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Chapter 22 Credit Risk 資管所 陳竑廷

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Agenda 22.1 Credit Ratings 22.2 Historical Data 22.3 Recovery Rate 22.4 Estimating Default Probabilities from bond price 22.5 Comparison of Default Probability estimates 22.6 Using equity price to estimate Default Probabilities

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Credit Risk –Arise from the probability that borrowers and counterparties in derivatives transactions may default.

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22.1 Credit Ratings S&P –AAA, AA, A, BBB, BB, B, CCC, CC, C Moody –Aaa, Aa, A, Baa, Ba, B, Caa, Ca, C Investment grade –Bonds with ratings of BBB (or Baa) and above bestworst

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22.2 Historical Data For a company that starts with a good credit rating default probabilities tend to increase with time For a company that starts with a poor credit rating default probabilities tend to decrease with time

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Default Intensity The unconditional default probability –the probability of default for a certain time period as seen at time zero 39.717 - 30.494 = 9.223% The default intensity (hazard rate) –the probability of default for a certain time period conditional on no earlier default 100 – 30.494 = 69.506% 0.09223 / 0.69506 = 13.27%

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Q(t) : the probability of default by time t (22.1)

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22.3 Recovery Rate Defined as the price of the bond immediately after default as a percent of its face value Moody found the following relationship fitting the data: Recovery rate = 59.1% – 8.356 x Default rate –Significantly negatively correlated with default rates

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Source : –Corporate Default and Recovery Rates, 1920-2006

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22.4 Estimating Default Probabilities Assumption –The only reason that a corporate bond sells for less than a similar risk-free bond is the possibility of default In practice the price of a corporate bond is affected by its liquidity.

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(22.1)

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1 1 R 1 λ 1-λ λ Taylor expansion

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A more exact calculation Suppose that Face value = $100, Coupon = 6% per annum, Last for 5 years –Corporate bond Yield : 7% per annum → $95.34 –Risk-free bond Yield : 5% per annum → $104.094 The expected loss = 104.094 – 95.34 = $ 8.75

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Q : the probability of default per year 288.48Q = 8.75 Q = 3.03% 0 1 2 3 4 5 e -0.05 *3.5

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22.5 Comparison of default probability estimates The default probabilities estimated from historical data are much less than those derived from bond prices

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Historical default intensity The probability of the bond surviving for T years is (22.1)

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Default intensity from bonds A-rated bonds, Merrill Lynch 1996/12 – 2007/10 –The average yield was 5.993% –The average risk-free rate was 5.289% –The recovery rate is 40% (22.2)

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0.11*(1-0.4)=0.066

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Real World vs. Risk Neutral Default Probabilities Risk-neutral default probabilities –implied from bond yields –Value credit derivatives or estimate the impact of default risk on the pricing of instruments Real-world default probabilities –implied from historical data –Calculate credit VaR and scenario analysis

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22.6 Using equity prices to estimate default probability Unfortunately, credit ratings are revised relatively infrequently. –The equity prices can provide more up-to-date information

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Merton’s Model If V T < D, E T = 0 ( default ) If V T > D, E T = V T - D

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V 0 And σ 0 can’t be directly observable. But if the company is publicly traded, we can observe E 0.

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Merton’s model gives the value firm’s equity at time T as So we regard E T as a function of V T We write Other term without dW(t), so ignore it

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Replace dE, dV by (*) (**) respectively We compare the left hand side of the equation above with that of the right hand side (22.4)

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Example Suppose that E 0 = 3 (million) r = 0.05 D = 10 σ E = 0.80 T = 1 Solving then getV 0 = 12.40 σ 0 = 0.2123 N(-d 2 ) = 12.7%

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Solving

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[F(x,y)] 2 +[G(x,y)] 2 =(D2)^2+(E2)^2 F(x,y) =A2*NORMSDIST((LN(A2/10)+(0.05+B2*B2/2))/B2) -10*EXP(-0.05)*NORMSDIST((LN(A2/10)+(0.05+B2*B2/2))/B2-B2) G(x,y) =NORMSDIST((LN(A7/10)+(0.05+B7*B7/2))/B7)*A7*B7 Excel Solver

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Initial V 0 = 12.40, σ 0 = 0.2123 Initial V 0 = 10, σ 0 = 0.1

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Thank you

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