Vista Windows 7 Windows 8 or later e Microsoft Excel XP 2003 2007 2010 or later e Microsoft Framework 2 0 or later versions 3 0 3 5 and so forth e 500 MB free space e 2GB RAM minimum
320. t as the regressors regression estimates are biased and inconsistent but can be fixed using ARIMA ARIMA p d q models are the extension of the AR model that uses three components for modeling the serial correlation in the time series data The first component is the autoregressive AR term The AR p model uses the lags of the time series in the equation An AR p model has the form y The second component is the integration d order term Each integration order corresponds to differencing the time series I 1 means differencing the data once I d means differencing the data d times The third component is the moving average MA term The MA q model uses the q lags of the forecast errors to improve the forecast An MA q model has the form y e bie byes Finally an ARIMA p q model has the combined form y az yu ej bg tig 80 Page Procedure Notes Results Interpretation RISK SIMULATOR e Start Excel and enter your data or open an existing worksheet with historical data to forecast the illustration shown next uses the file example file Time Series ARIMA e Select the time series data and select Simulator Forecasting ARIMA e Enter the relevant D parameters positive integers only enter the number of forecast period desired and click OK For ARIMA and Auto ARIMA you can model and forecast future periods by either using
321. t for details of the results Real Options Valuation www realoptionsvaluation com Figute 3 15 AUTO ARIMA Module 3 9 Basic Economettics Econometrics refers to a branch of business analytics modeling and forecasting techniques for modeling the behavior or forecasting certain business or economic variables Running the Basic Econometrics models is similar to regular regression analysis except that the dependent and independent variables are allowed to be modified before a regression is run The report generated and its interpretation is the same as shown in the Multivariate Regression section presented earlier Start Excel and enter your data or open an existing worksheet with historical data to forecast the illustration shown in Figure 3 16 uses the file example file Advanced Forecasting Models in the Examples menu of Risk Simulator Select the data in the Basic Econometrics worksheet and select Risk Simulator Forecasting Basic Econometrics Enter the desired dependent and independent variables see Figure 3 16 for examples and click OK to run the model and report or click on Sow Results to view the results before generating the report in case you need to make any changes to the model 86 Page RISK SIMULATOR Basic Econometrics Basic Econometrics Data Set This tool is used to run basic econometric models by first transforming the input variables before the multivariate regression analysis Y
322. the CDF computes the probability of getting no more than 2 heads or up to 2 heads or probabilities of 0 1 and 2 heads Taking the complement ie 1 0 00021 obtains 0 999799 or 99 9799 which is the probability of getting at least 3 heads or more 148 Page RISK SIMULATOR Distribution Analysis This tool generates the probability density function PDF cumulative distribution function CDF and the Inverse CDF ICDF of all the distributions in Risk Simulator including theoretical moments and probability chart Distribution Trials Probability 0 000181 0 001087 0 004621 0 014786 0 036964 0 073929 0 120134 0 160179 0 176197 0 160179 0 120134 0 073929 0 036964 0 014786 0 004621 0 001087 0 000181 0 000019 0 000001 Figure 5 35 Distributional Analysis Tool Binomial Distribution with 20 Trials Using this Distributional Analysis tool in Risk Simulator even more advanced distributions can be analyzed such as the gamma beta negative binomial and many others As further example of the tool s use in a continuous distribution and the ICDF functionality Figure 5 37 shows the standard normal distribution normal distribution with a mean of zero and standard deviation of one where we apply the ICDF to find the value of x that corresponds to the cumulative probability of 97 50 CDF That is a one tail CDF of 97 50 is equivalent to a two tail 95 confidence interval there is a 2 50 probabi
323. the Monte Carlo simulation approach as repeatedly picking golf balls out of a large basket with replacement The size and shape of the basket depend on the distributional input assumption e g a normal distribution with a mean of 100 and a standard deviation of 10 versus a uniform distribution or a triangular distribution where some baskets are deeper or more symmetrical than others allowing certain balls to be pulled out more frequently than others The number of balls pulled repeatedly depends on the number of trials simulated For a large model with multiple related assumptions imagine a very large basket wherein many smaller baskets reside Each small basket has its own set of golf balls that are bouncing around Sometimes these small baskets are linked with each other if there is a correlation between the vatiables and the golf balls are bouncing in tandem while other times the balls are bouncing independently of one another The balls that ate picked each time from these interactions within the model the large central basket ate tabulated and recorded providing a forecast output result of the simulation 11 Page RISK SIMULATOR 2 2 Getting Started with Risk Simulator 2 2 1 A High Level Overview of the Software The Risk Simulator software has several different applications including Monte Carlo simulation forecasting optimization and risk analytics e Simulation Module allows you to run simulations in your existing Exce
324. their mean If the number of data points is small micronumerosity it may be difficult to detect assumption violations With small sample sizes assumption violations such as non normality or heteroskedasticity of variances are difficult to detect even when they are present With a small number of data points linear regression offers less protection against violation of assumptions With few data points it may be hard to determine how well the fitted line matches the data or whether a nonlinear function would be more appropriate Even if none of the test assumptions are violated a linear regression on a small number of data points may not have sufficient power to detect a significant difference between the slope and zero even if the slope is nonzero The power depends on the residual error the observed variation in the independent variable the selected significance alpha level of the test and the number of data points Power decreases as the residual variance increases decreases as the significance level is decreased as the test is made more stringent increases as the variation in observed independent variable increases and increases as the number of data points increases Values may not be identically distributed because of the presence of outliers which are anomalous values in the data Outliers may have a strong influence over the fitted slope and intercept giving a poor fit to the bulk of the data points Outliers tend to increase
325. then this approach should be used The Stochastic Optimization process in contrast is similar to the dynamic optimization procedure with the exception that the entire dynamic optimization process is repeated T times That is a simulation with N trials is run and then an optimization is run with M iterations to obtain the optimal results Then the process is replicated T times The results will be a forecast chart of each decision variable with T values In other words a simulation is run and the forecast or assumption statistics ate used in the optimization model to find the optimal allocation of decision variables Then another simulation is run generating different forecast statistics and these new updated values are then optimized and so forth Hence the final decision variables will each have their own forecast chart indicating the range of the optimal decision variables For instance instead of obtaining single point estimates in the dynamic optimization procedure you can now obtain a distribution of the decision variables and hence a tange of optimal values for each decision variable also known as a stochastic optimization Finally an Efficient Frontier optimization procedure applies the concepts of marginal increments and shadow pricing in optimization That is what would happen to the results of the optimization if one of the constraints were relaxed slightly Say for instance the budget constraint is set at 1 million What wou
326. tion the resulting statistics are displayed but the accuracy of such statistics and their statistical significance are sometimes in question For instance if a simulation run s skewness statistic is 0 10 is this distribution truly negatively skewed or is the slight negative value attributable to random chance What about 0 15 0 20 and so forth That is how far is far enough such that this distribution is considered to be 131 Page RISK SIMULATOR negatively skewed The same question can be applied to all the other statistics Is one distribution statistically identical to another distribution with regard to some computed statistics or are they significantly different Suppose for instance the 90 confidence for the skewness statistic is between 0 0189 and 0 0952 such that the value O falls within this confidence indicating that on a 90 confidence the skewness of this forecast is not statistically significantly different from 0 or that this distribution can be considered as symmetrical and not skewed Conversely if the value 0 falls outside of this confidence then the opposite is true and the distribution is skewed positively skewed if the forecast statistic is positive and negatively skewed if the forecast statistic is negative Figure 5 17 illustrates some sample bootstrap results 8 Xx Standard Deviation X 100 1032 101 1032 5 i 21 9547 22 4547 229547
327. tion to find the best fitting distribution Probability Distributions Charts and Tables run 45 probability distributions their four moments CDF ICDF PDF charts and overlay multiple distributional charts and generate probability distribution tables Statistical Analysis descriptive statistics distributional fitting histograms charts nonlinear extrapolation normality test stochastic parameters estimation time series forecasting trendline projections etc ROV BIZSTATS over 130 business statistics and analytical models Absolute Values ANOVA Randomized Blocks Multiple Treatments ANOVA Single Factor Multiple Treatments ANOVA Two Way Analysis ARIMA Auto ARIMA Autocorrelation and Partial Autocorrelation Autoeconometrics Detailed Autoeconometrics Quick Average Combinatorial Fuzzy Logic Forecasting Control Chart C Control Chart NP Control Chart P Control Chart R Control Chart U Control Chart X Control Chart XMR Correlation Correlation Linear Nonlinear Count Covariance Cubic Spline Custom Econometric Model Data Descriptive Statistics Deseasonalize Difference Distributional Fitting Exponential J Curve GARCH Heteroskedasticity Lag Lead Limited Dependent Variables Logit Limited Dependent Variables Probit Limited Dependent Variables Tobit Linear Interpolation Linear Regression LN Log Logistic S Curve Markov Chain Max Median Min Mode Neural Network Nonlinear Regression
328. tions and the resulting simulated values have the same correlations 28 Page RISK SIMULATOR Price Quantity Price Quantity Positive Positive Negative Negative Correlation Correlation Correlation Correlation 1 95 0 91 1 89 1 06 1 92 0 95 1 98 1 05 2 02 1 04 Pearson s Correlation 1 89 1 09 Pearson s Correlation 2 04 1 03 1 88 1 04 1 89 0 91 0 80 1 96 0 93 0 80 1 98 1 05 2 02 0 93 2 05 1 03 2 00 1 02 1 87 0 91 1 86 1 04 1 84 0 91 1 96 1 02 2 06 1 03 1 90 1 02 1 98 1 01 1 92 1 10 Figure 2 16 Correlations Recovered 2 3 4 Precision and Error Control One vety powerful tool in Monte Carlo simulation is that of precision control For instance how many trials are considered sufficient to run in a complex model Precision control takes the guesswork out of estimating the relevant number of trials by allowing the simulation to stop if the level of prespecified precision is reached The precision control functionality lets you set how precise you want your forecast to be Generally speaking as more trials are calculated the confidence interval narrows and the statistics become more accurate The precision control feature in Risk Simulator uses the characteristic of confidence intervals to determine when a specified accuracy of a statistic has been reached For each forecast you can set the specific confidence interval for the precision level Make sure that you do not confuse three very different terms error precision and confide
329. tions than genetic algorithms given the same amount of computation time Therefore it is recommended that you first run the Genetic Algorithm and then rerun it by selecting the Apply Gradient Search Test option Figure 5 60 to check the robustness of the model This gradient search test will attempt to run combinations of traditional optimization techniques with Genetic Algorithm methods and return the best possible solution Finally unless there is a specific theoretical need to use Genetic Algorithm we recommend using Risk Simulator s Optimization module which allows you to run more advanced risk based dynamic and stochastic optimization routines for more robust results Objective Cell 8 Maximize Minimize Variables Add ColumnCell ColumnMin Column Max Figure 5 60 Genetic Algorithm 173 RISK SIMULATOR 5 27 ROV Decision Tree Module 5 27 1 Decision Tree Risk Simulator ROV Decision Tree runs the Decision Tree module Figure 5 61 ROV Decision Tree is used to create and value decision tree models Additional advanced methodologies and analytics ate also included Decision Tree Models Monte Carlo risk simulation Sensitivity Analysis Scenario Analysis Bayesian Joint and Posterior Probability Updating Expected Value of Information MINIMAX MAXIMIN Risk Profiles The following are some main quick getting started tips and procedures for using this intuitive tool There are 11
330. to worksheet cells is useful when the input parameters need to be visible or are allowed to be changed click on the link icon to link an input parameter to a worksheet cell e Enable Data Boundary These are typically not used by the average analyst but exist for truncating the distributional assumptions For instance if a normal distribution is selected the theoretical boundaries are between negative infinity and positive infinity However in practice the simulated variable exists only within some smaller range and this range can then be entered to truncate the distribution appropriately Correlations Pairwise correlations can be assigned to input assumptions here If correlations are required remember to check the Turn on Correlations preference by clicking on Simulator Edit Simulation Profile See the discussion on correlations later in this chapter for more details about assigning correlations and the effects correlations will have on a model Notice that you can either truncate a distribution or correlate it to another assumption but not both e Short Descriptions These exist for each of the distributions in the gallery The short descriptions explain when a certain distribution is used as well as the input parameter requirements See the section in Understanding Probability Distributions for Monte Carlo Simulation for details on each distribution type available in the software e Regular Input and Percentile Input Thi
331. tochastic optimizations 4 1 Optimization Methodologies Many algorithms exist to run optimization and many different procedures exist when optimization is coupled with Monte Carlo simulation In Risk Simulator there are three distinct optimization procedures and optimization types as well as different decision variable types For instance Risk Simulator can handle Continuous Decision Variables 1 2535 0 2215 etc as well as Integers Decision Variables 1 2 3 4 etc Binary Decision Variables 1 and 0 for go and no go decisions and Mixed Decision Variables both integers and continuous variables On top of that Risk Simulator can handle Linear Optimization i e when both the objective and constraints are all linear equations and functions as well as Nonlinear Optimizations i e when the objective and constraints are a mixture of linear and nonlinear functions and equations As far as the optimization process is concerned Risk Simulator can be used to run a Static Optimization that is an optimization that is run on a static model where no simulations ate run In other wotds all the inputs in the model are static and unchanging This optimization type is applicable when the model is assumed to be known and no uncertainties exist Also a discrete optimization can be first run to determine the optimal portfolio and its corresponding optimal allocation of decision variables before more advanced optimization procedures are applied F
332. tor provides the results of all four moments in its Statistics view in the forecast charts The first moment of a distribution measures the expected rate of return on a particular project It measures the location of the project s scenarios and possible outcomes on average The common statistics for the first moment include the mean average median center of a distribution and mode most commonly occurring value Figure 2 19 illustrates the first moment where in this case the first moment of this distribution is measured by the mean ot average value pa pa Figure 2 19 First Moment The second moment measures the spread of a distribution which is a measure of risk The spread or width of a distribution measures the variability of a variable that is the potential that the variable can fall into different regions of the distribution in other words the potential scenarios of outcomes Figure 2 20 illustrates two distributions with identical first moments identical means but very different second moments or risks The visualization becomes clearer in Figure 2 21 As an example suppose there are two stocks and the first stock s movements illustrated by the darker line with the smaller fluctuation is compared against the second stock s movements illustrated by the dotted line with a much higher price fluctuation Clearly an investor would view the stock with the wilder fluctuation as riskier because the
333. trapolation is fairly reliable relatively simple and inexpensive However extrapolation which assumes that recent and historical trends will continue produces large forecast errors if discontinuities occur within the projected time period That is pure extrapolation of time series assumes that all we need to know is contained in the historical values of the series that is being forecasted If we assume that past behavior is a good predictor of future behavior extrapolation is appealing This makes it a useful approach when all that is needed are many short term forecasts This methodology estimates the f x function for any arbitrary x value by interpolating a smooth nonlinear curve through all the x values and using this smooth curve extrapolates future x values beyond the historical data set The methodology employs either the polynomial functional form or the rational functional form a ratio of two polynomials Typically a polynomial functional form is sufficient for well behaved data however rational functional forms are sometimes more accurate especially with polar functions i e functions with denominators approaching zero Period Actual Forecast Fit Estimate Error Error Measurements 1 1 00 RMSE 19 6799 2 6 73 1 00 MSE 387 2974 3 20 52 1 42 8 15 MAD 10 2095 4 45 25 99 82 119 36 MAPE 31 56 5 83 59 55 92 46 67 Theil s U 1 1210 6 138 01 136 71 14 39 7 210 87 211 96 1 69 Function Type Rational 8 304 44 304 43 0 41 9 420 89
334. tribution is the most widely used distribution in hypothesis test This distribution is used to estimate the mean of a normally distributed population when the sample size is small to test the statistical significance of the difference between two sample means or confidence intervals for small sample sizes The mathematical constructs for the t distribution are as follows T r 1 2 Xrz TY r 2 Mean 0 this applies to all degrees of freedom r except if the distribution is shifted to another nonzero central location 1 t ry E Standard Deviation r 2 SRewness 0 this applies to all degrees of freedom r Excess Kurtosis forall r gt 4 r 4 X where t and 7 is the gamma function Degrees of freedom ris the only distributional parameter The t distribution is related to the F distribution as follows the square of a value of with r degrees of freedom is distributed as with 1 and r degrees of freedom The overall shape of the probability density function of the t distribution also resembles the bell shape of a normally distributed variable with mean 0 and variance 1 except that it is a bit lower and wider or is leptokurtic fat tails at the ends and peaked center As the number of degrees of freedom grows say above 30 the t distribution approaches the normal distribution with mean 0 and variance 1 Input requirements Degrees of freedom 2 1 and must be an integer 60 Pa
335. tric Mean 281 3247 Lower Confidence Interval for Standard Deviation 148 6090 Trimmed Mean 325 1739 Upper Confidence Interval for Standard Deviation 207 7947 Standard Error of Arithmetic Mean 24 4537 Variance Sample 29899 2588 Lower Confidence Interval for Mean 283 0125 Variance Population 29301 2736 Upper Confidence Interval for Mean 380 8275 Coefficient of Variability 0 5210 Median 307 0000 First Quartile Q1 204 0000 Mode 47 0000 Third Quartile Q3 441 0000 Minimum 764 0000 Inter Quartile Range 237 0000 Maximum 717 0000 Skewness 0 4838 Range Kurtosis 0 0952 Figure 5 30 Sample Statistical Analysis Tool Report 145 Page RISK SIMULATOR Hypothesis Test t Test on the Population Mean of One Variable Statistical Summary Statistics from Dataset Calculated Statistics Observations 50 t Statistic 13 5734 Sample Mean 331 92 P Value right tail 0 0000 Sample Standard Deviation 17291 P Value leftailed 1 0000 P Value two tailed 0 0000 User Provided Statistics Null Hypothesis Ho Hypothesized Mean Hypothesized Mean 0 00 Alternate Hypothesis Ha lt gt Hypothesized Mean Notes gt denotes greater than for right tail less than for left tail or not equal to for two tail hypothesis tests Hypothesis Testing Summary The one variable t test is appropriate when the population standard deviation is not known but the sampling distribution is assumed to be approximately normal the ttest is use
336. ts Probability of success gt 0 and lt 1 ie 0 0001 lt p lt 0 9999 Number of trials 2 1 or positive integers lt 1000 for larger trials use the normal distribution with the relevant computed binomial mean and standard deviation as the normal distribution s parameters The discrete uniform distribution is also known as the equally likely outcomes distribution where the distribution has a set of IN elements and each element has the same probability This distribution is related to the uniform distribution but its elements are discrete and not continuous The mathematical constructs for the discrete uniform distribution are as follows 1 N N 1 2 ranked value D N 1 Standard Deviation 12 ranked value SkRewness 0 Le the distribution is perfectly symmetrical 6 N 1 Excess Kurtosis NDN D ranged value Input requirements Minimum lt maximum and both must be integers negative integers and zero are allowed 39 Page Geometric Distribution Hypergeometric Distribution RISK SIMULATOR The geometric distribution describes the number of trials until the first successful occurrence such as the number of times you need to spin a roulette wheel before you win Conditions The three conditions underlying the geometric distribution are e The number of trials is not fixed trials continue until the first success e probability of success is th
337. ure e Select the data you wish to analyze e g B9 B28 and click on Rak Simulator Tools Seasonality Test Data Seasonality Test e Enter in the maximum seasonality period to test That is if you enter 6 the tool will test the following seasonality periods 1 2 3 4 5 and 6 Period 1 of course imply no seasonality in the data e Review the report generated for more details on the methodology application and resulting charts and seasonality test results The best seasonality periodicity is listed first ranked by the lowest RMSE error measure and all the relevant error measurements are included for comparison root mean squared error RMSE mean squared error MSE mean absolute deviation MAD and mean absolute percentage error MAPE A J K L M N 8 Procedure for Deseasonalizing and Detrending 1 Select the data you wish to analyze e g B9 B28 and click on Risk Simulator Tools Data ng Hn seasonality and trend to show only the absolute changes in values Deseasonalization and Detrending and to allow potential cyclical patterns to be identified after TS Ld endoncy acis bends onid 2 Select Deseasonalize Data and or Detrend Data seasonal cycles of a set of time series data select any detrending models you wish to run Dem jon 59828 and enter in the relevant orders e g Polynomial order moving average order difference order Deseasonalize Data and rate order and click OK Number
338. utcomes that are not deterministic in nature that is an equation or process that does not follow any simple discernible rule such as price will increase X percent evety year or revenues will increase by this factor of X plus Y percent A stochastic process is by definition nondeterministic and one can plug numbers into a stochastic process equation and obtain different results every time For instance the path of a stock price is stochastic in nature and one cannot reliably predict the stock price path with any certainty However the price evolution over time is enveloped in a process that generates these prices The process is fixed and predetermined but the outcomes are not Hence by stochastic simulation we create multiple pathways of prices obtain a statistical sampling of these simulations and make inferences on the potential pathways that the actual price may undertake given the nature and parameters of the stochastic process used to generate the time seties Three basic stochastic processes are included in Risk Simulator s Forecasting tool including geometric Brownian motion or random walk which is the most common and prevalently used process due to its simplicity and wide ranging applications The other two stochastic processes ate the mean reversion process and the jump diffusion process The interesting thing about stochastic process simulation is that historical data ate not necessarily required That is the model does not have to f
339. ution as it truncates the data set to ignore outliers The Inter Quartile Range is the difference between the third and first quartiles and is often used to measure the width ofthe center of a distribution Skewness is the third momentin a distribution Skewness characterizes the degree of asymmetry of a distribution around its mean Positive skewness indicates a distribution with an asymmetric tail extending toward more positive values Negative skewness indicates a distribution with an asymmetric tail extending toward more negative values Kurtosis characterizes the relative peakedness or flatness of a distribution compared to the normal distribution It is the fourth moment in a distribution A positive Kurtosis value indicates a relatively peaked distribution A negative kurtosis indicates a relatively flat distribution The Kurtosis measured here has been centered to zero certain other kurtosis measures are centered around 3 0 While both are equally valid centering across zero makes the interpretation simpler A high positive Kurtosis indicates a peaked distribution around its center and leptokurtic or fat tails This indicates a higher probability of extreme events e g catastrophic events terrorist attacks stock market crashes than is predicted in a normal distribution Summary Statistics Statistics Variable X1 Observations 50 0000 Standard Deviation Sample 172 9140 Arithmetic Mean 331 9200 Standard Deviation Population 171 1761 Geome
340. variables a Y or dependent variable and X or independent variable where X is also known as the regressor sometimes a bivariate regression is also known as a univariate regression as there is only a single independent variable X The dependent variable is so named because it depends on the independent variable for example sales revenue depends on the amount of marketing costs expended on a product s advertising and promotion making the dependent variable sales and the independent variable marketing costs An example of a bivariate regression is seen as simply inserting the best fitting line through a set of data points in a two dimensional plane as seen on the left panel in Figure 3 6 In other cases a multivariate regression can be performed where there are multiple or 7 number of independent X variables where the general regression equation will now take the form of Y f B X B X B X B X In this case the best fitting line will be within an 1 dimensional plane Figure 3 6 Bivariate Regression However fitting a line through a set of data points in a scatter plot as in Figure 3 6 may result in numerous possible lines The best fitting line is defined as the single unique line that minimizes the total vertical errors that is the sum of the absolute distances between the actual data points Y and the estimated line Y as shown on the right panel of Figure 3 6 To find the best fitting line that minimi
341. ve stock returns approach In Figure 4 9 column E Allocation Weights holds the decision variables which are the variables that need to be tweaked and tested such that the total weight is constrained at 100 cell E11 Typically to start the optimization we set these cells to a uniform value In this case cells E6 to E9 are set at 25 each In addition each decision variable may have specific restrictions in its allowed range In this example the lower and upper allocations allowed are 10 and 40 as seen in columns F and G This setting means that each asset class may have its own allocation boundaries Next column H shows the return to risk ratio which is simply the return percentage divided by the risk percentage for each asset where the higher this value the higher the bang for the buck The remaining parts of the model show the individual asset class rankings by returns risk return to risk ratio and allocation In other words these rankings show at a glance which asset class has the lowest risk or the highest return and so forth 111 Page Procedure alw Nne i3i3igieje e wo RISK SIMULATOR B C D E G H ASSET ALLOCATION OPTIMIZATION MODEL Required Required Asset Class Annualized Volatility Allocation Minimum Maximum Return to Description Returns Risk Weights Risk Ratio Allocation Allocation Asset 1 10 60 12 41 2 10 00 40 00 0 8544 Asset 2 11 21 16 16 10 00 40 00 0 6937 Asset 3 10 61
342. view multiple distributions use Risk Simulator s Overlay Chart tool Distribution Charts and Tables Change First Parameter Change Second Parameter Random x 02 05 From From To Series Custom To From To Step Chart Simulated Distributi Trials 1000 Seed 123 0 0716 0 0123 0 0040 0 0011 Figure 5 51 ROV Probability Distribution Distribution Tables 164 Page Procedure Notes RISK SIMULATOR 5 22 ROV BizStats This new ROV BizStats tool is a very powerful and fast module in Risk Simulator that is used for running business statistics and analytical models on your data It covers more than 130 business statistics and analytical models Figures 5 52 through 5 55 The following provides a few quick getting started steps on running the module and details on each of the elements in the software Run ROV BizStats at Simulator ROV BixSfats and click on Example to load a sample data and model profile A or type in your data or copy paste into the data grid D Figure 5 52 You can add your own notes or variable names in the first Notes row C e Select the relevant model F to run in Step 2 and using the example data input settings G enter in the relevant variables H Separate variables for the same parameter using semicolons and use a new line hit Enter to create a new line fo
343. weighted by the beta parameter and a seasonality component S weighted by the gamma parameter Several methods exist but the two most common are the Holt Winters additive seasonality and Holt Winters multiplicative seasonality methods In the Holt Winter s additive model the base case level seasonality and trend are added together to obtain the forecast fit The best fitting test for the moving average forecast uses the root mean squared errors RMSE The RMSE calculates the square root of the average squared deviations of the fitted values versus the actual data points Mean Squared Error MSE is an absolute error measure that squares the errors the difference between the actual historical data and the forecast fitted data predicted by the model to keep the positive and negative errors from canceling each other out This measure also tends to exaggerate large errors by weighting the large errors more heavily than smaller errors by squaring them which can help when comparing different time series models Root Mean Square Error RMSE is the square root of MSE and is the most popular error measure also known as the quadratic loss function RMSE can be defined as the average of the absolute values of the forecast errors and is highly appropriate when the cost of the forecast errors is proportional to the absolute size of the forecast error The RMSE is used as the selection criteria for the best fitting time series model Mean Absolute Percentage Er
344. y Standard Deviation of Errors 141 83 219 04 0 02 0 02 0 0612 0 0412 D Statistic 0 1036 202 53 0 02 0 04 0 0766 0 0366 D Critica at 196 0 1138 186 04 0 02 0 06 0 0948 0 0348 D Critica at 596 0 1225 174 17 0 02 0 08 0 1097 0 0297 D Critical at 10 0 1458 162 13 0 02 0 10 0 1265 0 0265 Nuli Hypothesis The errors are normally distributed 161 62 0 02 0 12 0 1272 0 0072 160 39 0 02 0 14 0 1291 0 0109 Conclusion The errors are normally distributed at the 145 40 0 02 0 16 0 1526 0 0074 1 alpha level 138 92 0 02 0 18 0 1637 0 0163 133 81 0 02 0 20 0 1727 0 0273 120 76 0 02 0 22 0 1973 0 0227 120 12 0 02 0 24 0 1985 0 0415 Figure 5 25 Test for Normality of Errors Sometimes certain types of time series data cannot be modeled using any other methods except for a stochastic process because the underlying events are stochastic in nature For instance you cannot adequately model and forecast stock prices interest rates price of oil and other commodity prices using a simple regression model because these variables are highly uncertain and volatile and they do not follow a predefined static rule of behavior in other words the process is not stationary Stationatity is checked using the Runs Test function while another visual clue is found in the autocorrelation report the ACF tends to decay slowly A stochastic process is a sequence of events or paths generated by probabilistic laws That is random events can occur over tim
345. y GJR TGARCH Runs the current analysis in Step 2 or selected Volatility Log Returns Approach saved analysis in Step 4 view the results charts Volatility T6ARCH and statistics copy the results and charts to Volatility TGARCH M or generate reports Yield Curve Bliss Statistics STEP 4 Save Optional You can save multiple analyses and notes in the profile for future retrieval vvvvvvyv Figure 5 53 ROV BizStats Data Visualization and Results Charts 167 Page RISK SIMULATOR File Data Language Help STEP 1 Data Manually enter your data paste from another application STEP 2 Analysis Choose an analysis and enter the or load an example dataset with analysis parameters required see example Wen Parameter inputs below 1 060 ANOVARandomizedBlocksMultipleTreatments VAR60 VAR61 VAR62 Stepwise Regression Backward VARG VAR7 VARS VARS VA VAR63 Stepwise Regression Correlation 2 062 ANOVATwoWayAnalysis VAR40 1 VAR42 VAR43 VAR45 Stepwise Regression Forward VAR46 VAR47 VAR48 VAR49 VAR50 VAR51 3 Regression in 3 061 ANOVASingleFactorMultipleTreatments VAR57 VAR58 59 Stochastic Process Exponential Brownian M 4 4 060 ANOVARandomizedBlocksMultipleTreatments VARGO VARG 1 VAR62 Stochastic Pr
346. y for reference Alpha captures the memory effect of the base level changes over time and beta is the trend parameter that measures the strength of the trend while gamma measures the seasonality strength of the historical data The analysis decomposes the historical data into these three elements and then recomposes them to forecast the future The fitted data illustrates the historical data and it uses the recomposed model and shows how close the forecasts are in the past a technique called backcasting The forecast values are either single point estimates or assumptions if the option to automatically generate assumptions is chosen and if a simulation profile exists The graph illustrates these historical fitted and forecast values The chart is a powerful communication and visual tool to see how good the forecast model is 69 Page Notes Historical Sales Revenues 2006 2006 2006 2006 2007 2007 2007 2007 2008 2008 2008 2008 2009 2009 2009 2009 2010 2010 2010 2010 RISK SIMULATOR Time Series Forecast Time Series Analysis is used to forecast time series variables by decomposing the historical data into 1 baseline trend and seasonality elements and replicating Sales these elements into the future forecasts This analysis assumes that the trend and seasonality will persist Quarter Period Auto Mode Selection Single Moving Average m Model Parameters Optimize 05
347. y and trend Por instance if the data variable has no trend or seasonality then a single moving average model or a single exponential smoothing model would suffice However if seasonality exists but no discernible trend is present either a seasonal additive or seasonal multiplicative model would be better and so forth No Seasonality With Seasonality 5 Single Moving Average d H 2 Single Exponential Seasonal Smoothing Multiplicative Double Moving Holt Winter s Average Additive Lr Double Exponential Holt Winter s z Smoothing Multiplicative Figure 3 3 The Eight Most Common Time Series Methods e Start Excel and open your historical data if required the example below uses the Time Series Forecasting file in the examples folder Select the historical data data should be listed in a single column e Select Rak Simulator Forecasting Time Series Analysis Choose the model to apply enter the relevant assumptions Figure 3 4 and click OK Figure 3 5 illustrates the sample results generated by using the Forecasting tool and a Holt Winter s multiplicative model The model fitting and forecast chart indicates that the trend and seasonality are picked up nicely by the Holt Winter s multiplicative model The time series analysis report provides the relevant optimized alpha beta and gamma parameters the error measurements fitted data forecast values and fitted forecast graph The parameters are simpl
348. y values cannot become negative Conditions The three conditions underlying the lognormal distribution are The uncertain variable can increase without limits but cannot fall below zero e The uncertain variable is positively skewed with most of the values near the lower limit natural logarithm of the uncertain variable yields a normal distribution Generally if the coefficient of variability is greater than 30 use a lognormal distribution Otherwise use the normal distribution The mathematical constructs for the lognormal distribution are as follows 1 2 e OF gt 0 gt gt 0 xJ2z In o 2 Mean exp Standard Deviation 2u Jexp o E Skewness 2 exp o Exvess Kurtosis expl4o 2 exp3c 3 expo 6 Mean and standard deviation 0 are the distributional parameters Input requirements Mean and standard deviation both gt 0 and can be any positive value 54 Page Lognormal 3 Distribution Normal Distribution RISK SIMULATOR Lognormal Parameter Sets By default the lognormal distribution uses the arithmetic mean and standard deviation For applications for which historical data are available it is more appropriate to use either the logarithmic mean and standard deviation or the geometric mean and standard deviation The Lognormal 3 distribution uses the same constructs as the origin
349. zes the errors a more sophisticated approach is required that is regression analysis Regression analysis therefore finds the unique best fitting line by requiring that the total errors be minimized or by calculating n Min Y Y i l where only one unique line minimizes this sum of squared errors The errors vertical distance between the actual data and the predicted line are squared to avoid the negative errors 72 Page Procedure RISK SIMULATOR canceling out the positive errors Solving this minimization problem with respect to the slope and intercept requires calculating a first derivative and setting them equal to zero d n d n 5 Y 0 and Y Y P 0 72 which yields the bivariate regression s least squares equations n n Sx Sy T3x X Y Yxy 5 5 iz a a n D Dix X x B yr 2 i l For multivariate regression the analogy is expanded to account for multiple independent variables where Y 8 B X B X and the estimated slopes can be calculated by A X DY Xs gt Xx x e DYXY 2 X In running multivariate regressions great care has to be taken to set up and interpret the results For instance a good understanding of econometric modeling is required e g identifying regression pitfalls such as structural breaks multicollinearity heteroskedasticity autocorrelation specific

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