Analytical solutions for the expected loss of a collateralized loan: a square root intensity process negatively correlated with collateral value

Abstract

This paper analytically evaluates the expected loss and the nth moment of the loss distribution for a collateralized loan by focusing on the negative correlation between default intensity and collateral value. To ensure a negative correlation and nonnegativity of intensity, we propose a square root process for default intensity and a negatively correlated affine diffusion process for collateral value. Given these settings, we derive an explicit solution for the integrand of the expected recovery value and the kth moment of the recovery value, using measure-changed survival probabilities. Finally, we analyze the expected loss and standard deviation of the loss based on the estimation of parameters for default intensity and the collateral value process.