1. Credit Derivatives HEG program in Risk Management Prof. Hugues Pirotte Professor of Finance Slides based upon some of my own notes, research and on Hull RMFI slides

2. Who am I?  PhD from HEC Lausanne with a dissertation on Credit Risk Analysis  Co-author of Advanced Credit Risk Analysis (Wiley, UK)  Co-founder of FinMetrics  Professor of Finance with courses on Asset Management, Derivatives, Risk Management and Governance, Corporate Finance  Co-founder of the Finance Club of Brussels  Member of the BEL20 Committee 2 Prof. Hugues Pirotte

3. Agenda 1.Refresher on Credit Risk 2.Forms of Credit Derivatives 3.Elements of pricing of Credit Derivatives 3 Prof. Hugues Pirotte

4. Credit risk  Where does credit risk play a role? 4 Prof. Hugues Pirotte

5. Credit risk  Why is credit risk different from other risks (and should not be traded the same way)? 5 Prof. Hugues Pirotte

6. H. Pirotte 6 What is credit risk?  Credit risk existence derives from the possibility for a borrower to default on its obligations to pay interest or to repay the principal amount. »As valued today... »We are valuing today a discontinuity in the future that may potentially happen but maybe not...  Consequence: »Cost of borrowing > Risk-free rate »Spread = Cost of borrowing – Risk-free rate (usually expressed in basis points) »Volume »Rating change Internal (for loans) External: rating agencies (for bonds)

7. Ratings & rating agencies  The traditional practice is to « rate » issuers and issuances... »Moody’s (www.moodys.com)www.moodys.com »Standard and Poors (www.standardandpoors.com)www.standardandpoors.com »Fitch/IBCA (www.fitchibca.com)www.fitchibca.com  Letter grades (qualitative score) to reflect safety of bond issue 7 H. Pirotte Long-termS&PMoody’s AAAAaa AAAa AA BBBBaa BBBa BB CCCCaa CCCa CC CI,R,SD,DWR,P NR = non-rated Short-termS&P A-1 A-2 A-3 B C D Moody’s P-1 P-2 P-3 NP A,B,C,D,E for banks

8. H. Pirotte 8 Credit Spreads by rating class

9. H. Pirotte 9 Inputs...  Used in probabilistic models and integrated in the regulation: »PD: probability of default »LGD: loss-given-default (may be in % or in value) »EAD: exposure-at-default (used by Basle II to separate the LGD in % from the real exposure beard by the firm).

10. H. Pirotte 10 Transition matrix of rating migrations Exhibit 15 - Average One-Year Letter Rating Migration Rates, 1920-2007* End-of-Period Rating Cohort RatingAaaAaABaaBaBCaaCa-CDefaultWR Aaa87.2927.4740.8410.1670.0240.0010.000 4.200 Aa1.26185.2046.4650.6870.1750.0370.0020.0040.0636.103 A0.0812.93485.0865.2980.6930.1080.0190.0080.0765.696 Baa0.0420.2934.61881.1405.1070.7760.1500.0160.2937.565 Ba0.0070.0820.4765.91773.6436.9770.5570.0511.32410.967 B0.0070.0540.1730.6306.29271.4595.0110.5023.91711.955 Caa0.0000.0280.0370.2160.9068.92062.7973.54912.00011.548 Ca-C0.000 0.1160.0000.4743.2407.69855.32319.87213.277 * Monthly cohort frequency Source: Moody’s, Corporate Default and Recovery Rates, 1920-2007, February 2008.

11. H. Pirotte 11 Cumulative default rates Exhibit 26 - Average Cumulative Issuer-Weighted Global Default Rates, 1920- 2007* RatingYear 1Year 2Year 3Year 4Year 5Year 6Year 7Year 8Year 9Year 10 Aaa000.0190.0770.1630.2550.3680.5310.7010.897 Aa0.0610.1810.2860.4460.7041.0131.3361.6511.9532.294 A0.0730.2370.50.8081.1161.4481.7962.1312.5042.901 Baa0.2880.851.5612.3353.1423.9394.7075.4756.2787.061 Ba1.3363.25.3157.499.58711.5613.36315.11116.73318.435 B4.0478.78613.49417.7221.42524.65627.59430.03732.15433.929 Caa-C13.72822.4629.02933.91637.63840.58442.87244.92146.99648.981 Investment-Grade0.1440.4310.8051.231.6872.1572.6263.0913.5784.076 Speculative-Grade3.597.23710.75213.91916.71419.17921.37223.33625.11426.827 All Rated1.4062.8784.3155.6266.8027.8548.8039.66710.48411.281 * Includes bond and loan issuers rated as of January 1 of each year. Source: Moody’s, Corporate Default and Recovery Rates, 1920-2007, February 2008.

12. H. Pirotte 12 Default rates by industry group Source: Moody’s, Corporate Default and Recovery Rates, 1920-2007, February 2008.

13. H. Pirotte 13 Recovery rates Source: Moody’s, Corporate Default and Recovery Rates, 1920-2007, February 2008.

14. H. Pirotte 14 Recovery rates Source: Moody’s, Corporate Default and Recovery Rates, 1920-2007, February 2008.

15. H. Pirotte 15 Recovery rates... and their volatility  A prior study

16. H. Pirotte 16 How do we try to quantify credit risk? 1)Historical stats »probabilities of default (PD) »recovery rates (R) or loss-given-default (1-R) 2)Scoring »Z-scores (Altman) »Ratings (Moody’s, S&P, Fitch): PIT and TTC 3)Model credit spreads »An exchange rate (Jarrow, Jarrow & Turnbull) »Reduced-form models (Duffie & Singleton, Lando) Calibration of PD and LGD to traded products »Through the option pricing model (Merton) »Strategic default (Anderson & Sundaresan) 4)Portfolio credit risk

17. Econometric scoring (2) Prof H. Pirotte 17

18. Modeling credit spreads (3) Prof H. Pirotte 18 Assets Debt Equity Assets Liabiities Modeling the value of shareholders and debtholders depending on the capital structure and against the asset value PD, LGD Credit spreads PD, LGD Credit spreads Strcutural Models – BOTTOM-UP approach Reduced-form Models – TOP-DOWN

19. H. Pirotte 19  A starting point  The credit spread being  The FX analogy (Jarrow & Turnbull)  If default is a possibility... The reduced-form approach(es)

20. H. Pirotte 20 The reduced-form approach(es) (2)  Therefore...  Or...  Which means...

21. H. Pirotte 21 Example

22. The structural approach (Merton) – step 1 22 Prof H. Pirotte Assets Debt Equity Assets Liabiities E Market value of equity F Face value of debt V Market value of company Bankruptcy D Market value of debt F Face value of debt V Market value of company F Loss given default Assets Liabiities Assets market value = 100K Debt F = 70K Equity...

23. Now, we know that...  Options can be valued in two ways »Continuous-time model: Black-Scholes(-Merton) formula »Binomial model Prof H. Pirotte 23 Increase the number to time steps for a fixed maturity The probability distribution of the firm value at maturity is lognormal Time Value Today Bankruptcy Maturity

24. 24 Prof. Hugues Pirotte

25. Continuous model (reminder)  From any options course, we know that... »Value at maturity of a call, e.g. »Thus, the value at t=0 The valuation difficulty is of course in the last step and was first demonstrated with the PDE approach and then with the equivalent martingale measure approach. Prof H. Pirotte 25

26. H. Pirotte 26 The (Merton) structural model (2)  Debt can be seen as...

27. H. Pirotte 27 Merton Model: example using binomial pricing Data: Market Value of Unlevered Firm: 100,000 Risk-free rate per period: 5% Volatility: 40% Company issues 1-year zero-coupon Face value = 70,000 Proceeds used to pay dividend or to buy back shares V = 100,000 E = 34,854 D = 65,146 V = 67,032 E = 0 D = 67,032 V = 149,182 E = 79,182 D = 70,000 ∆t = 1 Binomial option pricing: review Up and down factors: Risk neutral probability : 1-period valuation formula

28. Credit Derivatives  Derivatives where the payoff depends on the credit quality of a company or sovereign entity  The market started to grow fast in the late 1990s  By 2005 notional principal totaled $12 trillion  Main »CDS »CDOs »CLOs »TRSs

29. Credit Default Swaps (CDS)  Idea: »Buyer of the instrument acquires protection from the seller against a default by a particular company or country (the reference entity) »Example: Buyer pays a premium of 90 bps per year for $100 million of 5- year protection against company X »Premium is known as the credit default spread. It is paid for life of contract or until default »If there is a default, the buyer has the right to sell bonds with a face value of $100 million issued by company X for $100 million (Several bonds may be deliverable)  Advantages »Allows credit risks to be traded in the same way as market risks »Can be used to transfer credit risks to a third party »Can be used to diversify credit risks

30. CDS Structure  Payments are usually made quarterly or semiannually in arrears  In the event of default there is a final accrual payment by the buyer  Settlement can be specified as delivery of the bonds or in cash  Suppose payments are made quarterly in the example just considered. What are the cash flows if there is a default after 3 years and 1 month and recovery rate is 40%? Default Protection Buyer, A Default Protection Seller, B 90 bps per year Payoff if there is a default by reference entity=100(1-R) Recovery rate, R, is the ratio of the value of the bond issued by reference entity immediately after default to the face value of the bond

31. Credit Default Swaps and Bond Yields  Portfolio consisting of a 5-year par yield corporate bond that provides a yield of 6% and a long position in a 5-year CDS costing 100 basis points per year is (approximately) a long position in a riskless instrument paying 5% per year  What are the arbitrage opportunities in this situation if risk-free rate is 4.5%? What if it is 5.5%?

32. Credit Indices  CDX IG: equally weighted portfolio of 125 investment grade North American companies  iTraxx: equally weighted portfolio of 125 investment grade European companies  If the five-year CDS index is bid 65 offer 66 it means that a portfolio of 125 CDSs on the CDX companies can be bought for 66bps per company, e.g., $800,000 of 5-year protection on each name could be purchased for $660,000 per year. When a company defaults the annual payment is reduced by 1/125.

33. CDS Valuation  Suppose that conditional on no earlier default a reference entity has a (risk-neutral) probability of default of 2% in each of the next 5 years  Assume payments are made annually in arrears, that defaults always happen half way through a year, and that the expected recovery rate is 40%  Suppose that the breakeven CDS rate is s per dollar of notional principal

34. CDS Valuation (2)  See Excel file 34 Prof. Hugues Pirotte

35. Implying Default Probabilities from CDS spreads  Suppose that the mid market spread for a 5 year newly issued CDS is 100bps per year  We can reverse engineer our calculations to conclude that the default probability is 1.61% per year.  If probabilities are implied from CDS spreads and then used to value another CDS the result is not sensitive to the recovery rate providing the same recovery rate is used throughout

36. Binary CDS  The payoff in the event of default is a fixed cash amount  In our example the PV of the expected payoff for a binary swap is 0.0852 and the breakeven binary CDS spread is 207 bps

37. CDS Forwards and Options  CDS forward is an agreement to enter into a CDS with a specified spread at a future time  CDS option is an option to enter into a CDS with a specified spread at a future time  Both a CDS forward and a CDS option cease to exist if there is a default before the end of the life of the forward or option

38. Total Return Swap  Agreement to exchange total return on a corporate bond (or other portfolio of securities) for LIBOR plus a spread  At the end there is a payment reflecting the change in value of the bond  Usually used as financing tools by companies that want an investment in the corporate bond Total Return Payer Total Return Receiver Total Return on Bond LIBOR plus 25bps

39. First to Default Basket CDS  Similar to a regular CDS except that several reference entities are specified and there is a payoff when the first one defaults  This depends on “default correlation”  Second, third, and nth to default deals are defined similarly

40. The Impact of Correlation  Consider a basket of 20 names  What happens to the cost of first-to-default protection as the default correlation increases?  What happens to the cost of 15 th -to-default protection as the default correlation increases?

41. Collateralized Debt Obligation (CDO)  A pool of debt issues are put into a special purpose trust  Trust issues claims against the debt in a number of tranches »First tranche covers x% of notional and absorbs first x% of default losses »Second tranche covers y% of notional and absorbs next y% of default losses »etc  A tranche earn a promised yield on remaining principal in the tranche

42. Bond 1 Bond 2 Bond 3  Bond n Average Yield 8.5% Trust Tranche 1 1 st 5% of loss Yield = 35% Tranche 2 2 nd 10% of loss Yield = 15% Tranche 3 3 rd 10% of loss Yield = 7.5% Tranche 4 Residual loss Yield = 6% Cash CDO Structure (Figure 13.3, page 312)

43. 43 A CDO vs. other securitizations  A standard securitisation...  For a CDO... AssetsLiabilities BANK AssetsLiabilities SPV Loans Highly Senior Senior Junior Equity Mezzanine tranches AssetsLiabilities BANK AssetsLiabilities SPV Loans Same claims

44. Synthetic CDO  Instead of buying the bonds the arranger of the CDO sells credit default swaps.  Or...any exposure through the use of Indices...

45. Single Tranche Trading  This involves trading standard tranches of standard portfolios that are not funded  CDX IG (Aug 30, 2005):  iTraxx IG (Aug 30, 2005) Tranche0-3%3-7%7-10%10-15%15-30% Quote40%127bps35.5bps20.5bps9.5bps Tranche0-3%3-6%6-9%9-12%12-22% Quote24%81bps26.5bps15bps9bps

46. Technology (reminder)  First we need to define the default problem as a “stopping-time” problem

47. Technology (reminder 2)  How can we correlate these potential “stopping-time” functions for each counterpart?

48. Technology (reminder 3)  We can use the normal distribution applied to a factor model  Then we can compute the probability to mature after the stopping time (uing the Gaussian copula)  Hypotheses: »All companies assumed to have same constant default intensity that is consistent with the index, and same global correlation. »Define Q(T) as default probability by time T

49. Technology (reminder 4)  A simple binomial model applied on this can allow us to know what is the proba of more than “n” defaults  We can use this to know the probability of a tranche to be hit and therefore to put a price on it! (conditional on F)  Otherwise we use again Monte Carlo simulations!

50. Standard Market Model for nth to Defaults and CDOs  Conditional on F, defaults are independent so that the probability of exactly k defaults from N companies by time T is  This enables cash flows to be calculated conditional on F  We then integrate over F  Derivative dealers calculate an implied correlation from tranche quotes in the same way that they calculate an implied volatility from option quotes 50 Prof. Hugues Pirotte

51. Implied Correlations  Tranche correlation (or compound correlation) is the correlation that prices a particular tranche consistently with market quote  Base correlation is the correlation such that prices all tranches up to a certain level of seniority consistently with market quotes

52. 52 Consequence... Portfolio of real loans, or managed CDS Equity tranche Mezzanine tranche Senior tranche What do you think is the impact of correlation in the calculation? AssetsLiabilities