Chapter 24: Credit Risk

Credit Ratings

Credit rating agencies (S&P, Moody's, Fitch) assess the creditworthiness of companies:

Investment grade = BBB/Baa and above. Below = high yield/junk.

Historical Default Probabilities

Rating agencies publish transition matrices showing the probability of rating changes. Example cumulative default rates over 5 years:

These are real-world (physical) default probabilities, not risk-neutral.

Recovery Rates

The recovery rate is the proportion of the amount owed that is recovered in a default. Average recovery rates depend on seniority:

Recovery rates and default probabilities are negatively correlated: when defaults are high, recovery rates tend to be low ("default clustering with reduced recovery").

Estimating Default Probabilities from Bond Yields

The yield spread over the risk-free rate contains a credit spread:

$s=y-r$

where $y$ = corporate bond yield, $r$ = risk-free rate.

For a simple estimate, assuming zero recovery:

$\bar{\lambda }=\frac{s}{1-R}$

where $\bar{\lambda }$ is the average default intensity (hazard rate) per year and $R$ is the recovery rate.

Example: 5-year corporate bond yield = 5%, risk-free rate = 3%, recovery = 40%.

$s=2\%$, $\bar{\lambda }=0.02/(1-0.4)=0.0333=3.33\%$ per year risk-neutral default probability.

Key distinction: Bond-implied default probabilities are risk-neutral (higher than real-world because they include a risk premium for bearing credit risk).

Comparison of Estimates

The difference between the two reflects the risk premium for bearing credit risk and potential illiquidity effects. This is analogous to the difference between real-world equity returns ($\mu$) and the risk-free rate ($r$).

Using Equity Prices (Merton's Model)

In Merton's model (1974), a company's equity is a call option on its assets with strike price equal to the face value of debt:

$E=V_{0}N(d_1)-D{e}^{-rT}N(d_2)$

where $V_0$ = asset value, $D$ = debt face value, ${\sigma }_V$ = asset volatility. The risk-neutral probability of default is $N(-d_2)$.

The KMV model (now Moody's Analytics) extends this with a proprietary database. It uses the distance to default:

$\text{DD}=\frac{V_0-\text{Default Point}}{{\sigma }_V\cdot V_0}$

The expected default frequency (EDF) is derived from historical mapping of DD to actual defaults.

Credit Risk in Derivatives

Unlike loans, derivatives create bilateral credit exposure:

• The exposure varies over time (not constant like a loan)
• Only out-of-the-money derivatives create exposure for the in-the-money counterparty
• Collateral agreements (CSAs) mitigate exposure

Expected positive exposure (EPE) is the average positive exposure over time, crucial for CVA calculation.

Wrong-Way Risk

When exposure and counterparty credit quality deteriorate together. Example: buying a put option on oil from an oil company — when oil prices fall and the put is valuable, the oil company is more likely to be in financial distress.

Default Correlation

Default correlation measures the tendency of companies to default together. Two sources:

1. Systematic factors: Macroeconomic conditions affect all companies
2. Direct connections: One company's default weakens another (supply chain, interbank)

The Gaussian copula model (described in Chapter 25) is widely used for modeling default correlation in portfolios of credits.

Credit VaR

Credit VaR extends market VaR by modeling the impact of credit events (defaults and rating migrations) on portfolio value. The distribution is typically highly skewed with a long left tail — normal distribution approximations are inappropriate.