This document discusses credit risk and methods for estimating default probabilities. It covers the following key points: - Credit ratings agencies like S&P and Moody's provide ratings to indicate the creditworthiness of bonds from AAA to C. Investment grade bonds are BBB/Baa and above. - Historical default data shows that for good initial ratings, default probabilities increase over time, while for poor ratings they decrease over time. - Recovery rates are the price recovered after default, on average around 40-60% depending on default levels. - Bond prices can be used to estimate default probabilities using spreads over risk-free rates and assuming the spread is due to default risk adjusted by recovery rates.

1. Chapter 22
Credit Risk
資管所 陳竑廷

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

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

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

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. 1
λ (7) = − ln[1 − Q(7)]
7
1
= − ln[1 − 0.00759]
7
= 0.11%

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. 0.11*(1-0.4)=0.066

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. 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. Merton’s Model
If VT < D , ET = 0
( default )
If VT > D , ET = VT - D
ET = max(VT − D,0)

24. •
V0 And σ0 can’t be directly observable.
•
But if the company is publicly traded , we can observe E0.

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. 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. 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. 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. 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

31. • Initial V0 = 12.40 , σ0 = 0.2123
• Initial V0 = 10 , σ0 = 0.1

32. 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%

33. Thank you