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1、Credit Risk,Chapter 22,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,1,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 Moodys ratings are Aaa, Aa, A, Baa, Ba, B,Caa
2、, Ca, and C Bonds with ratings of BBB (or Baa) and above are considered to be “investment grade”,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,2,Historical Data,Historical data provided by rating agencies are also used to estimate the probability of
3、default,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,3,Cumulative Ave Default Rates (%) (1970-2006, Moodys, Table 22.1, page 490),Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,4,Interpretation,The ta
4、ble 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 0.181% of defaulting by the end of the first year, 0.506% by the end of the second year, and so on,Options, Futures, and Other Derivatives,
5、7th International Edition, Copyright John C. Hull 2008,5,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 t
6、ime,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,6,Default Intensities vs Unconditional Default Probabilities (page 490-91),The default intensity (also called hazard rate) is the probability of default for a certain time period conditional on no ear
7、lier 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 rate company in the third year?,Options, Futures, and Other Derivatives, 7th Internationa
8、l Edition, Copyright John C. Hull 2008,7,Default Intensity (Hazard Rate),The default intensity (hazard rate) that is usually quoted is an instantaneous If V(t) is the probability of a company surviving to time t,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hu
9、ll 2008,8,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,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,9,Recovery Rates (Moodys: 1982 to 2006, Table 22.2
10、, page 491),Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,10,Estimating Default Probabilities,Alternatives: Use Bond Prices Use CDS spreads Use Historical Data Use Mertons Model,Options, Futures, and Other Derivatives, 7th International Edition, Copy
11、right John C. Hull 2008,11,Using Bond Prices (Equation 22.2, page 492),Average default intensity over life of bond is approximately where s is the spread of the bonds yield over the risk-free rate and R is the recovery rate,Options, Futures, and Other Derivatives, 7th International Edition, Copyrigh
12、t John C. Hull 2008,12,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 cor
13、porate 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, 7th International Edition, Copyright John C. Hull 200
14、8,13,Calculations (Table 22.3, page 493),Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,14,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 w
15、e can use more parameters to describe the default probability distribution,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,15,The Risk-Free Rate,The risk-free rate when default probabilities are estimated is usually assumed to be the LIBOR/swap zero ra
16、te (or sometimes 10 bps below the LIBOR/swap rate) To get direct estimates of the spread of bond yields over swap rates we can look at asset swaps,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,16,Real World vs Risk-Neutral Default Probabilities,The d
17、efault 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, 7th International Edition, Copyright John C. Hull
18、 2008,17,A Comparison,Calculate 7-year default intensities from the Moodys data (These are real world default probabilities) Use Merrill Lynch data 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
19、 swap rate minus 10 basis point,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,18,Real World vs Risk Neutral Default Probabilities, 7 year averages (Table 22.4, page 495),Options, Futures, and Other Derivatives, 7th International Edition, Copyright Jo
20、hn C. Hull 2008,19,Risk Premiums Earned By Bond Traders (Table 22.5, page 496),Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,20,Possible Reasons for These Results,Corporate bonds are relatively illiquid The subjective default probabilities of bond tr
21、aders may be much higher than the estimates from Moodys 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
22、be priced by the market,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,21,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 e
23、stimates for calculating credit VaR and scenario analysis,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,22,Mertons Model (page 498-499),Mertons model regards the equity as an option on the assets of the firm In a simple situation the equity value is
24、max(VT D, 0) where VT is the value of the firm and D is the debt repayment required,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,23,Equity vs. Assets,An option pricing model enables the value of the firms equity today, E0, to be related to the value
25、 of its assets today, V0, and the volatility of its assets, sV,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,24,Volatilities,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,25,This equation together wit
26、h the option pricing relationship enables V0 and sV to be determined from E0 and sE,Example,A companys 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=2
27、1.23%,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,26,Example continued,The probability of default is N(-d2) 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
28、 rate is 91%,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,27,The Implementation of Mertons Model,Choose time horizon Calculate cumulative obligations to time horizon. This is termed by KMV the “default point”. We denote it by D Use Mertons model to
29、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, 7th International Edition, Copyright John C. Hull 2008,28
30、,Credit Risk in Derivatives Transactions (page 502-504),Three cases Contract always an asset Contract always a liability Contract can be an asset or a liability,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,29,General Result,Assume that default proba
31、bility is independent of the value of the derivative Consider times t1, t2,tn and default probability is qi at time ti. The value of the contract at time ti is fi and the recovery rate is R The loss from defaults at time ti is qi(1-R)Emax(fi,0). Defining ui=qi(1-R) and vi as the value of a derivativ
32、e that provides a payoff of max(fi, 0) at time ti, the cost of defaults is,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,30,If Contract Is Always a Liability (equation 22.9),Options, Futures, and Other Derivatives, 7th International Edition, Copyrigh
33、t John C. Hull 2008,31,Credit Risk Mitigation,Netting Collateralization Downgrade triggers,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,32,Default Correlation,The credit default correlation between two companies is a measure of their tendency to def
34、ault 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, 7th Intern
35、ational Edition, Copyright John C. Hull 2008,33,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, 7th International Edition, Copyright J
36、ohn C. Hull 2008,34,Binomial Correlation Measure (page 516),One common default correlation measure, between companies i and j is the correlation between A variable that equals 1 if company i defaults between time 0 and time T and zero otherwise A variable that equals 1 if company j defaults between
37、time 0 and time T and zero otherwise The value of this measure depends on T. Usually it increases at T increases.,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,35,Binomial Correlation continued,Denote Qi(T) as the probability that company A will defa
38、ult between time zero and time T, and Pij(T) as the probability that both i and j will default. The default correlation measure is,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,36,Survival Time Correlation,Define ti as the time to default for company
39、 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, 7th International
40、Edition, Copyright John C. Hull 2008,37,Gaussian Copula Model (page 514-515),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-1Q(ti) where N is the cumulative normal distribution function. This is a “percentile to percen
41、tile” 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
42、between xi and xj,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,38,Binomial vs Gaussian Copula Measures (Equation 22.14, page 516),The measures can be calculated from each other,Options, Futures, and Other Derivatives, 7th International Edition, Copy
43、right John C. Hull 2008,39,Comparison (Example 22.4, page 516),The correlation number depends on the correlation metric used Suppose T = 1, Qi(T) = Qj(T) = 0.01, a value of rij equal to 0.2 corresponds to a value of bij(T) equal to 0.024. In general bij(T) rij and bij(T) is an increasing function of
44、 T,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,40,Example of Use of Gaussian Copula (Example 22.3, page 515),Suppose that we wish to simulate the defaults for n companies . For each company the cumulative probabilities of default during the next 1,
45、 2, 3, 4, and 5 years are 1%, 3%, 6%, 10%, and 15%, respectively,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,41,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) =
46、-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, 7th International Edition, Copyright John C. Hull 2008,42,Use of Gaussian Copula continued,When sample for a company is less than -2.33, the company defaults in the first ye
47、ar 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 defaul
48、ts 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, 7th International Edition, Copyright John C. Hull 2008,43,A One-Factor Model for the Correlation Structure (Equation 22.10 , page 515),The correlation
49、between xi and xj is aiaj The ith company defaults by time T when xi N-1Qi(T) or Conditional on F the probability of this is,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,44,Credit VaR (page 517-519),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, 7th International