Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,26.1,Chapter 26Credit Risk,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,26.2,Credit Ratings,In the S&P rating system, AAA is the best rating. After that comes AA, A, BBB, BB, B, and CCCThe corresponding Moodys ratings are Aaa, Aa, A, Baa, Ba, B, and CaaBonds with ratings of BBB (or Baa) and above are considered to be “investment grade”,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,26.3,Information from Bond Prices,Traders regularly estimate the zero curves for bonds with different credit ratingsThis allows them to estimate probabilities of default in a risk-neutral world,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,26.4,Typical Pattern (See Figure 26.1, page 611),Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,26.5,The Risk-Free Rate,Most analysts use the LIBOR rate as the risk-free rateThe excess of the value of a risk-free bond over a similar corporate bond equals the present value of the cost of defaults,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,26.6,Example (Zero coupon rates; continuously compounded),Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,26.7,Example continued,One-year risk-free bond (principal=$1) sells forOne-year corporate bond (principal=$1) sells for or at a 0.2497% discountThis indicates that the holder of the corporate bond expects to lose 0.2497% from defaults in the first year,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,26.8,Example continued,Similarly the holder of the corporate bond expects to lose or 0.9950% in the first two yearsBetween years one and two the expected loss is 0.7453%,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,26.9,Example continued,Similarly the bond holder expects to lose 2.0781% in the first three years; 3.3428% in the first four years; 4.6390% in the first five yearsThe expected losses per year in successive years are 0.2497%, 0.7453%, 1.0831%, 1.2647%, and 1.2962%,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,26.10,Summary of Results (Table 26.1, page 612),Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,26.11,Recovery Rates(Table 26.3, page 614. Source: Moodys Investors Service, 2000),Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,26.12,Probability of Default,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,26.13,Reason Why This Analysis is Simplistic,Bonds are assumed to be zero-couponThe equation: Prob. of Def.(1-Rec. Rate)=Exp Loss%assumes that the claim in the event of default equals the no-default value of the bond,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,26.14,A More Complete Analysis: Definitions,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,26.15,Risk-Neutral Probability of DefaultPage 616, equations 26.3 to 26.5,PV of loss from defaultReduction in bond price due to defaultComputing ps inductively,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,26.16,Relaxing Assumptions,This analysis assumes constant interest rates, and known recovery rates and claim amountsIf default events, risk-free rates, and recovery rates are independent, results hold for stochastic interest rates, and uncertain recovery rates providing the recovery rate is set equal to its expected value in a risk-neutral world,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,26.17,The analysis can be extended to allow defaults at any timeIt is important to distinguish between the default probability density and the hazard rateThe default probability density, q(t) is defined so that q(t)dt as the probability of default between times t and t+dt as seen at time zero The hazard rate is the probability of default between times t and t+dt conditional on no earlier default,Extending the Analysis to Allow Defaults at Any Time,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,26.18,What Should We Use as the Claim Amount,The best assumption seems to be that the claim amount for a bond equals the face value plus accrued interest - not the no-default value,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,26.19,Sample Data (Risk-free Rate=5%; Expected Recovery Rate=30%),Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,26.20,Implied Default Probabilities Assuming That Default Can Happen on Bond Maturity Dates (Table 26.5, page 617),Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,26.21,Value Additivity,If claim amount equals no-default value, value of a coupon bond is sum of values of constituent zero-coupon bondsThe same is not true when claim amount equals face value plus accrued interest,