Chapter 17
Probability 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 N is the cumulative normal distribution function.

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

equation

where

equation

and

equation

The term c17-math-0011 can be interpreted as the i-th company exposure to a common factor x (such as the state of the economy), and the term c17-math-0012 represents the company's specific risks. Then

equation

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

equation

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.

..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.
Reset
18.188.143.21