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Credit risk (2)

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Credit risk (2)

  1. 1. Chapter 22 Credit Risk 資管所 陳竑廷
  2. 2. 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
  3. 3. • Credit Risk – Arise from the probability that borrowers and counterparties in derivatives transactions may default.
  4. 4. 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 best worst • Investment grade – Bonds with ratings of BBB (or Baa) and above
  5. 5. 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
  6. 6. 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%
  7. 7. λ (t ) : the default intensities at time t V (t ) : the cumulative probability of the company surviving to time t λ (t )∆t : the probability of default between time t and t + ∆t condtional on no earlier default [V (t ) − V (t + ∆t )] ÷ V (t ) = λ (t )∆t V (t + ∆t ) − V (t ) = −λ (t )V (t )∆t V (t + ∆t ) − V (t ) − λ (t )V (t )∆t = ∆t ∆t dV (t ) = −λ (t )V (t ) dt t V (t ) = e ∫ 0 − λ (τ ) dτ
  8. 8. • Q(t) : the probability of default by time t Q(t ) = 1 − V (t ) t = 1− e ∫ 0 − λ (τ ) dτ −λ (t ) t = 1− e (22.1)
  9. 9. 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
  10. 10. • Source : – Corporate Default and Recovery Rates, 1920-2006
  11. 11. 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.
  12. 12. λ : the average default intentisy per year s : the spread of the corporate bond yield R : the expected recovery rate s λ= 1− R R = 40 % s = 200 bp 0.02 λ = = 3.33% 1 − 0 .4 (22.1)
  13. 13. λ 1-λ λ 1 (1 − λ ) *1 + λ *1 e 1 rf R 1-λ 1 (1 − λ ) *1 + λ * R rf e e −rf [(1 − λ ) *1 + λ * R ] = 1* e (1 − λ ) + λ R = e −s λ ( R − 1) + 1 = 1 − s s = λ (1 − R ) −(rf + s ) Taylor expansion
  14. 14. 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
  15. 15. 0 1 2 3 4 5 Q : the probability of default per year 288.48Q = 8.75 Q = 3.03% e -0.05 *3.5
  16. 16. Comparison of default probability estimates 22.5 • The default probabilities estimated from historical data are much less than those derived from bond prices
  17. 17. Historical default intensity The probability of the bond surviving for T years is Q(t ) = 1 − e −λ (t )t 1 λ (t ) = − ln(1 − Q(t )) t (22.1)
  18. 18. 1 λ (7) = − ln[1 − Q(7)] 7 1 = − ln[1 − 0.00759] 7 = 0.11%
  19. 19. 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% s λ= (22.2) 1− R 0.05993 − 0.05298 = 1 − 0.4 = 1.16%
  20. 20. 0.11*(1-0.4)=0.066
  21. 21. 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
  22. 22. Using equity prices to estimate default probability 22.6 • Unfortunately , credit ratings are revised relatively infrequently. – The equity prices can provide more up-to-date information
  23. 23. Merton’s Model If VT < D , ET = 0 ( default ) If VT > D , ET = VT - D ET = max(VT − D,0)
  24. 24. • V0 And σ0 can’t be directly observable. • But if the company is publicly traded , we can observe E0.
  25. 25. Merton’s model gives the value firm’s equity at time T as ET = max(VT − D,0) So we regard ET as a function of VT We write dE = µ1dt + σ E dw(t ) ⇒ dE = Eµ1dt + Eσ E dw(t ) E dV = µ2 dt + σV dw(t ) ⇒ dV = Vµ2 dt + VσV dw(t ) V  E is a function of V ∴ By Ito' s Lemma ∂E dE = dV + ∂V Other term without dW(t) , so ignore it (*) (**)
  26. 26. Replace dE , dV by (*) (**) respectively ∂E Eµ1dt + Eσ E dw(t ) = (Vµ 2 dt + Vσ V dw(t )) + ∂V ∂E ∂E = Vµ 2 dt + Vσ V dw(t ) + ∂V ∂V We compare the left hand side of the equation above with that of the right hand side ∂E Eµ1dt = Vµ2 dt + ∂V and ∂E Eσ E dW (t ) = VσV dW (t ) ∂V ∂E ⇒ Eσ E = VσV (22.4) ∂V
  27. 27. Example • Suppose that E0 = 3 (million) σE = 0.80 r = 0.05 D = 10 T=1 Solving then get V0 = 12.40 σ0 = 0.2123 N(-d2) = 12.7%
  28. 28. Solving F (σ V , V0 ) : E0 = V0 N(d1 ) − De − rT N(d 2 ) G (σ V , V0 ) : σ E E0 = N(d1 )σ V V0 minimize F (σ V , V0 ) + G (σ V , V0 ) 2 2
  29. 29. Excel Solver 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 [F(x,y)]2+[G(x,y)]2 =(D2)^2+(E2)^2
  30. 30. • Initial V0 = 12.40 , σ0 = 0.2123 • Initial V0 = 10 , σ0 = 0.1
  31. 31. N(-d 2 ) = 12.7% The mark value of the debt = V0 − E0 = 12.40 − 3 = 9.4 The present value of the promised payment = 10e - 0 .05*1 = 9.51 The expected loss = ( 9.51-9.4 )/ 9.51 = 1.2% The recovery rate = (12.7 − 1.2) / 12.7 = 91%
  32. 32. Thank you

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