Understanding Bond Default Rates and Correlation

Distribution of Moody's Ratings

Historical Default Rates

• Rating Agencies
• Default rates from bonds
• Market Data

CAPM Model for Bond Returns

The usual CAPM has to be modified to include expected loss per year.

What must the β be to justify the excess spread for Baa Bonds?

Merton's Model

A Structural Model That Predicts Default Rates

How Do We Find V₀ and σ_V?

We know E₀ and σ_E, and

An application of Ito's lemma shows

These allow us to find V₀ and σ_V from E₀ and σ_E.

Default Correlation

Defaults: If Firm A Defaults, is Firm B More Likely to Default Too?

No Default Correlation

Consider two firms, X and Y:

Q_X = 0.1, Q_Y = 0.2

If X and Y are uncorrelated, then the probability that they both default is Q_X × Q_Y = 0.02

Building Default Correlation

If X and Y are correlated, we have to make the probability that they both default bigger.

We also have to keep Q_X = 0.1, Q_Y = 0.2

This is easy for 2 firms but hard for 3 or more.

More About the Factor Model

If we set the common factor in September, F, to some special value in the simulation, then all of the X_i values are determined by the Z_i's.

Because the Z_i's are all independent, defaults are uncorrelated.

All of the default correlation comes from the common factor, F.

Conditional Probability of Default

If the common factor, F, is set to F^*, we can also calculate the probability of default.

Company i defaults when X_i < x_i = N^-1[Q_i]

X_i < x_i when Z_i < Z_i^*

Given F = F*, the probability of default is

Probability of Two Firms Defaulting

The conditional probability that i and j default is

The unconditional probability that i and j default is

This is the procedure usually used to calculate the probability of multiple defaults.

Arbitrary Copulations

The one-factor Gaussian copula says the probability of default is itself random.

Q | F takes on many values, each with some probability, by the Gaussian copula model.

We can make up our own factor models.

Our factor, F, takes on K different values, F_k.

The probability that F = F_k is p_k.

Arbitrary Copulations - Continued

For each factor value, choose the conditional probability of default for each firm (Q_i | F = F_k) = Q_ik.

The only restriction is the average default probability has to equal the current probability.

The probability that both i and j default is