Recently because of Basel II and the subprime mortgage crisis, the quantification of recovery size and recovery rate for the debt of a defaulted company is a serious problem for financial institutions and their supervision, but there has been no study of structure of recovery process. Existent recovery models do not regard 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. Interest rate is explicitly used in our model. By our model, the relationship between cumulative recovery, the increment of recovery, the initial debt amount, the last recovery possible time and interest rate can be analyzed. We derive the expectation and the variance of the survival value of the debt and recovery rate, and also derive the probability distribution function and the expectation of the recovery completion time. For this paper we present the numerical methods of calculating the expectation and the variance based on Panjer recursion formula and the fast Fourier transformation, and give numerical result. The methods of calculating the transition density of an inhomogeneous compound Poisson process is necessary for calculating the expectation and the variance of those in the inhomogeneous compound Poisson model, however little attention has been given to such methods. Therefore we propose the new procedure for calculating it by a piecewise homogeneous compound Poisson process.
