Prices of Credit Default Swaps and the Term Structure of Credit Risk
Contents
1. derive the market perceived probability of default which can then be used to price credit default swaps or other credit derivative products 2 Tradable Assets and Risk Factors 2 1 Fixed Income Products A common form of investment is a fixed income security Fixed income securities come in many forms and differ from other variable income securities such as stocks in that all payments are known in advance A fixed income investor lends its money in exchange for a promise of a pre determined sequence of payments by the counterparty also known as the debt issuer Fixed income securities are also known as debt or credit instruments the investor credits 1ts money to the issuer who assumes the debt Fixed income products are structured based on the time value of money a dollar received today is different from a dollar to be received one year from today Risk Glossary 2007 2 1 1 Bonds A bond is a form of securitized debt which matures at a specified date in the future pays interest periodically in the form of coupon payments and repays its face value at maturity A zero coupon is a special kind of bond which provides only one payment at the bond s maturity date consisting of the accrued interest and the principal portion of the bond Bonds can be traded At any point in time the fair price of a bond is the present value of its future cash flows The price of a bond can fluctuate due to many factors the most important bein2. maturity dates Then go to the Premiums tab and you will need to use Solver in the same manner as described above to compute the hazard rates The only difference is that you will be solving for the cells in row 13 So for the premium payment for the first CDS with maturity 0 5 years you will use Solver and Set target cell C8 Equal to the Value of 0 By changing cells C13 Repeat this 10 times in increasing order until you solve for cell L13 The resulting premiums will be shown in the purple cells in row 13 Any time you change any data you need to run the Solver over again for every cell you wish to solve for 31 Bibliography Arvanitis Angelo and Jon Gregory Credit The Complete Guide to Pricing Hedging and Risk Management Risk Waters Group Ltd London 2001 Bond Basics Investopedia 2007 Investopedia Inc Feb Mar 2007 lt http www investopedia com university bonds default asp gt Bond Ratings Fidelity Investments 2005 FMR Corp Jan Feb 2007 lt http personal fidelity com products fixedincome bondratings shtml gt Bowers Gerber Hickman Jones and Nesbitt ed Actuarial Mathematics Illinois The Society of Actuaries 1997 2 edition Credit Derivatives Explained Market Products and Regulations Lehman Brothers International Europe March 2001 3 42 Galiani Stefano S Copula Functions and their Application in Pricing and Risk Managing Multiname Credit Derivative Produ
3. premiums 2 Price credit default swaps of different maturities using derived market hazard rates The workbook draws from market quoted premiums of credit default swaps of different maturities on the same reference credit to determine the implied hazard rate which models the default probability distribution The spreadsheet user can then use the market s perceived default probability distribution as a parameter to get the risk neutral price of credit default swaps Both spreadsheet applications require the use of Equation 4 and solving a set of non linear equations using the Solver Add In in Microsoft Excel See Appendix Worksheet User Manual for instructions on how to operate the spreadsheet 25 6 1 Computing the Hazard Rates Bootstrapping is a calibration procedure used by the workbook to solve for the hazard rates We began by gathering market premiums for current default swaps with different maturities on the same reference entity Then we assume the hazard rates ai for time intervals ti 1 ti are piecewise constant between the maturity dates of the individual market swaps We extract the hazard rates by solving for the appropriate ai using Equation 4 and a constant recovery rate 5 We solve for each a in order of increasing maturity using the data from the swap with the first maturity T to solve for a Consequently we know a and have the data from T to solve for a2 and so on This probability stripping procedure gives u
4. protection seller in exchange for the promise that if default occurs the protection seller will receive the defaulted security and repay the protection buyer a percentage of what was owed The premiums of the credit default swap contract are determined by the market s view of how likely it is that default will occur before the credit swap matures Time to default is a random variable which characterizes the term structure of credit risk and affects the price of credit derivative products This project quantifies the connection between the prices of the credit default swaps and the probability distribution of the time to default in both directions 1 We calculate the market perceived probabilities and timing of possible default by a particular borrower from the market prices of a series of traded credit default swaps referencing the same borrower s debt 2 We calculate the fair prices of the credit default swaps from the probability distribution of the default time and of the recovery rate The calculations are implemented in spreadsheets of a Microsoft Excel workbook The results of the project can also be used to determine prices of more complex credit derivates The market implied default probabilities determine the credit risk inherent in all securities depending on the same borrower They can then be used as input into more complicated models for multi name credit derivative products such as basket default swaps and collateralized debt o
5. the principal portion of the bond Depending on the terms set forth in the initial agreement the investor may be able to recover a percentage of their investment based the specific recovery rate involved The 10 recovery rate of a bond is the fraction of the outstanding obligation expected to be recovered through bankruptcy proceedings or some other form of settlement Risk Glossary 2007 Default risk can be assessed prior to purchasing a bond by investigating the credit ratings of the bond issuer Standard amp Poor s and Moody s Investors Service are two of the largest credit rating agencies which give companies credit ratings based on those companies abilities to pay back their outstanding debt These ratings reflect a company s risk of default on their obligations ultimately reflecting the company s overall credit risk Figure Bond Rating Codes Bond Ratings 2005 Rating S amp P Moody s Highest quality AAA Aaa High quality AA Aa Upper medium quality A A Medium grade BBB Baa Somewhat speculative BB Ba Low grade speculative B B ao H default CCC Caa Low grade partial cc Ca recovery possible Default recovery unlikely C C The credit ratings are based on the company s probability of default their average recovery rates on previous defaults and the quality and diversification of their assets The higher the risk of default or the lower the credit rating of a company the higher the yield the inv
6. which you want to determine the probability of default for as many different maturities that are available Also determine the following parameters that will be used size of time intervals between maturity dates the term structure of the risk free interest rate your reference entity s notional amount and the assumed constant recovery rate The workbook contains macros so before opening the workbook it is necessary to set the macro security to an appropriate level which allows for running these macros Upon opening the file the first page you should see is the Input sheet If this does not open up directly click on the Input tab EJ Microsoft Excel COS Default Probability xis Ja B ge Edt yes pom Format Toc Oo pr s teo 0x EU ECR S SVR Ree g u ceessas A Hazard Rates J Hazard Rate Step Function Z Smooth Hazard Rate Funcion Probably Deti lt gt f 7 um Tn De octet Osos wwe sucene peral snes unes acron these 2 Os Dette Probab EEE enm 28 Input the following parameters into the appropriate blue cells in column H size of time intervals between maturity dates risk free interest rate your reference entity s notional amount and the constant recovery rate Input the market premiums you previously gathered in the blue cells of Option 1 for the appropriate corresponding maturity dates To compute the hazard rates go to the Hazard Rates tab Open Solver by g
7. Prices of Credit Default Swaps and the Term Structure of Credit Risk by Mary Elizabeth Desrosiers A Professional Master s Project Submitted to the Faculty of the WORCESTER POLYTECHNIC INSTITUTE in partial fulfillment of the requirements for the Degree of Master of Science in Financial Mathematics May 2007 APPROVED Professor Domokos Vermes Advisor Executive Summary Investments in any form of financial product other than a bank deposit at the risk free interest rate involve some sort of risk due to the volatility of the economy Interest rate risk is the most critical risk factor affecting fixed income securities However the growing credit derivatives market is based primarily on credit or default risk This is the risk caused by the possibility that a company will have financial troubles and will have to default on payments which it owes to its lenders US treasury securities are considered to be free of credit risk because they are backed by the government In order to protect investors from this risk the credit derivatives market emerged with various products whose sole purpose is to hedge credit risk A credit derivative is a contract between a protection buyer and a protection seller to transfer the credit risk of an asset without the actual transfer of the asset The most fundamental credit derivative is the credit default swap In a credit default swap the protection buyer makes periodic premium payments to the
8. actual transfer of the asset The idea is to avoid direct ownership of the asset in the transaction in order to minimize losses in the event of default 12 3 1 Credit Derivative Products There are many different types of credit derivative products each based on the specific risk being transferred The two fundamental categories of credit derivatives are single name credit derivatives and multi name credit derivatives e Single name credit derivatives offer protection against the default risk of one particular borrower Examples are asset swaps credit linked notes and credit default swaps e Multi name credit derivatives are based on defaults of one or more borrowers from a group of borrowers These instruments depend not only on the credit risks posed by the individual borrowers but also on the correlation between them Examples of multi name credit derivatives are basket default swaps and CDOs A total return swap TRS also known as a total rate of return swap TRORS is a credit derivative intended to protect against depreciation of an asset The swap exchange is a combination of an underlying asset and an interest rate swap In the TRS agreement one party receives the total return or the generated income from the asset plus any capital gains while the other party receives payments based on a set rate as part of the interest rate swap The owner of the asset gets protection against any loss in value while the counterparty receive
9. ault Products 38 Portfolio CLOs 18 Asset Swaps 12 Total Return swaps 11 Credit Linked Notes 10 Baskets 6 Credit Spread products 5 An investor in a CDS only assumes the credit risk of default on the reference entity all other risks such as interest rate movements do not have an affect on the CDS agreement 5 1 CDS Pricing The main idea behind pricing models for credit default swaps is that they are completely independent from interest rate movements The only risk assumed is that of default or credit risk The price of the CDS is determined by setting the present value of the periodic premium payments equal to the present value of the reference entity at maturity or time of default It is common to think of a CDS as having two opposing legs the premium leg corresponding to the fixed premiums payments and the default leg corresponding to the contingent payment upon default Arvanitis 2001 The premium leg is a stream of discounted fixed cashflows at fixed times to t1 t gt tn These annualized premium payments X are paid until maturity T tn or 23 default t whichever occurs first This stream of cashflows is discounted back by the risk free discount factor B 0 t and weighted by the instantaneous probability of default h t or hazard rate to achieve the present value Equation 1 Present Value of Premium Leg PV PL K Y t t X B 0 t P t gt 1 where P r gt t exp h u du The
10. bligations Abstract The objective of this project is to investigate and model the quantitative connection between market prices of credit default swaps and the market perceived probability and timing of default by the underlying borrower We quantify the credit risk of a borrower in a two way relationship calculate the term structure of default probabilities from the market prices of traded CDSs and calculate prices of CDSs from the probability distribution of the time to default Table of Contents EXECUIVS SIMA E A An eb 2 A RO 4 Table of Comes SS AR 5 Mr A uausae totes nN RE AN ARA cy R 6 2 Tradable Assets and Risk Factors id dadas 6 21 Fixed Income Producida 6 DA W R A A E E AEAEE 7 Di LAO bass 7 2 1 3 Asse t backed SECURITIES i 8 22 A A 9 2 ZAS RALE RISK sigs hn oc shadeth dus decla n a a aac Cane au abide oaaeuee 9 li A a uals ote ed gah es alls see 10 A A aula AEEA 12 3 Credit Derivative Products A T a 13 4 Probability of Def lt nocsscinionicinnsiraiiirie ninoi edu aaien aiii 14 4 1 Typical default time distributions ii iia 16 4 1 1 Exponential Distribution a ii dia 16 4 1 2 a A O O seta coedaceMoudeaeueceone 17 4 1 3 Weibull DISTIN 19 4 2 Approximation of Default Time Distributions 0 cece eeeeseeeeeeeeceecneeeneeeneeerees 20 42 a AMA A N 21 4 2 2 Piecewise Constant Density FUN Ct ON miii cad canica 21 4 2 3 Piecewise Constant Hazard Rate Function oooocccnnonoccnononanonononanonocnnananononnnan
11. bond to the protection buyer if General Motors defaults on the bond The CDS will terminate either at the bond s maturity or the date a default event occurs whichever comes first If default never occurs General Motors continues to pay the periodic coupon payments and at maturity pays the principal portion of the bond to the investor and the investor pays the CDS premium payments to Morgan Stanley until the bond matures Morgan Stanley will never have to make any payments and profits for assuming the credit risk of the bond If a default event occurs before the set maturity Morgan Stanley instantly compensates the protection buyer for its loss and has no further obligations in the CDS contract In this event the investor would have minimized its losses by entering into the credit default swap The credit default swap is the basis for the credit derivatives market In 2001 credit default swaps accounted for 38 of the credit derivatives market which was more than two times that of the next highest contributor Today credit default swaps continue to dominate the market and are used as the foundation of newer more complicated 22 products For example a credit default swap index CDSI is a single product based on a basket of credit entities Investopedia 2007 Figure 1 Market Share of Outstanding Notional for Credit Derivative Products Credit Derivatives Explained 2001 Credit Derivative Instrument Type as pa 1999 Credit Def
12. cts King s College London September 2003 27 33 Gordy Michael B ed Credit Risk Modelling the Cutting Edge Collection London Risk Books 2003 Jackson Mary and Mike Staunton Advanced Modelling in Finance using Excel and VBA New York John Wiley amp Sons Ltd Li David X On Default Correlation A Copula Function Approach The RiskMetrics Group April 2000 1 11 Risk Glossary 1996 Current Contingency Analysis 13 Feb Mar 2007 lt http www riskglossary com gt Risk Investopedia 2007 Investopedia Inc Feb Mar 2007 lt http www investopedia com terms r risk asp gt Shimko David ed Credit Risk Models and Management London Risk Books 1999 32
13. default leg DL is the payment contingent upon default 1 5 where 4 is the assumed recovery rate discounted back using the risk free discount factor and the conditional probability of default at time t Equation 2 Present Value of Default Leg PV DL K 1 9 Y B 0 t P t lt t lt t K represents the notional amount and d is the recovery rate Equation 3 Present Value of Swap PV PV DL PV PL SO PVs K Y B 0 t Ple gt t 0 9 1 0 t 1 X 0 k 1 where h t a When a swap is initiated the premium payments are determined by setting the present value of the premium leg and default leg equal to zero in doing so neither party pays anything at the start of the swap contract K t t X BQ t P r gt t 0 K 1 9 Y B 0 t P t lt 7 lt t 0 24 Equation 4 Initial Present Value of CDS PV 0 K Y B0 4 PC gt ta I0 910 t X a k 1 with i l P t gt t le ka We now have an equation that we can use to solve for either the premium payments Xr or the hazard rates a depending on what data is known 6 CDS Spreadsheet The main goal of this project is to use credit default swaps to determine the market s perception of the risk neutral probability of default using a predetermined constant recovery rate The workbook constructed for this serves two major purposes 1 Derive the implied market hazard rates using market quotes for credit default swap
14. ent and vice versa Risks on investments can be grouped into two categories systematic and unsystematic Systematic risks are risks which affect the entire market or a whole market sector Unsystematic risk has an affect on a smaller specific group of investments or even one individual security While unsystematic risk can be reduced through methods such as diversification hedging and leveraging systematic risk can only be reduced by hedging Risk 2007 2 2 1 Interest Rate Risk Interest rates are a form of systematic or market risk because any change affects the entire market Interest rates are constantly changing due to the economy and market fluctuations Fixed and floating interest rates pose risks on investors An investment in a floating rate asset will depreciate if interest rates drop over time Risks also arise with fixed rate assets if maturities on assets and liabilities in a portfolio are mismatched Once an asset or a liability matures if interest rates have changed this has an affect on the overall portfolio value Interest rate risk is the most critical risk factor affecting fixed income securities It is the primary cause for market price fluctuations The varying level of exposure to interest rate risk is the cause for the difference between the interest rate spreads on short and long term bonds The interest rate spread is the difference between the interest rate available on a US treasury security of a given maturity a
15. es Hazard Rate Step Function Smooth Hazard Rate Function Probability Distri lt E gt Ready isc OS E Microsoft Outlook We Student Detail Sched wP1 Bectronic These cos Default Probabi ENE COS Default_Probabil TO 8 38PM You are going to want to solve for the appropriate present value equation in row 8 beginning with C8 and going in order to the right until you end on cell L8 In the Solver Parameters menu you want to Set target cell first to cell C8 Equal to the Value of 0 And in the box where it says By changing cells you want to select the cell of what you are solving for For a this is cell C19 Then click the solve button The corresponding hazard rate for the CDS with maturity 0 5 years should appear in the purple cell C19 You want to repeat this process using Solver 10 times to solve for the hazard rates in row 19 starting at C19 and moving to the right one cell at a time until you solve for the final hazard rate cell L19 30 The computed hazard rates and resulting hazard rate step function can be found by clicking the Hazard Rate Step Function tab A smoothed hazard rate function can be viewed on the Smooth Hazard Rate Function tab And the corresponding probability distribution can be viewed on the Probability Distribution tab To compute the Premiums of the CDS input the hazard rates into the blue cells of Option 2 next to the appropriate
16. ess property implies that the hazard rate is constant 4 1 2 Gamma Distribution The gamma distribution is the sum of k gt 0 independent exponentially distributed random variables The gamma distribution has two parameters k and B where k is the shape parameter and f is the scale parameter A special case of the gamma distribution is when k 1 we have the exponential distribution with 1 1 P ed CDF F t x Te P dx o 8 o gt 17 Gamma Cumulative Distribution Function alpha 1 beta 2 0 alpha 3 beta 2 0 alpha 9 beta 0 5 y PDF Det p x te tax 0 Gamma Probability Density Function 0 5 T T T T alpha 1 beta 2 0 alpha 3 beta 2 0 7 alpha 9 beta 0 5 0 45 0 35 F O 3F 0 25 0 2 0 15 aF y gt f oaos Hazard h t xt ter ar Gamma Hazard Rate Function 14 z z alpha 1 beta 2 alpha 3 beta 2 dia alpha 9 beta 0 5 yk ak y ost 0 6 dl A ae 02 wee LA A ol lt 4 The gamma distribution is suitable for default time modeling if it is perceived that a borrower has to go through a number of stages of crisis before it defaults As gets 18 large the gamma distributed default times behave similarly to exponentially distributed default times i e lim A t const ton 4 1 3 Weibull Distribution The Weibull d
17. estor should receive on the bond Yield is the annual rate of return of an investment The highest quality bonds for example AAA offer minimal credit risk and the lowest 11 yield As the quality decreases credit risk increases but lower quality bonds have much higher yields Higher risk should give higher returns Credit risk inherent in a debt instrument and consequently the credit spread depends on the following factors 1 The probability of a default by the issuer 2 The timing of a possible future default 3 The probability distribution of the recovery rate Assuming a constant and known recovery rate the term structure of the credit spread 1 e the credit risk has a one to one correspondence with the probability distribution of the time of default for the given issuer One of the main goals of this project is to use market prices of traded credit derivatives to recover the market s perception of the probability distribution of the time of default for the issuer 3 Credit Derivatives Upon purchasing a fixed income product the investor faces the risk of financial loss if the issuer defaults on the obligation In order to protect themselves or to hedge this risk investors have the option of buying a credit derivative A credit derivative is a contract between a protection buyer for example the owner of a bond and a protection seller a third party financial institution to transfer the credit risk of an asset without the
18. g interest rate sensitivity As market interest rates change the present value of future cash flows changes affecting the market price of the bond Another key factor in bond price movements is the perceived credit quality of the bond issuer Future payments are only certain once received so if the market senses an increased probability that the issuer will default on some or all of the future payments the value of the bond depreciates Quantifying this credit sensitivity of fixed income securities is the main focus of this project 2 1 2 Swaps A swap is an over the counter OTC financial derivative in which two parties enter into an agreement to exchange a series of cash flows based on the value of an underlying asset but that underlying asset is not directly traded The cash flows can be determined in any manner suitable to both parties objectives as long as the present values of both cash flows are equal Swaps have many uses such as hedging speculation and asset liability management and they are classified by the nature of the cash flow streams being exchanged The most important types are interest rate swaps foreign exchange swaps and credit related swaps An interest rate swap is useful for exchanging fixed rate future cash flows against variable rate future cash flows Foreign exchange swaps are agreements to exchange future cash flows of different currencies Credit related swaps are the main topic of this project and will be e
19. istribution is a three parameter distribution with a gt 0 as the shape parameter 2 gt 0 as the scale parameter and y as the location parameter with 00 lt y lt oo esl A CDF F t 1 e Weibull Cumulative Distribution Function a lambda 0 5 alpha 2 lambda 1 5 alpha 3 Es lambda 3 0 alpha 4 PDF f ae Az Weibull Probability Density Function lambda 0 5 alpha 2 lambda 1 5 alpha 3 H lambda 3 0 alpha 4 19 a l Hazard h t efez A A Weibull Hazard Rate Function lambda 0 5 alpha 2 lambda 1 5 alpha 3 lambda 3 0 alpha 4 The Weibull distribution is a form of extreme value distribution An extreme value distribution is a limiting distribution for the minimum and maximum of a large collection of random observations from the same distribution In terms of probability of default the Weibull distribution governs the time until default of the first to default from a collection of default times 4 2 Approximation of Default Time Distributions In real life not enough information is available about the time of default to determine its probability distribution at every time t Usually it is possible to estimate the probability that the default will happen in various time intervals of positive length e g 6 months 1 year etc In such cases the probabilit
20. ith constant hazard rate Piecewise constant hazard rate is the assumption used in this project and the method used in the spreadsheet for modeling time to default 5 Credit Default Swaps CDS A credit default swap is a contract indexed to a single reference asset which provides insurance against a default event on that asset There are three parties involved in a credit default swap The first is the protection buyer this is the investor and owner of the reference asset for example a General Motors bond The bond issuer General Motors in this example is the second party that plays a role indirectly in the CDS Based on the bond investment General Motors pays the investor periodic coupon payments and 21 promises to pay the principal portion of the bond at a set maturity date After purchasing the bond the investor becomes nervous that General Motors will suffer a credit event and default on its promised future payments So the bond owner purchases protection against the possibility of this credit event in the form of a credit default swap The CDS is a contract between the protection buyer and a protection seller The latter is typically an insurance company or a securities company e g Morgan Stanley In this agreement the protection buyer makes periodic premium payments periods are usually half year increments to the protection seller in this case Morgan Stanley and Morgan Stanley agrees to pay the entire face value of the
21. nd the risk free interest rate The interest rate spread graphed as a function of maturity time is known as the term structure of interest rates Interest rate risk inherent in a fixed income security can be reduced increased or even eliminated through hedging taking an offsetting position in a related security Commonly used hedging instruments are interest rate swaps interest rate options caps floors swaptions and other interest rate derivatives 2 2 2 Default Credit Risk Credit risk is the second most critical risk factor affecting debt instruments This is the risk caused by the possibility that the issuer of the bond may not be able to meet its obligations to pay interest or repay the principal of the loan US treasury securities are considered to be free of credit risk The difference between the interest rate offered on a bond of a particular issuer and the interest rate on the US treasury bond of the same maturity is called the credit spread The credit spread depends on the credit quality of the issuer and on the maturity of the bond The credit spread is the reward an investor receives for assuming the credit risk inherent in the security Default risk is an important factor to take into consideration when making an investment in a fixed income product such as a bond Default occurs when the bond issuer is unable to settle the remaining debt on a bond This leads to the investor losing the remainder of their future coupon payments and
22. oing to the Tools drop down menu and selecting Solver If Solver is not previously installed into your version of Excel you must first install it by clicking on the Tools drop down menu then select Add Ins Check the box next to the Solver Add In and click OK Click on Yes when prompted with the option to install solver now Solver should now be listed in the Tools drop down menu Once you open solver you will be prompted to enter the Solver Parameters 29 Ba Microsoft Excel COS Default_Probabitityd JE El ee Edit View Insert Format Tools Data Window Help Type a question for help _ a X EEE REEF PARAMETERS A 0 50 r 0 05 K 100 00 5 0 30 0 7 Pr P2 l P3 P4 P5 P6 P7 P8 P9 P10 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 10 Premiums Xi s B1 B2 B3 B4 BS B6 B7 B8 B9 B10 0 97531 0 951229 0 927743 0 904837 0 882497 0 860708 0 839457 0 818731 0 798516 0 778801 u 2 B td 5 6 tT t8 a 110 Alphas ai s 1 a 0 995007 0 987605 0 991834 0 987421 0 984425 0 98402 0 977426 0 980177 0 971428 0 975305 Solver Parameters X Set Target Cell ss aa Solve EqualTo Omax Omn Valueof 0 Close By Changing Cells c 19 Guess j Options Add Change Reset All Delete Help LA CS Input Z Premiums Hazard Rat
23. os 21 5 Credit Default Swaps UDS hit id 21 LEDS PACO sa 23 A IN 23 6 1 Computing the Hazard Rates e e ea e antl 26 6 2 Computing the PAS A rta 26 MESS AAA AE A A E E 27 Appendix Workbook User Mia e 28 A a a A a e a a tandem eos 32 1 Introduction Credit risk is becoming an increasingly important topic for evaluation in the financial industry Up until the recent growth of the credit derivatives market interest rate risk was one of the only risk factors taken into consideration when evaluating fixed income securities Interest rate risk still remains the most important risk factor to consider because it affects the entire market but credit risk is important when it comes to debt instruments based strictly on credit There are many different types of credit derivative products all falling into two categories single name credit derivatives and multi name credit derivatives Single name credit derivatives are based on the default risk of one particular company while multi name credit derivatives reference the correlation between the credit risks of various companies The most fundamental single name credit derivative and the basis for many more intricate credit products is the credit default swap A credit default swap provides insurance to the buyer against a credit event such as default Probability of default plays an important role in pricing credit default swaps but this probability is not always known This paper introduces methods to
24. s a step function for the hazard rates corresponding to the credit default swaps These piecewise constant hazard rates form a step function with jumps at the different maturity dates The workbook shows this step function and then takes this step function and smoothes it using cubic splines This smoothed curve represents a continuous hazard of default 6 2 Computing the Prices Assuming a constant recovery rate and using given or derived hazard rates the workbook prices credit default swaps of various maturities Similarly to the above hazard rate procedure the workbook uses Equation 4 and the corresponding hazard rates to compute the premiums 26 7 Conclusions The price of a credit default swap and the probability of default are directly connected Quantifying the default probability and term structure is useful for hedging out credit risk inherent in fixed income securities and is also helpful for calculating the risk neutral prices of credit derivatives other than CDSs These market perceived hazard rates which this pricing model computed can be integrated into more complicated models for multi name credit derivative products such as basket default swaps or CDOs Other extensions of this project could be to incorporate stochastic recovery rates or stochastic hazard rates 27 Appendix Workbook User Manual The first step in using the workbook is to gather market prices for credit default swaps on the reference entity for
25. s the benefits of the asset without having to put the asset on its balance sheet Investopedia 2007 An asset swap is quite similar to a total return swap in that it consists of a bond paired with an interest rate swap An investor purchases a bond and then hedges out the interest rate risk with an interest rate swap The major difference between an asset swap and a total return swap is that in the event of default 13 the total return swap terminates while the interest rate swap payments of the asset swap continue until maturity A credit linked note CLN or credit default note is a product issued by a Special Purpose Vehicle SPV offering investors periodic payments plus the par value of the reference entity at maturity unless default occurs The SPV also enters into a credit default swap with a third party which pays the SPV an annual fee This annual fee provides higher return to investors to compensate for the credit risk involved In the event of default the investors receive a portion of the par value based on the recovery rate and the SPV pays the third party the par value minus the recovery rate A collateralized debt obligation CDO is also a form of credit derivative In a cash flow CDO the investor faces credit risk based on the pool of underlying bonds or loans A CDO is a pool of assets packaged into one portfolio and then that portfolio is tranched It is split up into sections each corresponding to a different level of lo
26. ss The tranches provide the investor with some flexibility in choosing the amount of loss or credit risk to which they are willing to be exposed In a synthetic CDO a CDO made up of credit default swaps the investor faces credit risk based on the credit worthiness of the underlying companies There are many variations of these products but the most common and important credit derivative is the credit default swap CDS which will be explored in detail in Chapter 5 4 Probability of Default The premiums of the credit default swap contract are determined by the market s view of how likely it is that default will occur before the reference entity matures The 14 probability distribution of the time to default is the term structure of credit risk and is one of the driving factors behind the credit derivatives market Default is an event which is modeled using probability theory and statistics Time to default is a random variable t with non negative values which can be characterized by its cumulative probability distribution function F its probability density function f or hazard rate function A Common probability distributions that are used to model the probability of default are the exponential gamma and Weibull distributions The cumulative distribution function cdf F t gives the probability that the default occurs before time F t P lt t The probability density function pdf f t is the derivative of the cumulati
27. ve distribution function whenever F is differentiable d DOs FG or F t 5ds The probability that default will occur in a small time interval of length 4t around time t can be approximated as f t 4t f t Atx P it lt t lt t At The hazard rate h t is the conditional density function of the default time t conditioned on the event that no default has occurred before time t FU o O AGO RP 15 The probability density function can be recovered from the hazard rate function by the following formula t pO Im0 HO 4 1 Typical default time distributions 4 1 1 Exponential Distribution CDF Fi l e if t gt 0 Exponential Cumulative Distribution Function r T lambda 0 5 lambda 1 0 J lambda 1 5 T 7 2 A t PDF fH A e if t gt 0 Exponential Probability Density Function as T T T T T T T Lamda 0 5 Lamda 1 0 Lamda 1 5 16 Hazard h t A Exponential Hazard Rate Function The exponential distribution is characterized by a unique memoryless property In relation to probability of default memoryless indicates that at any given time the probability of default is distinct and does not depend on information from the past Memorylessness is a form of conditional probability that for any positive real numbers s and t we have P T gt t s T gt t P T gt s This memorylessn
28. xplained in further detail below 2 1 3 Asset backed Securities An asset backed security is a fixed income product based on a specified pool of underlying assets The assets or collateral are pooled together to form a single portfolio product that offers lower investment risk through diversification Typical asset backed securities are different combinations of highly illiquid assets such as bonds loans mortgages and credit instruments A common asset backed security is a collateralized debt obligation CDO A CDO is a broad term that encompasses various securities based on the specific type of debt by which they are backed Some examples of specific CDOs are Collateralized Bond Obligations CBOs Collateralized Loan Obligations CLOs Collateralized Mortgage Obligations CMOs etc CDO investors assume the credit risk of the pooled assets without assuming the credit risk of an individual provider Risk Glossary 2007 2 2 Risk Factors Risk requires uncertainty and exposure to that uncertainty The level of uncertainty and exposure determines the level of risk Risk Glossary 2007 Investments in any form of financial product other than a bank deposit at the risk free interest rate involve some sort of risk due to the volatility of the economy Risk comes in many forms and is a major factor involved in pricing financial products and in investor decision making Normally the more risk involved the better the return on the investm
29. y distribution of the continuous random variable t must be determined by some interpolation procedure The main assumption behind these techniques is that default occurs at particular discrete times This assumption is supported by the fact that borrowers usually declare bankruptcy when they are unable to meet an interest payment so defaults often occur on coupon payment dates Probabilistically this means that the time to default is a step function 20 4 2 1 Piecewise Constant CDF A piecewise constant CDF jumps from one step to the next at the discrete times when default is possible e g the semi annual coupon dates The size of each jump corresponds to the probability of default at that particular time Piecewise constant CDFs are not differentiable hence density and hazard functions are not defined in such cases 4 2 2 Piecewise Constant Density Function This approximation assumes that between discrete jump points of the probability density function the default time is uniformly distributed In other words within those intervals of constancy default is equally probable at any time Piecewise constant density functions imply piecewise linearly interpolated CDF The corresponding hazard rate graph consists of adjoining hyperbolic curves 4 2 3 Piecewise Constant Hazard Rate Function A piecewise constant hazard rate assumes that between discrete jump points of the step function the default time follows the exponential distribution w
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