The Probability Distribution of Fire Loss Amount

Abstract

Theoretical distributions frequently used to model fire loss amount are discussed. The problem of selecting models solely on the basis of statistics is addressed. Use of probabilistic arguments applied in Reliability theory to infer the type of probability distribution, is explored. The concept of failure rate of a fire is discussed and used to explore implications of the Pareto and Lognormal models as to the fire growth phenomenon. It is concluded that probabilistic arguments, regarding the nature of the fire growth process can aid analysts in their choice of an appropriate model for the probability distribution of fire loss amount. It is a basic assumption in all actuarial research and risk theory studies that there is a probability distribution of loss amount underlying the risk process. In other words, if a loss occurs, there is the probability S(x) that the loss will be for an amount less than or equal to x. In theoretical studies this distribution often is presented as continuous, having a derivative S'(x) s(x), which is called the probability density function of fire loss amount.' At a certain point in time, the results of the theoretical work have to be applied to practical situations. For example, the distribution of actual losses experienced by an insurer is then considered as a sample from an underlying model without defining the corresponding distribution, the characteristics of which are taken to agree with the corresponding statistics of the observed distribution. Many results can be obtained simply by using these sample statistics. However, it is often more desirable to work with analytically defined loss distributions, and the statistics are then used to establish suitable values of the parameters involved in the analytical distributions. When working with these analytical distributions, the researcher must make use of properties of the distributions other than those covered by the statistics observed. There are two main aspects of general insurance in which a knowledge David Shpilberg, Ph.D., is Associate Professor of Operations Research and Insurance at the Instituto de Estudios Superiores de Administraci6n (IESA), a Graduate School of Management in Caracas, Venezuela. The research for this paper was partly financed by a grant of the Factory Mutual Research Corporation. The paper was presented at the 1976 annual meeting of The American Risk and Insurance Association. Dr. Shpilberg received the 1975 Journal of Risk and Insurance Award for the best paper published in the 1975 issues. ( 103) This content downloaded from 157.55.39.163 on Wed, 21 Sep 2016 05:15:04 UTC All use subject to http://about.jstor.org/terms 104 The Journal of Risk and Insurance of the structure of the elements of risk variation is needed:2 first, in the rate making process; and second, in dealing with the question of financial stability (monetary risk). Traditional methods of rate making are based only on an estimate of the mean expected loss. Financial stability studies (e.g., studies addressing the probability of ruin of an insurer or evaluating the risk of unbearable monetary loss for a corporation which chooses not to insure its property) usually are based on an estimate of the variance of the possible loss. However, in the area of industrial fire losses, the probability distributions involved are markedly skewed in character (very small probabilities of a very large loss). Knowledge of its higher moments (in essence, the shape of the tail of the distribution) becomes essential if meaningful quantitative estimates of risk are to be made. Most often, this step involves assumptions regarding the behavior of losses larger than those observed in the sample of available loss experience. Thus, unless there is some theoretical support (not merely observed statistics) for an inference that a particular type of probability distribution is a more reasonable model for the distribution of fire loss amounts as a function of size, inferences derived for any region of the distribution outside the available data will be no better than a straight extrapolation on the data. This paper presents a sumary review of work in the area of modeling the probability distribution of fire loss amount, and attempts to illustrate how probabilistic arguments relating to the physical nature of the phenomenon (an approach extensively used in life testing of material failure and in reliability analysis of systems' components) can effectively aid in the choice of an appropriate model. Fire Loss As a Stochastic Process The total amount of fire losses in a given period can be modeled as a risk process characterized by two stochastic variables: the number of fires and the amount of the losses. If, Pr (t) Probability of r losses in the observed period, t S(x) =Probability that, given a fire loss, its amount is ? x S*r(x) rth convolution of the distribution function of fire loss amount, S (x), then the probability (see Figure 1) that total loss in a period of length t, is -? x can be expressed as

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