Ch6 Credit Risk

1. Credit Ratings
1-1. Rating Classification
- Investment-Grade:
AAA (Aaa)
AA (Aa)
A
BBB (Baa)
- Non-Investment-Grade(Speculative)
BB (Ba)
B
CCC (Caa-C)
D

**S&P vs. Moody's

1-2. Creteria used to rate corporate bonds
- The amount & composition of existing debt
- The stability of the issuer's cash flow
- The issuer's ability to meet scheduled payments of P&I obligations
- Asset protection
- Management capability


2. Default Rates
2-1. Default Intensities
**Average cumulative default rates: unconditional default probability
*negative relationship between default rates & post-default recovery values

e.g.) Caa-C cumulative default rates: 3rd year - 30.494, 2nd year - 39.717
==> uncoditional default during the 3rd year = 39.717 - 30.494 = 9.233%
probability that the bond will survive until the end of year 2 = 1 - 30.494 = 69.506%
hence, the default during the 3rd year conditional on no earlier default = 9.233%/69.506%= 13.27% (= default intensities or hazard rates)


3. Recovery Rates
The bond's market value immediately after a default as a percent of its face value. Recovery rate are significantly negatively correlated with default rates.

**Hamilton, Varma, Ou & Cantor's Recovery Rate
Recovery rate = 59.1 - 8.356 x Default rate


4. Probability of Default
The usual assumption is that the only reason a corporate bond sells for less than a similar risk-free bond is the possibility of default. => The probability of default for a company can be estimated from the proces of bonds

Average default intensity = s / (1 - R)
where:
s = spread of the corporate bond yield over ther risk-free rate
R = expected recover rate, (1-R)= (LGD)Loss Given Default

e.g.) 5-year 6% coupon bonds with 7%(continuous compounding) YTM, risk-free rate: 5%(continuous compounding), probability of default per year = Q, recovery: 40$
assumption that default can happen at times 0.5, 1.5, 2.5 and 4.5 years
=> semmi-annual rate = 2 x (exp(7%/2) - 1) = 7.1239% / 2 & 5.063% / 2
P.V of the the corporate bond = $95.34 & P.V. of the risk-free bond = $104.09
Expected Loss from default over the 5-year bonds = 104.09 - 95.34 = $8.75

The expected value of the corporate bond at time 0.5-year
= 3 + 3 e^(-5%*0.5) + 3 e^(-5%*1.0) + ...+103e^(-5%*5.5) = $106.73

LGD(Loss Given Default) = 106.73 - 40 = $66.73
P.V. of Expected Loss = 66.73 e^(-5%*0.5) = $65.08 x Q

==> Total expected loss = $288.48Q
Hence, Q = 8.75 / 288.48 = 3.03%


5. Risk-Fee Rate
Benchmark risk-free rate: the yield on similar Treasury bonds.
Traders usually use LIBOR/swap rate as proxies for risk-free rates when valuing derivatives & calculating default probabilities.
Credit default swaps can be used to imply the risk-free rate which is LIBOR/swap - 10 bp


6. Asset Swaps
Asset swap spreads are used as a way of extracting default probabilities from bond prices. (Asset swap spreads provide a direct estimate of the spread of bond yield over the LIBOR/swap curve)


7. Merton Model
The equity of the company at time T is expressed as a call option on the company's total assets with a strike equal to the debt repayment:

Equity Value = Max(VT - DT, 0)


8. Counterparty Credit Risk
Possible situations:
- Contract is always a liability (e.g., short option position): no credit risk
- Contract is always a asset (e.g., long option position): alway credit risk
- Contract can become either an assent or a liability (e.g., forward contract): positive to the financial institution - loss, negative to the financial institution - no loss


9. Adjusting for Counterparty Credit Risk

Risk-Neutral Expected Loss = PD x LGD x E[max(Fi, 0)]
where:
E[max(Fi, 0)] = the expected credit risk exposure amount

- right-way risk: counterparty is most likely to default when the financial institution has near zero exposure
- wrong-way risk: counterparty is most likely to default when their company has a big exposure

**the risk is wrong (right) way when the exposure tends to increase (decrease) when counterparty credit quality worsens.

e.g.)
- we sell credit protection on X to Y: general right-way
- we enter into oil receiver swap with oil producer: general wrong-way
- we buy a put option on X stock from Y: general wrong-way
- we buy a put option on X stock from X: specific wrong-way


10. Credit Migration Techniques
- Netting
- Collateralization
- Downgrade Triggers


11. Gaussian Copula Model
It assume that all companies will default eventually. Default correlation between companies is a measure of the tendency to default at approximately the same time, and the Gaussian copula attempts to model this correlation using either real worlk or risk-neutral default probabilities.