【正文】 a pany that starts with a poor credit rating default probabilities tend to decrease with time Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Which World Should We Use? ? We should use riskneutral estimates for valuing credit derivatives and estimating the cost of default ? We should use real world estimates for calculating credit VaR and scenario analysis Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull The Loss Given Default (LGD) ? For derivatives we need to distinguish between a) those that are always assets, b) those that are always liabilities, and c) those that can be assets or liabilities ? What is the loss in each case? Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Transition Matrix Consistent With Default Probabilities A B C Default A % % % % B % % % % C % % % % Default % % % 100% This is chosen to minimize difference between all elements of Mi1 d1 and the corresponding cumulative default probabilities implied by bond prices. Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Measure 1 vs Measure 2 n o r m a l tem u l t i v a r i a be to a ss u m e d be ca n t i m e s su r v i v a l dt r a n sf o r m e b e ca u se co n si d e r e d a r e co m p a n i e sm a n y w h e nu se to e a si e r m u ch is It 1. M e a su r e t h a n h i g h e rt l y si g n i f i ca nu su a l l y is 2 M e a su r ef u n ct i o n . ond i st r i b u t iy p r o b a b i l i t n o r m a l b i v a r i a t e cu m u l a t i v e t h e is w h e r ea n d:v e r sa v i ce a n d 2 M e a su r e f r o m ca l cu l a t e d be ca n 1 M e a su r eMTQTQTQTQTQTQTuTuMTTuTuMTPBBAABAABBAABABBAAB])()(][)()([)()(])。 20xx by John C. Hull Basing Credit VaR on Defaults Only (CSFP Approach) ? When the expected number of defaults is m, the probability of n defaults is ? This can be bined with a probability distribution for the size of the losses on a single default to obtain a probability distribution for default losses enn?mm!Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Pros and Cons ? Reduced form approach can be calibrated to known default probabilities. It leads to low default correlations. ? Structural model approach allows correlations to be as high as desired, but cannot be calibrated to known default probabilities. Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Use of Gaussian Copula continued ? We sample from a multivariate normal distribution for each pany incorporating appropriate correlations ? N 1() = , N 1() = , N 1() = , N 1() = , N 1() = Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Example Cumulative probability of default 1 2 3 4 5 A % % % % % B % % % % % C % % % % % Suppose there are three rating categories and riskneutral default probabilities extracted from bond prices are: Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Example continued ? The probability of default is N(d2) or % ? The market value of the debt is ? The present value of the promised payment is ? The expected loss is about % ? The recovery rate is 91% Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull RiskNeutral Probabilities The analysis based on bond prices assumes that ? The expected cash flow from the Arated bond is % less than that from the riskfree bond ? The discount rates for the two bonds are the same This is correct only in a riskneutral world Options, Futures, and Other Derivatives, 5th edition 169。P Report, January 20xx) 1 2 3 4 5 7 10 AAA 0 0 4 7 2 2 7 AA 1 4 0 8 9 2 6 A 4 2 1 6 7 1 6 BBB 4 5 9 5 3 0 0 BB 8 8 5 1 12 .57 18 .09 23 .86 B 4 13 .49 20 .12 25 .36 29 .58 36 .34 43. 41 C C C 25 .26 34 .79 42 .16 48 .18 54 .65 58 .64 62 .58 Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Value Additivity ? If claim amount equals nodefault value, value of a coupon bond is sum of values of constituent zerocoupon bonds ? The same is not true when claim amount equals face value plus accrued interest Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull A More Complete Analysis: Definitions Bj: Pric e today of b ond m at urin g at tjGj: Pric e today of b ond m at urin g at tj i f there we re noproba bili ty of defa ultFj( t ): Fo r ward pri ce a t ti m e t of Gj ( t tj)v ( t ): PV of $1 re ce ive d at