This paper deals with an insurance portfolio that covers two interdependent risks. The central model is a discrete-time bivariate risk process with independent claim increments. A continuous-time version of compound Poisson type is also examined. Our main purpose is to develop a numerical method for determining non-ruin probabilities over a finite-time horizon. The approach relies on, and exploits, the existence of a special algebraic structure of Appell type. Some applications in reinsurance to the joint risks of the cedent and the reinsurer are presented and discussed, under a stop-loss or excess of loss contract.