No. 1092

Title:

Expected loss with a negative correlation between hazard and recovery: Analytical evaluation with a square-root hazard process

Author(s):

Yamashita, Satoshi (Institute of Statistical Mathematics);
Yoshiba, Toshinao (Bank of Japan)

Key words:

stochastic recovery; hazard-based model; affine diffusion; survival probability

Abstract:

    In this paper, we derive an explicit solution for expected loss of a loan covered by some collateral whose value varies correlated with the hazard rate of the loan. We examine a negative correlation between the hazard rate and the recovery rate due to a common factor of a macro economic condition. We assume a square-root process for the hazard rate to keep its value non-negative. We also assume that logarithm of the collateral value has a diffusion process whose instantaneous volatility is proportional to the square root of the hazard rate. In those setting, we show the expected loss can be evaluated in an extended affine model and we derive an explicit solution for the expected loss. Numerical example shows that the expected loss is large if the absolute value of the negative correlation is large.