Chapter 23 Credit Risk Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Credit Ratings In the S&P rating system, AAA is the best rating. After that comes AA, A, BBB, BB, B, CCC, CC, and C The corresponding Moody’s ratings are Aaa, Aa, A, Baa, Ba, B,Caa, Ca, and C Bonds with ratings of BBB (or Baa) and above are considered to be “investment grade” Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Historical Data Historical data provided by rating agencies are also used to estimate the probability of default Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Cumulative Ave Default Rates (%) (1970-2009, Moody’s, Table 23 Cumulative Ave Default Rates (%) (1970-2009, Moody’s, Table 23.1, page 522) Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Interpretation The table shows the probability of default for companies starting with a particular credit rating A company with an initial credit rating of Baa has a probability of 0.176% of defaulting by the end of the first year, 0.494% by the end of the second year, and so on Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Do Default Probabilities Increase with Time? For a company that starts with a good credit rating default probabilities tend to increase with time For a company that starts with a poor credit rating default probabilities tend to decrease with time Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Hazard Rates vs Unconditional Default Probabilities (page 522-523) The hazard rate (also called default intensity) is the probability of default for a certain time period conditional on no earlier default The unconditional default probability is the probability of default for a certain time period as seen at time zero What are the default intensities and unconditional default probabilities for a Caa rated company in the third year? Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Hazard Rate The hazard rate that is usually quoted is an instantaneous rate If V(t) is the probability of a company surviving to time t Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Recovery Rate The recovery rate for a bond is usually defined as the price of the bond immediately after default as a percent of its face value Recovery rates tend to decrease as default rates increase Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Recovery Rates; Moody’s: 1982 to 2009 Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Estimating Default Probabilities Alternatives: Use Bond Prices Use CDS spreads Use Historical Data Use Merton’s Model Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Using Bond Prices (Equation 23.2, page 524) 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 Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

More Exact Calculation Assume 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) Price of risk-free bond is 104.09; price of corporate bond is 95.34; expected loss from defaults is 8.75 Suppose that the probability of default is Q per year and that defaults always happen half way through a year (immediately before a coupon payment. Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Calculations (Table 23.3, page 525) Time (yrs) Def Prob Recovery Amount Risk-free Value Loss given Default Discount Factor PV of Exp Loss 0.5 Q 40 106.73 66.73 0.9753 65.08Q 1.5 105.97 65.97 0.9277 61.20Q 2.5 105.17 65.17 0.8825 57.52Q 3.5 104.34 64.34 0.8395 54.01Q 4.5 103.46 63.46 0.7985 50.67Q Total 288.48Q Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Calculations continued We set 288.48Q = 8.75 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 Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

The Risk-Free Rate The risk-free rate when default probabilities are estimated is usually assumed to be the LIBOR/swap zero rate (or sometimes 10 bps below the LIBOR/swap rate) Asset swaps provide a direct estimates of the spread of bond yields over swap rates Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

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 real-world default probabilities Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

A Comparison Calculate 7-year default intensities from the Moody’s data, 1970-2009, (These are real world default probabilities) Use Merrill Lynch data to estimate average 7-year default intensities from bond prices, 1996 to 2007 (these are risk-neutral default intensities) Assume a risk-free rate equal to the 7-year swap rate minus 10 basis points Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Data from Moody’s and Merrill Lynch *The benchmark risk-free rate for calculating spreads is assumed to be the swap rate minus 10 basis points. Bonds are corporate bonds with a life of approximately 7 years. Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Real World vs Risk Neutral Hazard Rates (Table 23.4, page 527) 1 Calculated as−[ln(1-d)]/7 where d is the Moody’s 7 yr default rate. For example, in the case of Aaa companies, d=0.00245 and -ln(0.99755)/7=0.0004 or 4bps. For investment grade companies the historical hazard rate is approximately d/7. 2 Calculated as s/(1-R) where s is the bond yield spread and R is the recovery rate (assumed to be 40%). Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Average Risk Premiums Earned By Bond Traders 1 Equals average spread of our benchmark risk-free rate over Treasuries. 2 Equals historical hazard rate times (1-R) where R is the recovery rate. For example, in the case of Baa, 26bps is 0.6 times 43bps. Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Possible Reasons for These Results (The third reason is the most important) 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. 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 Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

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 Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Using Equity Prices: Merton’s Model (page 530-531) Merton’s model regards the equity as an option on the assets of the firm In a simple situation the equity value is max(VT −D, 0) where VT is the value of the firm and D is the debt repayment required Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Equity vs. Assets The Black-Scholes-Merton option pricing model enables the value of the firm’s equity today, E0, to be related to the value of its assets today, V0, and the volatility of its assets, sV Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Volatilities This equation together with the option pricing relationship enables V0 and sV to be determined from E0 and sE Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

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 V0=12.40 and sv=21.23% The probability of default is N(−d2) or 12.7% Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

The Implementation of Merton’s Model Choose time horizon Calculate cumulative obligations to time horizon. This is termed by KMV the “default point”. We denote it by D Use Merton’s model to calculate a theoretical probability of default Use historical data or bond data to develop a one-to-one mapping of theoretical probability into either real-world or risk-neutral probability of default. Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Credit Risk in Derivatives Transactions (page 531-534) Three cases Contract always an asset Contract always a liability Contract can be an asset or a liability Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

General Result Assume that default probability is independent of the value of the derivative Define t1, t2,…tn: times when default can occur qi: default probability at time ti. fi: The value of the transaction at time ti R: Recovery rate Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

General Result continued The expected loss from defaults at time ti is qi(1−R)E[max(fi,0)]. Defining ui=qi(1−R) and vi as the value of a derivative that provides a payoff of max(fi,0) at time ti, the PV of the cost of defaults is Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Applications If transaction is always an asset so that fi > 0 then vi = f0 and the cost of defaults is f0 times the total default probability, times 1−R If transaction is always a liability then vi= 0 and the cost of defaults is zero In other cases we must value the derivative max(fi,0) for each value of i Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Using Bond Yields for Instruments in the First Category All instruments that promise a (non-negative) payoff at time T should be reduced in price by the same amount for default risk where f0 and f0* are the no-default and actual values of the instrument; B0 and B0* are the no-default and actual values of a zero-coupon bond maturing at time T; y and y* are the yields on these zero coupon bonds Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Example A 2-year option has a Black-Scholes value of $3 Assume a 2 year zero coupon bond issued by the company has a yield of 1.5% greater than the risk free rate Value of option is 3e−0.015×2 = 2.91 Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Expected Exposure on Pair of Offsetting Interest Rate Swaps and a Pair of Offsetting Currency Swaps Maturity Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Interest Rate vs Currency Swaps The ui’s are the same for both The vi’s for an interest rate swap are on average much less than the vi’s for a currency swap The expected cost of defaults on a currency swap is therefore greater. Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Credit Risk Mitigation Netting Collateralization Downgrade triggers Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Netting Suppose a bank has three transactions worth of $10 million, $30 million, and −$25 million Without netting the exposure is $40 million With netting the exposure is $15 million Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Collateralization Transactions are marked to market periodically (e.g. every day) If total value of transactions Party A has with party B is above a specified threshold level it can ask Party B to post collateral equal to the excess of the value over the threshold level After that collateral can be withdrawn or must be increased by Party B depending on whether value of transactions to Party A decreases or increases Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Downgrade Triggers A downgrade trigger is a clause stating that a transaction can be closed out by Party A when the credit rating of the other side, Party B, falls below a certain level In practice Party A will only close out transactions that have a negative value to Party B When there are a large number of downgrade triggers they are likely to be counterproductive Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

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. Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Measurement There is no generally accepted measure of default correlation Default correlation is a more complex phenomenon than the correlation between two random variables Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Survival Time Correlation Define ti as the time to default for company i and Qi(ti) as the probability distribution for ti The default correlation between companies i and j can be defined as the correlation between ti and tj But this does not uniquely define the joint probability distribution of default times Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Gaussian Copula Model (page 538-540) Define a one-to-one correspondence between the time to default, ti, of company i and a variable xi by Qi(ti ) = N(xi ) or xi = N-1[Q(ti)] where N is the cumulative normal distribution function. This is a “percentile to percentile” transformation. The p percentile point of the Qi distribution is transformed to the p percentile point of the xi distribution. xi has a standard normal distribution We assume that the xi are multivariate normal. The default correlation measure, rij between companies i and j is the correlation between xi and xj Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Example of Use of Gaussian Copula (Example 23.3, page 539) Suppose that we wish to simulate the defaults for n companies . For each company the cumulative probabilities of default during the next 1, 2, 3, 4, and 5 years are 1%, 3%, 6%, 10%, and 15%, respectively Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Use of Gaussian Copula continued We sample from a multivariate normal distribution to get the xi Critical values of xi are N -1(0.01) = -2.33, N -1(0.03) = -1.88, N -1(0.06) = -1.55, N -1(0.10) = -1.28, N -1(0.15) = -1.04 Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Use of Gaussian Copula continued When sample for a company is less than -2.33, the company defaults in the first year When sample is between -2.33 and -1.88, the company defaults in the second year When sample is between -1.88 and -1.55, the company defaults in the third year When sample is between -1,55 and -1.28, the company defaults in the fourth year When sample is between -1.28 and -1.04, the company defaults during the fifth year When sample is greater than -1.04, there is no default during the first five years Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

A One-Factor Model for the Correlation Structure The correlation between xi and xj is aiaj The ith company defaults by time T when xi < N-1[Qi(T)] or Conditional on F the probability of this is Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Credit VaR (page 540-542) Can be defined analogously to Market Risk VaR A T-year credit VaR with an X% confidence is the loss level that we are X% confident will not be exceeded over T years Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Calculation from a Factor-Based Gaussian Copula Model (equation 23 Calculation from a Factor-Based Gaussian Copula Model (equation 23.13, page 540) Consider a large portfolio of loans, each of which has a probability of Q(T) of defaulting by time T. Suppose that all pairwise copula correlations are r so that all ai’s are We are X% certain that F is less than N−1(1−X) = −N−1(X) It follows that the VaR is Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Example (page 541) A bank has $100 million of retail exposures 1-year probability of default averages 2% and the recovery rate averages 60% The copula correlation parameter is 0.1 99.9% worst case default rate is The one-year 99.9% credit VaR is therefore 100×0.128×(1-0.6) or $5.13 million Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

CreditMetrics (page 541-542) Calculates credit VaR by considering possible rating transitions A Gaussian copula model is used to define the correlation between the ratings transitions of different companies Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012