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1. Figure 5 48 Probability Distribution Tool 45 Probability Distributions 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 rc 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 User Manual Risk Simulator Software 180 2005 2011 Real Options Valuation Inc types gridlines language and decimal settings M and try rerunning the distribution using theoretical versus empirically simulated values N amp 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 optio
2. Figure 5 50 ROV Probability Distribution Multiple Overlay Charts ROV PROBABILITY DISTRIBUTIONS ese Distributions 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 view multiple distributions use Risk Simulator s Overlay Chart tool Distribution Binomial gt Charts and Tables Chart Change First Parameter Change Second Parameter Theoretical Distribution Tiis 20 PDF SEN er Probabiity v Random x v Simulated Distribution poste 05 es fom 02 fom o Tiss 1000 ae cane 10s From ToSees O To 05 To 8 Seed 123 0 _ Resut Custom Step 005 Step ET E Eo 0 176197 z Run Table J 2 5 5 EE e Copy Chat Table P Row Variable Probability Column Variable Random X Type PDF rooe a Ca Cee Figure 5 51 ROV Probability Distribution Distribution Tables User Manual Risk Simulator Software 182 2005 2011 Real Options Valuation Inc 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 q
3. Input requirements Mean gt 0 Standard Deviation gt 0 Location can be any positive or negative value including zero Normal Distribution 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 The 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 1 f x e 7 forall values of x and u while o gt 0 V210 Mean u Standard Deviation o Skewness 0 this applies to all inputs of mean and standard deviation Excess Kurtosis 0 this applies to all inputs of mean and standard deviation User Manual Risk Simulator Software 67 2005 2011 Real Options Valuation Inc Mean u and standard deviation o are the distributional parameters Input requirements Standard deviation gt 0 and can be any positive value Mean can take on any value Parabolic Distribution The parabol
4. Options License About Risk Simulator Help Figure 1 1A Risk Simulator Menu and Icon Bar in Excel XP and Excel 2003 User Manual Risk Simulator Software 10 2005 2011 Real Options Valuation Inc Home Insert Page Layout Aix E d Formulas Data Review View Developer Risk Simulator Lor KBoebe amp Set Objective D Set Decision New Change Edit SetInput Set Output Copy Paste Remove Run RunSuper Step Reset Forecasting Analytical Options Help license Next Profile Profile Profile Assumption Forecast Speed X Optimization G Set Constraint Tools X X icon E New Simulation Profile ssumptions Forecasts Editing Simulation Run Forecasting Optimization Tools Options Help License _Icon l Edit Simulation Profile fe 2 Change Simulation Profile D E F G K L M N o P a R S Set Input Assumption Set Output Forecast oc Re Copy Parameter J Remove Parameter Close All Charts Minimize All Charts Run Simulation Step Simulation Run Super Speed Simulation a Reset Simulation Example Models d Forecasting d Optimization gt ROV BizStats z Options Languages gt Se License e e About Risk Simulator xy PRR PRRR HOO Check for Updates Resources gt fE Hands on Exercises S Probability Distribution Details s ea Bo User Manual Help ka ET 32 A 33 ro 34 35 36 37 38 39 a a m User Manual Risk Simula
5. 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 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 User Manual Risk Simulator Software 168 2005 2011 Real Options Valuation Inc Figure 5 39 Scenario Analysis Table User Manual Risk Simulator Software 169 2005 2011 Real Options Valuation Inc Rial B c D E F M oo ee d k m 24 Discounted Cash Flow ROI Model 3 4 Base Year 2009 Sum PV Net Benefits 4 762 09 Discount Type Discrete End of Year Discounting v 5 Start Year 2009 Sum PV Investments 1 634 22 EZ Market Risk Adjusted Discount Rate 15 00 Net Present Value 87 Model 1 Include Termina
6. Figure 2 17 Setting the Forecast s Precision Level User Manual Risk Simulator Software 38 2005 2011 Real Options Valuation Inc Income Risk Simulator Forecast Standard Deviation Variance Coefficient of Variation Maximum 25 Percentile 75 Percentile Percentage Eror Precision at 95 Confidence Figure 2 18 Computing the Error User Manual Risk Simulator Software 39 2005 2011 Real Options Valuation Inc 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 Simulator provides the results of all four moments in its Statistics view in the forecast charts Measuring the Center of the Distribution the First Moment 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 occ
7. Single Variable Distributional Fitting Statistical Summary Fitted Assumption 99 14 Fitted Distribution Normal Distribution Mu 99 28 Sigma 10 17 Koimogorov Smirnov Statistic 0 03 P Value 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 21 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 48 113 50 101 40 104 72 102 43 113 59 124 15 109 24 105 34 104 57 97 83 94 39 116 19 84 66 101 17 106 13 107 17 95 83 106 67 92 42 79 64 94 15 106 00 113 45 92 63 94 51 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 91 114 95 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 74 45 102 24 103 34 96 51 114 55 93 94 106 29 102 95 112 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
8. 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 Cauchy Distribution or Lorentzian or Breit Wigner Distribution 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 _l y 2 a x m y 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 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
9. Tools Data Seasonality Test 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 implies no seasonality in the data amp 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 This tool de seasonalizes and de trends your original data to take 1 Select the data you wish to analyze e g B9 B28 out any seasonal and trending components In forecasting models and click on Risk Simulator Tools Data the process of removing the effects of accumulating data sets from mig 3 seasonality and trend to show only the absolute changes in values Deseasonalization and Detrending and to allow potential cyclical patterns to be identified after a mad pase an raat DAME e of 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 Data Location B9528 and enter M the relevant ardei e g Polynomial order moving aver
10. Update Chart 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 Time Mean Stdev but are governed by specific statistical and probabilistic rules The main stochastic processes include Random Walk or Brownian 0 0000 100 00 0 00 Motion Mean Reversion and Jump Diffusion These processes can be used to forecast a multitude of variables that seemingly 0 1000 106 32 4 05 follow random trends but yet are restricted by probabilistic laws 0 2000 105 92 4 70 g N De 0 3000 105 23 8 23 The Random Walk Brownian Motion process can be used to forecast stock prices prices of commodities and other stochastic time 0 4000 109 84 11 18 series data given a drift or growth rate and a volatility around the drift path The Mean Reversion process can be used to reduce 0 5000 107 57 14 67 the fluctuations of the Random Walk process by allowing the path to target a long term value making it useful for forecasting time 0 6000 108 63 19 79 series variables that have a long term rate such as interest rates and inflation rates these are long term target rates by regulatory 2 i i i Se s EDAR i 0 7000 107 85 24 18 authorities or the market The Jump Diffusion process is useful for forecasting time series data when the variable can occasionally 0 8000 109 61 244
11. this click on Risk Simulator Run Super Speed Simulation or use the run super speed icon Notice how much faster the super speed simulation runs In fact for practice Reset Simulation and then Edit User Manual Risk Simulator Software 26 2005 2011 Real Options Valuation Inc 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 5 Interpreting the Forecast Results 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 Forecast Chart 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 occurring out of the total number of trials while the cumulative frequency smooth line shows the total probabilit
12. 139 70 290 10 292 30 data and then forecasting the future Advanced knowledge of Time Series Variable jeseo 140 70 29230 294 50 econometrics is required to properly model ARIMA Please 141 20 293 90 296 10 see the ARIMA example Excel model for more details However Exogenous Variable Le 141 70 29530 297 40 to get started quickly following the instructions below Autoregressive Order AR p 118 141 90 296 40 298 50 141 00 296 50 298 50 1 Risk Simulator Forecasting ARIMA Bierencing nis Ma 08 14050 29660 298 60 2 Click on the Time Series Variable link Moving Average Order MA q oe 140 40 297 20 299 20 icon and select the area B5 B440 aaae 106 140 00 297 80 299 80 3 Try different P D Q values and 140 00 298 30 300 30 select a Forecast Period of choice Forecast Periods 139 90 298 50 300 50 e g 1 0 0 for PDQ and 5 for Forecast Backcast 139 80 299 20 301 30 4 Click OK to run ARIMA and review the 139 60 300 10 302 20 ARIMA report for details of the results 139 60 301 00 303 00 139 60 30220 304 30 140 20 304 20 306 40 141 30 306 80 309 20 Auto ARIMA 141 20 308 20 310 70 Papa ee AUTO ARIMA Models 140 90 309 60 312 20 onder PDG cariene cd rade the i i 140 90 311 00 313 80 best fit using Adjusted R Squared Akaike Proper ARIMA modeling requires testing of the autoregressive and moving 140 70 312 30 315 30 ona Somer Crinan and ranks them average of the errors on the time series data in order to calibrate the correct 141 10 314 20 317 30 i
13. 4va 2 a 3 Excess Kurtosis _ UG 88 __ a 3 a 4 Input requirements Alpha Shape gt 0 Beta Scale gt 0 Pearson VI Distribution 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 7 x py FO B Blana l x By X rl Mean Bor a l 2 1 Standard Deviation P a a a 1 a 1 2 Skew 2 G2 20 a 1 a a a 1 3 Excess Kurtosis 3 a 2 Xa a 9 a 3 a 4 a a a lt 1 Input requirements Alpha 1 Shape 1 gt 0 Alpha 2 Shape 2 gt 0 Beta Scale gt 0 User Manual Risk Simulator Software 70 2005 2011 Real Options Valuation Inc PERT Distribution 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 goals across any number of project components using minimum most likely and maximum values but it is designed to gen
14. RISK SIMULATOR User Manual RISK SIMULATOR 2011 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 or 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 Founder and CEO Real Options Valuation Inc Written by Dr Johnathan Mun Written designed and published in the United States of America To purchase additional copies of this document contact Real Options Valuation Inc at the e mail address below Admin RealOptions Valuation com or visit www realoptionsvaluation com 2005 2011 by Dr Johnathan Mun All rights reserved Microsoft is a registered trademark of Microsoft Corporation in the U S and other countries Other product names mentioned herein may be trademarks and or registered trademarks of the respective holders TABLE OF CONTENTS L INTRODUCTION rere o EOE E ES E EESO EO eO ES EOE ES EES e 8 Welcome to the Risk Simulat
15. 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 i e 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 how confident the simulated statistics are In other words in a simple 10 000 trial simulation say the resulting 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
16. 0 Exponential 2 Distribution 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 User Manual Risk Simulator Software 61 2005 2011 Real Options Valuation Inc Extreme Value Distribution or Gumbel Distribution 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 construction 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 are as follows X a B 1 f x ze wherez e for gt 0 and any value of x and a Mean a 0 577215B 1 Standard Deviation r B 12V6 1 2020569 3 T Skewness 1 13955 this applies for all values of mode and scale Excess Kurtosis 5 4 this applies for all values of mode and scale Mode and scale are the dis
17. 10 23 24 25 26 116 119 icons 126 independent variable 154 155 156 159 inflation 155 158 inputs 126 installation 9 10 integer 8 22 49 51 58 64 73 84 112 113 interest 155 157 158 interest rate 155 157 158 investment 124 jump diffusion 90 Kolmogorov Smirnov test 145 kurtosis 43 57 lags 156 least squares 154 least squares regression 154 linear 154 155 157 159 160 Ljung Box Q statistics 156 logistic 65 Lognormal 66 67 lower 125 management 128 market 155 158 160 matrix 159 mean 154 157 Mean 65 66 68 mean reversion 90 mix 159 model 124 125 153 155 157 Model 131 models 155 Monte Carlo 18 37 46 47 multicollinearity 153 159 Multinomial SLS 8 multiple 124 128 159 161 multiple regression 159 multiple variables 161 multivariate 86 87 88 92 94 95 Mun 1 2 8 84 87 88 90 95 206 negative binomial 52 53 nonlinear 154 155 160 User Manual Risk Simulator Software normal 18 24 34 38 43 47 49 57 58 66 67 73 90 143 146 154 Normal 67 null hypothesis 154 156 157 objective 126 optimal 128 155 optimal decision 128 optimization 8 19 112 113 114 115 116 118 119 120 122 124 125 126 128 145 option 8 84 136 143 151 outliers 153 154 155 157 parameter 158 Parameter 67 pareto 68 Pareto 68 69 pause 26 Pearson 34 35 point estimate 128 Poisson 54 61 6
18. 138 2005 2011 Real Options Valuation Inc Notice that the ranking of critical success drivers similar to the tornado chart in the previous examples However if correlations are added between the assumptions 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 0 33 A Price 0 31 B Price 0 22 C Price 0 17 Tax Rate 0 05 Price Erosion 0 03 Sales Growth 0 0 0 1 0 2 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 Procedure amp Open or create a model define assumptions and forecasts and run the simulation the example here uses the Tornado and Sensitivity Charts Linear file amp Select Risk Simulator Tools Sensitivity Analysis amp Select the forecast of choice to analyze and click OK Figure 5 10 User Manual Risk Simulator Software 139 2005 2011 Real Options Valuation Inc 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
19. 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 a more accurate average value where the error is 2 User Manual Risk Simulator Software 37 2005 2011 Real Options Valuation Inc 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 taco 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 are 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 confidence interval will be between 18 and 22 units with this interval
20. 5 411 39 5 637 71 5 864 03 6 090 35 6 316 67 6 542 99 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 4 243 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 4698 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 4 394 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 35 4 0
21. 6p l Excess Kurtosis p l p Probability of success p is the only distributional parameter Also 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 1 e 0 0001 lt p lt 0 9999 Binomial Distribution 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 e The trials are independent what happens in the first trial does not affect the next trial e The probability of an event occurring remains the same from trial to trial The mathematical constructs for the binomial distribution are as follows P x _p 1 p _ forn gt 0 x 0 1 2 n and0 lt p lt 1 x n x Mean np Standard Deviation np 1 p User Manual Risk Simulator Software 48 2005 2011 Real Options Valuation Inc 1 2p ynp l p 6p 6p 1 np 1 p Skewness Excess Kurtosis Probability of success p and the integer number of total trials n are the distributional parameters The number of successful trials is denoted x It is important to
22. 7 00 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 Taxes 343 50 355 99 368 49 380 98 393 48 405 57 417 67 429 76 441 86 453 95 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 36 Investment Outlay 500 00 1 500 00 aq Figure 5 38 Scenario Analysis Tool SCENARIO ANALYSIS TABLE Output Variable G 6 Initial Base Case Value 3 127 87 Column Variable C 12 Min 10 Max 30 Steps 20 Stepsize Initial Base Case Value 10 00 Row Variable C 9 Min 0 3 Max 0 5 Steps Stepsize 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 00 3 827 14 4 053 46 4 279 78 4 506 10 4 732 42 4 958 74 5 185 06
23. Histogram Resolution Figure 2 9 Forecast Chart Global View User Manual Risk Simulator Software 30 2005 2011 Real Options Valuation Inc 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 are 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 Two Tail as the type entering the desired certainty value e g 90 and hitting TAB 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 Income Risk Simulator Forecast Figure 2 10 Forecast Chart Two Tail Confidence Interval Alternatively a one tail probability can be computed Figure 2 11 shows a left tail selection at 95 confidence i e choose Left Tail lt as the type enter 95 as the certainty level and hit TAB on the keyboard This means that there
24. In Figure 5 27 top P values less than 0 10 0 05 and 0 01 are highlighted in blue to indicate statistical 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 Pearson s R between two variables x and y is related to the covariance cov measure x y ee where R The benefit of dividing the covariance by the product of the two variables S 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 compare 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 correlations provide 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
25. Inc d Neural Network Forecast STEP 1 Data Manually enter your data paste from another application or load an example dataset with analysis VAR3 VAR4 Ey 459 11 460 71 460 34 460 68 460 83 461 68 461 66 461 64 465 97 469 38 wo N o ju S w N e On Din amp WNE gt STEP 2 Choose analysis type variable and forecast period to run 5 Cosine with Hyperbolic Tangent Hyperbolic Tangent Layers Testing Periods Forecast Periods Apply Multiphased Optimization Sum of Squared Errors Training 1 822044 RMSE Training 0 093820 Sum of Squared Errors Modified 59375 218349 RMSE Modified 16 814849 Forecasting indicates negative values Period Actual Y Forecast F Error E 211 581 5000 613 3528 31 8528 212 584 2200 613 5197 29 2997 213 589 7200 613 6203 23 9003 214 590 5700 613 7188 23 1488 215 588 4600 613 8520 25 3920 216 586 3200 614 0608 27 7408 217 591 7100 614 2046 22 4946 218 593 2600 614 3029 21 0429 219 592 7200 614 4223 21 7023 220 592 3000 614 5671 22 2671 221 589 2900 614 7154 25 4254 222 593 9600 614 8963 20 9363 223 597 3400 614 9954 17 6554 224 600 0700 615 0992 15 0292 225 596 8500 615 2115 18 3615 Figure 5 56 Neural Network Forecast 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 bina
26. Input requirements Minimum lt Most Likely lt Maximum and can be positive negative or zero Power Distribution The Power distribution is related to the exponential distribution in that the probability of small outcomes User Manual Risk Simulator Software 71 2005 2011 Real Options Valuation Inc is large but exponentially 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 f ax F x x Mean ao l a Standard Deviation a 2 a 2 2 a 1 Skew ere Ae a a 3 Excess Kurtosis is a complex function and cannot be readily computed Input requirements Alpha gt 0 Power 3 Distribution 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 Student s t Distribution The Student s t distribution is the most widely used distribution in hypothesis test This distribution is used to estimat
27. 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 User Manual Risk Simulator Software 134 2005 2011 Real Options Valuation Inc results of the model depend on these critical success drivers These variables are the ones that should be simulated Do not waste time simulating 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 Precedent Cel Downside Upside Range Downside Upside Value 360 00 1 620 00 1 980 00 1 800 00 219 73 3 43 16 71 23 18 30 53 40 15 48 05 138 24 116 80 90 59 95 08 97 09 96 16 96 63 96 63 5 00 Figure 5 5 Spider Chart User Manual Risk Simulator Software 135 2005 2011 Real Options Valuation Inc Sales Growth Depreciation Interest Amortization Capex Net Capital oF A ho eo N 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 nonlinearities
28. The two Variance measures are simply the squared values of the 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 distribution 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 of the 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
29. allows assumptions decision variables and forecasts to be copied and pasted 12 Profiling allows 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 13 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 14 Right Click Shortcuts access all of Risk Simulator s tools and menus using a mouse right click 15 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 16 RS Functions in Excel insert RS functions for setting assumptions and forecasts and right click support in Excel 17 Troubleshooter allows you to reenable the software check for your system requirements obtain the Hardware ID and others User Manual Risk Simulator Software 13 2005 2011 Real Options Valuation Inc 18 19 Turbo Speed Analysis runs forecasts and other analyses tools at blazingly fast speeds enhanced in version 5 2 The analyses and results remain the same but are now computed very quickly reports are generated very quickly as well Web Resources Case Studies and Videos download free mo
30. 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 confidence Although they sound similar the concepts are significantly different from one another A simple illustration is in order Suppose you are 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
31. 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 Error 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
32. skewness p 22 J p 8 3 excess kurtosis 6p p 6 2 P B 3X8 4 Shape and Location p are the distributional parameters Calculating Parameters There are 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 gt 0 05 Pearson V Distribution 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 are shown below x et De Bix OFT F x T a B x Tr a Mean Ea a l User Manual Risk Simulator Software 69 2005 2011 Real Options Valuation Inc B Standard Deviation a 1 a 2 Skew
33. the type color size zoom tilt 3D and other things in the forecast chart as well as to generate User Manual Risk Simulator Software 28 2005 2011 Real Options Valuation Inc overlay charts PDF CDF and run distributional fitting on your forecast data see the Data Fitting sections for more details on this methodology Global View versus Normal View 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 in 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 Display E Always Show Window On Top E Semitransparent When Inactive Histogram Resolution Faster P Higher Simulation i aad is ME iA Resolution Date Update Interval Faster y Update Figure 2 8A Forecast Chart Preferences Income Risk Simulator Forecast Global View Data Filter Show all data a ee i Show only data between nfinity and Infinity Show only data within standard deviation s Statistic Precision level used to calculate the error 9 4 Show the following statistic s on the histogram Mean Median F 1st Quartile
34. 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 i e 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 which 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 appears 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 User Manual Risk Simulator Software 140 2005 2011 Real Options Valuation Inc 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 fluct
35. 16 Note that only one variable is allowed as the Dependent Variable Y whereas multiple variables are allowed in the Independent Variables X section separated 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 report You can also automatically generate Multiple Models by entering a sample model and using the predefined INTEGERV N variable as well as Shifting Data up or down specific rows repeatedly For instance if you use the variable LAG VARI INTEGER and you set INTEGER to be between MIN 1 and MAX 3 then the following three models will be run LAG VARZ 1 then LAG VARI 2 and finally LAG VARI 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 i e the model will be run on months 1 100 4 100 7 100 etc Using this Multiple 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 User Manual Risk Simulator Software 101 2005 2011 Real Options Valuation Inc J
36. 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 J 5 Asset Class 7 15 53 14 25 5 00 35 00 1 0898 1 5 1 1 Asset Class 8 14 95 16 44 5 00 35 00 0 9094 2 g 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 00 35 00 0 8045 10 3 8 6 Portfolio Total 12 6920 4 52 Return to Risk Ratio Figure 4 3 Continuous Optimization Results 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 Start Real Options Valuation Risk Simulator Examples or accessed directly through Risk Simulator Example Models 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 ins
37. 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 Nonparametric Chi Square Goodness of Fit Nonparametric Chi Square Independence Nonparametric Chi
38. 60 4 363 00 4543 40 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 4 787 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 4322 41 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 4211 91 4 553 04 4 723 60 4894 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 4 435 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 4 318 92 4 482 92 4646 93 4810 93 4 974 93 5 138 93 5 302 93 5 466 93 5 630 93 Segmentation Clustering Tool A final analytical technique of interest is that of segmentation clustering Figure 5 40 illustrates a sample dataset You can select the data and run the tool through Risk Simulator Tools Segmentation Clustering Figure 5 40 shows a sample segmentation of two groups That is taking the original data set
39. 75 77 90 113 128 140 142 143 145 146 147 148 157 Distribution 46 48 49 50 52 54 55 61 62 63 65 66 67 68 73 74 75 144 145 distributional 125 distributions 124 e mail 2 9 206 equation 155 158 159 Erlang 63 64 error 8 9 21 26 28 37 68 83 86 92 94 96 146 errors 153 154 157 159 estimates 154 155 Excel 8 9 10 20 21 28 34 82 83 88 92 95 99 100 102 103 106 109 111 112 132 excess kurtosis 43 48 49 50 51 52 54 56 58 61 62 63 64 65 66 67 69 73 74 75 extrapolation 8 92 93 first moment 40 41 Fisher Snedecor 62 flexibility 128 fluctuations 153 158 forecast 18 21 25 26 27 28 31 33 36 37 40 47 77 78 83 84 90 92 94 95 99 100 112 113 130 139 140 146 147 148 149 150 forecast statistics 27 112 113 146 forecasting 153 155 158 Forecasting 77 forecasts 155 fourth moment 40 43 Frequency 45 46 functions 156 functions of 156 gallery 24 gamma 58 63 64 73 83 84 Gamma 56 63 64 75 geometric 49 50 52 67 114 Geometric 49 geometric average 125 goodness of fit 155 157 2005 2011 Real Options Valuation Inc goodness of fit tests 155 growth 124 158 growth rate 158 heteroskedasticity 153 154 155 157 Histogram 45 46 Holt Winter 83 85 hypergeometric 50 51 Hypergeometric 50 hypothesis 8 57 62 72 88 96 143 145 146 148 149 icon
40. 802 4484 82 9516 6 677 0000 863 9179 186 9179 7 1006 6000 971 7020 34 8980 R 1177 annn 1NR2 ANIR 3R 4072 m Figure 5 57 Fuzzy Logic Time Series Forecast 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 Risk Simulator Tools Goal Seek feature Note that 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 User Manual Risk Simulator Software 190 2005 2011 Real Options Valuation Inc One Variable Target Seek Set cell a3 E Result To value 300 By changing cell At E eee Figure 5 58 Goal Seek 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 are not sure what input value the formula needs to get that result use the Risk Simulator Tools Single Variable Optimizer feature Figure 5 59 Note that this tool runs very quickly but is only applicable to finding one
41. 86 49 68 4 3000 125 70 53 79 4 4000 126 72 49 70 4 5000 129 52 50 28 4 6000 132 28 49 70 4 7000 138 47 56 77 4 8000 139 69 66 32 4 9000 140 85 65 95 5 0000 143 61 68 65 Figure 3 10 Stochastic Forecast Result User Manual Risk Simulator Software 91 2005 2011 Real Options Valuation Inc Nonlinear Extrapolation Theory 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 helps 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 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 i
42. A Costs 15 75 16 07 I Depreci DCFMode C24 10 10 00 10 00 10 23 Operating Income EBITDA 873 74 837 82 8 IV Amottiza DCF Mode C25 3 10 00 10 00 10 24 Depreciation 10 00 10 00 I 7 Interest DCF Mode C27 2 10 00 1000 10 25 Amortization 3 00 3 00 m PredA DCFMode c15 50 1000 10 00 10 26 EBIT 860 74 82482 I 7 PiodB DCFMode C16 35 10 00 10 00 10 27 Interest Payments 2 00 2 00 M Pode DCFMode C17 20 10 00 1000 10 28 EBT 858 74 I m PodC DCFMode C14 1515 10 00 10 00 10 ixl 29 Taxes 343 50 329 13 Bure 30 Net Income 515 24 493 69 4 31 Depreciation 13 00 13 00 Show All Variables _ Use Cell Address 32 Chen e in Net Working Capital 0 00 0 00 e rs ble i ng g cap 5 Show Top m Variables T Ignore all possible integer values 33 Capital Expenditures 0 00 0 00 ie Dey rae 34 Free Cash Flow 528 24 506 69 s 7 Ignore zero or empty values Highlight possible integer values EE Tis E Use Global Setting nvestmen 800 T z Analyze This Worksheet Only Analyze All Worksheets 39 Financial Analysis 40 Present Value of Free Cash Flow 528 24 440 60 367 26 305 91 254 62 4 Present Value of Investment Outlay 1 800 00 0 00 0 00 0 00 0 00 42 Net Cash Flows 1 271 76 506 69 485 70 465 25 445 33 Figure 5 2 Running Tornado Analysis User Manual Risk Simulator
43. Biz Stats SS y 2 File Data Language Help STEP 1 Data Manually enter your data paste from another application A B or load an example dataset with analysis Choose an analysis and enter the parameters required see example Parameter inputs below X Absolute Values ABS VARS ANOVA Randomized Blocks Multiple Treatme p VARG VAR7 vAaR8 H 4 041 ANOVA Single Factor Multiple Treatments a 1 139 39 287 79 290 10 0 55 ANOVA Two Way Analysis 139 69 289 10 291 29 7 3 665 ARIMA F 139 69 290 10 292 29 2351 Auto ARIMA 140 69 292 29 294 5 29 76 Bin E Optional 0 1 Time Series Lags 141 19 293 89 296 10 3 294 Auto Econometrics Quid Optional 0 141 69 295 29 297 39 3 287 futocorretation sind Partal Autocorrelation 2 Nart gt Var2 Var3 Var4 141 89 296 39 298 5 0 666 rage ANG sor G 141 296 5 298 5 12 938 m gt 0 140 5 796 60 798 60 647R Control Chart C Control Chart NP Control Chart P Control Chart R Control Chart U Control Chart X u STEP 4 Save Optional You can save multiple analyses and notes in the profile for future retrieval 4 Dependent Variable Independent Variables P Value Threshold 14630 4008 38927 27377 11221 397 1163 4 993 2 477 Runs the current analysis in Step 2 or selected saved analysis in Step 4 view the results charts and statistics copy the results and char
44. 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 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 occurs for larger or smaller values of a variable and only a wider range will capture this nonlinear impact A tomado 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 0 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 resu
45. F 3rd Quartile Show Decimals ChartX Axis 4 H Confidence 4 HH Statistics 4 Figure 2 8B Forecast Chart Options User Manual Risk Simulator Software 29 2005 2011 Real Options Valuation Inc Sotho tnk O ea Chan Type Bar x Chan Overlay Continuous Title Income 1000 Trials Min Max Auto X Axis Y Axis E Char X Axis 4 Decimals Distribution Fitting Done Actual Theoretical Continuous Logistic Mean 0 86 0 85 Discrete Stdev 0 19 0 23 5 Fit Stats 0 03 Decimals Skew 0 12 0 00 2 P Value 0 2782 Kut 945 120 Fa Figure 2 8C Forecast Chart Controls R Income Risk Simulator Forecast p p p piele da da t D D Do Tipe Two Tal _ gt O diniy ay inty 100 0034 Percentage Eror Precision at 95 Confidence Char Type Bar Chart Overlay Continuous i Show all data Tue Show only data between nfinty and Infinity Chan XAxis 4 Decimals Show only data within 6s standard deviation s Statistic Theoretical Continuous Precision level used to calculate the error 9534 s5 ee Show the following statistio s on the histogram 0 00 2 ej Decimats Mean E Median E 1st Quartile E 3rd Quartile P Value 0 2782 J Show Decimals Chart X Axis 4 Confidence 4 Statistics Display Always Show Window On Top Semitransparent When Inactive
46. Forecasts ooeeeeeeseerssrssrrrsrersserssrerertestessrrssressreesseessee 103 Markov CHAINS cacti ten pa n RRO BR a aaa a a aaa 106 Limited Dependent Variables Logit Probit Tobit Using Maximum Likelihood Estimation107 Spline Cubic Spline Interpolation and Extrapolation cccccccccccccsccceetteeeceettt ee eteteeeeenaes 110 4 OPTIMIZATION scast cord cont cout tact lestsastrcust cosentasessacvoesusevenseacnslvasisastcbseubsevassuasacolunndasacosaabseute 112 Optimization Methodologies surunsa E e E AAEE E EE Manette 112 Optimization with Continuous Decision Variables 00000annooainssenanoeennesssnrssseeresseeeee 114 Optimization with Discrete Integer Variables 00annn00annooonnnnennneennnesenieserenessreresssee ene 119 Efficient Frontier and Advanced Optimization Settings cccccccccccsccccccsseeeectseesenteeeennaeees 123 Stochastic Optimization seaasseeaasenanseennseeeeesseeesssettssetetsssttttsreteessrtrtssreressrererererer esete 124 5 RISK SIMULATOR ANALYTICAL TOOLS eeseeseessessoesoesoessessessossosssessossossosssessessossosseee 130 Tornado and Sensitivity Tools in Simulation na0aaaaaoaaneennaeneeoeeeseneesseoeesssenessseressse rerent 130 Sensitivity ANALYSIS cccccccccccccccc cece cece eee e eee eee eee e LEE EEL EEE ECOG EE ECOG EE EE cna EEE e tb nade Ee tents 138 Distributional Fitting Single Variable and Multiple Variables ccccccccccesccteeteteteteees 142 Bootstrap Simul ationize
47. Manual Risk Simulator Software 92 2005 2011 Real Options Valuation Inc 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 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 Polynomial Growth Rates 3 Select the function type and extrapolation periods are required and click OK Year Month Period Sales Extrapolation 2010 1 1 1 00 2010 2 2 6 73 Nonlinear Extrapolation is used to make statistical mo s s a istorical trends It is us rical 2010 4 4 45 25 trends are nonlinear and well behaved The 2010 7 5 83 59 extrapolation is best used for short term forecasts 2010 6 6 138 01 Function Types 2010 7 7 210 87 2010 8 8 304 44 Automatic Selection Polynom
48. PDQ inputs Nonetheless you can use the AUTO ARIMA forecasts to automatically 141 60 316 60 320 00 Time Series Variable B5 B440 El test all possible combinations of the most frequently occurring PDQ values to find the 141 90 318 10 321 70 EAA Vae C best fitting ARIMA model To do so following these steps 142 10 319 90 323 80 142 70 32230 326 50 Maximum Iterations 100 1 Risk Simulator Forecasting AUTO ARIMA 142 90 324 10 328 70 Forecast Periods 5i 2 Click on the Time Series Variable link 142 90 325 70 330 60 icon and select the area B5 B440 143 50 327 60 332 60 Backcast 3 Click OK to run ARIMA and review the 143 80 329 30 334 50 ARIMA report for details of the results 144 10 331 20 336 60 144 80 333 50 339 00 P 145 20 335 50 341 00 gt Real Options 145 20 337 60 343 20 VY Valuation 145 70 340 20 346 20 ewniodtoptonevetetirizon Basic Econometrics Figure 3 15 AUTO ARIMA Module Theory 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 Procedure amp Start Exc
49. S Curve Forecasts Theory 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 period to another This model is typically used in forecasting biological growth and chemical reactions over time Procedure amp Start Excel and select Risk Simulator Forecasting JS Curves amp Select the J or S curve type enter 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 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 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 g
50. SC The Durbin Watson statistic measures the seria correlation in the residuals Generally DW less than 2 implies positive serial correlation Regression Results intercept AR MAG Coefficients 0 0626 1 0055 0 4936 Standard Error 0 3108 0 0006 0 0420 Statistic 0 2013 1691 1373 11 7633 pValue 0 8406 0 0000 0 0000 Lower 5 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 Statistic 99 confidence with af of 432 2 5873 Degrees of Freedom for Residual 432 Critical tStatistic 95 confidence with df of 432 1 9655 Total Degrees of Freedom 434 Critical Statistic 90 confidence with af of 432 1 6484 The Coefficients provide the estimated regression intercept and siopes For instance the coefficients are estimates of the true population b values in the foliowing regression equation Y Bo 8yX BaXa 8 X The Standard Error measures how accurate the predicted Coefficients are and the tStatistics 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 thatthe 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 tiest is very importa
51. Software 63 2005 2011 Real Options Valuation Inc Standard Deviation 4 a 2 Skewness Ja f 6 Excess Kurtosis a Shape parameter alpha and scale parameter beta are the distributional parameters and I 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 n as the number of these random variables the mathematical construct of the Erlang distribution is n 1_ x f x rer for all x gt 0 and all positive integers of n n 1 Input requirements Scale beta gt 0 and can be any positive value Shape alpha gt 0 05 and any positive value Location can be any value Laplace Distribution The Laplace distribution is also sometimes called the double exponential distribution 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 th
52. 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 User Manual Risk Simulator Software 79 2005 2011 Real Options Valuation Inc Custom Distributions 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 fitting routine GARCH 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 JCurve 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 v
53. a Skew 2 Va 6 Excess Kurtosis 3 a Alpha and Beta are the distributional parameters Input requirements Alpha Shape gt 0 and is an Integer Beta Scale gt 0 User Manual Risk Simulator Software 60 2005 2011 Real Options Valuation Inc Exponential Distribution 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 service 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 lifetime 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 The exponential distribution describes the amount of time between occurrences The mathematical constructs for the exponential distribution are as follows f x de forx gt 0 4 gt 0 Mean A Standard Deviation a Skewness 2 this value applies to all success rate A inputs Excess Kurtosis 6 this value applies to all success rate A inputs Success rate A is the only distributional parameter The number of successful trials is denoted x Input requirements Rate gt
54. 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 Aright tailed 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 a 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 ifthe 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 lefttailed 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 t
55. 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 User Manual Risk Simulator Software 9 2005 2011 Real Options Valuation Inc 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 A and B 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 A and B 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 E Microsoft Excel Book1 AE File Edit view Insert Format Tools Data Window Help Simulation Adobe PDF i i S13 7 cee hae Aes ee New Simulation Profile i Arial 710 z B 7 U 22 55 Edit Simulation Profile Change Simulation Profile Set Input Assumption Set Output Forecast Copy Parameter A h B tS Paste Parameter Remove Parameter Edit Correlations Run Simulation Step Simulation Reset Simulation a Forecasting gt New Risk Simulator Menu Optimization Tools
56. and their p values at lag k has the nuli hypothesis that there is no autocorrelation up to order k The dotted lines in the piots 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 5 significance level Forecasting Period Actual Y Forecast F Error E 2 139 399994 139 6056 0 2056 3 139 699997 140 0069 0 3069 4 139 699997 140 2586 0 5586 5 140 699997 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 g 141 142 3587 1 3587 10 1405 141 0466 0 5466 11 140 399994 140 9447 0 5447 12 140 140 8451 0 8451 13 140 140 2946 0 2946 14 139 899994 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 250 300 350 19 140 199997 140 0513 0 1487 20 141 300003 140 9862 0 3138 21 141 199997 142 1738 0 9738 22 140 899994 141 4377 0 5377 23 140 899994 141 3513 0 4513 24 140 699997 141 3939 0 6939 25 141 100006 141 0731 0 0270 26 141 600006 141 8311 0 2311 27 141 899994 142 2065 0 3065 28 142100006 142 4709 0 3709 29 142 699997 142 6402 0 0598 30 142 899994 143 4561 0 5561 31 142 899994 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 5
57. assume the properties of several other distributions For example depending on the shape parameter 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 are as follows a l x i af x 3 ponte PLE Mean BT a Standard Deviation B ra 2a7 T 1 a 2 A B 3F 0 Bd 28 4 T04 3p Piss ri Skewness Excess Kurtosis 6r 6 1217 1 BT 287 3r 267 4r A BT 38 Td 4B bisz rao Shape and central location scale are the distributional parameters and Iis the Gamma function Input requirements Shape Alpha gt 0 05 Scale Beta gt 0 and can be any positive value User Manual Risk Simulator Software 75 2005 2011 Real Options Valuation Inc Weibull 3 Distribution The Weibull 3 distribution uses the same constructs as the original Weibull distribution but adds a Location or Shift parameter The We
58. 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 the other Risk Simulator forecasting methodologies to build more robust models User Manual Risk Simulator Software 189 2005 2011 Real Options Valuation Inc a Combinatorial Fuzzy Logic Forecast STEP 1 Data Manually enter your data paste from another application or load an example dataset with analysis VARI VAR2 684 20 584 10 765 40 892 30 885 40 677 00 1006 60 1122 10 1163 40 10 1993 70 mT STEP 2 Enter required inputs and select the variable to forecast IO in uw N e gt Seasonality Forecast Periods Results RMSE 707 039492 Auto ARIMA RMSE 249 495091 Time Series Auto RMSE 287 252763 Trend Line Exponential RMSE 775 403678 Trend Line Linear RMSE 912 616213 Trend Line Logarithmic RMSE 1488 012692 Trend Line Moving Average RMSE 988 333906 Trend Line Polynomial RMSE 758 307610 Trend Line Power RMSE 1268 660480 indicates negative values Period Actual Y Forecast F 1 684 2000 2 584 1000 3 765 4000 4 892 3000 5 885 4000
59. 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 sample is 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 Test Result Errors Relative Observed Expected 0 E Regression Error Average 0 00 Frequency Standard Deviation of Errors 141 83 219 04 0 02 0 02 0 0612 0 0412 D Statisti
60. 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 Risk Simulator Edit Profile 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 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 zero 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 Turn on correlations If checked correlations between paired input assumptions will be computed Otherwise correlations will all be set to zer
61. cell B6 as well as the lower bound and upper bound values at cells F6 and G6 Then using Risk Simulator s copy copy this cell E6 decision variable and paste it to the remaining cells in E7 to E15 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 E17 and make it equal to 100 Click OK when done The final step in optimization is to set the objective function and start the optimization by selecting the objective cell C18 and Risk Simulator Optimization Run Optimization and then selecting the optimization of choice Static Optimization Dynamic Optimization or Stochastic Optimization To get started select Static Optimization Check to make sure the objective cell is set for C18 and select Maximize You can now review the decision variables and constraints if required or 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 colu
62. constructs for the negative binomial distribution are as follows P x Camden p forx r r 1 and0 lt p lt l r 1 x Mean UZA P P Standard Deviation ae P Skewness es a yrl p 2 6p Excess Kurtosis PON r l p Probability of success p and required successes R are the distributional parameters Input requirements Successes required must be positive integers gt 0 and lt 8000 Probability of success gt 0 and lt 1 that is 0 0001 lt p 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 User Manual Risk Simulator Software 52 2005 2011 Real Options Valuation Inc Pascal Distribution 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 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 Pascal distribution is related to the negative binomial distribution Negative binomial distribution computes the number of
63. correlate it to another assumption but not both 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 Regular Input and Percentile Input This 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 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 User Manual Risk Simulator Software 24 2005 2011 Real Options Valuation Inc 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 Triangular Ay Custom ei Bernouii Beta 3 CIE Binomial A chi square Ii Discrete Uniform AA Erang K
64. csser 200 TIPS SAVC a AA AAE ueiwetynverantuncedtedancednecubLesnessnvertacws bertedinvesneedsbeneeaverttes 200 TIPS Sampling and Simulation Techniques ccccccccsccccessecesseeeeneeseceeeenseseaecseeessueessesesenseesnaeenes 201 TIPS Software Development Kit SDK and DLL Libraries 201 TIPS Starting Risk Simulator with EXC ccccccccccecsccessecesseeeeneeseecseneeseseeesnsecseteessesessnsesenteennaeenss 201 TIPS Super Speed Simulation ccccccccccccccccscceececeeseeseneeeensecseeessceeseueeeseseseneeeseesseeseueesenreeeneeenss 201 LIPS FOrnado ANGI sts occ sc 2st obs es ok A Ae A E 202 TIPS Troublesh te tiie iiei cee teins EE hes ane cote ans Davee vtec RE R eins E a Oe 202 User Manual Risk Simulator Software 7 2005 2011 Real Options Valuation Inc 1 INTRODUCTION Welcome to the Risk Simulator Software The Risk Simulator is a Monte Carlo simulation Forecasting and Optimization software The software is written in Microsoft NET C 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 training courses and books written by the software s creator e g Dr Johnathan Mun s
65. data in chronological order past to present in a single column with multiple rows Data Location c8 c2428 E Generate a GARCH P Q model for Pi Q fi Periodicity 252 Base 1 Forecast Periods 10 E Apply Variance Targeting GARCH GARCH M TGARCH TGARCH M 5 EGARCH EGARCH T GJR GARCH GJRTGARCH Run All Models Figure 3 19 GARCH Volatility Forecast 104 2005 2011 Real Options Valuation Inc GARCH MODELS 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 distribution Z Normal Distribution Z T Distribution GAR CHEM y C 0 E y C 07 E Variance in Mean Equation ey Oat B Ons oO o a gt Bo Oo o a Bo GARCH M y ct do 6 y ctdo Standard Deviation i _ 0 2 0 2 in Mean Equation o 4aeE Bo o o aE Bo SARCHM y c Aln o7 e y c Aln o7 e Log Variance in Mean Equation ERTO gA SET Oe Oo ae Bo Oo ae po GARCH V X E y o ae po E ae eT oO o a Bo EGARCH y g y e 0 2 0 2 In o7 B In o In o o B In o7 alaei j r a 2 Ede r t t O O O O Ede b 2 Ele paa T v 1 2 VON i vV DE v 2 Vr GJR GARCH yee y 6 E 0 2 E 0 2 o a gt Oo a 7
66. 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 Hypothesis Testing Theory 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 Procedure amp Runa simulation amp Select Risk Simulator Tools Hypothesis Testing amp Select only two forecasts to test at a time select the type of hypothesis test you wish to run and click OK Figure 5 18 User Manual Risk Simulator Software 148 2005 2011 Real Options Valuation Inc MODELA MODEL B Revenue 200 00 Revenue 200 00 Cost 100 00 Cost 100 00 Income gt 100 Income 5 100 00 Hypothesis Testing 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 a another or their differences are due to random chance Please select the two forecasts on which to run a hypothesis test To replicate this model start by creating a Simulati Simulation New Profile then set the random
67. e g a correlation between sunspots and the stock market might be strong but one can surmise that there is 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 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 Procedure amp Open the example model Risk Simulator Examples Statistical Analysis go to the Data worksheet and select the data including the variable names cells C5 E55 amp Click on Risk Simulator Tools Statistical Analysis Figure 5 28 User Manual Risk Simulator Software 160 2005 2011 Real Options Valuation Inc amp 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 finishe
68. 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 User Manual Risk Simulator Software 78 2005 2011 Real Options Valuation Inc ARIMA Autoregressive integrated moving 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 Auto ARIMA 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 Basic Econometrics Econometrics refers to a branch of business analytics modeling and forecasting techniques for modeling the behavior of or forecasting certain business economic finance physics manu
69. 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 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 48 Reversion Rate 283 89 Jump Rate 20 41 Volatility 88 849 Long Term Value 327 72 Jump Size 237 89 Probability of stochastic model fit 46 48 A high fit means a stochastic mode is better than conventional models Runs 20 Standard Normal 1 7321 Positive 25 P Value 1 taif 0 0416 Negative 25 PValue 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 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 is more appropriate to use It is up to the user to make this determination For i
70. inputs are incorrect and the model is not correctly specified e g 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 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 realoptionsvaluation com download html or www rovdownloads com download html TIPS Hardware ID e Right Click HWID Copy in the nstall License user interface select or double click on the HWID to select its value right clic
71. 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 User Manual Risk Simulator Software 31 2005 2011 Real Options Valuation Inc Income Risk Simulator Forecast Figure 2 11 Forecast Chart One Tail Confidence Interval In addition to evaluating what the confidence interval is i e 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 Left Tail lt probability type enter 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 Income Risk Simulator Forecast Figure 2 12 Forecast Chart Probability Evaluation User Manual Risk Simulator Software 32 2005 2011 Real Options Valuation Inc For the sake of completeness you can select the Right Tail gt probability type enter the value 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
72. not forget the sign 5 Select cell C12 the objective to be maximized and make it the objective Risk Simulator Optimization Set Objective or click on the O icon 6 Run the optimization by going to Risk 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 Decision Variable Properties Decision Name Asset 1 E Decision Type Continuous e g 1 15 2 35 10 55 Integer e g 1 2 3 Lower Bound En Upper Bound E Binary 0 or 1 User Manual Risk Simulator Software 126 2005 2011 Real Options Valuation Inc Constraints E 11 100 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 allocation product mix project selection Run on static model without simulations Usually run to determine the intial optimal portfolio before more advanced optimizations are applied Dynamic Optimization 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 S
73. 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 requirements Probability of success gt 0 and lt 1 1 e 0 0001 lt p lt 0 9999 Number of trials gt 1 or positive integers and lt 1000 for larger trials use the normal distribution with the relevant computed binomial mean and standard deviation as the normal distribution s parameters Discrete Uniform The discrete uniform distribution is also known as the equally likely outcomes distribution where the distribution has a set of N 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 P x x Mean ranked value N 1 N 1 12 Skewness 0 i e the distribution is perfectly symmetrical 6 N 1 5 N 1N 1 Standard Deviation ranked value Excess Kurtosis ranked value Input requirements Minimum lt maximum and both must be integers negative integers and zero are allowed Geometric Distribution 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 User Manual Risk Simulator Software 49 2005 2011 Real Options
74. on Risk Simulator Optimization Constraints and select ADD to add a new constraint Then select the cell D17 and make it less than or equal to lt 5000 Repeat by setting cell J17 lt 6 The final step in optimization is to set the objective function and start the optimization by selecting cell C19 and Risk Simulator Optimization Set Objective Then run the optimization using Risk 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 or portfolio returns to risk 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 optimization for additional practice Ja B FS LT 5 a SS SN LY a R a a a 1 2 Return to Profitability Risk Ratio index 3 4 Project 1 54 96 8 33 1 26 5 Project 2 1 914 92 1 02 3 27 ES Project 3 1 551 03 1 03 1 87 7 Project 4 1 012 95 2 22 2 37 8 Project 5 925 44 0 92 2 85 g 10 11 12 Selection ENPY Cost Risk Risk Project 6 560 92 1 35 15 58 Project 7 5 633 10 0 51 4 75 Project 8 926
75. related to February s level which in turn is related to January s level etc Ignoring such blatant relationships will yield biased and less accurate forecasts In such events an autocorrelated regression model or an ARIMA model may be better suited Risk Simulator Forecasting User Manual Risk Simulator Software 155 2005 2011 Real Options Valuation Inc 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 is nonzero it means that the series is first order serially correlated If AC k 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 A 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 are the approximate two standard error bounds If the autocorrelation is within these bounds it is not signific
76. 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 Geometric 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 User Manual Risk Simulator Software 162 2005 2011 Real Options Valuation Inc Hypothesis Test t Test on the Population Mean of One Variable Sta
77. started with simulation one first needs to understand the concept of probability distributions To 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 60 50 Number of 40 Employees 30 20 10 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
78. the returns to risk ratio that is for the same amount of risk 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 Parolo Returns Risk PSUMA NO 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 User Manual Risk Simulator Software 118 2005 2011 Real Options Valuation Inc ASSET ALLOCATION OPTIMIZATION MODEL i one A Required Required Returns Risk Return to Allocation SEREI ae vou an Minimum Maximum Ken a Ranking Ranking Risk Ranking Ranking Allocation Allocation Hi Lo Lo Hi Hi Lo Hi Lo Asset Class 1 10 54 12 36 5 00 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 7 9 9 Asset Class 4 10 64 12 35 5 00 35 00 0 8615 8 1 5
79. 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 Drift Rate 5 86 Reversion Rate NWA Jump Rate 16 33 Volatility 7 04 Long Term Value N A Jump Size 21 33 Probability of stochastic model fit 4 63 Figure 5 33 Sample Statistical Analysis Tool Report Stochastic Parameter Estimation Distributional Analysis Tool The Distributional Analysis tool is a statistical probability tool in Risk Simulator that is useful in a variety 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 occurr
80. 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 restricted by probabilistic laws The process generating equation is known in advance but the actual results generated are 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 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 These processes can also 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 if the probability of fit similar to a goodness of
81. 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 a 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 Escape on the keyboard and then apply the Risk Simulator Paste User Manual Risk Simulator Software 194 2005 2011 Real Options Valuation Inc 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 equation 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 mul
82. 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 A B ue o E F G H al J K L 1 2 3 ASSET ALLOCATION OPTIMIZATION MODEL 4 P Required Required Returns Risk Return to Allocation Daai enning i Oaie Minimum Maximum Fiegen Ranking Ranking Risk Ranking Ranking 5 Allocation Allocation Hi Lo Lo Hi Hi Lo Hi Lo 6 Asset Class 1 10 54 12 36 5 00 35 00 0 8524 9 2 7 1 7 Asset Class 2 11 25 16 23 5 00 35 00 0 6929 7 8 10 1 8 Asset Class 3 11 84 16 64 5 00 35 00 0 7570 6 7 9 1 9 Asset Class 4 10 64 12 35 5 00 35 00 0 8615 8 1 5 1 1
83. 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 1 250 4501 lt lt A1 A2 J One Variable Quick Optimizer Objective Cell a3 E Maximize Minimize Variable Cell A1 Min 50 Max 250 Tolerance 0 000000001 Max Iterations fioo Optimized Variable 250 0000 Optimized Objective 450 0000 Figure 5 59 Single Variable Optimizer User Manual Risk Simulator Software 191 2005 2011 Real Options Valuation Inc 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 Risk 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 in
84. way you expect it to and click OK Figure 3 9 Results Interpretation 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 User Manual Risk Simulator Software 90 2005 2011 Real Options Valuation Inc Stochastic Process Forecasting 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 probabilistic rules They are useful for forecasting random events e g stock prices interest rates price of electricity Methods Brownian Motion Random Walk with Drift Exponential Brownian Motion Random Walk with Drift Mean Reversion Process with Drift Annualized Volatility 7 Jump Diffusion Process with Drift Forecast Horizon Years Starting Value Growth or Drift Rate Jump Diffusion Process with Drift and Mean Reversion Reversion Rate Long Term Value Jump Rate Jump Size Number of Steps Iterations Random Seed Show All Iterations
85. 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 Sample Ordered Data 1 1 00 2 1 00 3 2 00 4 3 00 5 2 00 6 4 00 7 15 00 8 16 00 9 14 00 10 15 00 11 125 00 12 126 00 176 13 128 00 14 129 00 Options 15 130 00 Showall 2 segmentation clusters 16 175 00 17 179 00 Show cluster number 2 18 474 00 Show cluster numbership for value Figure 5 40 Segmentation Clustering Tool and Results User Manual Risk Simulator Software 170 2005 2011 Real Options Valuation Inc SEGMENTATION AND CLUSTER ANALYSIS RESULT Groups 2 INOS RO RSET ROUND a ek h ad ek ck ah ei wh ad Risk Simulator 2011 New Tools Random Number Generation Monte Carlo versus Latin Hypercube and Correlation Copula Me
86. 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 no n 1 2 f x e for x gt 0 4 gt Oand ao gt 0 xvV2z In c Mean ct Standard Deviation J explo 742 ujexp co z 1 Skewness N explo tle exp o Excess Kurtosis exp 4o 2 expl3o 3 exp 20 6 Mean u and standard deviation o are the distributional parameters Input requirements User Manual Risk Simulator Software 66 2005 2011 Real Options Valuation Inc Mean and standard deviation both gt 0 and can be any positive value 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 Lognormal 3 Distribution The Lognormal 3 distribution uses the same constructs as the original 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
87. 0 1 104 64 Check ior Updates Distributional Analysis B 49 380 98 393 48 405 57 417 67 429 76 441 86 453 95 I Distributional Charts amp Tables 273 590 21 608 36 626 50 644 64 662 78 680 93 Resources 4 aoe f B 00 13 00 13 00 13 00 13 00 13 00 13 00 13 00 User Manual am DSR buBOret Desmer so oo sooo 0 00 so o0 0 00 sooo iiep amp Distributional Fitting Single Variable sooo so oo sooo sooof sooof sooof sooo 5 Distributional Fitting Multi Variable 584 47 603 21 621 36 639 50 ses7 64 675 78 5 444 64 36 Investment Outlay 3 Edit Correlations 2 00 1 37 A Hypothesis Testing z Net Free Cash Flow P T a 5 73 584 47 603 21 627 36 639 50 657 64 675 78 40 Financial Analysis 5s Overlay Charts 41 Present Value of Free Cai ga panai Compencnt DE L77 384 30 344 89 308 92 276 47 247 23 220 91 1 547 71 42 Present Value of Investme 22 0 00 0 00 0 00 0 00 0 00 0 00 0 00 Seasonality Test 43 Discounted Payback Perit te oi Ar xf e Segmentation Clustering 45 Risk Analysis 46 Base Case PV at Time 0 77 384 30 344 89 299 60 260 27 226 09 196 41 1 338 69 47 PV of Cash Flow at Time Scenario Analysis 94 441 94 396 62 355 26 317 94 284 32 254 05 1 779 86 48 Intermediate X Variable statistical Analysis Sensitivity Analysis 7 1 Structural Break Test L M 4 gt h Infor
88. 0 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 27 28 To run the Cubic Spline forecast click on 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 are the same Enter 32 the desired forecast periods e g Starting 1 Ending 50 Step Size 0 5 Click 33 OK and review the generated forecasts and chart z F Cubic Spline 36 The cublic spline polynomial interpolation and extrapolation model is used to fill in the 37 gaps of missing values and for forecasting time series data whereby the model can be used to both interpolate missing data points within a time series of data e 9 yield 38 curve interest rates macroeconomic variables like inflation rates commodity 39 prices or market returns and is also used to extrapolate outside of the given or known 40 range making it useful for forecasting 41 Known X Values kics E a Known Y Values 015 025 lel 44 Generate a spline curve based on the following X values 45 Starting 1 Ending 150 Step Size 0 5 46 4T Cancel 48 Figure 3 22 Cubic Spline Module User Manual Risk Simulator Software 110 2005 2011 Real Options Valuation Inc Procedure amp Start Excel and open the example file Advanced Fo
89. 0 Asset Class 13 25 13 28 5 00 35 00 0 9977 5 4 2 1 11 Asset Class 6 14 21 14 39 5 00 35 00 0 9875 3 6 3 1 12 Asset Class 7 15 53 14 25 5 00 35 00 1 0898 1 5 1 1 13 Asset Class 8 14 95 16 44 5 00 35 00 0 9094 2 9 4 1 14 Asset Class 9 14 16 16 50 5 00 35 00 0 8584 4 10 6 1 15 Asset Class 10 10 06 12 50 5 00 35 00 0 8045 10 3 8 1 16 17 Portfolio Total 12 6419 4 58 18 Return to Risk Ratio 19 20 21 Specifications of the optimization model 22 23 Objective Maximize Return to Risk Ratio C18 24 Decision Variables Allocation Weights E6 E15 25 Restrictions on Decision Variables Minimum and Maximum Required F6 G15 26 Constraints Portfolio Total Allocation Weights 100 E17 is set to 100 27 28 Additional specifications 29 30 1 One can always maximize portfolio total returns or minimize the portfolio total risk 31 2 Incorporate Monte Carlo simulation in the model by simulating the returns and volatility of each asset class 32 and apply Simulation Optimization techniques 33 3 The portfolio can be optimized as is without simulation using Static Optimization techniques Figure 4 1 Continuous Optimization Model User Manual Risk Simulator Software 114 2005 2011 Real Options Valuation Inc 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 c
90. 0 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 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 your model TIPS Troubleshooter e ROV Troubleshooter trun this troubleshooter to obtain your computer s HWID for licensing purposes to view your computer settings and prerequisites and to reenable Risk Simulator if it has been accidentally disabled User Manual Risk Simulator Software 202 2005 2011 Real Options Valuation Inc INDEX acquisition 155 allocation 124 125 126 alpha 154
91. 0 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 Trials Probability Type Formatting Single Value Value X Range of Values Lower Bound Upper Bound Step Size Figure 5 34 Distributional Analysis Tool Binomial Distribution with 2 Trials User Manual Risk Simulator Software 165 2005 2011 Real Options Valuation Inc 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 Type Formatting O Singe Vie f f 0 000019 Value X i j 0 000181 0 001087 ores 0 004621 Lower Bound 0 i 0 014786 0 036964 Upper Bauri L 0 073929 Step Size i 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 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 t
92. 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 i e each variable is arranged as a column and all the variables are fitted one at a time Procedure 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 Select Risk Simulator Tools Distributional Fitting Multi Variable Review the data choose the relevant types of distribution you want and click OK Kw ow 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 User Manual Risk Simulator Software 145 2005 2011 Real Options Valuation Inc Bootstrap Simulation Theory Bootstrap simulation is a simple technique that estimates the reliability or accuracy of for
93. 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 is running thereby reducing the memory required to visibly update the Excel spreadsheet and freeing up the memory to run the simulation The Clear All and Minimize All controls all the open forecast charts Options 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 error 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 is 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 Controls As shown in Figure 2 8C this tab has all the functionalities in allowing you to change
94. 2 367 1148 600 0 55 1 8 5 443 18068 372 3 665 32 3 5 7 365 7729 142 2 351 45 1 73 614 100484 432 29 76 190 8 7 5 385 16728 290 3 294 31 8 5 286 14630 346 3 287 678 4 6 7 Single Model Dependent Variable Independent Variables LN VAR1 LN VAR2 VAR3 VAR4 LAG VARS 1 DIFF VAR6 TIME ag LN VART Functions cg LOGIVAR2 VARS VART VARA LAG VARS 2 VARE RESIDUAL Show Result LNLOG LAG VART VAR TIME FORECAST VARS VARA DIFF VARS RATEIVARS Multiple Models F Econometrics Results 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 Max ANOVA F Statistic 9 2137 INTEGER2 Min Max ANOVA p Value 0 0000 INTEGER3 Min Max intercept _LN VAR2 VAR3 VAR4 LAG VARS 1 DIFF VAR6 TIME Coefficients 3 1049 0 2726 0 0000 0 0011 0 0219 0 0125 Standard Eror 0 8947 0 0974 0 0000 0 0003 0 0322 0 0049 t Statistic 3 4703 2 8001 0 7885 3 8576 0 6796 2 5234 pValue 0 0012 0 0077 0 4348 0 0004 0 5005 0 0155 Dependent Variable LN VAR1 cony Grose _ Figure 3 16 Basic Econometrics Module To run an econometric model simply select the data B5 G55 including headers and click on Risk Simulator Forecasting Basic Econometrics You can then type in the variables and their modifications for the dependent and independent variables Figure 3
95. 2 2 2 2 r _d _ Bo r _d _ Bo d 1 if e lt 0 d 1 ife lt 0 lo otherwise lo otherwise User Manual Risk Simulator Software 105 2005 2011 Real Options Valuation Inc For the GARCH M models the conditional variance equations are 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 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 z 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 variance 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 Markov Chains Theory A Markov chain exists when the probability of a future state depends on a
96. 2005 2011 Real Options Valuation Inc Results Interpretation The optimization s final results are shown in Figure 4 3 where the optimal allocation of assets 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 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 the 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
97. 25 1 33 11 74 Project 9 2 100 60 0 93 16 56 13 Project 10 1 912 50 1 18 5 94 14 Project 11 263 52 48 00 208 13 20 15 Project 12 309 75 1 69 6 00 16 17 Total 47 218 00 8 197 44 7 007 40 70 15 Goal MAX 5000 lt 19 Sharpe Ratio 2 4573 20 21 ENP V is the expected NPV of each credit line or project while Cost can be the total cost of 22 administration as well as required capital holdings to cover the credit line and Risk is the 23 Coefficient of Variation of the credit line s ENPY Figure 4 4 Discrete Integer Optimization Model User Manual Risk Simulator Software 120 2005 2011 Real Options Valuation Inc Decision Variable Properties Constraints Decision Name o E Current Constraints Decision Type D 17 lt 5000 Continuous e 1 15 2 35 10 55 SI 17 lt 6 Lower Bound H Upper Bound Integer fe g 1 2 3 Lower Bound Ej Upper Bound E Binary 0 or 1 ce 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 Objective Cell scsi9 W Optimization Objective Maximize the value in objective cell Minimize the value in objective cell Figure 4 5 Running Discrete Integer Optimization in Risk Simulator User Manual Risk Simulator Software 121 Real Options Valuatio
98. 26 19207729 8195 Critical F statistic 9096 confidence with af of 2 and 432 2 3449 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 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 Vaiue is calculated to test the nuli hypothesis Ho where ali the Coefficients are simultaneously equal to zero versus the alternate hypothesis Ha that they are all simultaneously different from zero indicating a significant overall regression model Ifthe 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 Siatistic with the critical F values at various significance levels User Manual Risk Simulator Software 97 2005 2011 Real Options Valuation Inc 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 3 0 9760 0 0109 0 0958 0 0958 1 275 4818 4 0
99. 3 population 154 157 portfolio 124 128 precision 8 21 26 28 37 prediction 154 155 price 90 probability 8 18 27 31 32 33 42 45 46 47 48 49 50 51 52 53 55 57 62 67 73 Probability 46 48 49 50 52 profile 20 21 22 23 35 84 116 119 125 143 p value 156 160 random 157 158 random number 18 22 46 range 24 40 41 55 63 113 115 125 128 130 155 rank correlation 160 rate 155 158 ratio 124 125 regression 8 86 87 88 92 94 95 Regression 86 regression analysis 153 154 155 relative returns 125 Reliability 130 report 22 83 88 90 92 95 133 140 143 152 156 157 return 124 125 returns 124 125 155 risk 124 125 Risk Simulator 126 running 155 157 sales 155 156 sample 154 157 2005 2011 Real Options Valuation Inc save 9 21 151 saving 151 seasonality 156 second moment 40 42 sensitivity 8 134 140 141 Sensitivity 130 139 significance 154 156 157 160 simulation 8 18 19 20 21 22 23 26 27 28 33 34 35 36 37 45 46 47 77 84 90 112 113 115 116 120 122 124 126 128 130 134 138 139 141 142 143 146 147 148 151 152 153 160 Simulation 18 45 130 146 148 151 152 153 160 164 168 170 171 172 174 176 178 179 183 187 190 191 192 single 128 155 161 Single Asset SLS 8 skew 40 42 Skew 42 skewness 42 43 48 49 50 51 52 54 56
100. 3 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 User Manual Risk Simulator Software 89 2005 2011 Real Options Valuation Inc Stochastic Forecasting Theory A stochastic process is nothing but a mathematically defined equation that can create a series of outcomes over time outcomes 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 every 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 simula
101. 5 60 Genetic Algorithm User Manual Risk Simulator Software 193 2005 2011 Real Options Valuation Inc 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 e 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 automatically 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
102. 57 58 61 62 63 64 65 66 67 69 73 74 75 147 SLS 8 Spearman 34 specification errors 153 spider 8 132 133 136 138 spread 36 40 standard deviation 18 28 36 38 41 42 43 47 49 51 54 57 58 61 62 63 64 66 67 68 73 74 90 112 113 143 148 149 157 160 static 157 statistics 27 28 36 37 40 41 95 112 113 143 146 147 148 User Manual Risk Simulator Software 205 stochastic 8 90 112 113 116 120 122 125 127 129 153 157 158 stochastic optimization 125 127 129 stock price 157 158 symmetric 154 t distribution 72 third moment 40 42 time series 8 78 83 84 90 92 93 94 95 155 157 158 time series data 155 157 158 title 20 21 toolbar 10 23 25 26 tornado 8 130 132 133 134 136 138 140 14 Tornado 130 132 133 134 138 139 trends 158 trials 18 21 22 26 27 37 47 48 49 50 51 52 53 61 112 113 126 146 triangular 18 47 73 Triangular 73 t statistic 159 types of 124 157 uniform 18 47 49 74 115 125 142 Uniform 74 upper 125 validity of 155 value 124 125 154 155 157 158 159 160 values 124 125 154 155 156 157 160 variance 153 154 volatility 158 Weibull 75 Yes No 47 2005 2011 Real Options Valuation Inc RISK SIMULATOR 2011 This manual and the software described in it are furnished under license and may only be used or copied in accordance with th
103. 6 exhibit random jumps such as oil prices or price of electricity discrete exogenous event shocks can make prices jump up or i y down Finally these three stochastic processes can be mixed and matched as required 0 9000 109 57 27 99 1 0000 110 74 30 81 The results on the right indicate the mean and standard deviation of all the iterations generated at each time step If the Show All 1 1000 111 53 35 05 Iterations option is selected each iteration pathway will be shown in a separate worksheet The graph generated below shows a 1 2000 111 07 34 10 sample set of the iteration pathways 1 3000 107 52 32 85 1 4000 108 26 37 38 Stochastic Process Brownian Motion Random Walk with Drift 1 5000 106 36 32 19 Start Value 100 Steps 50 00 Jump Rate N A 1 6000 112 42 32 16 Drift Rate 5 00 Iterations 10 00 Jump Size N A 1 7000 110 08 31 24 Volatility 25 00 Reversion Rate N A Random Seed 1720050445 1 8000 109 64 31 87 Horizon 5 Long Term Value N A 1 9000 110 18 36 43 2 0000 112 23 37 63 2 1000 114 32 33 10 2 2000 111 14 38 42 2 3000 111 03 37 69 2 4000 112 04 37 23 2 5000 112 98 40 84 2 6000 115 74 43 69 2 7000 115 11 43 64 2 8000 114 87 43 70 2 9000 113 28 42 25 3 0000 115 72 43 43 3 1000 120 05 50 48 3 2000 116 69 42 61 3 3000 118 31 45 57 3 4000 116 35 40 82 3 5000 115 71 40 33 3 6000 118 69 41 45 3 7000 121 66 45 34 3 8000 121 40 45 03 3 9000 125 19 48 19 4 0000 129 65 55 44 4 1000 129 61 53 82 4 2000 125
104. 658 0 0978 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 l 8 9 10 11 12 x1 0 8467 0 2045 0 3336 0 9105 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 User Manual Risk Simulator Software 156 2005 2011 Real Options Valuation Inc 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 informative test available and this could mean the difference
105. 692 0 3692 37 145 199997 145 7582 0 5582 38 145 699997 145 6649 0 0351 39 146 146 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 User Manual Risk Simulator Software 98 2005 2011 Real Options Valuation Inc AUTO ARIMA Box Jenkins ARIMA Advanced Time Series Theory 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 are automatically run and compared Procedure amp 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 amp Inthe Auto ARIMA worksheet select Risk 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 amp Click on the link icon and link to the existing time series data enter the number of forecast periods desired and click OK ARIMA and AU
106. 71 Forecast Precision Variance 16 9506 Precision Level Average Deviation 3 3389 f Error Level Maximum 9 3923 Minimum 9 7671 F Range 19 1594 Skewness 0 0494 Kurtosis 0 53994 a 25 Percentile 2 8924 75 Percentile 2 8015 Error Precision at 95 3 1644 Name iample Third Forecast Number of Datapoints 1000 Enabied Yes Mean 0 2864 Cell SESI4 Median 0 2624 Standard Deviation 0 4593 Forecast Precision Variance 0 0254 Precision Level Average Deviation 0 4305 Error Level es Maximum 0 8958 Minimum 0 0126 3 Range 0 8232 E Skewness 0 5797 Kurtosis 0 2064 sf 25 Percentile 0 4590 75 Percentile 0 3935 Error Precision at 95 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 User Manual Risk Simulator Software 152 2005 2011 Real Options Valuation Inc 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 i
107. 828 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 simultaneously 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 F Error E RMSE 140 4048 1 521 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 15
108. 93 519 107 231 105 253 110 750 72 306 104 638 114 671 82 774 100 455 113 540 116 882 102 387 101 451 118 545 99 574 93 431 109 074 99 901 110 392 104 347 114 534 98 788 90 383 84 614 74 349 101 032 102 992 99 822 102 005 102 582 114 762 100 853 88 833 86 101 101 915 109 511 84 912 93 900 105 235 97 832 96 564 98 365 95 603 91 974 106 448 100 588 112 635 102 622 100 571 R 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 ii Figure 5 43 Principal Component Analysis 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
109. 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 User Manual Risk Simulator Software 85 2005 2011 Real Options Valuation Inc Multivariate Regression Theory 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 B B X e where fh is the intercept p is the slope and is the error term It is bivariate as there are only two variables a Y or dependent variable and an 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 I
110. 96 71 4 300 07 4 503 43 4 706 79 4 910 15 5 113 51 5 316 88 1 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 4 205 97 4 406 06 4 606 14 4 806 22 5 006 30 5 206 38 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 4 505 48 4 702 28 4 899 08 5 095 88 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 5 372 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 54 684 65 4 874 89 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 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 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 2 919 79 3 100 20 3 280 60 3 461 00 3 641 40 3 821 80 4 00220 4 182
111. 9678 0 0142 0 0958 0 0958 1 688 5499 5 0 9594 0 0098 0 0958 0 0958 2 095 4625 6 0 9509 0 0113 0 0958 0 0958 2 4961572 n 7 OB 0 0124 0 0958 0 0958 2890 5594 A 8 0 9336 0 0147 0 0958 0 0958 3 278 5669 9 0 9247 0 0121 0 0958 00958 3 660 1152 a 10 0956 0 0139 0 0958 0 0958 4 0351192 7 11 0 9066 0 0049 0 0958 0 0958 4 4036117 142 08975 0 0068 0 0958 0 0958 4 7656032 13 0 8883 0 0097 0 0958 0 0958 5121 0697 M 14 087 0 0087 0 0958 0 0958 5 470 0032 15 0 8698 0 0064 0 0958 0 0958 5 812 4256 A 16 0 8605 0 0056 0 0958 0 0958 6 148 3604 m 17 0 8512 0 0062 0 0958 0 0958 6 4778620 18 0 8419 0 0038 0 0958 0 0958 6 800 9622 19 0 8326 0 0003 0 0958 0 0958 7117 7709 20 0 8235 0 0002 0 0958 0 0958 7 428 3952 if autocorrelation AC 1 is nonzero it means that the series Js first order serially correlated If AC K dies off more or jess geometrically with increasing Jag it implies thatthe series follows a low order autoregressive process If AC k drops to Zero after a smali number of lags it implies that the series follows a low order moving average process Partial correlation PAC kK 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 Liung Box Q statistics
112. 97 Range 0 3951 Skewness 0 1040 Kurtosis 0 3191 25 Percentile 1 8475 25 Percentile 1 9437 75 Percentile 2 1480 75 Percentile 2 0487 Percentage Eror Precision at 95 Confidence 0 5839 Percentage Error Precision at 95 Confidence 0 2224 Statistics Number of Trials Variance Coefficient of Variation 25 Percentile 75 Percentile Percentage Eror 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 User Manual Risk Simulator Software 36 2005 2011 Real Options Valuation Inc recovered in the simulation That is you enter 0 8 and 0 8 correlations and the resulting simulated values have the same correlations 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 Precision and Error Control One very powerful tool in Monte Carlo simulation is that of precision control For instance
113. E Exponential 2 D Gamma KK Gumbel Maximum Hil Hypergeometric PI Logistic E Lognormal 3 i Negative Binomial E Parabolic Pareto i Pascal lt Triangular Distribution a 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 Cost Assumption Name Revenue Location Sheet1 SAs2 You can enter pairwise correlations here when there are other assumptions available Minimum 1 LE Most Likely 2 Maximum 2s Regular Input Percentile Input m Enable Data Boundary Infinity E Infinity 5 Minimum Maximum E Enable Dynamic Simulations 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 sales show the minimum maximum and _ A short description of the selected distribution is provided Figure 2 4 Assumption Properties 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 Then proceed to defining the output forecasts in the next step 3 Defining Output Forecasts The next step is to define output f
114. 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 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
115. Location 10 Mean Alpha 1 5 DF Numerator Apply Global Inputs Maximum 20 Beta 5 Probability 0 5 Stdev Alpha2 5 DF Denominator MostLikely 15 Lambda 1 2 Factor 2 Successes Population 100 Pop Success Arcsine Bernoulli Beta Beta 3 Beta 4 Minimum Probability 1 Alpha Alpha 2 Alpha Maximum Beta Beta 5 Beta Location 10 Location Factor Random X 12 Random X 0 Random X 0 6 Random X 10 25 Random X 108 Percentile 05 Percentile 05 Percentile i Percentile 0 5 Percentile 05 PDF 0 7958 PDF 0 5000 PDF 0 4608 PDF 2 3730 PDF 1 5552 CDF 0 2952 CDF 0 5000 CDF 0 9590 CDF 0 4661 CDF 0 7667 ICDF 15 0000 ICDF 1 0000 ICDF 0 2644 ICDF 10 2644 ICDF 10 5289 Mean 15 0000 Mean 0 5000 Mean 0 2857 Mean 10 2857 Mean 10 5714 Stdev 3 5355 Stdev 0 5000 Stdev 0 1597 Stdev 0 1597 Stdev 0 3194 Skew 0 0000 Skew 0 0000 Skew 0 5963 Skew 0 5963 Skew 0 5963 Kurtosis 1 5000 Kurtosis 2 0000 Kurtosis 0 1200 Kurtosis 0 1200 Kurtosis 0 1200 Binomial Cauchy Chi Square Cosine Discrete Uniform Trials 20 Alpha DF Minimum 10 Minimum Probability 05 Beta Maximum 20 Maximum Random X 10 Random X Random X Random X 15 5 Random X Percentile 0 5 Percentile Percentile Percentile 0 5 Percentile PDF 0 1762 PDF PDF PDF 0 1551 PDF CDF 0 5881 CDF CDF CDF 0 5782 CDF ICDF 10 0000 ICDF ICDF ICDF 15 0000 ICDF Mean 10 0000 Mean Mean 15 0000 Mean Stdev 2 2361 Stdev Stdev 2 1762 Stdev Skew 0 0000 Skew Skew 0 0000 Skew Kurtosis 0 1000 Kurtosis Kurtosis 0 5938 Kurtosis Decimals 4 Language Run
116. MultipleTreatments VARGO VAR61 VAR62 VAR63 001_AutoEconometricsDetailed VAR5 VAR6 VAR7 VAR8 0 1 0 010_PrincipalComponentAnalysis VAR6 VAR7 VAR8 VAR9 VAR10 2 Runs the current analysis in Step 2 or selected saved analysis in Step 4 view the results charts and statistics copy the results and charts to dipboard or generate reports indicates negative values Standard Deviations 21685 9352 92 8151 5 4049 1 5207 0 0235 0 1681 0 0339 0 0813 0 0619 0 0909 0 1824 0 0683 0 0735 0 0869 0 5140 0 2595 0 6488 0 0150 0 1055 0 0459 0 1284 0 0438 0 3112 0 0311 0 1152 0 1010 0 1124 0 1061 0 0154 0 0241 0 0526 1 0000 0 3333 0 9590 0 2422 0 2374 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 Covariance Matrix 470279784 3284 670889 8820 11241n naa 670889 8820 8614 6500 175 7717 112410 0992 175 2712 70 2130 1222792 7730 6886 4692 247 11732 Choose an analysis and enter the parameters required see example parameter inputs below VARG VAR7 VARS VAR9 VA Stochastic Process Exponential Brownian M Stochastic Process Geometric Brownian Moti Stochastic Process Jump Diffusion Stochastic Process Mean Reversion and Ju Stochastic Process Mean Reversion S
117. Principal Component Analysis and click OK Review the generated report for the computed results VARL VAR2 VAR3 VAR4 VARS VAR6 VAR7 VAR8 VARS VAR10 Procedure 96 998 87 223 102 443 112 765 111 984 117 331 78 164 97 658 110 950 89 133 1 Select the data to analyze e g B11 K30 click on 93 098 83 096 81 531 90 224 92 265 78 821 94 321 95 960 101 349 96 345 Risk Simulator Tools Principal Component Analysis 96 730 96 298 113 426 99 147 98 138 94 868 119 722 108 657 123 757 93 451 and click OK 116 615 83 876 105 389 109 022 119 189 99 155 94 762 106 751 96 187 107 576 2 Review the generated report for the computed results 85 558 91 528 84 784 96 371 99 675 100 281 96 773 121 945 82 575 92 635 74 224 114 477 87 202 93 464 107 577 104 667 108 746 105 957 86 282 88 843 106 940 103 226 90 602 97 591 101 315 105 578 101 387 90 890 118 848 104 872 100 722 108 298 108 620 93 635 90 768 111 112 87 988 84411 107 113 106 384 122 057 114 438 113 039 101 130 100 020 104 537 99 745 89 453 82 252 108 283 104 442 106 179 102 135 989 731 112 382 96 888 91 601 91 789 95 710 95 466 94 762 108 494 105 132 93 917 113 050 82 391 105 506 98 837 100 417 93 459 94 504 108 493 108 030 104 564 106 914 116 306 103 039 105 890 118 528 96 644 110 383 101 435 111 410 98 517 92 202 110 760 94 182 105 339 105 458 96 836 95 592 86 340 119 930 94 335 100 861 97 657 128 354 112 520 108 809 113 322 101 879 105 420 97 504 87 789 112 667 97 111 86 941 107 643 107 843 104 282 104 039
118. 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 e 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 variables 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 Real Options SLS software is used for computing simple and complex options and includes the ability to create customizable option models This oftware has the following modules e Single Asset SLS for solving abandonment chooser contraction deferment and expansion options as well as for solving custo
119. Risk Simulator ROV BizStats Neural Network as well as in Risk Simulator Forecasting Neural Network Figure 5 56 shows the Neural Network forecast methodology Procedure amp Click on Risk Simulator Forecasting Neural Network amp 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 amp Select if you wish to 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 amp Click Run 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 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 User Manual Risk Simulator Software 187 2005 2011 Real Options Valuation
120. Running a Multivariate Regression User Manual Risk Simulator Software 88 2005 2011 Real Options Valuation Inc 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 to 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 partof the calculation to obtain the confidence interval of the estimates later Regression Results Intercept x1 X2 X3 X4 X5 Coeffici
121. 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 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 s
122. Software 132 2005 2011 Real Options Valuation Inc Results Interpretation Figure 5 3 shows the resulting tornado analysis report which indicates that capital investment has the largest impact on net present value followed by tax rate average sale price quantity demanded of the product lines and so forth The report contains four distinct elements 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 absolut
123. 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 Two 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 User Manual Risk Simulator Software 16 2005 2011 Real Options Valuation Inc 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 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 Brownian Motion Stochastic Processes Jump Diff
124. TO ARIMA Note 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 X Xz Xn 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 User Manual Risk Simulator Software 99 2005 2011 Real Options Valuation Inc ooN One wm A B c D Sample Historical Time Series Data Box Jenkins ARIMA Forecasts M1 M2 M3 FEMA Ae Paras modeling technique 138 90 286 70 289 00 Autoregressive Integrated Moving Average ARIMA used to and forecast time series 139 40 287 80 290 10 forecasts apply advanced econometric modeling tecniques oe Gee So ere eer 139 70 289 10 291 30 to forecast time series data by first back fitting to historical revenues gross domestic product
125. Valuation Inc The three conditions underlying the geometric distribution are The number of trials is not fixed The trials continue until the first success e The probability of success is the same from trial to trial The mathematical constructs for the geometric distribution are as follows P x p l p for 0 lt p lt land x 1 2 n Mean p3 P Standard Deviation 4 P J Skewness P y1 P 2 6p 6 Excess Kurtosis AB a op Probability of success p 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 lt 1 i e 0 0001 lt p 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 Hypergeometric Distribution The hypergeometric distribution is similar to the binomial distribution in that both describe the number of times a particular event occurs 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 rem
126. Value 596 63 Annualized Sales Growth Rate 2 00 Intemal Rate of Retum 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 10 00 9 50 9 03 8 57 8 15 Prod B Avg Price 12 25 Prod C Avg Price 15 15 Sensitivity Analysis Prod A Quantity 50 00 5 Prod B Quantity 35 00 an srati creates dynamic nate em mul e assumptions are perturbed Simultaneously to Prod C Quantitv eee identify the impact to the results It is used to identify Total T critical success factors of the forecast CostofG Net Present Value Risk Simulator Forecast Bian Histogram Statistics j Preferences Optio bed Controls Forecast Name SG amp A Cc Net Present Value DCF Model Opera p 10 Deprecia 80 Amortizal 70 EBIT a0 Ipea Interest F 2507 i EBT y Taxes 30 ie Net In 20 10 Deprecia Change 409 91 591 Capital E f Free Type Two Tail x Anfinity Infinity Certaj L Select All Clear All Chart Label Cell Add v InvestmMeims vrvuueuY scl Stes Financial Analysis Present Value of Free Cash Flow 528 24 440 60 367 26 305 91 254 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 Results Interpretation The results of the sensitivity analysis comprise a report and two key charts The first is a nonlinear rank correlation chart Figure 5
127. age order difference order M Deseasonalize Data and rate order and click OK Ninnber of Periods Per Seasonal Cycle a 3 Review the two reports generated for more details l on the methodology application and resulting IV Detrend Data charts and deseasonalized detrended data M Li Vv tial bad es Seasonality Test V Logarithmic IV Polynomial Order fs a Power IV Moving Average Order 3 Time Series Data B9 B28 Static Mean M Difference Order fi Maximum Seasonality Period to Test 6 2 IV Static Median IV Rate Order fi SSS SS Cancel Cancel 29 30 Figure 5 42 Deseasonalization and Detrending Data User Manual Risk Simulator Software 173 2005 2011 Real Options Valuation Inc 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 Select the data to analyze e g B11 K30 click on Risk Simulator Tools
128. al locations or two 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 appropriate 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 User Manual Risk Simulator Software 150 2005 2011 Real Options Valuation Inc 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 Procedure amp Open or create a model define assumptions and forecasts and run the simulation amp 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 e Raw data in a new worksheet where the simulated values both assumptions and forecasts can then be saved or further analyzed as required e Flat text file where the data can be exported into other data analysis
129. al 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 amp Start Excel and open the example file Advanced Forecasting Model go to the GARCH worksheet and select Risk Simulator Forecasting GARCH amp Click on the link icon select the Data Location enter the required input assumptions see Figure 3 19 and click OK to run the model and report Note The typical volatility forecast situation requires P 1 Q 1 Periodicity number of periods per 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 User Manual Risk Simulator Software 103 2005 2011 Real Options Valuation Inc 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 2nd Edition by Dr Johnathan Mun Wiley Finance 2010 on GARCH modeling for more details on what each specification is for x www realoptions valuation com Historical Data Inputs Days User Manual Risk Simulator Software Real Options Valuation Generalized Autoregressive Conditional Heteroskedasticity GARCH To run a GARCH model enter in the relevant ti
130. alpha beta and gamma parameters Refer to Dr Johnathan Mun s Modeling Risk Applying Monte Carlo Simulation Real Options Analysis Forecasting and Optimization Wiley Finance 2006 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 The maximum runtime is set at 300 seconds Typically no changes are 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 zero and one Certain User Manual Risk Simulator Software 84 2005 2011 Real Options Valuation Inc 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
131. 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 decision 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 J fo PX X where the larger the value of J the higher the probability of home ownership For each family a critical threshold exists where if exceeded the house is purchased otherwise no home is purchased and the outcome probability P 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 Simulator s Distributional Analysis tool by selecting N
132. alue generated Use Monte Carlo sampling when you want to simulate real world what if scenarios for your spreadsheet 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 Carlo simulation Discrete Distributions Bernoulli or Yes No Distribution 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 n total trials that computes the probability of x successes within this total number of trials e Geometric distribution a Bernoulli distribution with higher number of trials that computes the number of failures required before the first success occurs e Negative binomial distribution a Bernoulli distribution with higher number of trials that computes the number of failures before the Xth success occurs User Manual Risk Simulator Software 47 2005 2011 Real Options Valuation Inc The mathematical constructs for the Bernoulli distribution are as follows l p forx 0 P n p forx 1 or P n p 1 p Mean p Standard Deviation p l p Skewness e ms PU p 6p
133. alues will increase significantly from one period to another This model is typically used in forecasting biological growth and chemical reactions over time Markov Chain 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 Maximum Likelihood on Logit Probit and Tobit 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 oc
134. analysis 124 128 155 156 161 Anderson Darling test 145 annualized 125 approach 125 129 155 159 ARIMA amp 78 82 94 95 96 98 99 100 102 103 106 109 111 155 asset 124 125 asset classes 124 125 assumption 18 21 23 24 25 26 47 94 112 113 138 140 143 assumptions 124 125 154 157 autocorrelation 156 159 behavior 157 Beta 55 binomial 47 48 49 50 52 Binomial 48 52 bootstrap 8 146 147 148 Bootstrap 146 148 Box Jenkins 8 94 99 Brownian Motion 158 causality 160 center of 154 Chi Square test 145 coefficient of determination 154 confidence interval 31 32 37 72 146 148 constraints 126 Continuous 46 correlation 18 21 34 35 36 47 94 95 140 141 156 159 160 correlation coefficient 159 160 correlations 160 cross sectional 78 92 Crystal Ball 47 130 144 145 data 28 34 35 36 45 46 55 67 75 77 78 82 83 84 86 87 88 90 92 93 94 95 99 100 142 143 145 146 148 150 151 decision variable 112 113 115 116 119 120 decision variables 125 decisions 128 Delphi 77 142 Delphi method 142 dependent variable 154 155 156 User Manual Risk Simulator Software discrete 8 46 47 49 112 113 119 145 158 Discrete 46 distribution 18 23 24 25 27 34 36 40 42 43 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74
135. 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 variables 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 User Manual Risk Simulator Software 174 2005 2011 Real Options Valuation Inc 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 compared with 30 100 and 1 50 compared with 51 100 to see if there is a break in the 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 H 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 signif
136. antly 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 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 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 1
137. anual Risk Simulator Software 40 2005 2011 Real Options Valuation Inc 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 o variance coefficient of variation and percentiles Skew 0 KurtosisXS 0 Hi He Figure 2 20 Second Moment Stock prices Time Figure 2 21 Stock Price Fluctuations User Manual Risk Simulator Software 41 2005 2011 Real Options Valuation Inc Measuring the Skew of the Distribution the Third Moment 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 or left skew the tail of the distribution points to the left and Figure 2 23 illustrates a positive or right skew the tail of the distribution points to the right The mean is always skewed toward 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 con
138. 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 User Manual Risk Simulator Software 124 2005 2011 Real Options Valuation Inc The model shows the various asset classes Each asset class has 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 In Figure 4 9 column E Al
139. ast curve based on the following periods End Period 100 Figure 3 17 J Curve Forecast User Manual Risk Simulator Software 102 2005 2011 Real Options Valuation Inc 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 applications 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 VV Valuation 3 Clic
140. ater User Manual Risk Simulator Software 200 2005 2011 Real Options Valuation Inc TIPS Sampling and Simulation 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 e ROV Troubleshooter trun this troubleshooter to obtain your computer s HWID for licensing purposes to view your computer settings and prerequisites and to reenable 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 Simul
141. ation MLE The response or dependent variable Y is binary That is it can have only two possible outcomes that we denote as 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 will return nonsensical values of above or below 0 MLE analysis handles these problems using an iterative optimization routine to maximize a log likelihood function when the dependent variables 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 a 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 an
142. ation optimization and real options can be combined into a seamless analytical process User Manual Risk Simulator Software 122 2005 2011 Real Options Valuation Inc 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 J17 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 amp In an optimization model i e a model with Objective Decision Variables and Constraints already set up click on Risk Simulator Optimization Constraints and click on Efficient Frontier Select the constraint you want to change or step e g J17 enter in the parameters for Min Max and Step Size Figure 4 7 click ADD and then click OK and OK again You should deselect the D17 lt 5000 constraint before running Run Optimization as usual Risk Simul
143. ator Optimization Run Optimization You can choose static dynamic or stochastic The 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 Efficient Frontier Current Constraints D 17 lt 5000 Parameters MIN 4 MAX E STEP SIZE j Changing Constraints J 17 lt MIN 4 MAX 8 STEP 1 Figure 4 7 Generating Changing Constraints in an Efficient Frontier User Manual Risk Simulator Software 123 2005 2011 Real Options Valuation Inc Efficient Frontier R Optimization Complete Problem Parameters Number of variables 12 Number of functions 3 Objective function will be Maximized STEP1 D17 lt 5000 J17 lt 4 Functions Starting Values Final Results 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 2 45726 3 46137 2 G Aaa RNGE 3197 43710 1E 10 0 2 G 3197 43710 1472 56292 3 G met RNGE 8 00000 1E 10 o 3 G 8 00000 0 00000 TER Variables Efficient Frontier Analysis 2Step 1 Constraints are Starting Values Final Results oe Problem Parameters Number of variables is 12 Number of functions is 3 7 Variable Initial Lower Upper Variable Objective function will be maximized No Name Status Value Bound Bound No Name Initial Value Final Value S
144. ator Options menu TIPS Super Speed Simulation e Model Development if you wish to run super speed in your model 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 User Manual Risk Simulator Software 201 2005 2011 Real Options Valuation Inc 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 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 1
145. 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 dh S o g where Z is the error of 2 taco shells x is the sample average Z is the standard normal Z score vn obtained from the 90 precision level s is the sample standard deviation and n is the number of trials number of trials required to hit this precision is based on the sampling error equation of x Z 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 reflected in the Statistics tab as shown in Figure 2 18 Income Risk Simulator Forecast Histogram Statistics Preferences Options Controls Global View View Data Filter Show all data Show only data between Infinity and Infinity 5 Show only data within E standard deviation s Statistic Precision level used to calculate the error 94 Show the following statistic s on the histogram E Mean Median F 1st Quartile 3rd Quartile Show Decimals Chart X Axis 4 Confidence 4 Statistics 4
146. c 0 1036 202 53 0 02 0 04 0 0766 0 0366 D Critical at 1 0 1138 186 04 0 02 0 06 0 0948 0 0348 D Critical at 5 0 1225 174 17 0 02 0 08 0 1097 0 0297 D Critical at 10 0 4458 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 1435 40 0 02 0 16 0 4526 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 Stationarity 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 User Manual Risk Simulator Software 157 2005 2011 Real Options Valuation Inc can occur over
147. cally significantin 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 Coefficients 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 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 Sumsof Meanof Estatistic _p Value Squares Squares Hypothesis Test Regression 479388 49 95877 70 4 28 0 0029 Critical F statis tic 99 confidence with df of 5 and 44 3 4651 Residual 985675 19 22401 71 Critical F statis tic 95 confidence with df of 5 and 44 2 4270 Total 1465063 68 Critical F statis tic 90 confidence with df of 5 and 44 1 9
148. curring 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 User Manual Risk Simulator Software 80 2005 2011 Real Options Valuation Inc Multivariate Regression Multivariate regression is used to model the relationship structure and characteristics 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 Neural Network Forecast 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 Nonlinear Extrapolation 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
149. d amp 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 Spend 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 Variable X1 Variable X2 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 600 18068 372 7729 142 100484 432 16728 290 14630 346 4008 328 38927 354 22322 266 3711 320 3136 197 Data is from a single variable Data comprises multiple variables in columns Select the analyses to run Run E Tests MV Stochastic Process Parameter Estimation Descriptive Statistics Periodicity Annual Distributional Fitting Time series Autocorrelation Continuous Discrete V Time series Forecasting Histogram and Charts Seasonality Periods Cycle a 3 Hypothesis Testing Forecast Periods a Hypothesized Mean 0 I Trend Line Projection Nonlinear Extrapolation Forecast Periods Forecast Periods gag omea Figure 5 29 Statistical Tests User Manual Risk Simulator Software 161 2005 2011 Real Options Valuation Inc Descriptive Statistics Analysis of Statistics Almost all distributions can be described within 4 moments some distributions req
150. d 7 2 d z FSO L 5Y 0 andy 0 ap ap which yields the bivariate regression s least squares equations S 2A D xy e B i l i l n n 7 2 YX xX il lt 2 _ NA j l n B Y 6X For multivariate regression the analogy is expanded to account for multiple independent variables where Y P B X PX E and the estimated slopes can be calculated by gt So ae YX Mg Da ipa Sy x ce AG Xu Os es DOR OX Xs 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 specification tests nonlinearities etc before a proper model can be constructed See Modeling Risk Applying Monte Carlo Simulation Real Options Analysis Forecasting and Optimization Wiley Finance 2006 by Dr Johnathan Mun for more detailed analysis and discussion of multivariate regression as well as how to identify these regression pitfalls User Manual Risk Simulator Software 87 2005 2011 Real Options Valuation Inc Procedure amp Start Excel and open your historical data if required the illustration below uses the file Multiple Regression in the examples folder amp Check to make sure that the data is arranged in columns select the entire data area including the va
151. d Upper Intervals for the Mean Median is the data point where 50 of all data points fall above this value and 50 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 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
152. d 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 simplify 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 Applying Correlations in Risk Simulator Correlations can be applied in Risk Simulator in several ways When defining assumptions Risk Simulator Set Input Assumption simply enter the correlations into the correlation matrix grid in the Distribution Gallery With existing data run the Multi Fit tool Risk 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 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 graduat
153. d distribution every part of the distribution will be sampled when LHS is applied Options Random Number Generator T Minimize Excel and All Charts When Running ROV Risk Simulator Default V Start Risk Simulator with Excel E Always Show Forecast Windows on Top ao Cell on a Long Period Shuffle 5 its S orecasts and Decision Variables RAN Ha Pi Advanced Subtractive Random Shuffle Correlation Normal Copula Default T Copula DF Quasi Normal Copula DF 30 Simulation Monte Carlo Simulation Default Latin Hypercube Sampling LHS U Quick IEEE Hex Basic Minimal Portable Parameters Color Scheme Language English v LHS is not recommended when there are correlated assumptions Figure 5 41 Risk Simulator Options 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 series data For example a detrended data set may be necessary to see a more accurate account of a company s sales in a given year more clearly by shiftin
154. d models S You can double click on any of these models to run them and the results are 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 Click on the variable headers D to select one or multiple variables at once and then right click to add delete copy paste or visualize P the variables selected Models can also be entered using a Command console V W X To see how this works double click to run a model S and go to the Command console V You can replicate 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 The 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 User Manual Risk Simulator Software 183 2005 2011 Real Options Valuation Inc 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 Copy Delete or Paste data You can also click on and select multiple column headers to select multiple variables and right c
155. d the approach is simple User Manual Risk Simulator Software 107 2005 2011 Real Options Valuation Inc Specifically the Logit model is specified as Estimated Y LN P 1 P or conversely P EXP Estimated Y 1 EXP Estimated Y and the coefficients J are the log odds ratios So taking the antilog or EXP f we obtain the odds ratio of P J P This means that with an increase in a unit of fi the log odds ratio increases by this amount Finally the rate of change is the probability dP dX B P I P 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 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 e g 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 Y 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
156. dels getting started videos case studies whitepapers and other materials from our website Simulation Module 20 21 22 23 24 25 26 27 28 29 30 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 t2 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 Delphi method Distribution Truncation enabling data boundaries Excel Functions set assumptions and forecasts using functions inside Excel Multid
157. e 5 45 Trends can be linear or nonlinear such as exponential logarithmic moving average power polynomial or power Procedure 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 e g 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 Historical Sales Revenues Year Quarter Period Sales 2006 1 684 20 2006 2 584 10 2006 3 765 40 2006 4 892 30 Trendline 2007 1 885 40 2007 2 677 00 Selected Trendlines 2007 3 1 006 60 V Linear V Exponential 2007 4 1 122 10 SEALs s 2008 4 1 163 40 IV Logarithmic IV Polynomial Order 2 H 2008 2 993 20 V Power MV Moving Average Order 2 4 2008 3 1 312 50 2008 4 1 545 30 Generate forecasts je periods 2009 1 1 596 20 2009 2 1 260 40 i Cancel 2009 3 1 735 20 2009 4 2 029 70 2010 1 2 107 80 2010 2 1 650 30 2010 3 2 304 40 2010 4 2 639 40 I Figure 5 45 Trendline Forecasts 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 i
158. e beta distribution are as follows AA ason f x TENG for gt 0 B gt 0 x gt T a PB Mean a at p Standard Deviation oh a f8 U a Pf 2 B a jl at Bp 2 a Bap 3 a B Dlapla B 6 2 a B ap a BP 2 a B 3 Skewness Excess Kurtosis Alpha q and beta are the two distributional shape parameters and Tis the Gamma function Conditions The two conditions underlying the beta distribution are The uncertain variable is a random value between 0 and a positive value 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 Beta 3 and Beta 4 Distributions 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 Bbeta 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 User Manual Risk Simulator Software 56 2005 2011 Real Options Valuation Inc
159. e 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 J 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 e The language can be changed in the Language menu Note that currently there are 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 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 e 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 1 000 50 and so forth The decimal settings can 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 User Manual Risk Simulator Software 184 2005 2011 Real Options Valuation Inc SE EXAMPLE ROV
160. e 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 User Manual Risk Simulator Software 64 2005 2011 Real Options Valuation Inc f x zol Mean a Standard Deviation 1 41428 Skewness is always 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 Logistic Distribution 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 mathematical constructs for the logistic distribution are as follows a x e B f x for any value of a and u a x Bllt e Mean a Standard Deviation 5 m p Skewness 0 this applies to all mean and scale inputs Excess Kurtosis 1 2 this applies to all mean and scale inputs Mean and scale f are the distributional parameters Calculating Parameters Th
161. e model These usually are events such as totals net profit or gross expenses Simplistically think of 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 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 v
162. e 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 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 all the time 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 matrix 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 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 User Manual Risk Simulator Software 35 2005 2011 Real Options Valuation Inc example folder The calculation for
163. e 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 or 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 Founder and CEO Real Options Valuation Inc Written by Dr Johnathan Mun Written designed and published in the United States of America To purchase additional copies of this document contact Real Options Valuation Inc at the e mail address below Admin RealOptions Valuation com or visit www realoptionsvaluation com 2005 2011 by Dr Johnathan Mun All rights reserved Microsoft is a registered trademark of Microsoft Corporation in the U S and other countries Other product names mentioned herein may be trademarks and or registered trademarks of the respective holders User Manual Risk Simulator Software 206 2005 2011 Real Options Valuation Inc
164. e the latest additions to version 2011 General Capabilities 1 Available in 10 languages English French German Italian Japanese Korean Portuguese Spanish Simplified Chinese and Traditional Chinese Books analytical theory application and case studies are supported by 10 books 3 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 4 Detailed Example Models 24 example models in Risk Simulator and over 300 models in Modeling Toolkit 5 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 8 Flexible Requirements works in Window 7 Vista and XP integrates with Excel 2010 2007 2003 and works in MAC operating systems running virtual machines 9 Fully customizable colors and charts tilt 3D color chart type and much more 10 Hands on Exercises detailed step by step guide to running Risk Simulator including guides on interpreting the results 11 Multiple Cell Copy and Paste
165. e 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 User Manual Risk Simulator Software 72 2005 2011 Real Options Valuation Inc T f 1 2 Jra T r 2 Mean 0 this applies to all degrees of freedom r except if the distribution is shifted to another nonzero a t fry fO central location r r 2 Skewness 0 this applies to all degrees of freedom r Standard Deviation 6 Excess Kurtosis F for allr gt 4 r X X where t and T is the gamma function S Degrees of freedom r is 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 F 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 gt 1 and must be an integer Triangular Distributio
166. e value of the slope indicates the 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 effects 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 User Manual Risk Simulator Software 133 2005 2011 Real Options Valuation Inc Tornado and Spider Charts Statistical Summary One of the powerful simulation tools is the tornado chart ilit captures the static impacts of each variable on the outcome of the model That is the too automatically perturbs each precedent variable in the model a user specified preset amount captures the fluctuation on the model i 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
167. ecast 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 error or confidence interval relatively easy However when a statistic s sampling distribution is not normally distributed or easily found these classical methods are difficult to use or are invalid In contrast bootstrapping analyzes sample statistics empirically by repeatedly sampling the data and creating distributions of the different statistics from each sampling Procedure Run a simulation Select Risk Simulator Tools Nonparametric Bootstrap Select only one forecast to bootstrap select the statistic s to bootstrap enter the number of bootstrap trials and click OK Figure 5 16 MODEL A MODEL B Revenue 200 00 Revenue 200 00 Cost 100 00 Cost 100 00 Income 100 01 Income gt 100 00 Nonparametric Bootstrap Nonparametric bootstrap simulation is a distribution _ free technique used to estimate the reliability or accuracy of forecast statistics i e to compute the ar L forecast intervals of each of the statistics To replicate this model start by creating a Simulation Profi Simulation New Profile then set the rand
168. el 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 amp Select the data in the Basic Econometrics worksheet and select Risk Simulator Forecasting Basic Econometrics amp 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 Show Results to view the results before generating the report in case you need to make any changes to the model User Manual Risk Simulator Software 100 2005 2011 Real Options Valuation Inc Basic Econometrics Data Set N Basic Econometrics o 8 amp This tool is used to run basic econometric models by first transforming the input variables before running the multivariate regression analysis You can enter in multiple econometric model specifications to test Each model is on a new line and within each line the first variable is the dependent variable followed by at least one or more independent variables separated by semi colons In the following example LN VAR1 and VAR3 are dependent variables in two models and the remaining items are independent variables in the two econometric models LN VAR1 LN VAR2 VAR3 VAR4 TIME VAR3 LAG VAR2 3 DIFF VAR1 RESIDUAL VAR3 VAR4 VARI VAR2 VAR3 VAR4 VARS VARG 521 18308 185 4 041 79 6 7
169. elations and correlation significance Distributional Fitting Single Kolmogorov Smirnov and chi square tests on continuous distributions complete with reports and distributional assumptions User Manual Risk Simulator Software 15 2005 2011 Real Options Valuation Inc 70 71 72 73 74 75 76 Tis 78 79 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 chart types Principal Component Analysis tests the best predictor variables and ways to reduce the data array Scenario Analysis hundreds and thousands of static two dimensional scenarios Seasonality Test tests for various seasonality lags Segmentation Clustering groups 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 Statistics and BizStats Module 80 81 82 83 Percentile Distributional Fitting using percentiles and optimization to find the best fitting distribution Probability Distributions Charts and Tables run
170. ell 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 returns 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 0 R 0 R Ro R p Where Rp is the return on the portfolio R42 cp are the individual returns on the projects and 4g cp are the respective weights or capital allocation across each project In addition the portfolio s diversified risk in cell D17 is computed by taking i n m Op Zoo D y2 20 0 P 0 0 Here p j are the respective cross correlations between the i l i l j l asset classes hence if the cross corre
171. embles 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 errors 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 Jenkins 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
172. ence 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 Risk Simulator Tools Distributional Analysis As an example of its use Figure 5 34 shows the computation of a binomial distribution i e 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 0 50 the probability of success of getting Heads Selecting the PDF and setting User Manual Risk Simulator Software 164 2005 2011 Real Options Valuation Inc 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 are 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 2
173. encing 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 User Manual Risk Simulator Software 94 2005 2011 Real Options Valuation Inc forecast errors to improve the forecast An MA q model has the form y e bye bye Finally an ARIMA p q model has the combined form y a 7 A pVip er bierni bg Cg Procedure amp 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 amp Select the time series data and select Risk Simulator Forecasting ARIMA amp Enter the relevant P D and Q parameters positive integers only enter the number of forecast period desired and click OK ARIMA and AUTO ARIMA Note 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 X X2 Xn 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 variab
174. enscesseesseessueeseneeeeeeessieeeenteseneees 26 5 Interpreting the Forecast ReSUlts cccccccccscccsceceescsessecenseceeeeseceeseueessusecseseeseteeseeesensesenesenaeenes 27 Correlations and Precision COntrol ccccccccccccceeccetetceeceteeeeeneetenceeteneeeeseeteasaeeeneeesnneeeeneeees 34 The Basics of Correlations eiieeii reto oiire e tae iE E A EE E a ii taai 34 Applying Correlations in Risk Simulator eeeeeeerserisrersrrrsrrrsrrerrresrresrresrreseresees 35 The Effects of Correlations in Monte Carlo Simulation ccccccccccccccceccececeteseceetieeeeeeneeeeenseeesenaets 35 Precision and Error Control sis ccs ie ve Gans a tige tae ae aaa Tee ages aes ee ees ee A 37 Understanding the Forecast Statistics cccccccccccccccccssccceceeee teense eee ee eee e cent eee cn ae deecnieeeeeniaas 40 Measuring the Center of the Distribution the First Moment cccccccccceeceeceeeceeeeeetteeeeeeteeeeenees 40 Measuring the Spread of the Distribution the Second Moment 0cccccceeeeeeceeeceeteeeeeeteeeeeees 40 Measuring the Skew of the Distribution the Third Moment cccccccccccccecseceeteseeeeeteeeeeeteesenes 42 Measuring the Catastrophic Tail Events in a Distribution the Fourth Momentt c001cc00 43 The Functions Of Moments secara a sued a a sie aaae Aaa aaora Sairis dunedoncudoaseesesucsowsds 44 Understanding Probability Distributions for Monte Carlo Simulation c0ccccccceesseceeeees 45 Discrete Di
175. ents 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 estimates of the true population b values in the following regression equation Y b0 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 At 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 statisti
176. erate 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 are 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 The mathematical constructs for the PERT distribution are shown below x min max x 47 FQ Al A2 1 B A1 A2 max min min 4 likely max an max int 4 likely max where Al 6 6 and A2 6 O oo a max min max min and B is the Beta function Min 4Mode Max Mean 6 Standard Deviation Ue lt 2M Skew 7 Maen te Min Max u 4
177. ere 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 User Manual Risk Simulator Software 65 2005 2011 Real Options Valuation Inc Input requirements Scale Beta gt 0 and can be any positive value Mean Alpha can be any value Lognormal Distribution 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 property values cannot become negative Conditions The three conditions underlying the lognormal distribution are e 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 The natural logarithm of the uncertain variable
178. ets 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 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 I lt Real Options 2 Select Exponential J Curve and enter in the desired inputs V a l uat Lon e g Starting Value of 100 Growth Rate of 5 percent End Period of 100 eee 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 5 Saturation Level Generate forec
179. 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 distributional parameters Conditions The three conditions underlying the negative binomial distribution are The number of trials is not fixed e The trials continue until the rth success e The probability of success is the same from trial to trial The mathematical constructs for the Pascal distribution are shown below x 1 f x 5 x s s 1 0 otherwise Sd p gt forall x gt s ko x D s Mp F x Zoae l p forallx gt s 0 otherwise Mean a P Standard Deviation s 1 p p Skewness eee 9 vr l p _ p 6p 6 Excess Kurtosis r l p Successes Required and Probability are the distributional parameters Input requirements User Manual Risk Simulator Software 53 2005 2011 Real Options Valuation Inc Successes required gt 0 and is an integer 0 lt Probability lt 1 Poisson Distribution 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 Poi
180. f 67 70 and 32 30 is of course 100 the total probability under the curve Income Risk Simulator Forecast Histogram Statistics Preferences Options_ Controls ncome 1000 Trials 4 Beeg o Type Right Tail gt _ 1000 Infinity Certainty 32 3044 Figure 2 13 Forecast Chart Probability Evaluation The forecast window is resizable by clicking on and dragging the bottom right corner of the forecast window It is also advisable that the current simulation be reset Risk 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 You can also hit the spacebar on the keyboard repeatedly to cycle among the histogram to statistics preferences options and control tabs In addition if you click on Risk Simulator Options you can access several different options for Risk Simulator including allowing Risk Simulator to start each time Excel starts or to only start when you want it to by going to Start 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 playi
181. facturing 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 Auto Econometrics Similar to basic econometrics but Auto Econometrics allows thousands of linear nonlinear interacting 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 Combinatorial Fuzzy Logic 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 Cubic Spline Curves 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
182. g 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 User Manual Risk Simulator Software 172 2005 2011 Real Options Valuation Inc Procedure Deseasonalization and Detrending Select the data you wish to analyze e g B9 B28 and click on Risk Simulator Tools Data Deseasonalization and Detrending 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 amp Review the two reports generated for more details on the methodology application and resulting charts and deseasonalized detrended data Procedure Seasonality Test Select the data you wish to analyze e g B9 B28 and click on Risk Simulator
183. ge p values the variances are statistically identical to one another User Manual Risk Simulator Software 149 2005 2011 Real Options Valuation Inc Hypothesis Test on the Means and Variances of Two Forecasts Statistical Summary 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 tiest with equal variances the population variance of forecast 1 is expected to 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 Hest 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 tw
184. gs of the forecast errors to Improve the forecast An MA q mode has the form y e t b 1 e t 1 b q e t q Finally an ARMA p q model has the combined form yit a 1 V t 1 a p Vitp e b t et b Qe t 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 Muitipie Correlation Coefficient Multiple R measures the correlation between the actual dependent variable Y and the estimated or fitted based on the regression equation This correlation is aiso 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 The AIC and SC are ofen used in model selection SC imposes a greater penalty for additional coefficients Generally the user should select a mode with the lowest value of the AIC and
185. guage 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 925 271 4438 or visit our website at www realoptionsvaluation com 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 or installation issues and fixes 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 Start Excel XP 2003 2007 2010 click on the License icon or click on 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 copy or click on the e mail Hardware ID link 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 or click on the License icon
186. he 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 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 i e 1 0 00021 obtains 0 999799 or 99 9799 which is the probability of getting at least 3 heads or more User Manual Risk Simulator Software 166 2005 2011 Real Options Valuation Inc 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 Type Formatting Single Value FA Value X 0 000201 Range of Values ooma Lower Bound f i 0 020695 0 057659 0 131588 Step Size a i 0 251722 0 411901 0 588099 0 748278 0 868412 0 942341 0 979305 0 994091 0 998712 0 999799 Upper Bound Figure 5 36 Distributional Analysis Tool Binomial Distribution s CDF 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 other
187. he sample dataset Using a 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 ttest is more conservative and does not require a known population standard deviation as in the Z test we only use this ttest Figure 5 31 Sample Statistical Analysis Tool Report Hypothesis Testing of One Variable Test for Normality The Normality test is a form of nonparametric test which makes no assumptions about the specific shape of the population from which the sample is drawn allowing for smaller sample data sets to be analyzed This test evaluates the null hypothesis of whether the 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 accept the 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
188. heet 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 search 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 e Ignore 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 Highlight 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
189. iables 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 Procedure amp Click on Risk 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 User Manual Risk Simulator Software 178 2005 2011 Real Options Valuation Inc 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 can 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
190. ial Function Rational Function 2010 9 9 420 89 2010 10 10 562 34 Number of Extrapolation Periods 34 2010 11 11 730 85 2010 12 12 928 43 ok Cad f gt Real Options Valuation www realoptionsvaiuation 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 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 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 foreca
191. iation Max Min 8 Skewness 0 for all inputs 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 Beta Distribution 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 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 User Manual Risk Simulator Software 55 2005 2011 Real Options Valuation Inc The value of the beta distribution lies in the wide variety of shapes it can assume when you vary 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 are near the maximum value The mathematical constructs for th
192. ibull distribution starts from a minimum value of 0 whereas this Weibull 3 or Shifted Weibull distribution shifts the starting location to any other value Alpha Beta and Location or Shift are the distributional parameters Input requirements Alpha Shape gt 0 05 Beta Central Location Scale gt 0 and can be any positive value Location can be any positive or negative value including zero User Manual Risk Simulator Software 76 2005 2011 Real Options Valuation Inc 3 FORECASTING Forecasting 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 QUANTITATIVE a Management Assumptions Market Research CROSS SECTIONAL Use Risk Simulator to run Monte Carlo Simulations use distributional fitting or nonparametric custom distributions Classical Decompositi Muli te Regressi SSIONS 5 Monte Carlo Simulation Multiple Regression Figure 3 1 Forecasting Methods Different Types of Forecasting Techniques Generally forecasting can be divided into quantitative and qualitative approaches Qualitative forecasting is used when little to n
193. ibution are shown below a pal for min lt x lt max F x 5 2b b 0 otherwise min max max min where a e ae fE Jormin lt x lt a 2 2b b Ta x a i in bral ET swe 2 2b b Mea Min Max 2 _ Miny Standard Deviation Max Skewness is always equal to 0 User Manual Risk Simulator Software 59 2005 2011 Real Options Valuation Inc 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 Erlang Distribution 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 application 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 carrier 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 J _ B f x pe forx 0 0 otherwise an Sxl BY l e x gt 0 F x e 2 a for x 0 otherwise Mean af Standard Deviation
194. ic 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 a l 8 1 1 roy Os ra T a PB Where the functional form above is for a Beta distribution and for a Parabolic function we set Alpha fora gt 0 B gt 0 x gt 0 Beta 2 and a shift of location in Minimum with a multiplicative factor of Maximum Minimum Wons Min Max 2 Standard Deviation ae 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 Pareto Distribution 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 in communication circuits The mathematical constructs for the Pareto are as follows User Manual Risk Simulator Software 68 2005 2011 Real Options Valuation Inc TP IO a forx gt L mean BL 1 standard deviation p B D 8 2
195. icantly different at the 1 5 and 10 significance levels High p values indicate that there is no statistically significant structural break Procedure Select the data you wish to analyze e g B15 D34 click on Risk Simulator Tools Structural Break Test enter in the relevant test points you wish to apply on the data e g 6 10 12 and click OK 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 Structural Break Test Time Series Data B15 034 iEn Test Breakpoints 6 10 12 zi eg 15 20 23 separate muge brespoms with commas 185 60 D 42 346 328 354 266 320 266 ES EZA m E A allefall fatela fsele 18 14630 38927 22322 3136 13035 19235 Figure 5 44 Structural Break Analysis User Manual Risk Simulator Software 175 2005 2011 Real Options Valuation Inc Trendline Forecasts Trendlines can be used to determine if a set of time series data follows any appreciable trend Figur
196. ick OK Figure 5 13 Review the results of the fit choose the relevant distribution you want and click OK Figure 5 14 User Manual Risk Simulator Software 142 2005 2011 Real Options Valuation Inc Student s T Triangular Uniform 47 56 185 86 53 30 i 49 71 204 77 53 09 50 24 145 61 52 09 50 36 219 85 45 81 R Single Fit 97 32 Distribution fitting takes existing raw data and statistically 87 25 finds the best fitting distribution i e by optimizing the 90 68 parameters of each distribution and performing statistical 3 5 56 hypotheses tests 98 74 Distribution Type 97 70 Fit to Continuous Distributions Fit to Discrete Distributions 90 05 Select Distributions to Fit 106 63 fa ea ww 66 48 104 38 Cauchy Distribution ChiSquare Distribution Exponential Distribution 123 26 103 65 ae ll al 86 04 F Distribution Gamma Distribution Gumbel M aximum Distribution 102 26 m 105 36 97 64 SelectAll Clear Al 109 15 110 98 52 25 128 85 49 08 108 09 49 01 166 19 52 81 95 38 50 51 197 52 50 74 93 21 49 72 279 06 47 98 Figure 5 13 Single Variable Distributional Fitting 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 distribu
197. idered 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 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 a 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 User Manual Risk Simulator Software 137 2005 2011 Real Options Valuation Inc 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 e The Analyze This Works
198. ies of all values at and below x occurring in the forecast Forecast Statistics 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 60 50 gt Bes Type Two Tal imiy Infinity Certainty 100 00 Figure 2 6 Forecast Chart User Manual Risk Simulator Software 27 2005 2011 Real Options Valuation Inc Income Risk Simulator Forecast Histogram Statistics Preferences Options Controls iNumber of Trials Mean Median Standard Deviation Variance Coefficient of Variation Maximum Minimum Range Skewness Kurtosis 25 Percentile 75 Percentile Percentage Eror Precision at 95 Confidence Figure 2 7 Forecast Statistics Forecast Chart Tabs Preferences The preferences tab in the forecast chart Figure 2 8A allows you to change the look and feel of the charts For instance if Always 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
199. iles e Multiple Profiles create and switch among multiple profiles in a single model This allows you to run scenarios on simulation by being able to change input parameters or distribution types in your model to see the effects on the results e 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 e Active Profile the last profile used when you save Excel will be automatically opened the next time the Excel file is opened User Manual Risk Simulator Software 199 2005 2011 Real Options Valuation Inc e Multiple Excel Files when switching between several opened Excel models the active profile will be from the current and active Excel model e 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 e 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 e Profile Location the profiles you create contai
200. imensional Simulation simulation of uncertain input parameters Precision Control determines if the number of simulation trials run is sufficient Super Speed Simulation runs 100 000 trials in a few seconds Forecasting Module 31 32 33 34 35 36 3T 38 39 40 41 42 ARIMA autoregressive integrated moving average models ARIMA P D Q Auto ARIMA runs the most common combinations of ARIMA to find the best fitting model Auto Econometrics tuns thousands of model combinations and permutations to obtain the best fitting model for existing data linear nonlinear interacting lag leads rate difference Basic Econometrics econometric and linear nonlinear and interacting regression models Combinatorial Fuzzy Logic Forecasts time series forecast methods Cubic Spline nonlinear interpolation and extrapolation GARCH volatility projections using generalized autoregressive conditional heteroskedasticity models GARCH GARCH M TGARCH TGARCH M EGARCH EGARCH T GJR GARCH and GJR TGARCH J Curve exponential J curves Limited Dependent Variables Logit Probit and Tobit Markov Chains two competing elements over time and market share predictions Multiple Regression regular linear and nonlinear regression with stepwise methodologies forward backward correlation forward backward Neural Network Forecasts linear nonlinear logistic hyperbolic tangent and cosine User Manual Risk Simu
201. imulation Trials 500 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 5002 Number of Optimization Runs 2044 Figure 4 10 Setting Up the Stochastic Optimization Problem User Manual Risk Simulator Software 127 2005 2011 Real Options Valuation Inc Viewing and Interpreting Forecast Results 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 Notes Super Speed Simulation with Optimization 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 checkb
202. in the model For instance Figure 5 7 shows another spider chart where nonlinearities are fairly evident the lines on the graph are not straight but curved 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 User Manual Risk Simulator Software 136 2005 2011 Real Options Valuation Inc Dividend Yield 0 0 60 00 40 00 20 00 0 00 20 00 40 00 60 00 Figure 5 7 Nonlinear Spider Chart Additional Notes on Tornado 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 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 cons
203. ine 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 are 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 5 The cubic 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 can 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 13 14 Years Spot Yields 15 0 0833 4 55 These are the yields 16 0 2500 4 47 that are known and 17 0 5000 4 52 are used as inputs in 18 1 0000 4 39 the Cubic Spline 19 2 0000 4 13 Interpolation and 2
204. ing the Right Probability Distribution 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 e 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 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 User Manual Risk Simulator Software 46 2005 2011 Real Options Valuation Inc a forecast Forecasts are events usually with formulas or functions that you define as important outputs of th
205. ings show at a glance which asset class has the lowest risk or the highest return and so forth Running an Optimization To run this model simply click on Risk Simulator Optimization Run Optimization Alternatively and for practice you can set up the model using the following steps illustrated in Figure 4 10 Start a new profile Risk Simulator New Profile 1 For stochastic optimization set distributional assumptions on the risk and returns for each asset class That is select cell C6 set an assumption Risk Simulator Set User Manual Risk Simulator Software 125 2005 2011 Real Options Valuation Inc Input Assumption and designate your own assumption as required Repeat for cells C7 to D9 2 Select cell E6 and define the decision variable Risk Simulator Optimization Set Decision or 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 G6 3 Then use Risk Simulator s copy on cell E6 select cells E7 to E9 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 4 Next set up the optimization s constraints by selecting Risk Simulator Optimization Constraints selecting ADD and selecting the cell E11 and making it equal 00 total allocation and do
206. ion 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 occurring are higher for a high kurtosis distribution e g stock market returns are leptokurtic or have high kurtosis Ignoring a project s kurtosis may be detrimental Typically a higher excess kurtosis value indicates that the downside risks are higher e g the Value at Risk of a project might be significant 0 02 Skew 0 ze Kurtosis gt 0 y Hi h2 Figure 2 24 F
207. ion Size gt 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 User Manual Risk Simulator Software 51 2005 2011 Real Options Valuation Inc Negative Binomial Distribution The negative binomial distribution 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 p and number of successes required R are the distributional parameters It is essentially a superdistribution of the geometric and binomial 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 e The trials continue until the rth success e The probability of success is the same from trial to trial The mathematical
208. ion 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 Show Forecast Window Allows the user to show or not show a particular forecast window The default is to always show a forecast chart il Forecast Properties Forecast Name income E Forecast Precision Precision Level Confidence Error Level of Mean or from the Mean Options V Show Forecast Window OK Cancel Figure 2 5 Set Output Forecast 4 Running the Simulation If everything looks right simply click on Risk Simulator Run Simulation or click on the Run icon on the Risk Simulator toolbar and the simulation will proceed You may also reset a simulation after it has run to rerun it Risk Simulator Reset Simulation or the reset simulation icon on the toolbar or to pause it during a run Also the step function Risk Simulator Step Simulation or the step simulation icon on the toolbar allows you to simulate a single trial one at a time useful for educating others on simulation i e 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
209. is considered to be nonlinear these data points are from a squared function S Curve 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 occurring phenomenon Stochastic Processes 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 Time Series Analysis and Decomposition 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 seasona
210. it a z lt var name VAR95 notes data gt Nonparametric Chi Square Independence lt var name VAR96 notes data gt Nonparametric Chi Square Population Varia ere aes lt var name VAR97 notes data gt Nonparametric Friedman s Test 2 lt var name VAR98 notes data gt Nonparametric Kruskal Wallis Test lt var name VAR99 notes data gt Nonparametric Liliefors Test lt var name VAR100 notes data gt Nonparametric Runs Test lt data gt Nonparametric Wilcoxon Signed Rank One lt analysis gt Nonparametric Wilcoxon Signed Rank Two lt model name Absolute Values notes id 114 pareameter VAR77 gt Parametric One Variable T Mean lt model name ANOVA Randomized Block notes id 60 parameter VAR60 VAR61 VAR62 VAR63 gt Parametric One Variable Z Mean lt model name ANOVA Single Factor Multiple Treatments notes id 61 parameter Parametric One Variable Z Proportion seer Sona ae Way notes id 62 parameter mea aie OaE VAR4O VAR41 VAR42 VAR43 VAR44 VAR45 VAR46 VAR47 VAR48 VAR49 VAR5O VAR51 earpenetric wo a Mens 110 3 gt STEP 4 Save Optional You can save multiple analyses and notes in the profile for future retrieval 111 lt model name ARIMA 1 0 1 notes id 17 parameter VAR1 1 fa o Name Auto Econometrics Detailed a Notes This is a test model running AE methodology inside ROV BizStats 115 lt model name ARIMA 1 0 2 n
211. ither efficient nor accurate One quick test of the presence 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 Correlation Matrix CORRELATION x2 x3 x4 KS x1 0 333 0 959 0 242 0 237 X2 1000 0 349 0 319 0 120 x3 1000 0 196 0 227 x4 1 000 0 290 Variance Inflation Factor VIF x2 x3 x4 KS x1 1 12 12 46 1 06 1 06 x2 WA 1 14 1 11 1 01 x3 WA 1 04 1 05 x4 WA 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 User Manual Risk Simulator Software 159 2005 2011 Real Options Valuation Inc 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 values are listed
212. k OK and review the forecast report www realoptionsvaluation com 7 000 JSCurves 6 000 s ssssscesscssonscocssg3 P The J S curves stand for J curve exponential growth and S curve Maturity and logistic growth 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 environment becomes saturated at capacity S curve 5 000 4 000 Value Exponential J Curve Logistic S Curve Growth Phase Starting Value 200 2 000 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 i End Period 100 Period Figure 3 18 S Curve Forecast 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 nature changes over time according to some econometric characteristics The theoretic
213. k Simulator Software 163 2005 2011 Real Options Valuation Inc Stochastic Process Parameter Estimations 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 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 and other stochastic time series data givena 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
214. k to copy or click on the E mail HWID link to generate an e mail with the HWID e Troubleshooter trun the Troubleshooter from the Start Programs Real Options Valuation Risk Simulator folder and run the Get HWID tool to obtain your computer s HWID User Manual Risk Simulator Software 198 2005 2011 Real Options Valuation Inc 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 e Books Getting Started Videos Models White Papers resources available on our website www realoptionsvaluation com download html or www rovdownloads com download html TIPS Optimization e 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 TIPS Prof
215. ket are correlated but there is no causation between the two There are two general types of correlations parametric and nonparametric correlations 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 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 User Manual Risk Simulator Software 34 2005 2011 Real Options Valuation Inc function the correlation inputs require
216. l Valuation v 7 Private Risk Discount Rate 5 00 Internal Rate of Return 55 68 Terminal Period Growth Rate 2 00 Return on Investment 191 40 9 Effective Tax Rate 40 00 Profitability Index 2 91 11 2009 2010 20114 ES Scenario Analysis ama eee Pon 7 2018 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 Product B Avg Price Unit 12 25 12 50 12 75 Location of Output Variable G6 14 50 Product C Avg Price Unit 15 15 15 30 15 45 Soe 16 50 Product A Sale Quantity 000s 50 50 50 First Input Variable to Test c9 Second Input Variable to Test C12 50 Product B Sale Quantity 000s 35 35 351 Next enter the starting value ending value and number of steps or the step size to test 35 itv i yee 1 231 E 1 268 A 1 305 al Peni liez 1 562 A evenues 231 268 305 562 k Starting Value j Starting Value Direct Cost of Goods Sold 184 76 190 28 195 79 oS 10 234 38 20 Gross Profit 1 046 99 1 078 23 1 109 46 Endina var 1 328 13 Operating Expenses 157 50 157 50 157 50 C Steps 157 50 2 Sales General and Admin Costs 15 75 15 75 15 75 Step Size 0 01 E C Step Size 15 75 Operating Income EBITDA 873 74 904 98 936 21 l 1 154 88 Depreciation 10 00 10 00 10 00 Cancel 10 00 Amortization 3 00 3 00 3 00 3 00 EBIT 860 74 891 98 923 21 1 141 88 Interest Payments 2 00 2 00 2 00 2 00 2 00 3 00 4 00 5 00 6 00
217. lat Tes eat latees ade 196 LIPS Forecast Charts iterator see eek acta Pes ee cai Fata a tet ie Be ee aod rh ole oe Ne es ie 196 TIPS Forecasting oreren tan e siecle sae cos E E ace oe bin Big te eg bse eg leg tea aa gts E Bee uae 197 TIPS Forecasting ARIMA vs sicsces siesccesnlesetevedepectvwiete ivesete Suv A E REEE E E AEE A 197 TIPS Forecasting Basic Econometrics cccccceeccc cee nteeeteeee cee teneeeecesnseceseseeessseaeessnnieeesenaeesenea 197 TIPS Forecasting Logit Probit and Tobit cccccccccccccceeececeeceseeeeeneeeeeeeeeeeceseeeesneeseeeeeneeeensa 197 TIPS Forecasting Stochastic PrOCCSSCS cccccecseseeeseetenseteseeseeeseceesenseesesecseieesseeesenteseneeeenaeenes 197 TIPS Forecasting Trendlines ntie id i a a A S N Ea 198 TIPS FURCHON Calls sirnana a a a a eg Was e E S 198 TIPS Getting Started Exercises and Getting Started Videos ee 198 User Manual Risk Simulator Software 6 2005 2011 Real Options Valuation Inc FIPS HOLA WALCHID otic Koctaa tia be a Batu a aa aeae ae a ea de nee Buc cede tbudtites 198 TIPS Latin Hypercube Sampling LHS vs Monte Carlo Simulation MCS n se 199 TLS OMMNCER CSOULCES tos inti Batata e a e Nia Be Ciel Bee doa lain te fe Bitoni balls 199 PIPS OPEN ZAtiOn oo iiaeie oe ea cael nts cantata Matas ols otal ntaa cates clued te Staten ate vuln ate tae aA 199 LIPS PHO UOS ores s sooo eta Sacer sree T os gusputh oda tues E 199 TIPS Right Click Shortcut and Other Shortcut Keys
218. lations 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 Objective 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 User Manual Risk Simulator Software 115 2005 2011 Real Options Valuation Inc Procedure Ke Ke Open the example file and start a new profile by clicking on Risk Simulator New Profile and provide it a name The first step in optimization is to set the decision variables Select cell E6 set the first decision variable Risk Simulator Optimization Set Decision and click on the link icon to select the name
219. lator Software 14 2005 2011 Real Options Valuation Inc 43 44 45 46 Nonlinear Extrapolation nonlinear time series forecasting S Curve logistic S curves Time Series Analysis 8 time series decomposition models for predicting levels trends and seasonalities Trendlines forecasting and fitting using linear nonlinear polynomial power logarithmic exponential and moving averages with goodness of fit Optimization Module 47 48 49 50 51 52 53 54 55 56 57 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 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 additional statistics for stochastic optimization including percentiles as well as conditional means which are critical in computing condi
220. le 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 Results Interpretation In interpreting the results of an ARIMA model most of the specifications are identical to the multivariate regression analysis see Modeling Risk Applying Monte Carlo Simulation Real Options Analysis Stochastic Forecasting and Portfolio Optimization 2nd 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 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 Fo
221. lick and select Visualize to chart the data e Ifacell 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 Ctr Home to jump to the top left cell Ctr 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 your notes short and simple e 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 th
222. liers 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 prior 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 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 independent variable to the second or third power and run a regression or forecast using the nonlinearly transformed data Diagnostic Results Heteroskedasticity Micronumerosity Outlier
223. lity 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 User Manual Risk Simulator Software 8l 2005 2011 Real Options Valuation Inc performs a back fitting backcast of historical data before it provides estimates of future values forecasts Trendlines 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 Running the Forecasting Tool in Risk Simulator In general to create forecasts several quick steps are required Start Excel and enter in or open your existing historical data 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 Start Real Options Valuation Risk Simulator Examples or accessed directly through Risk Simulator Example Models ar Orecas
224. location 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 A B E D E F G H 1 2 3j ASSET ALLOCATION OPTIMIZATION MODEL 4 Asset Class Annualized Volatility Allocation Ragdulred Required Return to Description Returns Risk Weights Minimum ae Maximum Risk Ratio 5 Allocation Allocation 6 Asset 1 10 60 12 41 10 00 40 00 0 8544 fi Asset 2 11 21 16 16 10 00 40 00 0 6937 8 Asset 3 10 61 15 93 10 00 40 00 0 6660 z Asset 4 10 52 12 40 10 00 40 00 0 8480 10 11 Portfolio Total 10 7356 7 17 100 00 12 Return to Risk Ratio Figure 4 9 Asset Allocation Model Ready for Stochastic Optimization 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 rank
225. lts Result Base Value 96 6261638553219 Input Changes ae Output 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 36 00 44 00 40 00 3 43 189 83 z 9 00 11 00 10 00 16 71 176 55 11 03 13 48 12 25 23 18 170 07 K 45 00 55 00 50 00 30 53 162 72 5 31 50 38 50 35 00 40 15 153 11 7 13 64 16 67 15 15 48 05 145 20 18 00 22 00 20 00 138 24 57 03 a 13 50 16 50 15 00 116 80 76 64 i 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 0 00 0 00 0 00 96 63 96 63 i 0 00 0 00 0 00 me s I Price 13 cof 6 665 cous oS 2 Discount Rate 0 165 I o 55 Price Erosion 0 055 B 0 045 Sales Growth oral 0 022 Depreciation 9 11 interest 22 18 Amortization 27 33 Capex 0 Net Capital o Figure 5 3 Tornado Analysis Report Notes 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
226. lts 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 Risk Simulator Change Simulation 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 Risk 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 Book 1 2010 10 14 Second Profile Book 1 2010 10 14 N A Third Profile Book 1 2010 10 14 N A V View simulation profiles in all workbooks baa oie Figure 2 2 Change Active Simulation User Manual Risk Simulator Software 22 2005 2011 Real Options Valuation Inc 2 Defining Input Assumptions The next step is
227. m x z Maximum infinity Ej maximum and most likely values to occur ar For example you could describe the Ate F number of cars sold per week when past IE Enable Dynamic Simulations sales show the minimum maximum and En seam K aaan __CANCE Figure 2 3 Setting an Input Assumption Note that you can also set assumptions by selecting the cell you wish 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 User Manual Risk Simulator Software 23 2005 2011 Real Options Valuation Inc 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 2 4 there are several key areas in the Assumption Properties worthy of mention 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 Distribution Gallery This area to the left shows all of the different distributions available in the software To change the
228. mation Model Ready 2J SS Tornado Analysis Figure 5 46 Model Checking Tool User Manual Risk Simulator Software 177 2005 2011 Real Options Valuation Inc Percentile Distributional 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 method is suitable for use when there are insufficient data only when percentiles and moments are 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 fit 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 var
229. 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 set 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 p lags of the time series in the equation An AR p model has the form y ayy ApYi p The second component is the integration d order term Each integration order corresponds to differ
230. me series data then click on Risk Simulator Forecasting GARCH and click on on the data location link 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 conditional 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 9 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 a 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 Make sure to arrange your raw price
231. mized options e Multiple Asset and Multiple Phase SLS for solving multiphased sequential options options with multiple underlying assets and phases combination of multiphased 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 User Manual Risk Simulator Software 8 2005 2011 Real Options Valuation Inc Installation Requirements and Procedures To install the software follow the on screen instructions The minimum requirements for this software are e Pentium IV processor or later dual core recommended e Windows XP Vista or Windows 7 e Microsoft Excel XP 2003 2007 2010 or later e Microsoft NET Framework 2 0 or later versions 3 0 3 5 and so forth e 350 MB free space e 1GB RAM minimum 24GB 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 NET 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 lan
232. mn amp Select Risk Simulator Forecasting Time Series Analysis amp Choose the model to apply enter the relevant assumptions and click OK Results Interpretation 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 simply 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 User Manual Risk Simulator Software 83 2005 2011 Real Options Valuation Inc assumptions if the option to automatically generate assumptions is chosen and if a simulation profile exists The graph illustrates these historical fitted and forecast val
233. mns C and D and apply the dynamic optimization and stochastic optimization for additional practice Decision Variable Properties Decision Name Asset Class 1 E Decision Type Continuous e g 1 15 2 35 10 55 Lower Bound 0 05 Upper Bound 0 35 E Integer e g 1 2 3 Lower Bound E Upper Bound E Binary 0 or 1 a User Manual Risk Simulator Software 116 2005 2011 Real Options Valuation Inc Constraints MSES17 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 Optimization 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 1004 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 100 Number of Optimization Runs 24 Figure 4 2 Running Continuous Optimization in Risk Simulator User Manual Risk Simulator Software 117
234. n 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 The minimum number of items is fixed The maximum number of items is fixed e 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 User Manual Risk Simulator Software 73 2005 2011 Real Options Valuation Inc 2 x Min Max Min Likely min 2 Max x Max Min Max Likely for Min lt x lt Likely f x for Likely lt x lt Max Mean Min Likely Max Standard Deviation a Min Likely Max Min Max Min Likely Max Likely V2 Min Max 2 Likely 2Min Max Likely Min 2Max Likely 5 Min Max Likely MinMax MinLikely MaxLikely Excess Kurtosis 0 6 this applies to all inputs of Min Max and Likely Skewness Minimum value Min most likely value Likely and maximum value Max are the distributional parameters I
235. n Inc Results Interpretation 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 Return to Profitability R Optimization Complete ENPV Cost Risk Rk RiskRatio Index Selection Project 1 458 00 1 732 44 54 96 12 00 8 33 1 26 Project 2 1 954 00 859 00 1 914 92 96 00 1 02 327 Project 3 1 599 00 1 845 00 1 551 03 97 00 1 03 1 87 Project 4 2 251 00 1 645 00 1 012 95 45 00 222 237 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 00 756 00 5 633 10 198 00 0 51 475 Project amp 1 235 00 115 00 926 25 75 00 1 33 11 74 Project 9 1 945 00 125 00 2 100 60 108 00 0 93 16 56 Project 10 2 250 00 456 00 1 912 50 85 00 1 18 591 Project 11 549 00 45 00 263 52 48 00 2 08 13 20 P
236. n 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 case 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 Procedure amp Open a spreadsheet with existing data for fitting amp Select the data you wish to fit data should be in a single column with multiple rows amp Select Risk Simulator Tools Distributional Fitting Single Variable amp Select the specific distributions you wish to fit to or keep the default where all distributions are selected and cl
237. n 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 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 ROV PROBABILITY DISTRIBUTIONS Distributions 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 view multiple distributions use Risk Simulator s Overlay Chart tool Distribution Arcsine A Charts and Tables Chart Change First Parameter Change Second Parameter Theoretical Distribution Minimum 10 B Parameter x l Simulated Distribution Maximum 20 CDF Fo 0 From 0 Trials 10000 RandomX 12 From To Series Ta 1 To 1 0 Custom Step 01 Step 0 Chart Table SESH BttEeeOOkENUGGoa
238. n 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 range 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 would 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
239. n other cases a multivariate regression can be performed where there are multiple or n number of independent X variables where the general regression equation will now take the form of Y P B X B8 X 2 X 8 X e In this case the best fitting line will be within an n 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 as shown on the right panel of Figure 3 6 To find the best fitting line that minimizes 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 User Manual Risk Simulator Software 86 2005 2011 Real Options Valuation Inc Min 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 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
240. 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 courses 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 answers to the more mathematically elegant methods So what is Monte Carlo simulation and how does it work 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 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 Ca
241. ncorrect run this tool from Risk 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 properly User Manual Risk Simulator Software 176 2005 2011 Real Options Valuation Inc RY ed o s I DCE ROT and Volatiity xls Compatibility Mode Microsoft Excel non Home Insert Page Layout Formulas Data Review View Developer Risk Simulator E WN allii T i f A rs ib ia A Set Objective D Set Decision New Change Edit SetInput SetOutput Copy Paste Remove Run RunSuper Step Reset Forecasting Run p Analytical Profile Profile Profile Assumption Forecast Speed 5 Optimization G Set Constraint Tools New Simulation Profile Assumptions Forecasts Editing Simulation Run Forecasting Optimization Tools 3 Edit Simulation Profile Discounted Cash Flow ROI Model amp Change simutation Prote pE Sethi it A ti li aiaa Discounted Cash Flow ROI Model l d Set Output Forecast o Copy Parameter 2009 Sum PV Net Benefits 4 762 09 Discount Type Discrete End of Y ear Discounting X 2009 Sum PV Investmen
242. ne Power Trend Line Rate Detrended Trend Line Static Mean Detrended Trend Line Static Median Detrended Variance Population Variance Sample Volatility 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 100 0 05 0 25 10 lt gt Initial Value Drift Rate ey STEP 4 Save Optional You can save multiple analyses and notes in the profile for future retrieval Stdev Population Stdev Sample Stepwise Regression Forward Stepwise Regression Backward Stepwise Regression Correlation Stepwise Regression Forward Backward Figure 5 53 User Manual Risk Simulator Software 185 2005 2011 Real Options Valuation Inc ROV BizStats Data Visualization and Results Charts Zieaee rovecssts I _ _ File Data Language Help STEP 1 Data Manually enter your data paste from another application or load an example dataset with analysis Dataset Visualize Vv 1 060_ANOVARandomizedBlocksMultipleTreatments VARGO VARG1 VAR62 VAR63 062_ANOVATwoWayAnalysis VAR40 VAR41 VAR42 VAR43 VAR44 VAR45 VAR46 VAR47 VAR48 VAR49 VAR5O VAR51 3 061_ANOVASingleFactorMultipleTreatments VAR57 VAR58 VAR59 060_ANOVARandomizedBlocks
243. ned during the simulation run Procedure amp Open or create a model define assumptions and forecasts and run the simulation Select Risk Simulator Create Report Figure 5 21 Simulation Example Profile General Number of Trials 1000 Stop Simulation on Error No Random Seed 423456 Enable Correlations Yes Assumptions Name aple First Assumption Name e Second Assumption Name iple Third Assumption Enabled Yes Enabled Yes Enabled Yes Ceil SESE Celt SEBO Cell E 10 Dynamic Simulation No Dynamic Simulation No Dynamic Simulation No Range Range Range Minimum intinity Minimum infinity Minimum infinity Maximum infinity Maximum infinity Maximum infinity Distribution Normal Distribution Triangular Distribution Beta Mean 100 Minimum 10 Alpha 2 Standard Deviation 10 Most Likely 0 Beta 5 Maximum 10 0 00 A2 At 77AG 975R 1N7 44 172 34 137 1 ARR 597 197 Forecasts Name Sample First Forecast Number of Datapoints 1000 Enabled ves Mean 100 0400 Celt E 12 Median 99 8427 Standard Deviation 9 8331 Forecast Precision Variance 96 6903 Precision Level Average Deviation 7 8397 E Enor Level Maximum 134 5452 A Minimum 66 9132 I Range 67 6320 E Skewness 0 1121 i Kurtosis 0 1401 ig 25 Percentile 93 3563 75 Percentile 106 3153 eet ae Error Precision at 95 0 0064 Name nple Second Forecast Number of Datapoints 1000 Enabled Yes Mean 0 0806 Celt E 13 Median 0 0755 Standard Deviation 4 141
244. ng around with the forecast chart outputs and various bells and whistles especially the Controls tab User Manual Risk Simulator Software 33 2005 2011 Real Options Valuation Inc Correlations and Precision Control 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 7 of two variables x and y using p nix 9x dy Se O x fry Sy 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 mar
245. ning the assumptions forecasts decision variables objectives 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 e 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 e 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 e Saving and Extracting Simulated Data in Excel extracts a simulated run s assumptions and forecasts the Excel file itself will still have to be saved in order to save the data for retrieval later e 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 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 are 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 l
246. nput requirements Min lt Most Likely lt Max and can take any value However Min lt Max and can take any value Uniform Distribution 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 The minimum value is fixed The maximum value is fixed e All values between the minimum and maximum occur with equal likelihood The mathematical constructs for the uniform distribution are as follows f x a for all values such that Min lt Max Me Min ax Mean Min Max 2 Max Min Standard Deviation Skewness 0 this applies to all inputs of Min and Max User Manual Risk Simulator Software 74 2005 2011 Real Options Valuation Inc Excess Kurtosis 1 2 this applies to all inputs of Min and Max Maximum value Max and minimum value Min are the distributional parameters Input requirements Min lt Max and can take any value Weibull Distribution Rayleigh Distribution 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
247. nstance if User Manual Risk Simulator Software 158 2005 2011 Real Options Valuation Inc we see a 283 reversion rate chances are a mean reversion 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 use e g stock prices tend to follow a Brownian motion random walk whereas inflation rates follow a mean reversion process or a jump diffusion process is more appropriate should you be forecasting the price of electricity 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 is ne
248. nt as it calculates if each of the coefficients is statistically significant in the presence of the other regressors This means that the 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 Statistic exceeds the Critical Statistic at the relevant degrees of freedom df The three main confidence levels used to test for significance are 90 9596 and 99 If a Coefficient s tStatistic exceeds the Critical level it is considered statistically significant Alternatively the p Value calculates each tStatistic 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 99 confidence levels The Coefficients with their p Vaiues 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 of Mean of Squares Squares ranse p Vahie Hypothesis Test Regression 38415447 5277 19207723 7638 3171851 1034 0 0000 Critical F statistic 99 confidence with df of 2 and 432 4 6546 Residual 2616 0549 6 0557 Critical F statistic 95 confidence with af of 2 and 432 3 0466 Total 38418063 58
249. nterpolate Observation KnownX Known Y 55 4 29 interpolate 1 0 0833 4 55 6 0 4 32 interpolate 2 0 2500 447 65 4 35 interpolate 3 0 5000 452 7 0 4 38 Interpolate 4 1 0000 4 39 7 5 441 interpolate 5 2 0000 4 13 8 0 444 interpolate 6 3 0000 4 16 8 5 447 interpolate T 5 0000 4 26 9 0 450 Interpolate 8 7 0000 4 38 9 5 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 User Manual Risk Simulator Software Ill 2005 2011 Real Options Valuation Inc 4 OPTIMIZATION This 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 stochastic optimizations 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 t
250. ntiGl 2 Distributiot anueg eae e e EEE E eE e AETA ne co everett 61 Extreme Value Distribution or Gumbel Distribution c 62 F Distribution or Fisher Snedecor Distribution ccccccccccessecesseeeneeeseseeseneeseeecenseeeeseeseaeeseeseate 62 Gamma Distribution Erlang Distribution 0 cccccccccecececeeeessecenseesnseeseeeseeesenseseuseecnseesneesetenate 63 GD IGCED I SUIDULL ON oreert eens Oe ean aad Maree tie che Mati otes Lie dala nod a atte babi h 64 Logistic DisttibutiOn orriren oain bia ineine Sond barbie suse Salt sag cde sane bates nbd tobe a edie def obuacnee sododeds 65 L gnormal Distribution nimre oinas eiee aaae eaa eaaa aiaa aaaea eaaa aa aasa naie eadteussnedonss 66 L gn rmal 3 Distribution sn sesine ake Bia tne Sede Coie batind thn nde sulie dafie dade Sede dnhc bathe Son toibasihd shed vn Sevtadeendstiedubess 67 NOP IAL DISEPIDULION coc sis ices EAEE E auc daves owe bes Shwe ate cpus dvs ceuntsoad eas tabs ceandahe cosnedbedessesondouade 67 P raboli Distributiom sarren aae aaeegan dees dawstseed ui E AAAA AE AEE RARER PI EES NESER SAE aeia 68 PGPCTO DIS UIDULION EERE AEA E NA E A TAE N 68 Pearson A DIiStri Buti Oh AEE EA EE E 69 Pearson A E BIIN 01017111 PAA E A OE E E ious 70 PERT Distributi On oeeo O E E ass A E A Aan aba Oca sR CER 71 Power Distribution srecen E E E NE a O ag REA A A E 71 Power 3 Distribution e reee etat eer EEA es oa ina TAE e EE E O EE AARRE ea EE NE EEE event 72 Student s t Distribution cosines
251. o OH Kew E a Minimum 10 0000 Maximum 20 0000 11 2669 12 3338 13 4007 14 4675 15 5344 16 6013 17 6682 18 7351 19 8020 Language English X Chart Type 2D Area Run Figure 5 49 ROV Probability Distribution PDF and CDF Charts User Manual Risk Simulator Software 181 2005 2011 Real Options Valuation Inc Bomoa OOOO a Distributions 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 view multiple distributions use Risk Simulator s tool Overlay Chart tool Distribution Beta Charts and Tables Chart Change First Parameter H Change Second Parameter Theoretical Distribution Apa 2S PDF G Porras er Alpha Beta Simulated Distribution 5 cDF po Beta D Fom 0 Fom o Trials 1000 RandomX 06 ICDF Fom To Series oE l o 0 Res Custom Step 01 sep 01 0 0 460800 2 5 5 J i5 3 5 e g Choose Gamma distribution set Alpha and Beta as parameters to change and enter 2 3 and 5 9 in the two custom input Copy Chat Table boxes for generating Gamma 2 5 and Gamma 3 9 charts SOSH BE he oO ORE UU AGoOS WORK ROMO e i v L Cor Language Engish v M Chat Type 2D Line x Gidines Run Ghose
252. o and a simulation is run assuming no cross 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 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 User Manual Risk Simulator Software 21 2005 2011 Real Options Valuation Inc 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 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 resu
253. o ENRE EEE REE AEE RE R E 72 Triangular Distrib ti n so sonete i E E aian Ea rie ao aiaia AAA aea 73 Uniform Distribution roseis iea Goce ENAERE e EEEE EE ENE E AEE E E 74 Weibull Distribution Rayleigh Distribution ccccccccccceccccceeeceeceetneeeeeenseeeececeeeeeseeeesuaeeeensseeees 75 Weibull 3 Distribution ssi etecsel yes 4 Bei a Le ae geo cae ayaa gles ge a ne aes 76 User Manual Risk Simulator Software 4 2005 2011 Real Options Valuation Inc 3 FORECASTING ccsscsssssssssscsccccserscsseseescssessesssseseesecesssesssssseeseesessessesseseessessesssessesessenes 77 Different Types of Forecasting Techniques cccccccccceccccceceeeeee tee e escent escent ee ennaeeeeetneeeeees 77 Running the Forecasting Tool in Risk Simulator cccccccccccscccceceeectee tees scence ee eeteeeeenneeeeees 82 TAME SCVICSANGLYSIS aAA T E taht decidir A icin dah ceeibadl sh untangle 83 Multivariate RESTession aver istetsse acti Ep EURER RURU Cede haa alate RER EE E N E 86 Stochastics FOV CCAS UNG Sec Nise EE EEEE AEE ET EEO EEEE 90 Nonlinear Extrapolation seeaaaaeeaaanenaeseeeneseeeneessneesreeressreetssrretssstrsssrtntssrentssreressse reeset 92 Box Jenkins ARIMA Advanced Time Series cccccccccccceecseeeeseeeeueeteneeetsueeteaeeeseueeeeneeeenneees 94 AUTO ARIMA Box Jenkins ARIMA Advanced Time Series o on 99 BaSiC Econometrics 33 ei eae hk ARR AS 100 JS Curve F OFCCOSIS ineo ie a e T E E E E E 102 GARCH Volatility
254. o reliable historical contemporaneous or comparable data are 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 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 User Manual Risk Simulator Software 77 2005 2011 Real Options Valuation Inc 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
255. o tailed hypothesis test is performed on the nuli 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 07 0 05 or 0 10 this means thatthe hypothesis is rejected which implies that the forecast means are statistically significantly different at the 196 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 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 t statistic 0 32947 P value for t statistic 0 74184 Computed F statistic 4 026723 P value for F statistic 0 351212 Figure 5 19 Hypothesis Testing Results Notes 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 geographic
256. 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 off manually or you can use the Ignore Possible Integer Values function to turn all of them off simultaneously Sensitivity Analysis Theory 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 are 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 User Manual Risk Simulator Software
257. 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 Chi Square Distribution The chi square distribution is a probability distribution used predominatly in hypothesis testing and is User Manual Risk Simulator Software 57 2005 2011 Real Options Valuation Inc related to the gamma and standard normal distributions For instance the sum of independent normal distributions is distributed as a chi square 7 with k degrees of freedom d Z Z Z The mathematical constructs for the chi square distribution are as follows f x 0 5 k 2 1 x 2 T k 2 Mean k Standard Deviation 42k Skewness E k Excess Kurtosis forallx gt 0 12 T is the gamma function Degrees of freedom k is the only distributional parameter The chi square distribution can also be modeled using a gamma distribution by setting the k shape parameter equal to a and the scaleequal to 2S where S is the scale Input requirements Degrees of freedom gt 1 and must be an integer lt 300 Cosine Distribution The co
258. om 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 dj Income A Simulation Model D10 each of the cost cells Finally define forecast outputs for thy C Income B Simulation Model G10 the simulation Bootstrap Select a forecast to run the nonparametric bootstrap Income A Risk Simulator Forecast Histogram Statistics Preferences Options Controls Statistics to Bootstrap v Mean Median V Standard Deviation Variance Skewness Kurtosis 25 Percentile 75 Percentile Number of Bootstrap Trials Figure 5 16 Nonparametric Bootstrap Simulation User Manual Risk Simulator Software 146 2005 2011 Real Options Valuation Inc Results Interpretation In essence nonparametric bootstrap simulation can be thought of as simulation based on a simulation Thus after running a simulation 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 f
259. op 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 Discrete Optimization that is an optimization that is run on a discrete or static model where no simulations are run In other words 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 For 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
260. 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 model e g computing the Sharpe ratio in asset allocation and optimization problems where we have mean divided by standard deviation of the portfolio then this approach should be used User Manual Risk Simulator Software 112 2005 2011 Real Options Valuation Inc 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 are used in the optimization model to find the optimal allocatio
261. or Software nnossanneennneenanseennneeennesseenessseeresseeressererssrereesseee 8 Installation Requirements and Procedures nnneoanneennnnenensennnesseenesereressrenessrereseserresreee 9 LT CONSTNG en e sis a EE a RE EE E E A E A tales al allel cal A A og 9 WHAT S NEW IN VERSION 2011 csscsscssssssecsssssseccsssscscssssescssssscesescsssesessssessesscssessoes 13 A Comprehensive List of Risk Simulator s Capabilities ccccccccccccccceeceeeeteneeeecenseeeeeenaeees 13 2 MONTE CARLO SIMULATION sssscssssssscsssssseccssssseecsssssesessssssescsssssesesssssssesssseesones 18 Whatls Monte Carlo Simulation sccicet casicasisoeisast cuss sales Sonulst catacasisasgensacabasageda soontacaganasecssoe3s 18 Getting Started with Risk Simulator nnooaaaneenaanenneenenneeenneessenessseessseeressererssseressserrrsseee 19 A High Level Overview of the Software sssseeeeeeeeeserserisrersrrrsrersrrstresrtesrresrresrrssereserssees 19 Running a Monte Carlo Simulation c cccccecccsecceeseesececeeneeseneeseneeeesseesseesseecseessureseneesseeesteeaes 20 1 Starting a New Simulation Profile ccccccccccccceeceseteeesnsecensesenseesseesseeseueeseesessneeesneseneesseesaes 20 2 Defining Input ASSUMPTIONS cesses terein entien Eo EE E EEE EE EE CEE AEEA 23 3 Defining Output Forecasts a nnter ee aaae o e a eae a o eee aE e eaae 25 4 Running the Simulation ccccccccccccccceccseseceentecenseeseecsseeseceesensec
262. orecasts in the model Forecasts can only be defined on output cells with equations or functions The following describes the set forecast process amp Select the cell you wish to set a forecast e g cell G10 in the Basic Simulation Model example amp Click on Risk Simulator and select Set Output Forecast or click on the set output forecast icon on the Risk Simulator icon toolbar Figure 1 3 amp Enter 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 Simulator menu to set an output forecast Figure 2 5 illustrates the set forecast properties 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 User Manual Risk Simulator Software 25 2005 2011 Real Options Valuation Inc 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 simulat
263. ormal distribution and setting the mean to be 0 and standard deviation to be Finally to obtain a Probit or probability unit measure set J 5 because whenever the probability P lt 0 5 the estimated J 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 describe 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 model assumes that there is a latent unobservable variable Y This variable is linearly dependent on the X variables via a vector of p 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 User Manual Risk Simulator Software 108 2005 2011 Real Options Valuation Inc 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 inte
264. ornado Analysis To follow 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 User Manual Risk Simulator Software 130 2005 2011 Real Options Valuation Inc Disc
265. orth That is how far is far enough such that this distribution is considered to be 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 0 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 of amp Standard Deviation e 8 Global View Statistics Preferences Options Controls Global View U 100 1032 101 1032 102 1032 21 9547 22 4547 Skewness Histogram Statistics Preferences Options Controls Global View Type Two Tai ioos26 9 02842 Figure 5 17 Bootstrap Simulation Results User Manual Risk Simulator Software 147 2005 2011 Real Options Valuation Inc Notes
266. ot 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 User Manual Risk Simulator Software 154 2005 2011 Real Options Valuation Inc 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 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 out
267. otes id 17 parameter VAR1 1161 ADD 1170 a8 2 gt EDIT Parametric 2 Var T Test for Independent Unequal Variances 119 lt model name Auto ARIMA notes id 18 parameter VAR1 gt Parametric 2 Var Z Test for Independent Means 120 lt model name Auto Econometrics Detailed notes id 1 parameter VAR5 DEL Parametric 2 Var Z Test for Independent Proportions A 121 VAR6 VART VARB Power ia 220 1 123 0 gt Relative LN Returns ail 24 lt model name Auto Econometrics Quick notes _id 2 parameter VAR5 Relative Returns v Save E Hide basic XML tags Seasonality Segmentation Clustering Semi Standard Deviation Lower a Pont eroded ns isin MEN Figure 5 55 ROV BizStats XML Editor User Manual Risk Simulator Software 186 2005 2011 Real Options Valuation Inc 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
268. ounted Cash Flow Model Base Year 2005 Sum PV Net Benefits 1 896 63 Market Risk Adjusted Discount Rate 15 00 Sum PV Investments 7 800 00 Priate 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 50 00 1 00 52 02 53 068 54 12 Amortization EBIT Interest Payments EBT Taxes Net Income Noncash Depreciation Amortization Noncash Change in Net Working Capital Noncash Capital Expenditures Free Cash Flow Investment Outlay Financial Analysis Present Value of Free Cash Flow 528 24 Present Value of Investment Outlay 7 800 00 Net Cash Flows 1 271 76 User Manual Risk Simulator Software 131 440 60 0 00 506 69 Figure 5 1 Sample Model 1go000 CT o 367 26 305 91 254 62 0 00 0 00 0 00 485 70 465 25 445 33 2005 2011 Real Options Valuation Inc Procedure Select the single output cell i e a cell with a function or equation in an Excel model e g cell G6 is selected in our example Selec
269. ourth Moment User Manual Risk Simulator Software 43 2005 2011 Real Options Valuation Inc The Functions of Moments 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 four First Moment Arithmetic Average or Simple Mean Sample y The Excel equivalent function is AVERAGE c X n 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 I n 2 i7 8 The Excel equivalent function is SKEW skew Fourth Moment Kurtosis Sample n n 1 x x 3 n 1 n 1 n 2 n 3 S n 2 n 3 The Excel equivalent function is KURT kurtosis User Manual Risk Simulator Software 44 2005 2011 Real Options Valuation Inc Understanding Probability Distributions for Monte Carlo Simulation This section demonstrates the power of Monte Carlo simulation but to get
270. ove 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 probabilities would have remained the same and the process would have satisfied the conditions for a binomial distribution Conditions User Manual Risk Simulator Software 50 2005 2011 Real Options Valuation Inc The three conditions underlying the hypergeometric distribution are The 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 The sample size the number of trials represents a portion of the population e The known initial probability of success in the population changes after each trial The mathematical constructs for the hypergeometric distribution are as follows N N N Fe aa oe ALa eae iN n N n Mean Natt N N N n N n N N 1 N I Skewness V V N JN nN n Excess Kurtosis complex function Standard Deviation The number of items in the population or Population Size N trials sampled or Sample Size n 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 Populat
271. ox for Run 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 Simulation Statistics for Stochastic and Dynamic Optimization Notice that if there are input simulation assumptions in the optimization model i e these input assumptions are 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 User Manual Risk Simulator Software 128 2005 2011 Real Option
272. 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 marketing expenses and market share exist to help to model the forecast predictions The Risk Simulator software provides the user several forecasting methodologies ARIMA 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 ee ea on ee a e p ie ono N Neural Network Forecasts A Nonlinear Extrapolation p Wn S Curve p ea Stochastic Processes N Time Series Analysis and Decomposition lo 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 by Dr Johnathan Mun Wiley Finance 2006 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 examples in using the software More detailed descriptions and
273. 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 Procedure amp Start Excel and select Risk Simulator Forecasting Markov Chain amp Enter in the required input assumptions see Figure 3 20 for an example and click OK to run the model and report Note 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 User Manual Risk Simulator Software 106 2005 2011 Real Options Valuation Inc 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 inst
274. puts 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 Notes In many problems genetic algorithms may have a tendency to converge towards local optima or 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 solutions 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 User Manual Risk Simulator Software 192 2005 2011 Real Options Valuation Inc Figure
275. r 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 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 The Ljung Box Q statistics and their p values at lag k are also provided where the null User Manual Risk Simulator Software 95 2005 2011 Real Options Valuation Inc hypothesis being tested is such that there is no autocorrelation up to order k 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 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 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 infla
276. racy 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 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 Econometrics e Variable Separation with Semicolons separate independent variables using a semicolon TIPS Forecasting Logit Probit and Tobit e Data Requirements 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 User Manual Risk Simulator Software 197 2005 2011 Real Options Valuation Inc e Stochastic Process Model sometimes if the stochastic process user interface hangs for a long time chances are your
277. ral Break Test Tornado Analysis ARIMA Auto ARIMA Auto Econometrics Basic Econometrics Combinatorial Fuzzy Logic Cubic Spline GARCH J S Curves Markov Chain MLE LIMDEP Neural Network Nonlinear Extrapolation Regression Analysis Stochastic Processes Time Series Analysis Trendline English Simplified Chinese pp Traditional Chinese Rp French Francais German Deutsch Italian Italiano Japanese A48 Korean Gt Oh Portuguese Portugu s Spanish Espanol Figure 1 1B Risk Simulator Menu and Icon Bar in Excel 2007 2010 Risk Stmulator uation Inc 05 All rights reserved Figure 1 2 Risk Simulator Splash Screen 2005 2011 Real Options Valuation Inc Edit Profile Set Input Multiple Nonlinear Stochastic Hypothesis Sensitivity Regression Extrapolation Processes Testing Analysis pos ae Tornado New Simulation recast Simulation Step Simulation Analysis Optimization Analysis Profile Simulation i Nonparameteric Bootstrap Figure 1 3A Risk Simulator Icon Toolbar in Excel XP and Excel 2003 Figure 1 3B Risk Simulator Icon Toolbars in Excel 2007 2010 User Manual Risk Simulator Software 12 2005 2011 Real Options Valuation Inc WHAT S NEW IN VERSION 2011 A Comprehensive List of Risk Simulator s Capabilities The following lists the main capabilities of Risk Simulator where the highlighted items indicat
278. rcept 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 regression analysis Procedure amp Start Excel and open the example file Advanced Forecasting Model go to the MLE worksheet select the data set including the headers and click on Risk Simulator Forecasting Maximum Likelihood amp 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 amp PROBIT SAMPLE DATA will yield incorrect and biased results including the violation of normality requirements and e probabilities or values exceeding 100 Only these LIMDEP models are use when dependent variables are limited 1 3 0 1 0 1 0 1 1 2 0 2 0 1 0 1 1 1 4 Figure 3 21 Maximum Likelihood Module User Manual Risk Simulator Software 109 2005 2011 Real Options Valuation Inc Spline Cubic Spline Interpolation and Extrapolation Theory 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 Spl
279. recasting Model go to the Cubic Spline worksheet select the data set excluding the headers and click on Risk Simulator Forecasting Cubic Spline amp 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 data 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 x Filed Y Notes Real Options 2 oe I lt Valuation 2 0 4 13 interpolate wwverealoptionsvaluation com 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 4 22 Interpolate model 45 4 24 Interpolate 5 0 4 26 I
280. 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 Correlation Model Without Positive Negative Correlation Correlation Correlation 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 compared 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 risk r gt gt E Revenue Positive Correlation Risk Simulator Forecast I 52 E Revenue Negative Correlation Risk Simulator Forec 53 Histogram Statistic Preferences Options Histogram Statistic Preferences Options Statistics Number of Trials Number of Trials Mean Median Standard Deviation V arance Coefficient of Variation Maximum Minimum 1 81
281. riable name and select Risk Simulator Forecasting Multiple Regression amp Select the dependent variable and check the relevant options lags stepwise regression nonlinear regression etc and click OK Results Interpretation Figure 3 8 illustrates a sample multivariate regression result report The report comes complete 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 Applying Monte Carlo Simulation Real Options Analysis Forecasting and Optimization Wiley Finance 2006 by Dr Johnathan Mun for more detailed analysis and discussion of multivariate regression as well as the interpretation of regression reports Multivariate Regression key peal Opina www realoptionsvaluation com 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 i variables YP through a series of lags or nonlinear transformations or regressedin G A A a stepwise fashion starting with the most correlated variable Non linear Regression Show All Steps Figure 3 7
282. rlo 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 variables and the golf balls are bouncing in tandem while other times the balls are bouncing independently 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 User Manual Risk Simulator Software 18 2005 2011 Real Options Valuation Inc Getting Started with Risk Simulator A High Level Overview of the Software The Risk Simulator software has several different applications including Monte Carlo simulation forecasting optimization and risk analytics The Simulation Module allows you
283. rofiles 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 amp 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 Risk Simulator New Simulation Profile MS K Specify a title for your simulation as well as all other pertinent information Figure 2 1 User Manual Risk Simulator Software 20 2005 2011 Real Options Valuation Inc Enter a relevant tile for Simulation Properties Enter the desired number of simulation p Profile Name New Simulation Profile trials default is 1 000 Simulation Settings this simulation Number of trials 1 0002 Select if you want the Select if you want correlations to be simulation to stop Pause simulation on error when an error is WA encountered default is considered inthe V Turn on correlations ease d simulation default is V Specify random number sequence Seed checked Select and enter a seed 9994 value if you want the simulation to follow a specified random number sequence default is unchecked Figure 2 1 New Simulation Profile Title Specifying a simulation title allows you to create multiple simulation profiles in a single Excel model Thus you
284. roject 12 525 00 105 00 309 75 59 00 1 69 6 00 Total 5 776 00 3 694 44 1 539 26 64 Goal MAX lt 5000 12345 6 7 8 9 1011121314 15 16 17 1819 20 21 22 23 24 25 26 27 28 Sharpe Ratio 3 7543 Number of Iterations ENPV is the expected NPV of each credit line or project while Cost can be the total cost of REDE an Panametare administration as well as required capital holdings to cover the credit line and Risk is the Number of variables is 12 i Number of functions is 3 Coefficient of Variation of the credit line s ENPV Bbyeckive funccion wiil be Maximized starting values Functions Function Initial Lower Upper Name value Bound Bound No Status Type G 087 2 4573 2 G eiris RNGE 3197 4371 1 000000E 010 0 000000E 000 3 G iaioa RNGE 6 0000 1 000000E 010 0 000000E 000 variables variable Initial Lower upper Name No 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 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 2nd Edition Wiley Finance 2005 by Dr Johnathan Mun That case study illustrates how an efficient frontier can be generated and how forecasting simul
285. ructions 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 ercents and click OK switching behavior between one state of nature versus another and p A eventually settling on a long term steady state equilibrium e g 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 purchasing Product A will continue to purchase Product A in the next period or switch to a competitive brand B Real Options YW V a l ua t 10n Probability of Staying at State 1 if Starting at State 1 90 b am es Probability of Staying at State 2 if Starting at State 2 80 eee Figure 3 20 Markov Chains Switching Regimes Limited Dependent Variables Logit Probit Tobit Using 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 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 estim
286. ry 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 series data that has seasonality and trend This methodology is found inside the ROV BizStats module in Risk Simulator at Risk Simulator ROV BizStats Combinatorial Fuzzy Logic as well as in Risk Simulator Forecasting Combinatorial Fuzzy Logic User Manual Risk Simulator Software 188 2005 2011 Real Options Valuation Inc Procedure amp Click on Risk Simulator Forecasting Combinatorial Fuzzy Logic amp 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 amp Select the variable you wish to run the analysis on from the drop down list and enter in the seasonality period e g 4 for quarterly data 12 for monthly data etc and the desired number of Forecast Periods e g 5 amp Click Run 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
287. s 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 probability 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 User Manual Risk Simulator Software 167 2005 2011 Real Options Valuation Inc 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 273 0 74 0 74 Formatting Single Value Probability 5 Range of Values Lower Bound Upper Bound tep Size Figure 5 37 Distributional Analysis Tool Normal Distribution s ICDF and Z Score Scenario Analysis Tool
288. s 5 48 through 5 51 Note that there are three similar tools in Risk Simulator but each does very different things e Distributional Analysis 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 e Distributional Charts and Tables 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 another e Overlay Charts 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 User Manual Risk Simulator Software 179 2005 2011 Real Options Valuation Inc Procedure Run ROV BizStats at Risk 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 ROV PROBABILITY DISTRIBUTIONS Distributions Charts and Tables This tool lists all the probability distributions available in Real Options Valuation Inc s suite of products Minimum 10 Alpha 2
289. s 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 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 TTA 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 revenues 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
290. s Valuation Inc 25 Percentile 75 Percentile Percentage Eror Precision at 95 Confidence Figure 4 11 Simulated Results from the Stochastic Optimization Approach User Manual Risk Simulator Software 129 2005 2011 Real Options Valuation Inc 5 RISK SIMULATOR ANALYTICAL TOOLS This chapter covers Risk Simulator s analytical tools providing detailed discussions of 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 risk analysis Tornado and Sensitivity Tools in Simulation Theory 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 Risk Simulator Tools T
291. s 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 User Manual Risk Simulator Software 195 2005 2011 Real Options Valuation Inc TIPS Distributional Analysis Charts and Probability Tables e 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 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 comparison TIPS Efficient Frontier e Efficient Frontier Variables to access the frontier variables first set the model s Constraints before setting efficient frontier
292. s 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 Procedure amp Start Excel and open your historical data if required the illustration shown next uses the file Nonlinear Extrapolation from the examples folder amp Select the time series data and select Risk Simulator Forecasting Nonlinear Extrapolation amp 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 Results Interpretation 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 compare the forecast quality and accuracy of extrapolation versus time series analysis Notes User
293. 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 each of the cost cells Finally define forecast outp the simulation Simulation Mode Simulation Model Income A Risk Simulator Forecast Assumptions Independent Samples With Unequal Variances Independent Samples With Equal Variances Type Two Ta Diy ty Certainty 700 003 L Figure 5 18 Hypothesis Testing Report Interpretation A two tailed hypothesis test is performed on the null hypothesis Ho such that the two variables population means are statistically identical to one another The alternative hypothesis Ha is such 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 null hypothesis is rejected which implies that the forecast means are 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 are statistically different from one another otherwise for lar
294. shows 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 User Manual Risk Simulator Software 45 2005 2011 Real Options Valuation Inc 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 four 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 exactly Select
295. sine 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 occurrence 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 f zl b 0 otherwise a for min lt x lt max min max max min where a _ and b 2 mT User Manual Risk Simulator Software 58 2005 2011 Real Options Valuation Inc 3 resin 59 for min lt x lt max F x 42 1 for x gt max Mean Min Max 2 Max Min nr Standard Deviation Max Min a 8 An Skewness is always equal to 0 4 Excess Kurtosis ae S x 6 Minimum and maximum are the distributional parameters Input requirements Maximum gt minimum either input parameter can be positive negative or zero Double Log Distribution 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 distr
296. software e Risk Simulator file where the results both assumptions and forecasts can be 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 R Data Extraction 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 S Forecast J item s oo ccsssusannnsntnin Sample Second Sheet E13 Sample Third Sheeti gt E14 Assumption 3 item s ccs Sample First Sheet E8 Sample Second Sheetl E9 4 Sample Third Sheet E10 Extraction Format New Excel Worksheet New Excel Worksheet Risk Simulator Data risksim Select All Text File txt Figure 5 20 Sample Simulation Report User Manual Risk Simulator Software 151 2005 2011 Real Options Valuation Inc 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 obtai
297. ss scssseteressgetegegeg eke n n n eS E RER R BRT REE 146 Hypothesis TSENG nara aSa a E EEE A O TEE lat ata ON E 148 Data Extraction and Saving Simulation Results cccccccccccccccceccceeteecneeecetceeeeseneeeesnnseeeees 151 SAKEA 0 A AAEE AEA A AA EAEE EE EE EEE EEEE EEEE EEEE 152 Regression and Forecasting Diagnostic Tool 00annn00nnneonannonnesseneeseeenesseenrssseeess seee 153 Statistical Analysis To linsssrsarenenn n i e A S ate eet 160 Distributional Analysis Tool aooaaaaeaaaneenasenneessenessseeiessseresssenessrseesssreressrerrssserersreert gt 164 User Manual Risk Simulator Software 5 2005 2011 Real Options Valuation Inc SCCNATIOANGLVSIS LOO tec ctnaitoectenccnw odorant tictnedteattencdiestbenddtemb amet Dinb calicentiedlirendndbicedenaht ibis 168 Segmentation Clustering Tool ccccccccccccccccccesee cect e teen e eect eee teen nE Ecco dat eect atte eens 170 Risk Simulator 2011 New Tools sccssccccssssscccssvsccccvevsssccsssscsccssssccccssecsscessccececsbecsevedssescvssesescecvecess 171 Random Number Generation Monte Carlo versus Latin Hypercube and Correlation Copula Methods spa n a inn Set a a as BA a ER N Ea a a a 171 Deseasonalizing and Detrending Data 00a nnooannneenannennanssennesseeoeesseoersseeresssentesserrssseeet gt 172 Principal Component Analysts siene a aha Naa aka a aloe ee Rae e aa 174 Structural Break Analysis cccccccccccccccccceee cece eee eee e eee eee ECOG E Eee cade eec
298. sson distribution are The number of possible occurrences in any interval is unlimited e The occurrences are independent The number of occurrences in one interval does not affect the number of occurrences in other intervals e The average number of occurrences must remain the same from interval to interval The mathematical constructs for the Poisson are as follows 4x P x l for xand A gt 0 x Mean Standard Deviation Ja Skewness VA 1 Excess Kurtosis L Rate or Lambda A is the only distributional parameter Input requirements Rate gt 0 and lt 1000 i e 0 0001 lt rate lt 1000 User Manual Risk Simulator Software 54 2005 2011 Real Options Valuation Inc Continuous Distributions Arcsine Distribution The arcsine distribution is U shaped and is a special case of the bBeta 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 F x e jfor0 lt sxs1 f x 4 ayx x 0 otherwise 0 x lt 0 F x sin for 0 lt x lt 1 l x gt l eama O U 2 Standard Dev
299. stant If the third moment is not considered then looking only at the expected returns e g median or mean and risk 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 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 Skew lt 0 KurtosisXS 0 Hy u2 U Fb Figure 2 22 Third Moment Left Skew 0 02 tS Skew gt 0 os KurtosisXS 0 H M2 Hy W2 Figure 2 23 Third Moment Right Skew User Manual Risk Simulator Software 42 2005 2011 Real Options Valuation Inc Measuring the Catastrophic Tail Events in a Distribution the Fourth Moment The fourth moment or kurtosis measures the peakedness of a distribut
300. stributions eee eeseeesrrrsrrrsrerssrrsstetstttstttsttisstisstesteesttessreeerrestee 47 Bernoulli or Yes No Distributions erraniorinngaotiraeora niaaa aa ai ra a 47 Binomial Distributii om ce enan E EAA EE AE E E E E 48 User Manual Risk Simulator Software 3 2005 2011 Real Options Valuation Inc Discrete UnYOrM oe ee Bod ots Sales Bedale thn ae attotea aas 49 GOOMEIIIC Distribution eoccicee cud iesesen Stcee hed duccse a en ce dduiahcutastant a PBageees 49 Fypergeometric Distribution ccccccccceccccescceessecessecesecseeesseecsucecsnseecusecseeesseecseeessteceeaeeestessaee 50 Negative Binomial Distribution se daiiris eeii derana aiaa 52 Pasat Distribution eiee A TO E E E TE a 53 Poisson Distribution orei EO E R A A T 54 Continuous Distributions ccccccccccccceecceeeeceeesneeeeneetesceeteneeceeeesenceeeeueeeesueeseaeeeeeieeesineeeeneees J3 Aresine Distribuo oan ese E ORE E E ues ta eu Daa E E Ga R 55 Beta Distributions morerei a es ea ee ae Ie ae eRe 55 Beta S and Beta 4 Distributions ece nee n aes eee aaa E E A E E A 56 Cauchy Distribution or Lorentzian or Breit Wigner Distribution ee 57 ChisSquare Distribution oiris Sis ess Ae EE E E ee AERE A none sen i ia 57 Cosine Distibutions seoa cess EE ese ae ea Re i ES 58 Do ble Log Distributiot serere een a KERRE OEE AEE E E E EAN EE 59 Erlang Distribution reei oere eia TE edi E EE EASE A AE TNA N EE 60 Exponential DISH IDULI ON meiege aae EESE E TE E N E E A EE E TEN 61 EXpPONe
301. sts 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 89 420 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 User Manual Risk Simulator Software 93 2005 2011 Real Options Valuation Inc Box Jenkins ARIMA Advanced Time Series Theory One very powerful advanced times series forecasting tool is the ARIMA or Auto Regressive Integrated Moving Average approach ARIMA forecasting ass
302. svaluation 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 Running a Monte Carlo Simulation Typically to run a simulation in your existing Excel model the following steps have to be performed Start a new simulation profile or open an existing profile Define input assumptions in the relevant cells Define output forecasts in the relevant cells Run simulation Ne as ee 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 Start Real Options Valuation Risk Simulator Examples or accessed directly through Risk Simulator Example Models 1 Starting a New Simulation Profile To start a new simulation you will first need to create a simulation profile A simulation profile contains a complete set of instructions 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 p
303. t Risk Simulator Tools Tornado Analysis 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 BA B re D E D o l 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 00 Sum PV Investments 1 800 00 E Private Risk Discount Rate 5 00 Net Present Value 96 63 ee Annualized Sales Growth Rate 2 00 internal Rate of Return 18 80 8 Price Erosion Rate 5 00 Return on Investment 5 37 a Effective Tax Rate 40 00 10 Tornado Analysis 11 2005 2006 12 Prod A Avg Price 10 00 9 50 Tomado analysis creates static perturbations i e each precedent 13 Prod BAvg Price a22 snoa et ee 14 Prod C Avg Price 15 15 14 39 simulations 15 Prod A Quantity 60 00 51 00 16 Prod B Quantity 35 70 Review the precedents below and make any necessary changes 17 Prod C Quantity 20 00 Selection Name Worksheet Cell Base Value Upside Downside Test Points 18 Total Revenues 4 234 75 I 4 Maket DCF Mode C5 0 15 10 00 10 00 10 19 Cost of Goods Sold 184 76 Investm DCFMode C36 1800 10 00 10 00 10 20 Gross Profit 1 046 99 i I Capital DCF Mode C33 0 10 00 1000 10 21 Operating Expenses 157 50 160 65 si I Changei DCF Mode C32 0 10 00 10 00 10 22 SG amp
304. t ante eens 174 Trendline Forecasts i a E allt cacaeesdadbaae nda dbecesde dashed cnatceeasnacabactaaccbscneaale casdacses 176 Model Checking TOO bse crie ernest eee cena estate Suan ana eal le ea Canta a ala aca a a aha hal at 176 Percentile Distributional Fitting TOOL ccccccccccsccccecesseeee cnet ete ence eee e cnet ee cnaee eee enseeeessnaaeees 178 Distribution Charts and Tables Probability Distribution Tool cccccccccceeseteesseeeettteees 179 ROV ANI 183 Neural Network and Combinatorial Fuzzy Logic Forecasting Methodologies 00 187 Optimizer Goal Seek mirnninnibp ncnian unnn RE REG NUn AER EO h n nni 190 SineleVariable OPUmMiZeT aie a EN E E cates OEN OAT IER 191 Genetic Algorithm Optimization aoaasaeenaanenaeeeeeesennessseeesssoesssetrsssreressrenessrererssereste 192 Helpful Tips and Techniques cccccssssssescssscccssscccccssccesecsssscccscccceseesssscccsssesessssssscoocess 194 TIPS Assumptions Set Input Assumption User Interface ccce 194 TIPS Copy and Paste riene eee EEEE Eae AERE ae aE aaa Ma TA OAAR EAOa Era TART NaS 194 TIPS Correlations eiei A EE O E A E EA E NES 195 TIPS Data Diagnostics and Statistical Analysis eeeeeeeerreeresrreerreerrererereerrrsrre 195 TIPS Distributional Analysis Charts and Probability Tables 196 TIRS Efficient FVOntercccsscetetetz chia Aari ai aioli eee ea et es EE ote oe 196 IIPS Forecast CUS vic een Pet yt e los ite Tost las et Telecast a les tie Ta
305. t 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 i e 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 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 n
306. ta 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 User Manual Risk Simulator Software 153 2005 2011 Real Options Valuation Inc 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 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 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 detec
307. tance 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 Procedure amp Open the example file and start a new profile by clicking on Risk Simulator New Profile and provide it a name CW The first step in optimization is to set up the decision variables Set the first decision variable by selecting cell J4 select Risk Simulator Optimization Set Decision click on the link icon to select the name cell B4 and select the Binary variable Then using Risk Simulator s copy copy this cell J4 decision variable and paste the decision variable to the remaining cells in J5 to J15 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 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 User Manual Risk Simulator Software 119 2005 2011 Real Options Valuation Inc not exceed 6 So click
308. tarting val 1 X UL 1 00000 0 1 1 x 1 00000 1 00000 Fonctions arting values 2 X UL 1 00000 0 1 2 x 1 00000 0 00000 Function Initial Lower Upper 3 x UL 1 00000 0 1 3 x 1 00000 0 00000 pe bo SES es VEINS aan aa x UL 1 00000 0 1 4 x 1 00000 1 00000 OB 2 4573 5 x UL 100000 o 1 5 x 4 00000 0 00000 s222 RNGE 3197 4371 1 000000E 010 0 000000E 000 _ 6 x UL 1 00000 0 1 6 x 1 00000 0 00000 7 Xx UL 1 00000 0 1 7 x 1 00000 0 00000 Optimal values have been found Do you wish to replace the existing decision variables with the optimized values or 8 x UL 1 00000 0 1 8 X 1 00000 0 00000 revert to the original inputs 9 Xx UL 1 00000 0 1 9 x 1 00000 0 00000 10 xX uL 1 00000 0 1 10 x 1 00000 0 00000 11 x UL 1 00000 0 1 11 x 1 00000 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 _Cond No Size Step 1 3205 43710 0 12 2 0 57590 1 0 2 3 55285 0 11 1 0 28146 1 1 3 2 88211 0 10 1 0 34697 1 0 061 Figure 4 8 Efficient Frontier Results 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 buck or returns to risk ratio is maximized That is the goal is to allocate 100 of an individual s investment among several different
309. the relevant inputs Triangular Minimum MostLikely Percentile Triangular Percentile Percentile Maximum Triangular Percentile MostLikely Percentile Triangular Minimum Percentile Percentile Triangular Percentile Percentile Percentile Triangular Mean Stdev Percentile Uniform Uniform Minimum Percentile Uniform Percentile Maximum Uniform Percentile Percentile Uniform Mean Stdev Weibull Weibull Alpha Percentile Weibull Percentile Beta Weibull Percentile Percentile Parameter Percentile Percentile 10 Percentile l 45 Percentile Step 3 Run curve fit and review the empirical versus theoretical distributions FittedR Square SUH Alpha Beta Location Empirical Weibull Mean Stdev Weibull 3 Weibull 3 Percentile Beta Location Weibull 3 Alpha Percentile Location Weibull 3 Alpha Beta Percentile Weibull 3 Percentile Percentile Location Weibull 3 Percentile Beta Percentile Weibull 3 Alpha Percentile Percentile Weibull 3 Percentile Pe tile Percentile Weibull 3 Mean Stdev Percentile nono Ea Figure 5 47 Percentile Distributional Fitting Tool 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 Figure
310. 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 5 iE Data Average 331 92 pan Frequency A E 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 1 0 1150 87 00 0 02 0 06 0 0783 0 0183 D Critical at 5 0 1237 96 00 0 02 0 08 0 0862 0 0062 D Critical at 10 0 1473 10200 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 11400 0 02 0 14 0 1038 0 0362 Conclusion The sample data is normally distributed at 127 00 0 02 0 16 0 1180 0 0420 the 1 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 22200 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 User Manual Ris
311. thods Starting with version 2011 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 Risk 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 and most conservative results The t copula provides for extreme values in the tails of the simulated distributions whereas the quasi normal copula returns res
312. timization Module 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 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 The Real Options Super Lattice Solver is another standalone software that complements Risk Simulator used for solving simple to complex real options problems The following sections walk you through the basics of the Simulation Module in Risk Simulator while future 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 User Manual Risk Simulator Software 19 2005 2011 Real Options Valuation Inc In fact it is highly recommended that you first watch the getting started videos on the web www realoption
313. tin ie ft ARIMA amp amp Auto ARIMA Auto Econometrics Basic Econometrics E Combinatorial Fuzzy Logic Z Cubic Spline Z GARCH J S Cures Markov Chain MLE LIMDEP Neural Network Nonlinear Extrapolation Regression Analysis Stochastic Processes Time Series Analysis i i fs F3 3 pt f Trendline Figure 3 2 Risk Simulator s Forecasting Methods User Manual Risk Simulator Software 82 2005 2011 Real Options Valuation Inc Time Series Analysis Theory Figure 3 3 lists the eight most common time series models segregated by seasonality and trend For 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 discernable trend is present either a seasonal additive or seasonal multiplicative model would be better and so forth No Seasonality With Seasonality 3 Seasonal nod 5 Single Moving Average Unies o z Single Exponential Seasonal Smoothing Multiplicative Double Moving Holt Winter s w Average Additive D S Double Exponential Holt Winter s Smoothing Multiplicative Figure 3 3 The Eight Most Common Time Series Methods Procedure amp Start Excel and open your historical data if required the example below uses the Time Series Forecasting file in the examples folder amp Select the historical data data should be listed in a single colu
314. tion 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 series 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 are the mean reversion process and the jump diffusion process The interesting thing about stochastic process simulation is that historical data are not necessarily required That is the model does not have to fit 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 2nd Edition Wiley Finance 2006 by Dr Johnathan Mun for more details on how each of the inputs are computed e g mean reversion rate jump probabilities volatility etc Procedure amp Start the module by selecting Risk Simulator Forecasting Stochastic Processes amp Select the desired process enter the required inputs click on Update Chart a few times to make sure the process is behaving the
315. tion sales revenues gross domestic product Time Series Variable B5 B440 E Exogenous Variable Autoregressive Order AR p Differencing Order I d UE Moving Average Order MA opl Maximum Iterations Forecast Periods Backcast Figure 3 13 Box Jenkins ARIMA Forecast Tool User Manual Risk Simulator Software 96 2005 2011 Real Options Valuation Inc ARIMA Autoregressive Integrated Moving Average Regression Statistics R Squared Coefficient of Determination 0 9999 Akaike information Criterion AIC 4 6213 Adjusted R Squared 0 9999 Schwarz Criterion SC 4 6632 Multipie R Multipie Correlation Coefficient 1 0000 Log Likelihood 1005 1340 Standard Error of the Estimates SEy 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 model that use 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 p lags of the time series in the equation An AR p model has the form y a t y 1 a p yitp e The second component is the integration d order term Each integration order corresponds to differencing the time series i f means differencing the data once Ifd means differencing the data d times The third component is the moving average MA term The MA q model uses the q la
316. tion is the wrong 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 i e 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 User Manual Risk Simulator Software 143 2005 2011 Real Options Valuation Inc ee ab b ab od d d d ad ad 00 00 00 00 00 00 00 00 00 00 00 Cann Normal Mean 100 67 Standard Deviation 10 40 Kolmogorov Smirnov Test Statistic K Test Statistic 0 02 e P Value 99 96 Actual Theoretical 100 61 100 67 ke 10 31 10 40 0 01 0 00 has i 0 13 0 00 Automatically Generate Assumption Figure 5 14 Distributional Fitting Result User Manual Risk Simulator Software 144 2005 2011 Real Options Valuation Inc
317. tional 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 Analytical Tools Module 58 59 60 61 62 63 64 65 66 67 68 69 Check Model tests for the most common mistakes in your model Correlation Editor allows large 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 heteroskedasticity 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 retrieves previous simulation run results Deseasonalization and Detrending deasonalizes and detrends your data Distributional Analysis computes exact PDF CDF and ICDF of all 42 distributions and generates probability tables Distributional Designer allows you to create custom distributions Distributional Fitting Multiple runs multiple variables simultaneously accounts for corr
318. tiple 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 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 proces
319. tistical 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 172 91 P Value left tailed 1 0000 P Value two tailed 0 0000 User Provided Statistics Null Hypothesis Ho u Hypothesized Mean Hypothesized Mean 0 00 Alternate Hypothesis Ha u lt gt Hypothesized Mean Notes lt 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 ttest is appropriate when the population standard deviation is not known but the sampling distribution is assumed to be approximately normal the ttest is used when the sample size is less than 30 butis also appropriate and in fact provides more conservative results with larger data sets This ttest 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
320. to run simulations in your existing Excel 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 or applying expert opinion forecasts The Forecasting Module can be used to generate automatic time series forecasts with and without seasonality and trend multivariate regressions 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 Op
321. 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 exists Do the following to set new input assumptions in your model amp Make sure a Simulation Profile exists open an existing profile or start a new profile Risk Simulator New Simulation Profile amp Select the cell you wish to set an assumption on e g cell G8 in the Basic Simulation Model example amp Click on Risk Simulator Set Input Assumption or click on the set input assumption icon in the Risk Simulator icon toolbar amp Select the relevant distribution you want enter the relevant distribution parameters e g Triangular distribution with J 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 Minimum Normal Triangular a 1 Tee te 1 3 Most Likely l L i 5 0 30 n U a Maximum Uniform Custom k 25 amp m i Regular Input Arcsine Bernoulli D Percentile Input E Enable Data Boundary Triangular Distribution i The triangular distribution describes a Minimum Infinity Ej situation where you know the minimu
322. tor Software Il Run Optimization PROP PORE HB OB Re oD 01 Advanced Forecasting Models Set Objective D Set Decision Constraints 02 Basic Simulation Model 03 Correlated Simulation 04 Correlation Risk Effects Model 05 Cost Estimation Model 06 Data Fitting 07 DCF ROI and Volatility Genetic Algorithm Goal Seek Single Variable Optimizer 08 Hypothesis Testing and Bootstrap Simulation 09 Multiple Regression Check Model 10 Nonlinear Extrapolation Create Forecast Statistics Table 11 Optimization Continuous Create Report 12 Optimization Discrete Data Deseasonalization amp Detrending 13 Optimization Stochastic Data Extraction Export 14 Overlay Charts Data Open Import 15 Queuing Models Diagnostic Tool 16 Regression Diagnostics Distributional Analysis 17 Retirement Funding with VBA Macros Distributional Charts amp Tables 18 Statistical Analysis Distributional Designer Distributional Fitting Single Variable Distributional Fitting Multi Variable 19 Stochastic Processes 20 Time Series ARIMA 21 Time Series Forecasting 22 Tornado and Sensitivity Charts Linear 23 Tornado and Sensitivity Charts Nonlinear Distributional Fitting Percentiles Edit Correlations Hypothesis Testing 24 Tools on Data Behavior 2222888288882 E i Overlay Charts Principal Component Analysis Seasonality Test Segmentation Clustering Scenario Analysis Statistical Analysis Structu
323. tributional 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 F Distribution or Fisher Snedecor Distribution 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 and denominator degree of freedom m is related to the chi square distribution in that 2 d tn ln 4 n m x m User Manual Risk Simulator Software 62 2005 2011 Real Options Valuation Inc Mean m 2 2m m n 2 n m 2 m 4 2 m 2n 2 2 m 4 Skewness m 6 n m n 2 12 16 20m 8m m 44n 32mn 5m n 22n 5mn n m 6 m 8 n m 2 for all m gt 4 Standard Deviation Excess Kurtosis The numerator degree of freedom n and denominator degree of freedom m are the only distrib
324. tructural Break Sum Time Series Analysis Auto Time Series Analysis Double Exponential Sm Time Series Analysis Double Moving Average Time Series Analysis Holt Winter s Additive Time Series Analysis Holt Winter s Multiplica Time Series Analysis Seasonal Additive STEP 4 Save Optional You can save multiple analyses and notes in the profile si for future retrieval Data gt Vari Var2 Var3 Auto Econometrics Detailed This is a test model running AE methodology inside ROV BizStats Parametric 2 Var T Test for Independent Unequal Variances Parametric 2 Var Z Test for Independent Means Parametric 2 Var Z Test for Independent Proportions Power Relative LN Returns Relative Returns Seasonality Segmentation Clustering Semi Standard Deviation Lower ERL E ES ese AN Data Grid Configuration from another application Bempe STEP 2 Analysis Choose an analysis and enter the th analwsis parameters required see example Decimal Settings IN North America 1 000 50 View Alphabetical 7 Parameter inputs below 3 Europe and Latin America 1 000 50 Aao ft Cokunns li nasai 2 Multiple Regression Linear A VARG VAR7 VARS VARI VA a Multiple Regression Nonlinear Nonlinear Regression z ae Nonparametric Chi Square Goodness of F
325. ts 1 634 22 bo G lunt Rate 15 00 Net Present Value 3 127 87 Model Include Terminal Valuation x Remove Parameter 5 00 intemal Rate of Return 55 68 2 00 Return on Investment 191 40 Close All Charts 40 00 Profitability Index 2 91 Minimize All Charts E i 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 k Bun simuation 1250 13 00 13 50 E 2 Run Super Speed Simulation 12 50 12 75 13 00 13 25 13 50 13 75 14 00 i425 14 50 gt step Simulation 15 75 15 90 16 05 16 20 16 35 16 50 a sof sof o Rasika ES e E E EA Example Models gt 20 a F 1 378 75 1 415 50 1 452 25 1 489 00 1 562 50 Forecasting gt 201 30 206 81 212 33 217 84 223 35 228 86 234 38 3 46 1 140 70 1 171 94 1 203 18 1 234 41 1 265 65 1 296 89 _ 1 328 13 Optimization Create Forecast Statistics Table 157 50 157 50 157 50 157 50 tos Create Report sis7s s575 s1575 515 75 515 75 i ie Data Deseasonalization amp Detrending 2 967 45 998 69 1 029 93 1 061 16 1 092 40 1 123 64 1 154 88 amp Options p 00 10 00 10 00 10 00 10 00 10 00 10 00 Languages it Data Extraction Export 3 00 3 00 3 00 3 00 3 00 3 00 B ucene W Data Open Import 985 69 1 016 93 1 048 16 1 079 40 1 110 64 1 141 88 i License 3 ___s200 5300 ssoo ssoo seoo s700 About Risk Simulator ponian 1 013 93 1 044 16 1 074 4
326. ts respective report in the model Procedure Open the example model Risk Simulator Examples Regression Diagnostics go to the Time Series Data worksheet and select the data including the variable names cells C5 H55 Click on Risk Simulator Tools Diagnostic Tool Check the data and select from the Dependent Variable Y drop down menu Click OK when finished Figure 5 22 Multiple Regression Analysis Data Set Dependent vVariablext Variablex2 Variablex3 Variable xi Variable X5 Variable 521 18308 185 4 041 79 6 7 2 367 1148 600 0 55 1 6 5 443 18068 372 3 665 32 3 5 7 365 7729 ina i ORRE i 614 100484 I Diagnostic Tool oD a oS fos This tool is used to diagnose forecasting problems in a set of multiple variables 397 4008 Dependent Variable Dependent Variable Y x 764 38927 427 22322 Dependent Variable Y i Variable X1 Variable X2 Variable x3 Yariat 4 153 3711 521 18308 185 4 041 796 _ 231 3136 1148 600 0 55 1 524 50508 18068 372 3 665 32 3 328 28886 7729 142 2 351 45 1 240 16996 100484 432 29 76 190 8 286 43035 16728 290 3 294 31 8 285 12973 14630 346 3 287 678 4 569 16309 4008 328 0 666 340 8 96 5227 38927 12 938 239 6 498 19235 22322 6 478 111 9 481 44487 y gt 468 44213 177 23619 198 9106 7 z Fg 5 6 456 24917 189 5 117 74 3 6 6 108 3872 196 0 799 5 5 69 246 8945 183 1 578 20 5 Qe 291 2373 417 1 202 10 9 5 5 68 7128 233 1 109 123 7 7 2 Figure 5 22 Running the Da
327. ts to dipboard or generate reports K Number of Dependent Variables Tested 3 Number of Econometric Models Tested 61 Number of Best Models Shown 20 Auto Econometrics Detailed L VAR 1 VAR2 LN VAR3 This is a test model running AE methodology inside ROV BizStats LN VAR2 LN VAR3 VAR2 LN VAR3 LN VAR 1 LN VAR3 LN VAR2 LN VAR2 N VAR3 LN VAR 1 LN VAR1 4LN VAR3 VAR2 LN VAR 1 LN VAR3 LN VAR 1 LN VAR2 LN VAR1 VAR2 LN VAR 1 LN VAR2 LN VAR2 LN VAR3 LN VAR 1 LN VAR2 LN VAR3 LN VAR3 LN VAR2 VAR3 LN VAR 1 LN VAR3 VAR2 VAR3 VAR 1 LN VAR2 Absolute Values ANOVA Randomized Block ANOVA Single Factor Multiple Treatments ANOVA Two Way ARIMA 1 0 1 ARIMA 1 0 2 Auto ARIMA GE EXAMPLE ROV Biz Stats _ File Data Language Help STEP 1 Data Manually enter your data paste from another application P or load an example dataset with analysis Visualize Command Choose an analysis and enter the parameters required see example Parameter inputs below R Alphabetical Dataset Bthee POR bP eoe sO Ek Runs the current analysis in Step 2 or selected saved analysis in Step 4 view the results charts and statistics copy the results and charts to dipboard or generate reports Trend Li
328. uations in the forecast how much of the variation can be explained by each 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 0 17 Tax Rate 0 05 Price Erosion 0 03 Sales Growth 0 0 0 1 0 2 0 3 0 4 0 5 0 6 Figure 5 11 Rank Correlation Chart 4 84 C Price 3 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 User Manual Risk Simulator Software 141 2005 2011 Real Options Valuation Inc Distributional Fitting Single Variable and Multiple Variables Theory 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 questio
329. uch 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 User Manual Risk Simulator Software 113 2005 2011 Real Options Valuation Inc 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 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 linear optimization model its objective and all of its constraints are linear Therefore these two examples encapsulate all of the procedures aforementioned 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
330. ues The chart is a powerful communication and visual tool to see how good the forecast model is z R Time Series Forecast Historical Sales Revenues Time Series Analysis is used to forecast time series variables by decomposing the historical data into baseline trend and seasonality elements and replicating these elements into the future forecasts This analysis assumes that the trend and seasonality will persist Year Quarter Period Sales 2006 2006 2006 2006 2007 2007 2007 2007 2008 2008 2008 2008 2009 2009 2009 2009 2010 2010 2010 2010 Auto Model Selection Single Moving Average Single Exponentiz ri m Model Parameters Optimize Alpha o Vv Seasonality Periods Cycle Quarters 4 v Beta OS IV Number of Forecast Periods DO Gamma o Vv Periodicity 4 Vv Maximum Runtime sec 300 I Automatically Generate Assumption Allow Polar Parameters RONDA A HWHAARWONHDAAONDA KRW NDY Figure 3 4 Time Series Analysis Notes 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 optimize checkboxes and enter your own
331. uick getting started steps on running the module and details on each of the elements in the software Procedure Notes Run ROV BizStats at Risk Simulator ROV BizStats 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 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 for different parameters 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 If required you can provide a 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 or deleted M and rearranged in order of appearance N and all the changes can be saved O into a single profile with the file name extension bizstats 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 variable The menu also allows you to change the language settings and decimal settings for your data To get started it is always a good idea to load the example file A that comes complete with some data and precreate
332. uire 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 i e 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 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 for an infinitely large sample size the error approaches zero indicating that the population parameter has been estimated Due to sampling errors the 95 Confidence Interval for the Mean is provided Based on an analysis of the sample data points the actual population mean should fall between these Lower an
333. ults 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 in one area of the distribution as compared to a more User Manual Risk Simulator Software 171 2005 2011 Real Options Valuation Inc uniformly sample
334. urring value Figure 2 19 illustrates the first moment where in this case the first moment of this distribution is measured by the mean u or average value Skew 0 KurtosisXS 0 by U W Ls Figure 2 19 First Moment Measuring the Spread of the Distribution the Second 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 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 more 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 User M
335. usion 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 User Manual Risk Simulator Software 17 2005 2011 Real Options Valuation Inc 2 MONTE CARLO SIMULATION Monte Carlo simulation
336. utional parameters Input requirements Degrees of freedom numerator and degrees of freedom denominator must both be integers gt 0 Gamma Distribution Erlang Distribution 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 are The number of possible occurrences in any unit of measurement is not limited to a fixed number e The occurrences are 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 x m E ok f r a g Mean af with any value of gt Oand gt 0 User Manual Risk Simulator
337. 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 e TAB 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 e Copy copies the forecast chart or the entire global view depending on whether you are in the normal or global view User Manual Risk Simulator Software 196 2005 2011 Real Options Valuation Inc 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 e Forecast RMSE use as the universal error measure on multiple forecast models for direct comparisons of the accu
338. views right click anywhere in the gallery and select large icons small icons or list There are over two dozen distributions available 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 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 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 Risk 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

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