Written in 1987; printed in Derivatives Pricing: The Classic Collection, P. Carr (ed.), London: Risk Books, 2004.

Consider a portfolio consisting of n loans in equal dollar amounts. Let the probability of default on any one loan be p, and assume that the values of the borrowing companies' assets are correlated with a coefficient ρ for any two companies. We wish to calculate the probability distribution of the percentage gross loss L on the portfolio, that is,

Let be the value of the i-th company's assets, described by a logarithmic Wiener process

where are Wiener processes with

The company defaults on its loan if the value of its assets drops below the contractual value of its obligations D_i payable at time T. We thus have

where

and N is the cumulative normal distribution function.

Because of the joint normality and the equal correlations, the processes z_i can be represented as

where

and

The term can be interpreted as the i-th company exposure to a common factor x (such as the state of the economy), and the term represents the company's specific risks. Then

In terms of the original parameters p and ρ, we have

Note that the integrand is the conditional probability distribution of the portfolio loss given the state of the economy, as measured by the market increase or decline in terms of its standard deviations.