Discussion Paper Series 2010-E-10

Analytical Solution for Expected Loss of a Collateralized Loan: A Square-root Intensity Process Negatively Correlated with Collateral Value

Satoshi Yamashita, Toshinao Yoshiba

In this study, we derive an explicit solution for the expected loss of a collateralized loan, focusing on the negative correlation between default intensity and collateral value. Three requirements for the default intensity and the collateral value are imposed. First, the default event can happen at any time until loan maturity according to an exogenous stochastic process of default intensity. Second, default intensity and collateral value are negatively correlated. Third, the default intensity and collateral value are non-negative. To develop an explicit solution, we propose a square-root process for default intensity and an affine diffusion process for collateral value. Given these settings, we derive an explicit solution for the integrand of the expected recovery value within an extended affine model. From the derived solution, we find the expected recovery value is given by a Stieltjes integral with a measure-changed survival probability.

Keywords: stochastic recovery; default intensity model; affine diffusion; extended affine; survival probability; measure change

Views expressed in the paper are those of the authors and do not necessarily reflect those of the Bank of Japan or Institute for Monetary and Economic Studies.

Copyright © 2010 Bank of Japan All Rights Reserved.

Home Japanese Home