Recovery rates 26.11 (Table 26.3, page 614. Source: Moody's Investors Service, 2000) Class Mean (%)sd(%) Senior secured 52.31 25.15 Senior Unsecured 48.84 25.01 Senior Subordinated 3946 24.59 Subordinated 33.71 20.78 Junior subordinated 19.69 13.85 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 26.11 Recovery Rates (Table 26.3, page 614. Source: Moody’s Investor’s Service, 2000) Class Mean(%) SD (%) Senior Secured 52.31 25.15 Senior Unsecured 48.84 25.01 Senior Subordinated 39.46 24.59 Subordinated 33.71 20.78 Junior Subordinated 19.69 13.85
26.12 Probability of Default Prob of Def. x(1-Rec Rate)=Exp. LosS% Prob of def Exp LosS% 1-Rec Rate if Rec Rate=0.5 in our example, probabilities of default in years 1, 2, 3, 4, and 5 are 0.004994 0.014906,0.021662.0.025294,and0.025924 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 26.12 Probability of Default 0.014906, 0.021662, 0.025294, and 0.025924 of default i n years 1, 2, 3 , 4, and 5 are 0.004994, If Rec Rate 0.5 i n our example, probabilities 1-Rec.Rate Exp. Loss% Prob of Def Prob. of Def. (1-Rec.Rate) Exp. Loss% = = =
26.13 Reason hy This analysis is Simplistic Bonds are assumed to be zero-coupon The equation Prob of Def. X(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
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-coupon • The 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
A More Complete Analysis 26.14 Definitions Bi: Price today of bond maturing at t G: Price today of bond maturing at t; if there were no probability of default F(t: Forward price at time t of G;(t< t) v(t PV of$l received at time t with certainty c(t: Claim made if there is a default at time t< R,(t): Recovery rate in the event of a default at time t<t, air: PV of loss from a default at time ti relative to G Pi: The risk-neutral probability of default at time ti Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull A More Complete Analysis: 26.14 Definitions Bj: Price today of bond maturing at tj Gj: Price today of bond maturing at tj if there were no probability of default Fj(t): Forward price at time t of Gj (t < tj) v(t): PV of $1 received at time t with certainty Cj(t): Claim made if there is a default at time t < tj Rj(t): Recovery rate in the event of a default at time t< tj ij: PV of loss from a default at time ti relative to Gj pi: The risk-neutral probability of default at time ti
26.15 Risk-Neutral Probability of Default Page 616, equations 26.3 to 26.5 P∨ of loss from defau|t ()F()-R()C() Reduction in bond price due to default G-B P Computing p's inductively ∑P Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 26.15 Risk-Neutral Probability of Default Page 616, equations 26.3 to 26.5 • PV of loss from default • Reduction in bond price due to default • Computing p’s inductively 1 j j j i ij i G B p = − = 1 1 j j j i ij i j jj G B p p − = − − = ij i j i j i j i = − v t F t R t C t ( ) ( ) ( ) ( )