Written in 1989; printed in Derivatives Pricing: The Classic Collection, P. Carr (ed.). London: Risk Books, 2004.

The cumulative probability that the percentage loss on a portfolio of n loans does not exceed is

where are given by an integral expression in Oldrich Vasicek's memo, “Probability of Loss on Loan Portfolio,” February 1987 (Chapter 17 of this volume). The substitution

in the integral gives as

where

By the law of large numbers,

and therefore the cumulative distribution function of loan losses on a very large portfolio is

This is a highly skewed distribution. Its density is

Its mean, median, and mode are given by

The α-quantile, P[L < L_α] = α, is given by