The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 722 Wednesday, February 4, 2004 1:12 PM

722 The Mathematics of Financial Modeling and Investment Management

How to Model Correlated Default Processes^43

Default correlation is not an easy concept to define or measure. Put in

simple terms, it is a measurement of the degree to which default of one

asset makes more or less likely the default of another asset. One can

think of default correlation as being jointly due to (1) a macroeconomic

effect which tends to tie all industries into the common economic cycle;

(2) a sector specific effect, and (3) a company specific effect.

The first contribution implies that default correlation should in gen-

eral be positive even between companies in different sectors. Within the

same sector we would expect companies to have an even higher default

correlation since they have more in common. For example, the severe

fall in oil prices during the 1980s resulted in the default of numerous

oil-producing industries. On the other hand, the fall in the price of oil

would have made the default of oil-using industries less likely as their

energy costs fell, thereby reducing their likelihood of default and reduc-

ing the default correlation. However the sheer lack of default data

means that such assumptions are difficult to verify with any degree of

certainty.

It is simple enough to define pure default correlation. Basically, this

number must correspond to the likelihood that should one asset default

within a certain time period, how more or less likely is another asset to

also default. In the case of default correlation, it is important to specify

the horizon which is being considered.

The pairwise default correlation between two assets A and B is a

measure of how more or less likely two assets are to default than if they

were independent.

Specifying Directly Joint Default Distribution

Let two firms, A and B, follow the following joint Bernoulli distribution

(letting superscripts denote complement sets):

Firm A

0 1

Firm B 0

1

pA( C ∩BC)

pAC ∩( B)

1 – pA()

pA ( ∩BC)

pA B∩( )

pA() 1

1 – pB()

pB()

(^43) This discussion draws from Ren-Raw Chen and Ben J. Sopranzetti, “The Valua-
tion of Default-Triggered Credit Derivatives,” Journal of Financial and Quantitative
Analysis (June 2003).