1. 1 Credit risk References: see course outline

2. 2 Outline Management of credit risk (“Credit risk I”) Modelling of credit risk intro (“Credit risk II”) –Merton (1974) Credit and counterparty risk in derivatives transactions (“Credit risk III”)

3. 3 Credit risk I (Credit Derivatives & Structured Finance) References: see course outline

4. 4 Managing credit risk FIs and creditors in general have a variety of tools to manage credit risk: –Credit risk mitigation –Loan sales –Credit derivatives –Securitization

5. 5 Credit Risk Mitigation Netting Collateralization Downgrade triggers (embedded put options which give the investor the right to force early redemption at face value) Bond insurance, guarantees, and letters of credits Termination or reassignment

6. 6 Credit Derivatives Derivatives where the payoff depends on the credit quality of a company or country The market started to grow fast in the late 1990s Mostly OTC By 2003 notional principal totaled $3 trillion

7. Credit Default Swaps (page 352) 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) 7

8. 8 CDS Structure 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

9. 9 Other Details 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 (but there are also “binary CDSs”) 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%?

10. Attractions of the CDS Market 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 10

11. 11 Using a CDS to Hedge a Bond 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

12. Credit Default Swaps and Bond Yields (page 357-358) 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 arbitrage opportunities in this situation is risk-free rate is 4.5%? What if it is 5.5%? 12

13. 13 Securitization Securitization is a structured finance process, which involves pooling and repackaging of cash-flow producing financial assets into securities that are then sold to investors. The name is derived from the fact that the form of financial instruments used to obtain funds from the investors are securities. All assets can be securitized, so long as they are associated with cash flow. Hence, the securities, which are the outcome of securitization processes, are termed asset-backed securities (ABS). So, securitization could also be defined as a process leading to an emission of ABS. Securitization has evolved from tentative beginnings in the late 1970s to a vital funding source –estimated total aggregate outstanding of $8.06 trillion (as of the end of 2005, by the Bond Market Association) –new issuance of $3.07 trillion in 2005 in the U.S. markets alone.

14. 14 Securitization Suppose an FI has $100 million worth of new mortgages and these are financed by short term deposits. It faces a variety of costs and risks –Capital requirements, etc. –Duration mismatch –Illiquidity The FI can securitize by issuing collateralized mortgage obligations (special case of collateralized debt obligations)

15. 15 Collateralized Debt Obligation A pool of debt issues are put into a special purpose vehicle (SPV), a 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

16. 16 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% CDO Structure

17. 17 Why is (was!) the sum of the parts greater than the whole? Diversification benefits Higher liquidity Segmentation/tailoring to specific capital market needs Regulatory capital arbitrage See Oldfield’s (JFI, 1997) “The Economics of Structured Finance”

18. 18 Credit Risk II ( Estimating Default Probabilities and pricing ) References: see course outline

19. 19 Estimating Default Probabilities Alternatives: –Use Historical Data –Use Bond Prices or spreads implied by other traded securities (e.g. asset swaps or CDS spreads) –Use Option Pricing Models (Merton’s Model)

20. 20 Credit Ratings Historical data provided by rating agencies are often used as estimates of the probability of default In the S&P rating system, AAA is the best rating. After that comes AA, A, BBB, BB, B, and CCC The corresponding Moody’s ratings are Aaa, Aa, A, Baa, Ba, B, and Caa Bonds with ratings of BBB (or Baa) and above are considered to be “investment grade”

21. Cumulative Average Default Rates % (1970-2010, Moody’s) Table 16.1, page 350 21

22. Recovery Rate The recovery rate for a bond is usually defined as the price of the bond 30 days after default as a percent of its face value 22

23. Recovery Rates; Moody’s: 1982 to 2010 (Table 16.2, page 352) 23 ClassAve Rec Rate (%) First lien bank loan65.8 Second lien bank loan29.1 Senior unsecured bank loan47.8 Senior secured bond50.8 Senior unsecured bond36.7 Senior subordinated bond30.7 Subordinated bond31.3 Junior subordinated bond24.7 But see Altman, Resti and Sironi (JoB 2004, BIS 2005, etc) for evidence on cyclical time-variation of the recovery rate….

24. Recovery Rates Depend on Default Rates Moody’s best fit estimate for the 1982 to 2007 period is Ave Recovery Rate = 59.33 − 3.06 × Spec Grade Default Rate R 2 of regression is about 0.5 24

25. 25 Using Bond Prices Denoting by d the cumulative default probability over T years, then 1 – d is the probability of the bond issuer surviving for T years and average default intensity over life of a bond is defined as λ such that Average default intensity over life of bond is approximately where s is the spread of the bond’s yield over the risk-free rate and R is the recovery rate

26. 26 TED

27. 27 Asset Swaps Asset swaps are used by the market as an estimate of the bond yield relative to LIBOR Under LOOP, the present value of the asset swap spread is the present value of the RN expectation of the cost of default

28. 28 Asset Swaps Suppose asset swap spread for a particular corporate bond is 150 basis points One side pays coupons on the bond; the other pays LIBOR+150 basis points. The cash flows (coupons on the bond and the LIBOR + spread) are paid regardless of whether there is a default on the underlying bond. In addition, there is an initial exchange of cash reflecting the difference between the bond price and $100 –i.e., the party that pays the coupons of the bond also pays the difference between $100 and the bond price (this is because the counterparty pays the LIBOR+150bps stream on a notional principal of $100) The PV of the asset swap spread is the amount by which the price of the corporate bond is exceeded by its present value when the flat LIBOR/swap curve is used for discounting (i.e., if LIBOR flat  risk free rate, approx. the price of a risk-free bond with same coupon and maturity).

29. 29 Asset Swap Example Suppose that the LIBOR/swap curve is flat at 6% with continuous compounding and a five-year bond with a coupon of 5% (paid semiannually) sells for 90.00. You hold the bond for a face value of $100 and want to enter an asset swap to finance it at a fixed rate so as to remove funding rate risk. How would the asset swap be structured? What is the asset swap spread that would be calculated in this situation? The principal is $100. The asset swap is structured so that $10 is paid initially by you (so the initial cost of the overall strategy for you is $90 + $10 = $100). After that, you pay out the coupon of the bond, i.e. $2.50 every six months. In return, you receive LIBOR plus a spread on the principal of $100. The present value of the fixed and LIBOR floating payments, excluding the spread, is The spread over LIBOR must therefore have a present value of $5.3579. The present value of $1 received every six months for five years is $8.5105. The spread cash flow received every six months must therefore be $1  5.3579/8.5105 = $1  0.6296 = $0.6296. The asset swap spread is therefore $0.6296  2 / $100= 1.2592% per annum.

30. PV of asset swap spread 30

31. PV of asset swap spread 31

32. 32 Merton’s Model Merton’s (1974) model regards the equity and debt of a firm as a long call and (a risk-free bond plus) a short put, respectively, on the assets of the firm with payoffs  max(V T - D, 0)  max(-V T, -D) = D - max(D - V T, 0) where V T is the value of the firm and D is the debt repayment required in T

33. 33 All Equity Firm In all equity firm, shareholders have the firm’s assets that are worth V. Just like a payoff on a stock, payoff on firm’s assets can be replicated with a bond that pays a risk-free amount D in T and a long call and a short put on firm’s assets with strike D and maturity T  V = De -rT + C – P  On expiry: V T = D + C T – P T = + D + Max(V T – D, 0) – Max(D – V T, 0)

34. 34 Debt as Short Put When firm issues (risky) debt for an amount D and maturity T, shareholders sell risk-free zero with face value D and buy a put from debt holders with strike D. They now have V – De -rT + P = (De -rT + C – P) – De -rT + P = C Debt holders have De -rT – P Default probability is probability that the put option ends up in the money

35. 35 Equity vs. Assets An option pricing model (e.g. B&S) enables the value of the firm’s equity today, E 0, to be related to the value of its assets today, V 0, and the volatility of its assets,  V

36. 36 Volatilities This equation together with the option pricing relationship enables V 0 and  V  to be determined from E 0 and  E

37. 37 Distance From Default Default probability is then N(-d 2 ), i.e. B&S probability of exercise of put The quantity d 2 is the (standardized) ‘distance from default’

38. 38 Merton’s Model Here F denotes the s.c. default point, this is in theory D but realistically might be a bit lower, e.g. short term debt + half long term debt (holders of long term debt are stuck with it, and typically it takes them so long to force bankruptcy on a delinquent debtor firm that in the meantime the value of the firm asset has dropped to around half or less of the face value of debt (hence, the typical 50% or less recovery rate).

39. 39 Example A company’s equity is $3 million and the volatility of the equity is 80% The risk-free rate is 5%, the debt is $10 million and time to debt maturity is 1 year Solving the two equations yields V 0 =12.40 and  v =21.23%

40. 40 Example continued The probability of default is N(-d 2 ) or 12.7% The market value of the debt is 9.40 The present value of the promised payment is 9.51 The expected loss is about 1.2% The recovery rate is 91%

41. 41 The Implementation of Merton’s Model (e.g. Moody’s KMV) Choose time horizon Calculate cumulative obligations to time horizon. This is termed by KMV the “default point”. We denote it by F Calculate ‘distance from default’ d 2 Then, two possible routes –Use d 2 and Merton’s model to calculate a theoretical RN probability of default –Use historical data to develop a one-to-one mapping of d 2 into real- world probability of default, e.g. could estimate the expected default frequency (EDF) typical of firms with given distance from default. Assumption is that the rank ordering of probability of default given by the model is the same as that for real world probability of default

42. 42 Risk Capital Merton’s model provides the conceptual framework for interpreting equity as ‘risk capital’, i.e. the capital buffer against adverse fluctuations in the value of business –Given equity, riskiness of assets determines probability of default –Given acceptable probability of default, riskiness of assets determines required equity, i.e. risk capital –See my article

43. 43 Default Correlation The credit default correlation between two companies is a measure of their tendency to default at about the same time Default correlation is important in risk management when analyzing the benefits of credit risk diversification It is also important in the valuation of some credit derivatives, eg a first-to-default CDS and CDO tranches.

44. 44 Measurement There is no generally accepted measure of default correlation, but Gaussian copula has become benchmark model Default correlation is a more complex phenomenon than the correlation between two random variables

45. 45 Real World vs Risk-Neutral Default Probabilities The default probabilities backed out of bond prices or credit default swap spreads are risk- neutral default probabilities The default probabilities backed out of historical data are estimates of real-world default probabilities See appendix for some empirical comparison

46. 46 Credit risk III (Credit and counterparty risk in derivatives transactions) References: see course outline

47. Credit Risk in Derivatives Transactions and loss given default (page 531-534) 47 Three cases –Contract always an asset –Contract always a liability –Contract can be an asset or a liability

48. General Result Assume that default probability is independent of the value of the derivative Define t 1, t 2,…t n : times when default can occur q i : default probability at time t i. f i : The value of the transaction at time t i R: Recovery rate 48

49. General Result continued The expected loss from default at time t i is q i (1−R)E[max(f i,0)] Defining u i = q i (1−R) and letting v i denote the value at t 0 of a derivative that provides a payoff of max(f i,0) at time t i, the PV of the cost of defaults is 49

50. Applications If transaction is always an asset so that f i > 0, then v i = f 0  i and the cost of defaults is f 0 times the total default probability,, times 1−R If transaction is always a liability, then v i = 0 and the cost of defaults is zero In other cases we must value the derivative max(f i,0) for each value of i 50

51. Expected Exposure on Pair of Offsetting Interest Rate Swaps and a Pair of Offsetting Currency Swaps Exposure Maturity Currency swaps Interest Rate Swaps 51

52. Interest Rate vs Currency Swaps The u i ’s are the same for both The v i ’s for an interest rate swap are on average much less than the v i ’s for a currency swap The expected cost of defaults on a currency swap is therefore greater. 52

53. Appendix 53

54. 54 Loan Sales What can a bank do if its loans are too heavily concentrated in one sector? Loan sales - sales of loans from one FI to another. The loan may be sold with or without recourse. This market has grown substantially in the last 20 years or so.

55. 55 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 ( = 0.0511  (1 – 40%)) and the breakeven binary CDS spread is 207 bps ( = 124 bps / (1 – 40%))

56. Credit Indices (page 356-357) 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 165 offer 166 it means that a portfolio of 125 CDSs on the CDX companies can be bought for 166bps per company, e.g., $800,000 of 5-year protection on each name could be purchased for $1,660,000 per year. When a company defaults the buyer receives $800,000 and the annual payment that she must make is reduced by 1/125. 56

57. 57 Total Return Swap Agreement to exchange total return on a corporate bond 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

58. 58 Other Credit Derivatives Credit default option Etc.

59. 59 Synthetic CDOs Instead of creating an ABS from corporate bonds it can be created from CDSs, i.e. instead of buying the bonds the arranger of the CDO sells credit default swaps. This is referred to as a synthetic CDO –Tranche 1 might be responsible for the first $5 million of losses –Tranche 2 might be responsible for the next $10 million of losses –And so on

60. 60 CMOs Say the FI issues GNMA (Government National Mortgage Association, ‘Ginnie Mae’) pass-through securities. Another bank or another CMO issuer such as FHLMC could buy up the entire issue, place it in a trust, and issue n new classes of bonds backed by the GNMA securities and other assets as collateral. Normally, the sum of the value of these n classes will (used to!) exceed the value of the GNMA bond and the other collateral. Why?

61. Hazard Rate vs. Unconditional Default Probability The hazard rate or default intensity is the probability of default over a short period of time conditional on no earlier default The unconditional default probability is the probability of default as seen at time zero 61

62. Properties of Hazard rates 62 Suppose that (t) is the hazard rate at time t The probability of default between times t and t+  t conditional on no earlier default is (t)  t The probability of default by time t is where is the average hazard rate between time zero and time t Suppose that (t) is the hazard rate at time t The probability of default between times t and t+  t conditional on no earlier default is (t)  t The probability of default by time t is where is the average hazard rate between time zero and time t

63. 63 More Exact Calculation for Bonds Default Probabilities (page 262) Suppose that a five year corporate bond pays a coupon of 6% per annum (semiannually). The yield is 7% with continuous compounding and the yield on a similar risk-free bond is 5% (with continuous compounding) The expected loss from defaults is 8.738. This can be calculated as the PV of asset swap spreads or as the difference between the market price of the bond and its risk-free price Suppose that the unconditional probability of default is Q per year and that defaults always happen half way through a year (immediately before a coupon payment).

64. 64 Calculations

65. 65 Calculations continued We set 288.48Q = 8.738 to get Q = 3.03% This analysis can be extended to allow defaults to take place more frequently With several bonds we can use more parameters to describe the default probability distribution

66. Credit VaR (page 321) Can be defined analogously to Market Risk VaR A one year credit VaR with a 99.9% confidence is the loss level that we are 99.9% confident will not be exceeded over one year 66

67. Risk-free Rate The risk-free rate used by bond traders when quoting credit spreads is the Treasury rate The risk-free rate traditionally assumed in derivatives markets is the LIBOR/swap rate By comparing CDS spreads and bond yields it appears that in normal market conditions traders are assuming a risk-free rate 10 bp less than the LIBOR/swap rates In stressed market conditions the gap between the LIBOR/swap rate and the “true” risk-free rate is liable to be much higher 67

68. Real World vs. Risk-Neutral Probabilities of Default Calculate 7-year hazard rates from the Moody’s data (1970-2010). These are real world default probabilities) Use Merrill Lynch data (1996-2007) to estimate average 7-year default intensities from bond prices (these are risk-neutral default intensities) Assume a risk-free rate equal to the 7-year swap rate minus 10 basis points 68

69. Real World vs Risk Neutral Default Probabilities (7 year averages) Table 16.4, page 363 69

70. Risk Premiums Earned By Bond Traders (Table 16.5, page 364) 70

71. 71 Possible Reasons for These Results Corporate bonds are relatively illiquid The subjective default probabilities of bond traders may be much higher than the estimates from Moody’s historical data Bonds do not default independently of each other and of the economy. This leads to systematic risk that cannot be diversified away. Bond returns are highly skewed with limited upside. The non-systematic risk is difficult to diversify away and may be priced by the market.

72. 72 Which World Should We Use? We should use risk-neutral estimates for valuing credit derivatives and estimating the present value of the cost of default We should use real world estimates for calculating credit VaR and scenario analysis