Harry H. Panjer

Recursive Evaluation of a Family of Compound Distributions

Compound dlstributmns such as the compound Pmsson and the compound negative binomial are used extensively m the theory of risk to model the distributmn off the total claims incurred m a fixed period of time The usual method of evaluating the dlqtributmn functmn requires the computatmn of many convolutions of the conditional d~atnbutmn of the amount of a claim given that a clmm has occurred When the expected number of claims is large, the computatmn can become unwmldy even with modern large scale electronic computers In tlus paper, a recurs|xe definitmn of the distribution of total clmms is developed for a family of claml numbel distnbutmns and arbitrary claim amount distributions When the clam1 amount is discrete, the recursive dehnitmn can be used to compute the distribution of total claims without the use of convolutions. This can reduce the number of required computations by several orders of magnitude for sufhcmntlv large portfolios Results for some spemfic dlatnbutmna have been prevmusly obtained using generating functions and Laplace transforms (see PANJER (1980) including dlscussmn). The simple algebraic proof of this paper yields all the previous results as special case~

Axiomatic characterization of insurance prices

Abstract In this paper, we take an axiomatic approach to characterize insurance prices in a competitive market setting. We present four axioms to describe the behavior of market insurance prices. From these axioms it follows that the price of an insurance risk has a Choquet integral representation with respect to a distorted probability (Yaari, 1987). We propose an additional axiom for reducing compound risks. This axiom determines that the distortion function is a power function.

Stochastic Modelling of Interest Rates with Applications to Life Contingencies. Part II

This paper extends the results of a previous paper [4], by conditioning the stochastic process of interest rates on current and past values. Conditional autoregressive interest rate models are developed and applied to interest, insurance and annuity functions. Numerical results are also given.

On the Stability of Recursive Formulas

Based on recurrence equation theory and relative error (rather than absolute error) analysis, the concept and criterion for the stability of a recurrence equation are clarified. A family of recursions, called congruent recursions, is proved to be strongly stable in evaluating its non-negative solutions. A type of strongly unstable recursion is identified. The recursive formula discussed by PANJER (1981) is proved to be strongly stable in evaluating the compound Poisson and the compound Negative Binomial (including Geometric) distributions. For the compound Binomial distribution, the recursion is shown to be unstable. A simple method to cope with this instability is proposed. Many other recursions are reviewed. Illustrative numerical examples are given.

Direct Calculation of Ruin Probabilities

This paper gives a simple recursive method for calculating ultimate ruin probabilities. The method is especially easy to apply in practical situations of discrete claim size distributions for which a numerical illustration is given.

Difference equation approaches in evaluation of compound distributions

Abstract Panjer (1981), Sundt and Jewell (1981), and Panjer and Willmot (1982) derive computational formulae for the density of the compound distribution when the frequency distribution satisfies certain difference equations. In many instances restrictions are placed on the severity distribution. In this paper it is shown how the results may be adapted to a wider class of severity distributions, including the inverse Gaussian and the Pareto distribution (among others). The computational techniques are clarified and in some cases simplified, and an additional class of frequency distributions is considered, which contains some other well-known distributions. It is then shown how the results may be extended to some well-known contagious frequency distributions such as the Neyman class.

Computational aspects of recursive evaluation of compound distributions

Abstract This paper addresses some computational problems that may arise when using recursive methods for evaluating the distribution of aggregate claims for portfolios with a large number of expected claims. Some methods for overcoming the problem are suggested and compared.

Practical aspects of stop-loss calculations

Abstract This paper examines various methods of ‘arithmetizing’ the claim size distribution so that stop-loss premiums can be recursively calculated. Claim frequencies are assumed to be Poisson. A decision strategy for choosing a method when both error and computer costs are constrained is developed.

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.

Recursions for Compound Distributions

Various methods for developing recursive formulae for compound distributions have been reported recently by Panjer (1980, including discussion), Panjer (1981), Sundt and Jewell (1981) and Gerber (1982) for a class of claim frequency distributions and arbitrary claim amount distributions. The recursions are particularly useful for computational purposes since the number of calculations required to obtain the distribution function of total claims and related values such as net stop-loss premiums may be greatly reduced when compared with the usual method based on convolutions. In this paper a broader class of claims frequency distributions is considered and methods for developing recursions for the corresponding compound distributions are examined. The methods make use of the Laplace transform of the density of the compound distribution. Consider the class of claim frequency distributions which has the property that successive probabilities may be written as the ratio of two polynomials. For convenience we write the polynomials in terms of descending factorial powers. For obvious reasons, only distributions on the non-negative integers are considered.