Credit Risk

Yiling Lai2008/10/3

1

Outline

• Introduction• Credit Ratings• Historical Default Probabilities• Recovery Rates• Estimating Default Probability from Bond

Prices• Comparison of Default Probability Estimates

2

Introduction

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

3

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…

4

Non-investment grade bond (high yield bond, speculative grade bond or

junk bond)

Investment grade bond

Historical Default Probabilities• Average cumulative default rates(%), 1970-2006.

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 0.000 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 70.870

5

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 0 0 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 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 …0.352=0.506-0.181

Increasing

Decreasing6

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 …

The probability that the bond will survive until the end of year

2 is 100-30.949=69.506%

Unconditional default probability

Term 1 2 3 4

Caa-C 19.476 11.018 9.223 …

Conditional default probability is 9.223/69.506 = 13.27%(for 1-year time period) 7

Default Intensities

8

9

Default Intensities

• Q(t): the probability of default by time t.

10

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. 1982-2003 (Source: Moody’s)

Class Average recovery rate (%)

Senior secured 54.44

Senior unsecured 38.39

Senior Subordinated 32.85

Subordinated 31.61

Junior subordinated 24.4711

Recovery Rates

• Recovery rates are significantly negatively correlated with default rates.

• Recovery rate = 59.1 - 8.356 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.

12

Default rate = 0.1%=> Recovery rate = 58.3%

Default rate = 3%=> Recovery rate = 34.0%

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:

13

S = 200 base points and R = 40% => h=0.02/(1-0.4)=3.33%

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: 104.09

• The expected loss from default over the 5-year life of the bond is 104.9-95.34=$8.75

14

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

15

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 104.34-40=64.34– The PV of this loss is

16

0 1 2 3 4 5years

Cash flow 3 3 3 103

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.

17

Time(years)

DefaultProbabilities

RecoveryAmount ($)

Risk-freeValue($)

Loss givenDefault($)

Discountfactor

PV of expected 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

288.48Q=8.75 => Q = 3.03%

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.

18

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.

19

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.

20

Fixed Rates to Floating Rates

• The investor transforms the yield on its fixed rate asset into a floating rate asset.

21

Currency Transformation

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

22

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

23

Asset Swaps Spread

• The sum of PV is $6.55 per $100 of principal.• 288.48Q = 6.55 => Q=2.27%

24

0 1 2 3 4 5years

Cash flow 0.75 0.750.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75

PV

Comparison of Default Probability Estimates (part I)

25

• From historical data:– Based on – Consider t=7:

– Consider an A-rated company in table 20.1: Q(7) =0.00759

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 = 5.398-0.1=5.298%

26

s = 0.05993-0.05298 = 0.00695

Comparison of Default Probability Estimates (part I)

27

Seven-year average default intensities (% per annum)

Rating Historical defaultintensity

Default intensityfrom 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 5.50decline increase

Comparison of Default Probability Estimates (part II)

• Another way of looking at these results: excess return over the risk-free rate

28

Expected excess return on bonds (bps) = (A)-(B)-(C)

Rating Bond yield spreadover Treasuries

(A)

Spread of risk-free rateover Treasuries (B)

Spread for historical defaults (C)

Excess return

Aaa 78 42 2 34

Aa 87 42 3 42

A 112 42 7 63

Baa 170 42 26 102

Ba 323 42 129 151

B 521 42 366 112

Caa 1132 42 784 305Historical default intensity x (1-Recovery rate)

increase

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.

29

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.

30

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

31