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Finance, Economics, and Mathematics by Robert C. Merton, Oldrich A. Vasicek

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Chapter 18Limiting Loan Loss Probability Distribution

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 c18-math-0001 is

equation

where c18-math-0003 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

equation

in the integral gives c18-math-0005 as

equation

where

equation

By the law of large numbers,

equation

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, ...

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