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

2. 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

3. 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

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

5. 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

6. 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

7. 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

8. 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

9. 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

10. Recovery Rates; Moody’s: 1982 to 2009
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11. 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

12. 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

13. 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 ; 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

14. 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

15. Calculations continued
We set Q = 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

16. 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

17. 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

18. A Comparison
Calculate 7-year default intensities from the Moody’s data, , (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

19. 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

20. 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= and -ln( )/7= 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

45. 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

46. 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

47. 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 and -1.88, the company defaults in the second year
When sample is between 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 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

48. 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

49. Credit VaR (page )
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

50. 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

51. 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

52. 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