Estimating Default Probabilities e Alternatives Use bond prices a Use CDs spreads Use historical data Use merton's model Options, Futures, and Other Derivatives, 8th Edition Copyright@ John C. Hull 2012
Estimating Default Probabilities Alternatives: Use Bond Prices Use CDS spreads Use Historical Data Use Merton’s Model Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 11
Using Bond prices (Equation 23.2 page 524) Average default intensity over life of bond is approximately S 1-R 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, 8th Edition, Copyright O John C. Hull 2012 12
Using Bond Prices (Equation 23.2, page 524) Average default intensity over life of bond is approximately where s is the spread of the bond’s yield over the risk-free rate and R is the recovery rate R s 1− Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 12
More exact calculation e 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) e Price of risk-free bond is 104.09; price of corporate bond is 95. 34; expected loss from defaults is 8.75 e Suppose that the probability of default is o per year and that defaults always happen half way through a year(immediately before a coupon payment Options Futures, and other Derivatives, 8th Edition Copyright o John C. Hull 2012 13
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 corporate 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, 8th Edition, Copyright © John C. Hull 2012 13
Calculations (Table 23.3, page 525) Time Def Recovery Risk-free Loss given Discount PV of Exp (yrs) Prob Amount Value Default Factor LOSS 05Q 106.73 66.73 0.9753 65080 15Q 40 105.97 6597 0.92776120 25Q 0 105.17 65.17 088255752Q 3.5 O 40 104.34 64.34 0.839554010 4.5 40 10346 6346 0798550679 Tota 28848Q Options, Futures, and Other Derivatives, 8th Edition, Copyright O John C. Hull 2012
Calculations (Table 23.3, page 525) Time (yrs) Def Prob Recovery Amount Risk-free Value Loss given Default Discount Factor PV of Exp Loss 0.5 Q 40 106.73 66.73 0.9753 65.08Q 1.5 Q 40 105.97 65.97 0.9277 61.20Q 2.5 Q 40 105.17 65.17 0.8825 57.52Q 3.5 Q 40 104.34 64.34 0.8395 54.01Q 4.5 Q 40 103.46 63.46 0.7985 50.67Q Total 288.48Q Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 14
Calculations continued o We set 288.480=8. 75 to get 2=3.03% e This analysis can be extended to allow defaults to take place more frequently e With several bonds we can use more parameters to describe the default probability distribution Options, Futures, and Other Derivatives, 8th Edition Copyright@ John C. Hull 2012 15
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 we can use more parameters to describe the default probability distribution Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 15