Fundamentals of Actuarial Mathematics, 3rd Edition

21 Compound distributions

21.1 Introduction

In earlier parts of this book, we concentrated on the present value of the benefits paid on a single insurance or annuity contract. The insurer, of course, is interested in the total benefits paid on an entire portfolio of policies. An obvious way to handle this is simply to obtain the present value of the total amount paid on all policies in the portfolio, as the sum of the individual random variables. This is known as the individual risk model. There is another method for estimating the total amount paid on a group of policies, known as the collective risk model, which has advantages in certain cases. In this chapter, we deal with a static version of this model, covering a 1-year time period. Chapter 23, concerned with ruin theory, will involve a dynamic multi-period version of the collective risk model. The combined subject matter of these chapters has traditionally been referred to as risk theory in the actuarial literature. The collective risk model is particularly useful for casualty insurance such as automobile, home, or health policies. The following are three main ways in which such contracts differ from life insurance:

1. In a given period, there can be several claims under a single policy. Clearly, you can have several accidents or several visits to the doctor, even in a relatively short period. However, no matter how long the period, you can only die once.
2. The amount of each claim can vary substantially. A collision claim ...