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.

In this chapter we consider the Bayesian approach in updating the prediction for future losses. We consider the derivation of the posterior distribution of the risk parameters based on the prior distribution of the risk parameters and the likelihood function of the data. The Bayesian estimate of the risk parameter under the squared-error loss function is the mean of the posterior distribution. Likewise, the Bayesian estimate of the mean of the random loss is the posterior mean of the loss conditional on the data.

This book is about modeling the claim losses of insurance policies. Our main interest is nonlife insurance policies covering a fixed period of time, such as vehicle insurance, workers compensation insurance and health insurance. An important measure of claim losses is the claim frequency, which is the number of claims in a block of insurance policies over a period of time. Though claim frequency does not directly show the monetary losses of insurance claims, it is an important variable in modeling the losses.

We consider models for analyzing the surplus of an insurance portfolio. Suppose an insurance business begins with a start-up capital, called the initial surplus. The insurance company receives premium payments and pays claim losses. The premium payments are assumed to be coming in at a constant rate. When there are claims, losses are paid out to policyholders. Unlike the constant premium payments, losses are random and uncertain, in both timing and amount. The net surplus through time is the excess of the initial capital and aggregate premiums received over the losses paid out. The insurance business is in ruin if the surplus falls to or below zero. The main purpose of this chapter is to consider the probability of ruin as a function of time, the initial surplus and the claim distribution. Ultimate ruin refers to the situation where in ruin occurs at finite time, irrespective of the time of occurrence.

Short-term insurance policies often take multiple years before the final settlement of all losses incurred is completed. This is especially true for liability insurance policies, which may drag on for a long time due to legal proceedings. To set the premiums of insurance policies appropriately so that the premiums are competitive and yet sufficient to cover the losses and expenses with a reasonable profit margin, accurate projection of losses cannot be overemphasized. Loss reserving refers to the techniques to project future payments of insurance losses based on policies in the past.

We have discussed the limited-fluctuation credibility method, the Bühlmann and Bühlmann–Straub credibility methods, as well as the Bayesian method for future loss prediction. The implementation of these methods requires the knowledge or assumptions of some unknown parameters of the model. For the limited-fluctuation credibility method, Poisson distribution is usually assumed for claim frequency.

While the classical credibility theory addresses the important problem of combining claim experience and prior information to update the prediction for loss, it does not provide a very satisfactory solution. The method is based on arbitrary selection of the coverage probability and the accuracy parameter. Furthermore, for tractability some restrictive assumptions about the loss distribution have to be imposed.

The main focus of this chapter is the estimation of the distribution function and probability (density) function of duration and loss variables. The methods used depend on whether the data are for individual or grouped observations, and whether the observations are complete or incomplete.

Some problems arising from loss modeling may be analytically intractable. Many of these problems, however, can be formulated in a stochastic framework, with a solution that can be estimated empirically. This approach is called Monte Carlo simulation. It involves drawing samples of observations randomly according to the distribution required, in a manner determined by the analytic problem.

In this chapter, we discuss some applications of Monte Carlo methods to the analysis of actuarial and financial data. We first revisit the tests of model misspecification introduced in Chapter 13.

Some models assume that the failure-time or loss variables follow a certain family of distributions, specified up to a number of unknown parameters. To compute quantities such as average loss or VaR, the parameters of the distributions have to be estimated. This chapter discusses various methods of estimating the parameters of a failure-time or loss distribution.