The RiskMetrics Group
Working Paper Number 99-07
On Default Correlation: A Copula Function Approach
David X. Li
This draft: April 2000
First draft: September 1999
44 Wall St.
New York, NY 10005
david.li@riskmetrics.com
www.riskmetrics.com
On Default Correlation: A Copula Function Approach
David X. Li
April 2000
Abstract
This paper studies the problem of default correlation. We first introduce a random variable called “time-
until-default” to denote the survival time of each defaultable entity or financial instrument, and define the
default correlation between two credit risks as the correlation coefficient between their survival times.
Then we argue why a copula function approach should be used to specify the joint distribution of survival
times after marginal distributions of survival times are derived from market information, such as risky
bond prices or asset swap spreads. The definition and some basic properties of copula functions are
given. We show that the current CreditMetrics approach to default correlation through asset correlation
is equivalent to using a normal copula function. Finally, we give some numerical examples to illustrate
the use of copula functions in the valuation of some credit derivatives, such as credit default swaps and
first-to-default contracts.
1 Introduction
The rapidly growing credit derivative market has created a new set of financial instruments which can be
used to manage the most important dimension of financial risk - credit risk. In addition to the standard
credit derivative products, such as credit default swaps and total return swaps based upon a single underlying
credit risk, many new products are now associated with a portfolio of credit risks. A typical example is the
product with payment contingent upon the time and identity of the first or second-to-default in a given credit
risk portfolio. Variations include instruments with payment contingent upon the cumulative loss before a
given time in the future. The equity tranche of a collateralized bond obligation (CBO) or a