Introduction

Although we have previously used the term premium, we have not formally defined it. A premium is the payment that a policyholder makes for complete or partial insurance cover against a risk. In this chapter we describe and discuss some ways in which premiums may be calculated, but we consider premium calculation from a mathematical viewpoint only. In practice, insurers have to take account not only of the characteristics of risks they are insuring, but other factors such as the premiums charged by their competitors.

We denote by ПX the premium that an insurer charges to cover a risk X. When we refer to a risk X, what we mean is that claims from this risk are distributed as the random variable X. The premium ПX is some function of X, and a rule that assigns a numerical value to ПX is referred to as a premium calculation principle. Thus, a premium principle is of the form ПX = φ(X) where φ is some function. In this chapter we start by describing some desirable properties of premium calculation principles. We then list some principles and consider which of the desirable properties they satisfy.

Properties of premium principles

There are many desirable properties for premium calculation principles. The following list is not exhaustive, but it does include most of the basic properties for premium principles.