In a risk exchange, participants trade a privately owned risk for a share in a pool. If participants agree on a valuation rule, it can be decided whether or not, according to the given rule, these trades take place at equal value. If equality of values holds for all participants, then the exchange is said to be “financially fair”. It has been shown by Bühlmann and Jewell (1979) that, under mild assumptions, the constraint of financial fairness singles out a unique solution among the set of all Pareto efficient risk exchanges. In this paper, we find that an analogous statement is true if we limit ourselves to linear exchanges. Conditions are provided for existence and uniqueness of linear sharing rules that are both financially fair and Pareto efficient among all linear sharing rules. The performance of the linear rule is compared to that of the general (nonlinear) rule in a number of specific cases.

To use Bayesian analysis to model insurance losses, one usually chooses a parametric conditional loss distribution for each risk and a parametric prior distribution to describe how the conditional distributions vary across the risks. A criticism of this method is that the prior distribution can be difficult to choose and the resulting model may not represent the loss data very well. In this paper, we apply techniques from nonparametric density estimation to estimate the prior. We use the estimated model to calculate the predictive mean of future claims given past claims. We illustrate our method with simulated data from a mixture of a lognormal conditional over a lognormal prior and find that the estimated predictive mean is more accurate than the linear Bühlmann credibility estimator, even when we use a conditional that is not lognormal.

The recursive algorithm of Hesselager (1994) is extended to a more general class of counting distributions, which includes Sundt's (1992) class as well as all the mixed Poisson distributions discussed by Willmot (1993).

The literature on Bayesian chain ladder models is surveyed. Both Mack and cross-classified forms of the chain ladder are considered. Both cases are examined in the context of error terms distributed according to a member of the exponential dispersion family. Tweedie and over-dispersed Poisson errors follow as special cases. Bayesian cross-classified chain ladder models may randomise row, column or diagonal parameters. Column and diagonal randomisation has been largely absent from the literature until recently. The present paper allows randomisation of row and column parameters. The Bayes estimator, the linear Bayes (credibility) estimator, and the MAP estimator are shown to be identical in the Mack case, and in the cross-classified case provided that the error terms are Tweedie distributed. In the Mack case the variance structure is generalised considerably from the existing literature. In the cross-classified case the model structure differs somewhat from the existing literature, and a comparison is made between the two. MAP estimators for the cross-classified case are often given by implicit equations that require numerical solution. Recursive formulas are given for these in the general case of error terms from the exponential dispersion family. The connection between the cross-classified case and Bornhuetter-Ferguson prediction is explored.

We shall define a mutual insurance firm as a firm whose stockholders are the bearers of the insurance contracts issued by the firm. The firm's insurance is then viewed as a collective process of say N persons seeking to protect themselves against claims that may occur to any one of them. For example, large employers protecting their employees by pooling risks and deducting for protection given amounts from salaries may be a case in point. In this latter case, the employer may match withdrawals from employees salaries and provide in the process a fringe benefit and increase employees loyalty to the firm. Alternatively, agricultural collectives have in some cases established mutual insurance firms whose purposes are to protect them, at a cost, from the uncertainty implicit in their production processes and the fluctuations of agricultural markets. Since these firms do not work for profit, contingent payments, or fund reimbursement in case of excess cash holdings are typical control policies which help cover extraordinary claims and at the same time are assumed the best investment policies. To further protect themselves against extraordinary claims, mutual insurers can turn to reinsurance firms, “selling” for example the excess claims of, say, a given amount R (e.g., see Tapiero and Zuckerman (1982)). The purpose of this paper is to consider such a mutual insurance firm facing a jump stochastic claims process, as is often assumed in the insurance literature (e.g., Feller (1971), Borch (1974)). For example, Poisson and Compound Poisson processes are typical jump processes treated in this paper, although other processes could be considered as well (see Srinivasan (1973) and Srinivasan and Mehata (1976), Bensoussan and Tapiero (1982)). First we define the mutual insurer problem as a jump process stochastic control problem which we solve analytically under the assumption of a gamma density claim sizes distribution. A solution is obtained by applying arguments from stochastic dynamic programming and by solving the resultant functional equation by application of Laplace transforms (e.g., Colombo and Lavoine (1972), Miller (1956), Tapiero, Chapter V). Subsequently, the effects of reinsurance are introduced and preliminary results obtained. As in Tapiero and Zuckerman (1982), we assume that the mutual insurance firm is a direct underwriter and that the reinsurer is a leader in a Stackleberg game (Stackleberg (1952), Simman and Cruz (1973), Luce and Raiffa (1957)).

Much attention has been focused recently on the issue of valuing guaranteed minimum death benefits embedded in annuity contracts. These benefits resemble a sequence of put options and their value should obey a differential equation similar to the Black-Scholes equation for simple put options. This paper derives a number of analytic solutions to this equation for a number of simple mortality laws.

The number of people who receive a stable income for life from a closed pooled annuity fund is studied. Income stability is defined as keeping the income within a specified tolerance of the initial income in a fixed proportion of future scenarios. The focus is on quantifying the effect of the number of members, which drives the level of idiosyncratic longevity risk in the fund, on the income stability. To do this, investment returns are held constant, and systematic longevity risk is omitted. An analytical expression that closely approximates the number of fund members who receive a stable income is derived and is seen to be independent of the mortality model. An application of the result is to calculate the length of time for which the pooled annuity fund can provide the desired level of income stability.

The premiums for a bonus-malus system which stays in financial equilibrium over the years are calculated. This is done by minimizing a quadratic function of the difference between the premium for an optimal BMS with an infinite number of classes and the premium for a BMS with a finite number of classes, weighted by the stationary probability of being in a certain class, and by imposing various constraints on the system.

To evaluate a large portfolio of variable annuity (VA) contracts, many insurance companies rely on Monte Carlo simulation, which is computationally intensive. To address this computational challenge, machine learning techniques have been adopted in recent years to estimate the fair market values (FMVs) of a large number of contracts. It is shown that bootstrapped aggregation (bagging), one of the most popular machine learning algorithms, performs well in valuing VA contracts using related attributes. In this article, we highlight the presence of prediction bias of bagging and use the bias-corrected (BC) bagging approach to reduce the bias and thus improve the predictive performance. Experimental results demonstrate the effectiveness of BC bagging as compared with bagging, boosting, and model points in terms of prediction accuracy.

There is a duality between the surplus process of classical risk theory and the single-server queue. It follows that the probability of ruin can be retrieved from a single sample path of the waiting time process of the single-server queue. In this paper, premiums are allowed to vary. It has been shown that the stationary distribution of a corresponding storage process is equal to the survival probability (with variable premiums). Thus by simulation of the corresponding storage process, the probability of ruin can be obtained. The special cases where the surplus earns interest and the premiums are charged by layers are considered and illustrated numerically.

Together with the Chain Ladder (CL) method, the Bornhuetter/Ferguson (BF) method is one of the most popular claims reserving methods. Whereas a formula for the prediction error of the CL method has been published already in 1993, there is still nothing equivalent available for the BF method. On the basis of the BF reserve formula, this paper develops a stochastic model for the BF method. From this model, a formula for the prediction error of the BF reserve estimate is derived.

Moreover, the model gives important advice on how to estimate the parameters for the BF reserve formula. E.g. it turns out that the appropriate BF development pattern is different from the CL pattern. This is a nice add-on as it makes BF to a standalone reserving method which is fully independent from CL. The other parameter required for the BF reserve is the well-known initial estimate for the ultimate claims amount. Here the stochastic model clearly shows what has to be meant with ‘initial’.

In order to apply the formula for the prediction error, the actuary must assess his uncertainty about both sets of parameters, about the development pattern and about the initial ultimate claims estimates. But for both, much guidance can be drawn from the estimates itself and from the run-off data given. Finally, a numerical example shows how the resulting prediction error compares to the one of the CL method.

We consider a classical surplus process where the insurer can choose a different level of reinsurance at the start of each year. We assume the insurer’s objective is to minimise the probability of ruin up to some given time horizon, either in discrete or continuous time. We develop formulae for ruin probabilities under the optimal reinsurance strategy, i.e. the optimal retention each year as the surplus changes and the period until the time horizon shortens. For our compound Poisson process, it is not feasible to evaluate these formulae, and hence determine the optimal strategies, in any but the simplest cases. We show how we can determine the optimal strategies by approximating the (compound Poisson) aggregate claims distributions by translated gamma distributions, and, alternatively, by approximating the compound Poisson process by a translated gamma process.

In the classical definition skewness is departure from symmetry. It was therefore natural to measure skewness by using a normalized third moment μ3/σ3. This condensed measure, however, is not refined enough to be used as an operational instrument for studying various functions which might be used to describe actual claim distributions. This is true especially when the interest is concentrated towards the higher values of the variate.

In their paper (1) Benktander-Segerdahl have suggested that the average excess claim m(x) as a function of the priority x should be used to reveal the characteristics of the tail of the distribution

where P(x) = 1 — H(x) denotes the distribution function.

This statistic is very apt when comparing actual claim distributions with possible theoretical models. It is also useful when classifying these models.

If, however, emphasis mainly is laid on classifying distributions according to their skewness, another statistic might be preferable. Let μ(x)dx denote the probability that a stochastic variable which is known to be at least equal to x, does not exceed x+dx. In other words, μ(x)dx represents the probability that a claim or the corresponding stochastic variable, which, when observed from the bottom, is “alive” at x, “dies” in the interval (x, x + dx). The lower this claims rate of mortality, the skewer and more dangerous is the claim distribution.

A statistical analysis is performed on natural events which can produce important damages to insurers. The analysis is based on hurricanes which have been observed in the United States between 1954 et 1986.

At first, independence between the number and the amount of the losses is examined. Different distributions (Poisson and negative binomial for frequency and exponential, Pareto and lognormal for severity) are tested. Along classical tests as chi-square, Kolmogorov-Smirnov and non parametric tests, a test with weights on the upper tail of the distribution is used: the Anderson – Darling test.

Confidence intervals for the probability of occurrence of a claim and expected frequency for different potential levels of claims are derived. The Poisson Log-normal model gives a very good fit to the data.

Mixed Poisson distributions are widely used in various disciplines including actuarial applications. The family of mixed Poisson distributions contains several members according to the choice of the mixing distribution for the parameter of the Poisson distribution. Very few of them have been studied in depth, mainly because of algebraic intractability. In this paper we will describe an EM type algorithm for maximum likelihood estimation for mixed Poisson distributions. The main achievement is that it reduces the problem of estimation to one of estimation of the mixing distribution which is usually easier. Variants of the algorithm work even when the probability function of the mixed distribution is not known explicitly but we have only an approximation of it. Other discrete distributions are treated as well.

In this article, we study the problem of optimal index insurance design under an expected utility maximization framework. For general utility functions, we formally prove the existence and uniqueness of optimal contract and develop an effective numerical procedure to derive the optimal solution. For exponential utility and quadratic utility functions, we obtain analytical expression of the optimal indemnity function. Our results show that the indemnity can be a highly nonlinear and even non-monotonic function of the index variable in order to align with the actual loss variable so as to achieve the best reduction in basis risk. Due to the generality of model setup, our proposed method is readily applicable to a variety of insurance applications including index-linked mortality securities, weather index agriculture insurance, and index-based catastrophe insurance. Our method is illustrated by numerical examples where weather index insurance is designed for protection against the adverse rice yield using temperature and precipitation as the underlying indices. Numerical results show that our optimal index insurance significantly outperforms linear-type index insurance contracts in terms of basis risk reduction.

Much attention has been paid in the last ten years to the necessary solvency margin especially as influenced by risk theory considerations. In these calculations, by which the solvency margin is determined in such a way that the probability of ruin remains under a specified norm, such factors as type of insurance, size of portfolio, reinsurance etc. play an essential role.

In these calculations, based on pure risk theory, there is thus no attention paid to risks in the investment field, in the area of costs, etc. Yet one must not forget that the solvency margin must also act as a buffer against these uncertainties, in other words the solvency margin must be large enough to cover all risks to which the concern is liable, within stipulated limits of certainty.

One drawback of a pure risk theoretical approach, as interpreted above, is that the bases of the risk theory are often difficult to translate into practical terms, and furthermore will differ from company to company. The latter is also the case with the size of portfolio, reinsurance, etc. However, if this could all be translated into practical terms in the right way, the results would upset relations between competitors, so that it seems natural to make the required solvency margin the same for all companies.

Seen in this light Campagne's approach is of great importance—already in 1948 he realized that a scientific basis for the size of the solvency margin was necessary. He confined himself to the life insurance industry, and based his calculations on the idea that the total results of the companies working in a country should be such that, ceteris paribus, the chance of bankruptcy remains below a stipulated norm.

The statistical theory of extreme values well described by Gumbel [1] has been fruitfully applied in many fields, but only in recent times has it been suggested in connection with fire insurance problems. The idea originally stemmed from a consideration of the ECOMOR reinsurance treaty proposed by Thepaut [2]. Thereafter, a few papers appeared investigating the usefulness of the theory in the calculation of an excess of loss premium. Among these, Beard [3, 4], d'Hooge [5] and Jung [6] have made contributions which are worth studying. They have considered, however, only the largest claims during a succession of periods. In this paper, generalized techniques are presented which enable use to be made of all large losses that are available for analysis and not merely the largest. These methods would be particularly useful in situations where data are available only for large losses.

In the present paper we present two credibility regression models for the classification of passenger cars. As regressors we use technical variables like price, weight, etc. In both models we derive credibility estimators and find expressions for their estimation errors. Estimators for structure parameters are proposed. A numerical example based on real data is given. The second model is hierarchical with a level for make of car.

We present a methodology to forecast mortality rates and estimate longevity and mortality risks. The methodology uses generalized dynamic factor models fitted to the differences in the log-mortality rates. We compare their prediction performance with that of models previously described in the literature, including the traditional static factor model fitted to log-mortality rates. We also construct risk measures using vine-copula simulations, which take into account the dependence between the idiosyncratic components of the mortality rates. The methodology is applied to forecast mortality rates for a population portfolio for the UK and to estimate longevity and mortality risks.