This paper considers the expected penalty functions for a
discrete semi-Markov risk model, which includes several existing risk models such
as the compound binomial model (with time-correlated claims) and the compound Markov
binomial model (with time-correlated claims) as special cases. Recursive formulae
and the initial values for the discounted free penalty functions are derived in the
two-state model by an easy method. We also give some applications of our results.