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

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Chapter 17Probability of Loss on Loan Portfolio

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,

equation

Let c17-math-0002 be the value of the i-th company's assets, described by a logarithmic Wiener process

equation

where c17-math-0004 are Wiener processes with

equation

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

equation

where

equation

and ...

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