1. Credit Risk
Yiling Lai
2008/10/3

2. Outline Introduction Credit Ratings Historical Default Probabilities
Recovery Rates
Estimating Default Probability from Bond Prices
Comparison of Default Probability Estimates

3. Introduction
Credit risk raises from the possibility that borrowers and counterparties in derivatives transactions may default.

4. Credit Rating
Credit Rating assesses the creditworthiness of corporate bonds.
Default Risk
S&P
Moody’s
中華信評
Low
AAA
Aaa
twAAA
AA+ , AA, AA-
Aa1, Aa2, Aa3
twAA+, twAA, twAA-
A+, A , A-
A1, A2, A3
twA+, twA, twA-
BBB+, BBB, BBB-
Baa1, Baa2, Baa3
twBBB+, twBBB, twBBB-
BB+, BB, BB-
Ba1, Ba2, Ba3
twBB+, twBB, twBB-
B+, B, B-
B1, B2, B3
twB+, twB, twB-
High
CCC…
Caa…
twCCC…
Investment grade bond
Non-investment grade bond
(high yield bond, speculative grade bond or junk bond)

5. Historical Default Probabilities
Average cumulative default rates(%), Source: Moody’s
From this table, we can calculate unconditional default probability and conditional default probability.
Term 1 2 3 4 5 7 10 15 20
Aaa
0.000
0.026
0.099
0.251
0.521
0.992
1.191 Aa
0.008
0.019
0.042
0.106
0.177
0.343
0.522
1.111
1.929 A
0.021
0.095
0.220
0.344
0.472
0.759
1.287
2.364
4.238
Baa
0.181
0.506
0.930
1.434
1.938
3.959
4.637
8.244
11.362 Ba
1.205
3.219
5.568
7.985
10.215
14.005
19.118
23.380
35.093 B
5.236
11.296
17.043
22.054
26.794
34.771
43.343
52.175
54.421
Caa-C
19.476
30.494
39.717
46.904
52.622
59.938
69.178
70.870

6. Historical Default Probabilities
Unconditional default probability
The probability of a bond defaulting during a particular year as seen at time 0.
Term 1 2 3 4 5 7
Aaa
0.026
0.073 … Aa
0.008
0.011
0.023
0.064
0.071 A
0.021
0.074
0.125
0.124
0.128
Baa
0.181
0.325
0.424
0.504 Ba
1.205
2.014
2.349
2.417
2.23 B
5.236
6.06
5.747
5.011
4.74
Caa-C
19.476
11.018
9.223
7.187
5.718
Increasing
0.352 =
Decreasing

7. Historical Default Probabilities
Conditional default probability (default intensity or hazard rate)
The probability that the bond will default during a particular year conditional on no earlier default.
Average cumulative default rates
Term 1 2 3 4
Caa-C
19.476
30.494
39.717 …
Unconditional default probability
Term 1 2 3 4
Caa-C
19.476
11.018
9.223 …
The probability that the bond will survive until the end of year 2 is =69.506%
Conditional default probability is 9.223/ = 13.27%
(for 1-year time period)

8. Default Intensities

10. Default Intensities
Q(t): the probability of default by time t.

11. Average recovery rate (%)
Recovery Rates
The recovery rate for a bond is normally defined as the bond’s market value immediately after a default, as a percent of its face value.
Recovery rates on corporate bonds as a percentage of face value (Source: Moody’s)
Class
Average recovery rate (%)
Senior secured
54.44
Senior unsecured
38.39
Senior Subordinated
32.85
Subordinated
31.61
Junior subordinated
24.47

12. Recovery Rates
Recovery rates are significantly negatively correlated with default rates.
Recovery rate = x Default rate
The recovery rate: the average recovery rate on senior unsecured bonds in a year measured as %
The default rate: the corporate default rate in the year measured as %
A bad year for the default rate is usually double bad because it is accompanied by a low recovery rate.
Default rate = 0.1%
=> Recovery rate = 58.3%
Default rate = 3%
=> Recovery rate = 34.0%

13. Estimating Default Probability from Bond Prices
Assumption: The only reason a corporate bond sells for less than a similar risk-free bond is the possibility default.
An approximate calculation:
這個假設並不完美，因為公司債的價格會受到“流動性”的影響。liquidity
S = 200 base points and R = 40% => h=0.02/(1-0.4)=3.33%

14. A More Exact Calculation
The corporate bond:
Period: 5 years
Coupon: 6% per annum (paid semiannually)
Yield: 7% per annum (with continuous compounding)
Price: 95.34
A similar risk-free bond
Yield: 5% per annum (with continuous compounding)
Price:
The expected loss from default over the 5-year life of the bond is =$8.75

15. A More Exact Calculation
Assumption: defaults can happen at times 0.5, 1.5, 2.5, 3.5, and 4.5 years (immediately before coupon payment dates).
Notional Principal=$100
Risk-free rates: 5% (with continuous compounding)
Recovery rate = 40% => 100*40%=$40

16. A More Exact Calculation
Consider the 3.5 years:
The expected value of the risk-free bond at time 3.5 years:
The loss given default is =64.34
The PV of this loss is 1 2 3 4 5
years
Cash flow
103

17. A More Exact Calculation
Suppose that the probability of default per year (assumed to be the same each year) is Q.
Calculating of loss from default on a bond in terms of the default probabilities per year, Q.
Time
(years)
Default
Probabilities
Recovery
Amount ($)
Risk-free
Value($)
Loss given
Default($)
Discount
factor
PV of expected
loss ($)
0.5 Q 40
106.73
66.73
0.9753
65.08Q
1.5
105.97
65.97
0.9277
61.20Q
2.5
105.17
65.17
0.8825
57.52Q
3.5
104.34
64.34
0.8395
54.01Q
4.5
103.46
63.46
0.7985
50.67Q
Total
288.48Q
288.48Q=8.75
=> Q = 3.03%

18. A More Exact Calculation
We can extend this calculations by changing some assumptions.
Example:
Defaults can take place more frequently
A constant default intensity
A particular pattern for the variation of default probabilities with time
With several bonds we can estimate several parameters describing the term structure of default probabilities.

19. The Risk-Free Rate
The benchmark risk-free rate that is usually used in quoting corporate bond yields is the yield on similar Treasury bonds.
In fact, those derivative traders usually use LIBOR rates as short-term risk-free rates.
They regard LIBOR as their opportunity cost of capital.
Treasury rates are too low to be used as risk-free rates.

20. Asset Swaps Asset swaps = asset + interest rate swap or
Asset swaps = asset + currency swap
An asset swap transforms the character of an end user’s assets.
Repacking an issue paying fixed rates into floating rates (or vice versa)
Converting cash flow stated in one currency to another.
It does not eliminate the asset from an investor’s portfolio.

21. Fixed Rates to Floating Rates
The investor transforms the yield on its fixed rate asset into a floating rate asset.

22. Currency Transformation
The investor converts with the swap counterparty the FFr500mm principal cost of the asset and $100mm, the investor’s base currency.

23. Asset Swaps Spread
In practice, traders often use asset swap spreads as a way of extracting default probabilities from bond prices.
Asset swap spreads provide a direct estimate of the spread of bond yields over the LIBOR/swap curve.
Example:
An asset swap spread for a particular bond: 150 bps
The LIBOR/swap zero curve is flat at 5%
The amount by which the value of the risk-free bond exceeds the value of the corporate is the present value of 150bps per year for 5years.
Assuming semiannual payments

24. Asset Swaps Spread The sum of PV is $6.55 per $100 of principal.
288.48Q = 6.55
=> Q=2.27% 1 2 3 4 5
years
Cash flow
0.75 PV

25. Comparison of Default Probability Estimates (part I)
From historical data:
Based on
Consider t=7:
Consider an A-rated company in table 20.1:
Q(7) =

26. Comparison of Default Probability Estimates (part I)
From bond prices:
Based on
Recovery rate (R): 40%
Risk-free rate: the 7-year swap rate minus 10 bps.
For A-rated bond, the average Merrill Lynch yield was 5.993%
The average swap rate was 5.398%
=> risk-free rate = =5.298%
s = =

27. Comparison of Default Probability Estimates (part I)
Seven-year average default intensities (% per annum)
Rating
Historical default
intensity
Default intensity
from bonds
Ratio
Difference
Aaa
0.04
0.60
16.7
0.56 Aa
0.05
0.74
14.6
0.68 A
0.11
1.16
10.5
1.04
Baa
0.43
2.13
5.0
1.71 Ba
2.16
4.67
2.2
2.54 B
6.10
7.97
1.3
1.98
Caa and lower
13.07
18.16
1.4
5.50
decline
increase

28. Comparison of Default Probability Estimates (part II)
Another way of looking at these results: excess return over the risk-free rate
Expected excess return on bonds (bps) = (A)-(B)-(C)
Rating
Bond yield spread
over Treasuries (A)
Spread of risk-free rate
over Treasuries (B)
Spread for historical
defaults (C)
Excess
return
Aaa 78 42 2 34 Aa 87 3 A
112 7 63
Baa
170 26
102 Ba
323
129
151 B
521
366
Caa
1132
784
305
increase
Historical default intensity x (1-Recovery rate)

29. Real-World vs. Risk-Neutral Probabilities
The default probabilities implied from bond yields are risk-neutral probabilities of default.
The risk-neutral valuation principle states that the price of a derivative is given by the expectation of the discounted terminal payoff under the risk neutral measure.
This means that the default
probability Q must be a risk-neutral probability.
By contrast, the default probabilities implied from historical data are real-world probabilities of default.
risk neutral valuation principle states that the price of a derivative is given
by the expectation of the discounted terminal payoff under the risk neutral measure,

30. Why do we see such big differences between real-world and risk-neutral default probabilities?
Corporate bonds are relatively illiquid.
The subjective default probabilities of bond traders may be much higher than those given in Tables 20.1.
Bonds do not default independently of each other because of systematic risk.
Unsystematic risk: It is much more difficult to diversify risks in a bond portfolio than in an equity portfolio.

31. Which one is better?
The answer depends on the purpose of the analysis.
Risk-neutral default probabilities:
To value credit derivatives
To estimate the impact of default risk
Real-world default probabilities:
To calculate potential future losses from default when carrying out scenario analyses