Claim severity refers to the monetary loss of an insurance claim. Unlike claim frequency, which is a nonnegative integer-valued random variable, claim severity is usually modeled as a nonnegative continuous random variable. Depending on the definition of loss, however, it may also be modeled as a mixed distribution, i.e. a random variable consisting of probability masses at some points and continuous otherwise.

We begin this chapter with a brief review of some statistical tools for analyzing continuous distributions and mixed distributions. The use of the survival function and techniques of computing the distribution of a transformed random variable are reviewed. Some standard continuous distributions for modeling claim severity are summarized. These include the exponential, gamma, Weibull, and Pareto distributions. We discuss methods for creating new claim-severity distributions such as the mixture-distribution method. As losses that are in the extreme right-hand tail of the distribution represent big losses, we examine the right-hand tail properties of the claim-severity distributions. In particular, measures of tail weight such as limiting ratio and conditional tail expectation are discussed. When insurance loss payments are subject to coverage modifications such as deductibles, policy limits, and coinsurance, we examine their effects on the distribution of the claim severity.

Learning objectives

1. Continuous distributions for modeling claim severity

2. Mixed distributions

3. Exponential, gamma, Weibull, and Pareto distributions

4. Mixture distributions

5. Tail weight, limiting ratio, and conditional tail expectation

6. Coverage modification and claim-severity distribution

Review of statistics

In this section we review some results in statistical distributions relevant for analyzing claim severity.