1. Copyright © John Hull 2008 1 Dynamic Models of Portfolio Credit Risk: A Simplified Approach John Hull Princeton Credit Conference May 2008

2. 7 th Edition Out! Copyright © John Hull 2008 2

3. 3 January 30, 2007 Data Table 1 iTraxx CDO tranche quotes January 30, 2007. aLaL aHaH 3 yr5 yr7 yr10 yr 00.03n/a 10.25%24.25%39.30% 0.030.06n/a 42.00106.00316.00 0.060.09n/a 12.0031.5082.00 0.090.12n/a 5.5014.5038.25 0.120.22n/a 2.005.0013.75 Index15.00 23.0031.0042.00 CDX IG CDO tranche quotes January 30, 2007. aLaL aHaH 3 yr5 yr7 yr10 yr 00.03n/a 19.63%38.28%50.53% 0.030.07n/a 63.00172.25427.00 0.070.10n/a 12.0033.7596.00 0.100.15n/a 4.5014.5043.25 0.150.30n/a 2.006.0013.75 Index19.0031.0043.0056.00

4. Copyright © John Hull 2008 4 CDO Models Standard market model is one-factor Gaussian copula model of time to default Alternatives that have been proposed: t-, double-t, Clayton, Archimedian, Marshall Olkin, implied copula All are static models. They provide a probability distribution for the loss over the life of the model, but do not describe how the loss evolves Dynamic models are needed to value options and structured deals such as leveraged super seniors

5. Copyright © John Hull 2008 5 Dynamic Models for Portfolio Losses: Prior Research Structural: Albanese et al; Baxter (2006); Hull et al (2005) Reduced Form: Duffie and Gârleanu (2001), Chapovsky et al (2006), Graziano and Rogers (2005), Hurd and Kuznetsov (2005), and Joshi and Stacey (2006) Top Down: Sidenius et al (2004), Bennani (2005), Schonbucher (2005), Errais, Giesecke, and Goldberg (2006), Longstaff and Rajan (2006), Putyatin et al (2005), and Walker (2007)

6. Copyright © John Hull 2008 6 Our Objective Build a simple dynamic model of the evolution of losses that is easy to implement and easy to calibrate to market data The model is developed as a reduced form model, but can also be presented as a top down model

7. Copyright © John Hull 2008 7 Specific vs General Models Specific models track the evolution of default risk on a single name or portfolio that remains fixed (e.g. describes how credit spread for a particular company evolves) General models track the evolution of default risk on a single name or portfolio that is updated so that it always has certain properties (e.g. describes how the average spread for an A-rated company evolves) We focus on specific models

8. Copyright © John Hull 2008 8 CDO Valuation Key to valuing a CDO lies in the calculation of expected tranche principal on payment dates Expected payment on a payment date equals spread times expected tranche principal on that date Expected payoff between payment dates equals reduction in expected tranche principal between the dates Expected accrual payments can be calculated from expected payoffs Expected tranche principal at time t can be calculated from the cumulative default probabilities up to time t and recovery rates of companies in the portfolio

9. The Model Instead of modeling the hazard rate, h ( t ) we model This is –ln[ S ( t )] where S ( t ) is the survival probability calculated from the path followed by the hazard rate between times 0 and t Filtration: We assume that at time t we know the path followed by S between time zero and time t and the number of defaults up to time t Copyright © John Hull 2008 9

10. 10 The Model (Homogeneous Case) where dq represents a jump that has intensity and jump size H  and are functions only of time and H is a function of the number of jumps so far.  > 0, H > 0. X +  t X +  t + H tt  t X

11. The Hazard Rate Process The hazard rate process is where dI is an impulse that has intensity Copyright © John Hull 2008 11

12. Copyright © John Hull 2008 12 Analytic Results and Binomial Trees Once  (t), (t), and the the size of the jth jump, H j, have been specified the model can be used to value analytically CDOs Forward CDOs Options on CDOs For other instruments a binomial tree can be used

13. Copyright © John Hull 2008 13 Binomial Tree for Model

14. Copyright © John Hull 2008 14 Simplest Version of Model Jump size is constant and  (t), is zero Jump intensity, (t) is chosen to match the term structure of CDS spreads There is then a one-to-one correspondence between tranche quotes and jump size Implied jump sizes are similar to implied correlations

15. Copyright © John Hull 2008 15 Comparison of Implied Jump Sizes with Implied Tranche Correlations

16. Copyright © John Hull 2008 16 Possible Extensions to Fit All Market Data Multiple processes each with its own jump size and intensity Intensity and jump size changing in different intervals: 0 to 5 yrs, 5 to 7 yrs, and 7 to 10 yrs Model can (in principle) fit all quotes simultaneously We have chosen to focus on extensions where there are relatively few parameters

17. Copyright © John Hull 2008 17 Extension Involving Three Parameters The size of the Jth jump is H J = H 0 e  J The three parameters are The initial jump size The growth in the jump size The jump intensity The model reflects empirical research showing that correlation is higher in adverse market conditions

18. Copyright © John Hull 2008 18 Variation of best fit jump parameters, H 0 and  across time. (10-day moving averages)

19. Copyright © John Hull 2008 19 Variation of jump intensity (10-day moving averages)

20. Copyright © John Hull 2008 20 Evolution of Loss Distribution on January 30, 2007 for 3-parameter model.

21. Copyright © John Hull 2008 21 Valuation of Forward Contracts on CDOs that End in Five Years Using 3-Parameter Model on January 30, 2007 Table 4 Breakeven tranche spread for forward start CDO tranches on iTraxx on January 30, 2007. The tranches mature in five years. Results are based on the three-parameter model in Section IIIC calibrated to the market data in Table 1. Tranche Start1.02.03.04.04.5 aLaL aHaH Breakeven Tranche Spreads 00.03 11.511.311.16.43.4 0.030.06 54.070.193.2124.4144.2 0.060.09 14.719.426.135.240.7 0.090.12 5.87.710.614.817.5 0.120.22 2.02.63.75.36.3 Index 25.329.136.741.443.7

22. Copyright © John Hull 2008 22 Valuation of At-The-Money Options (in basis points) on CDOs that End in Five Years Using 3- Parameter Model on January 30, 2007 Table 5 Prices in basis points of at-the-money European options on iTraxx CDO tranches on January 30, 2007. The tranches mature in five years. Results are based on the three-parameter model in Section IIIC calibrated to the market data in Table 1. Option Expiry in Years aLaL aHaH 1.02.03.04.04.5 0.030.0667.891.389.768.341.4 0.060.0923.229.830.723.113.5 0.090.129.712.213.310.06.1 0.120.223.74.45.03.82.4

23. Copyright © John Hull 2008 23 Implied Volatilities from Black’s Model as Option Maturity is Changed Table 6 Implied volatilities of at-the-money European style options on iTraxx CDO tranches on January 30, 2007. The tranches mature in five years. Results are based on the three-parameter model in Section IIIC calibrated to the market data in Table 1. Option Expiry in Years aLaL aHaH 1.02.03.04.04.5 0.030.0696.1%100.9%96.8%104.3%107.5% 0.060.09123.2%122.6%125.6%137.2%135.3% 0.090.12133.0%128.6%139.4%144.8%148.9% 0.120.22149.3%137.3%160.5%160.6%181.8%

24. Copyright © John Hull 2008 24 Implied Volatilities for 2 yr Option from Black’s Model as Strike Price is Changed Table 7 Implied volatilities for 2-year European options on iTraxx CDO tranches for strike prices between 75% and 125% of the forward spread on January 30, 2007. The tranches mature in five years. Results are based on the three-parameter model in Section IIIC calibrated to the market data in Table 1. CDO Tranche K / F3 to 6%6 to 9%9 to 12%12 to 22% 0.75100.1%125.6%132.9%143.5% 0.80100.6%125.2%132.1%142.3% 0.85100.9%124.7%131.3%141.1% 0.90101.1%124.1%130.5%139.9% 0.95101.0%123.4%129.5%138.6% 1.00100.9%122.6%128.6%137.3% 1.05100.6%121.7%127.5%136.0% 1.10100.3%120.9%126.5%135.2% 1.1599.8%119.9%125.4%135.3% 1.2099.3%119.0%125.0%136.0% 1.2598.8%118.0%124.8%137.1%

25. Copyright © John Hull 2008 25 Leverage Super Senior with Loss Trigger Total exposure of seller of protection is limited to a fraction x of the tranche notional When losses reach some level the buyer of protection can cancel the deal and seller has to pay the value of the tranche to the buyer. Define n * as the number of losses triggering cancellation

26. Copyright © John Hull 2008 26 Breakeven LSS spread on Jan 30, 2007 as a function of the maximum percentage loss by protection seller, x%, and the number of defaults triggering close out, n *

27. Copyright © John Hull 2008 27 Extensions Model can be extended so that Different companies have different CDS spreads The recovery rate is negatively correlated with the default rate iTraxx and CDX jumps are modeled jointly

28. Bespokes Calibrate homogeneous model to iTraxx and CDX IG If all names are North American, use a non- homogeneous model where underlying companies have the the CDX IG jumps and their own drifts. If there are a mixture of European and North American names use a non-homogeneous model where the iTraxx and CDX IG jumps are modeled jointly Copyright © John Hull 2008 28

29. Copyright © John Hull 2008 29 Conclusions It is possible to develop a simple dynamic model for losses on a portfolio by modeling the cumulative default probability for a representative company The only way of fitting the market appears to be by assuming that there is a small probability of a series of progressively bigger jumps in the cumulative probability. As credit market deteriorates default correlation becomes higher