Property-Casualty insurance liabilities, related to claims from automobile accidents, house fires, liability claims, etc., are characterized by reporting and settlement lags which can be several years long. As a result, the liabilities and loss payments from a given set of insurance policies evolve over time, with payments gradually increasing to their ultimate settlement values. Actuaries use aggregate loss distributions (random sums) to model ultimate settlement values but there is no established way of decomposing ultimate losses into losses paid each year. This talk will explain how the negative multinomial distribution can be used to decompose ultimate losses into losses by year, and show that the resulting decomposition has empirically desirable properties. Next, we will discuss a Markov-chain model of claim complexity, which can be combined with the decomposition result, in order to produce a model with increasing average claim severity over time, a phenomenon observed in most lines of insurance. The Markov-chain model is an interesting departure from traditional actuarial analyses because it uses detailed cross-sectional data rather than long-term summary data.