Recovery Process Model for Two Companies

The quantification of the recovery rate for the debt of the defaulted small company is one of the most important problems for banks and their supervisors. However, the data of the real recovery rates is seldom available for academic study. Therefore, there have been a few studies for the recovery rate for the debt. The recovery process model for a single company is introduced by Itoh (Asia Pac Financ Mark 2008). In this paper, we extend the model of Itoh (Asia Pac Financ Mark 2008) to two defaulted companies, and we model the recovery processes using an inhomogeneous bivariate compound Poisson process with the delays. In other word, we assume that the recoveries are occurred by the shocks, and that there are individual shocks that affect only one company and common shocks that affect two companies. Moreover, we assume that there are delays between the shock points and the recovery points. Therefore, the recovery points of two companies are correlated, but the recoveries do not occur synchronously almost surely. We derive the correlation of the recovery rates of the debts of two defaulted companies, and the expected value and the standard deviation of the recovery rate for the defaulted debt portfolio. Furthermore, we present two methods based on the Vernic recursion formula and the Monte Calro simulation for calculating the probability distribution function of the recovery process, and illustrate several numerical results.