Encyclopedia Of Actuarial Science

Aggregate Loss Modeling One of the primary goals of actuarial risk theory is the evaluation of the risk associated with a portfolio of insurance contracts over the life of the contracts. Many insurance contracts (in both life and non-life areas) are short-term. Typically, automobile insurance, homeowner’s insurance, group life and health insurance policies are of one-year duration. One of the primary objectives of risk theory is to model the distribution of total claim costs for portfolios of policies, so that business decisions can be made regarding various aspects of the insurance contracts. The total claim cost over a fixed time period is often modeled by considering the frequency of claims and the sizes of the individual claims separately. Let X1 , X2 , X3 , . . . be independent and identically distributed random variables with common distribution function FX (x). Let N denote the number of claims occurring in a fixed time period. Assume that the distribution of each Xi , i = 1, . . . , N , is independent of N for fixed N. Then the total claim cost for the fixed time period can be written as S = X1 + X2 + · · · + XN , with distribution function FS (x) = ∞
pn FX∗n (x), (1) n=0 where FX∗n (·) indicates the n-fold convolution of FX (·). The distribution of the random sum given by equation (1) is the direct quantity of interest to actuaries for the development of premium rates and safety margins. In general, the insurer has historical data on the number of events (insurance claims) per unit time period (typically one year) for a specific risk, such as a given driver/car combination (e.g. a 21-year-old male insured in a Porsche 911). Analysis of this data generally reveals very minor changes over time. The insurer also gathers data on the severity of losses per event (the X ’s in equation (1)). The severity varies over time as a result of changing costs of automobile repairs, hospital costs, and other costs associated with losses. Data is gathered for the entire insurance industry in many countries. This data can be used to develop models for both the number of claims per time period and the number of claims per insured. These models can then be used to compute the distribution of aggregate losses given by equation (1) for a portfolio of insurance risks. It should be noted that the analysis of claim numbers depends, in part, on what is considered to be a claim. Since many insurance policies have deductibles, which means that small losses to the insured are paid entirely by the insured and result in no payment by the insurer, the term ‘claim’ usually only refers to those events that result in a payment by the insurer. The computation of the aggregate claim distribution can be rather complicated. Equation (1) indicates that a direct approach requires calculating the n-fold convolutions. As an alternative, simulation and numerous approximate methods have been developed. Approximate distributions based on the first few lower moments is one approach; for example, ga