Loss Models, 5th Edition

8Frequency and Severity with Coverage Modifications

8.1 Introduction

We have seen a variety of examples that involve functions of random variables. In this chapter, we relate those functions to insurance applications. Throughout this chapter, we assume that all random variables have support on all or a subset of the nonnegative real numbers. At times in this chapter and later in the text, we need to distinguish between a random variable that measures the payment per loss (so zero is a possibility, taking place when there is a loss without a payment) and a variable that measures the payment per payment (the random variable is not defined when there is no payment). For notation, a per-loss variable is denoted and a per-payment variable is denoted . When the distinction is not material (e.g. setting a maximum payment does not create a difference), the superscript is omitted.

8.2 Deductibles

Insurance policies are often sold with a per-loss deductible of d. When the loss, x, is at or below d, the insurance pays nothing. When the loss is above d, the insurance pays . In the language of Chapter 3, such a deductible can be defined as follows.

Definition 8.1

An ordinary deductible