The present paper investigates the error committed by using an infinite time horizon Markov renewal program as an approximation of the (often more realistic) Markov renewal program with a finite time horizon t(0). Under weak assumptions the error is shown to converge to zero exponentially fast when t(0) --> infinity. The convergence is based on explicit error bounds. Improved error bounds hold when the (transformed) transition law has a nontrivial stochastic lower bound. Some bounds use the discounted renewal function. For the latter, monotone upper and lower bounds are obtained by an iterative method combined with an extrapolation. Several examples demonstrate the applicability of the results.