An efficient approach for calculating default probabilities for cash-flow based project finance with reserve account

Abstract

The quantitative assessment of risks associated with several types, eg, rating methods for cash-flow driven projects, can be reduced to determining the probability that a random variable, for instance representing a cash-flow, drops below a given threshold. That probability can be derived in an analytic closed form, if the underlying distribution is not too complex. However, in practice there is often a reserve account in place, which saves excess cash to reduce the volatility of the cash-flow available for debt service. Due to the reserve account, the derivation of a solution in an analytic closed form is even in the case of rather simple underlying distributions, eg, independent Gaussian distribution, not feasible. In this paper, we present two very efficient approximation methodologies for calculating the probability that a random variable falls under a threshold allowing the presence of a reserve account. The first proposed approach is derived using transition probabilities. The resulting recursive scheme can be implemented easily and yields fast and stable results even in the case of dependent cash-flows. The second methodology uses the similarity of the considered stochastic processes with convection-diffusion processes and combines the stochastic transition probabilities with the finite volume method, which is well known for solving partial differential equations. We present numerical results for some realistic test problems demonstrating convergence of order h for the transition probability based approach and $h^2$ for the combination with the finite volume method for sufficiently smooth probability distributions.