A practical method is developed for computing moments of insurance functions when interest rates are assumed to follow an autoregressive integrated moving average process.

In an earlier paper the author derived a recursion formula which permits the exact computation of the aggregate claims distribution in the individual life model. To save computing time he also proposed an approximative procedure based on the exact recursion.

In the present contribution the exact recursion formula and the related approximations are generalized to the individual risk theory model with arbitrary positive claims. Error bounds for the approximations are given and it is shown that they are smaller than those of the Kornya-type approximations.

We present some relatively simple structural ideas about how probabilistic modeling, and in particular, the modern theory of point processes and martingales, can be used in the estimation of claims reserves.

This paper presents applications of stochastic control theory in determining an insurer's optimal reinsurance and rating policy. Optimality is defined by means of variances of such variables as underwriting result of the insurer, solvency margins of the insurer and reinsurer and the premiums paid by policy-holders.

An IBNYR event is one that occurs randomly during some fixed exposure interval and incurs a random delay before it is reported. Both the rate at which such events occur and the parameters of the delay distribution are unknown random quantities. Given the number of events that have been reported during some observation interval, plus various secondary data on the dates of the events, the problem is to estimate the true values of the unknown parameters and to predict the number of events that are still unreported. A full-distributional Bayesian model is used, and it is shown that the amount of secondary data is critical. A recursive procedure calculates the predictive density; however, an explicit formula for the predictive mode can be obtained. The main computational work is the evaluation of an integral involving the prior density of the delay parameters, but this can be simplified in the exponential case using Gammoid approximations.

For the infinite time ruin probability in the classical risk process, efficient estimators are proposed in cases in which the claim amount distribution is unknown. Confidence intervals are computed which are based on normal approximations or on the bootstrap method. The procedures are checked in a Monte-Carlo study.

This article summarizes some main results in modern portfolio theory. First, the Markowitz approach is presented. Then the capital asset pricing model is derived and its empirical testability is discussed. Afterwards Neumann–Morgenstern utility theory is applied to the portfolio problem. Finally, it is shown how optimal risk allocation in an economy may lead to portfolio insurance.

This paper derives several formulas for the probability of eventual ruin in a discrete-time model. In this model, the number of claims process is assumed to be binomial. The claim amounts, premium rate and initial surplus are assumed to be integer-valued.

We consider a risk generating claims for a period of N consecutive years (after which it expires), N being an integer valued random variable. Let Xk denote the total claims generated in the kth year, k ≥ 1. The Xk's are assumed to be independent and identically distributed random variables, and are paid at the end of the year. The aggregate discounted claims generated by the risk until it expires is defined as where υ is the discount factor. An integral equation similar to that given by Panjer (1981) is developed for the pdf of SN(υ). This is accomplished by assuming that N belongs to a new class of discrete distributions called annuity distributions. The probabilities in annuity distributions satisfy the following recursion:

where an is the present value of an n-year immediate annuity.

It is widely acknowledge that there has been a major breakthrough in the mathematical theory of option trading. This breakthrough, which is usually summarized by the Black–Scholes formula, has generated a lot of excitement and a certain mystique. On the mathematical side, it involves advanced probabilistic techniques from martingale theory and stochastic calculus which are accessible only to a small group of experts with a high degree of mathematical sophistication; hence the mystique. In its practical implications it offers exciting prospects. Its promise is that, by a suitable choice of a trading strategy, the risk involved in handling an option can be eliminated completely.

Since October 1987, the mood has become more sober. But there are also mathematical reasons which suggest that expectations should be lowered. This will be the main point of the present expository account. We argue that, typically, the risk involved in handling an option has an irreducible intrinsic part. This intrinsic risk may be much smaller than the a priori risk, but in general one should not expect it to vanish completely. In this more sober perspective, the mathematical technique behind the Black–Scholes formula does not lose any of its importance. In fact, it should be seen as a sequential regression scheme whose purpose is to reduce the a priori risk to its intrinsic core.

We begin with a short introduction to the Black–Scholes formula in terms of currency options. Then we develop a general regression scheme in discrete time, first in an elementary two-period model, and then in a multiperiod model which involves martingale considerations and sets the stage for extensions to continuous time. Our method is based on the interpretation and extension of the Black–Scholes formula in terms of martingale theory. This was initiated by Kreps and Harrison; see, e.g. the excellent survey of Harrison and Pliska (1981,1983). The idea of embedding the Black-Scholes approach into a sequential regression scheme goes back to joint work of the first author with D. Sondermann. In continuous time and under martingale assumptions, this was worked out in Schweizer (1984) and Föllmer and Sondermann (1986). Schweizer (1988) deals with these problems in a general semimartingale model.