One of the key quantities of interest to an insurance company is the total amount to be paid out on a particular portfolio of policies over a fixed time interval, such as an accounting period. This quantity may be approached in various ways, and we mention two popular models below. We refer to both these models as examples of short term risk models because they model a risk over a fixed time period. This is in contrast to the classical risk model in Chapter 6, where the stochastic evolution of the flow of claim payments and premium income is modelled over time, and properties of this evolution over an infinite time period are derived. As might be expected, the techniques and results of Chapter 6 are deeper and more complex than those in this chapter, but they build on the foundations that we develop here for short term models.

One short term model is the individual risk model, where we consider the portfolio to consist of a fixed number, n, of independent policies, and the total amount claimed on the portfolio in a fixed time period is modelled as a random variable T, given by

where Yi is the total amount claimed on policy i, and Y1, …, Yn are assumed to be independent, but not necessarily identically distributed. It turns out that it is more difficult than might be expected at first sight to deal with this apparently simple quantity in terms of numerical calculations and in terms of obtaining analytical expressions for the distribution of T. This model is considered in §3.8.