【正文】 20xx by John C. Hull Credit VaR (page 630) Credit VaR asks a question such as: What credit loss are we 99% certain will not be exceeded in 1 year? Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Use of Gaussian Copula continued ? When sample for a pany is less than , the pany defaults in the first year ? When sample is between and , the pany defaults in the second year ? When sample is between and , the pany defaults in the third year ? When sample is between 1,55 and , the pany defaults in the fourth year ? When sample is between and , the pany defaults during the fifth year ? When sample is greater than , there is no default during the first five years Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Matrix Implied Default Probability ? Let M be the annual rating transition matrix and di be the vector containing probability of default within i years ? d1 is the rightmost column of M ? di = M di1 = Mi1 d1 ? Number of free parameters in M is number of ratings squared Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull The Implementation of Merton’s Model (. KMV Moody’s) ? 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 onetoone mapping of theoretical probability into either realworld or riskneutral probability of default. Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull The RealWorld Probability of Default ? The expected cash flow from the Arated bond is % less than that from the riskfree bond ? But we still get the same price if we discount at about 38 bps per year more than the riskfree rate ? If riskfree rate is 5%, it is consistent with the beta of the Arated bond being Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Interpretation ? The table shows the probability of default for panies starting with a particular credit rating ? A pany with an initial credit rating of BBB has a probability of % of defaulting by the end of the first year, % by the end of the second year, and so on Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Asset Swaps (page 618) ? An asset swap exchanges the return on a bond for a spread above LIBOR ? Asset swaps are frequently used to extract default probabilities from bond prices. The assumption is that LIBOR is the riskfree rate Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull RiskNeutral Probability of Default Page 616, equations to ? PV of loss from default ? Reduction in bond price due to default ? Computing p’s inductively 1jj j i ijiG B p ???? ?11jj j i ijijjjG B pp??????? ?? ? ? ? ? ? ? ?i j i j i j i j iv t F t R t C t? ??????Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Example continued ? Similarly the bond holder expects to lose % in the first three years。P rating system, AAA is the best rating. After that es AA, A, BBB, BB, B, and CCC ? The corresponding Moody’s ratings are Aaa, Aa, A, Baa, Ba, B, and Caa ? Bonds with ratings of BBB (or Baa) and above are considered to be “investment grade” Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Information from Bond Prices ? Traders regularly estimate the zero curves for bonds with different credit ratings ? This allows them to estimate probabilities of default in a riskneutral world Options, Futures, and Other Derivatives, 5th edition 169。 % in the first four years。 20xx by John C. Hull Relaxing Assumptions ? This analysis assumes constant interest rates, and known recovery rates and claim amounts ? If default events, riskfree 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 riskneutral world, ?ROptions, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Asset Swaps: Example 1 ? An investor owns a 5year corporate bond worth par that pays a coupon of 6%. LIBOR is flat at %. An asset swap would enable the coupon to be exchanged for LIBOR plus 150bps ? In this case Bj=100 and Gj= (The value of 150 bps per year for 5 years is .) Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Do Default Probabilities Increase with Time? ? For a pany that starts with a good credit rating default probabilities tend to increase with time ? For