Recovery Process Model

Recently, because of Basel II and the subprime mortgage crisis, the quantification of the recovery size and the recovery rate for the debt of a defaulted company is a serious problem for financial institutions and their supervisors, but there has been no study of structure of the recovery process which is the relationship between time and the cumulative recovery size. Existent recovery models do not regard the recovery progress before the time of achievement of recovery. We directly model recovery process for the debt of a single defaulted company. We model the recovery process by a homogeneous compound Poisson process and extend our model to an inhomogeneous compound Poisson process. The interest rate is explicitly used in our model. By our model, the relationship between the cumulative recovery, the increment of recovery, the initial debt amount, the last recovery possible time and the interest rate can be analyzed. We derive the expected value and the variance of the survival value of the debt and the recovery rate, and also derive the probability distribution function and the expected value of the recovery completion time. Moreover we present the numerical methods for calculating the expected value and the variance based on Panjer recursion formula and the fast Fourier transformation, and show numerical results. Also we propose a new method of calculating the transition density of an inhomogeneous compound Poisson process. Our method is based on approximating an inhomogeneous compound Poisson process by a piecewise homogeneous compound Poisson process. This method is used to compute the expected value and the variance of an inhomogeneous compound Poisson process.