Stuart A. Klugman

Fitting bivariate loss distributions with copulas

Abstract Various processes in casualty insurance involve correlated pairs of variables. A prominent example is the loss and allocated loss adjustment expenses on a single claim. In this paper the bivariate copula is introduced and an approach to conducting goodness-of-fit tests is suggested. A large example illustrates the concepts.

Loss models : further topics

An essential resource for constructing and analyzing advanced actuarial models Loss Models: Further Topics presents extended coverage of modeling through the use of tools related to risk theory, loss distributions, and survival models. The book uses these methods to construct and evaluate actuarial models in the fields of insurance and business. Providing an advanced study of actuarial methods, the book features extended discussions of risk modeling and risk measures, including Tail-Value-at-Risk. Loss Models: Further Topics contains additional material to accompany the Fourth Edition of Loss Models: From Data to Decisions, such as: Extreme value distributions Coxian and related distributions Mixed Erlang distributions Computational and analytical methods for aggregate claim models Counting processes Compound distributions with time-dependent claim amounts Copula models Continuous time ruin models Interpolation and smoothing The book is an essential reference for practicing actuaries and actuarial researchers who want to go beyond the material required for actuarial qualification. Loss Models: Further Topics is also an excellent resource for graduate students in the actuarial field.

Credibility Models with Time-Varying Trend Components

Traditional credibility models have treated the process generating the losses as stable over time, perhaps with a deterministic trend imposed. However, there is ample evidence that these processes are not stable over time. What is required is a method that allows for time-varying parameters in the process, yet still provides the shrinkage needed for sound ratemaking. In this paper we use an automobile insurance example to illustrate how this can be accomplished.

Bayesian modelling of mortality catastrophes

Abstract When a mortality catastrophe is at its beginning data is scarce and estimates of its consequences are likely to be unreliable. When performing financial calculations (e.g., reserve setting) it is necessary to account for variability due to both estimation error and the random insured event itself. In this paper a Bayesian approach and an example using AIDS data will be presented.

The Credibility Problem

The problem is estimation of the amount or number of claims to be paid on a particular insurance policy in a future coverage period. This is a random quantity whose ultimate value will be affected by a number of factors: the individual characteristics of the insured, the characteristics of a larger group to which the insured belongs, external factors (mostly economic quantities), and the random nature of the insured event. Recognizing that no amount of information will allow us to exactly predict future claims, we settle for either the probability distribution of this amount or properties of this distribution such as the mean and variance. Of greatest interest is the mean, which (under squared error loss) would be our best guess as to what the future claims might be. For the most part we will ignore the economic variables, or equivalently, assume they are accounted for outside the credibility analysis.

Computational Aspects of Bayesian Analysis

To complete a Bayesian analysis it is often necessary to perform integrations and/or maximizations with respect to functions of many variables. In this Chapter, five approaches will be presented for solving these problems. They all have advantages and disadvantages. Often, but not always, the ones that take the smallest amount of time will be the least accurate. Programs for implementing these procedures are given in the Appendix.

Prediction with Parameter Uncertainty

The model is the same one that introduced the Bayesian paradigm in Chapter 2. Observations have been obtained from a random variable with known general form, but unknown parameters. Of interest is the value of a future observation whose distribution also depends on these parameters. Of course, this is the traditional actuarial problem. The observations are the benefits paid in the past to policyholders and we desire to predict the payments that will be made in the future.

The Hierarchical Bayesian Approach

It is essential at the outset to be clear about what is and what is not a Bayesian approach. In particular, none of the credibility methods being used at this time qualify as true Bayesian analyses. The requirements as introduced in Chapter 2 are few — a prior probability distribution that is determined before the data are collected and a model probability distribution. What we need to do for the credibility problem is identify just where these two items come in.

The Hierarchical Normal Linear Model

In this Chapter one more restriction to the normal model of Chapter 6 will be imposed: linearity in the parameters. Within this model most all standard situations involving severity, pure premiums, or loss ratios can be handled. The only reasonable case that cannot be handled is the Poisson model for frequency. This will be covered in Chapter 9.

Bayesian Statistical Analysis

As discussed in Chapter 1, it is not the intention of this monograph to provide a convincing philosophical justification for the Bayesian approach. Excellent discussions of these matters can be found, for example, in Berger (1985, Chapters 1 and 4) and Lindley (1983). In the latter paper the Bayesian paradigm is described in its simplest form. Of interest is a quantity θ whose value is unknown. What is known is a probability distribution π(θ) that expresses our current relative opinion as to the likelihood that various possible values of θ are the true value. For additional discussion of the merits of expressing uncertainty by probability see Lindley (1982 and 1987). This is called the prior distribution as it represents the state of our knowledge prior to conducting the experiment.