This paper examines a class of premium functionals which are (i) comonotonic additive and (ii) stochastic dominance preservative. The representation for this class is a transformation of the decumulative distribution function. It has close connections with the recent developments in economic decision theory and non-additive measure theory. Among a few elementary members of this class, the proportional hazard transform seems to stand out as being most plausible for actuaries.

Compound distributions such as the compound Poisson and the compound negative binomial are used extensively in the theory of risk to model the distribution of the total claims incurred in a fixed period of time. The usual method of evaluating the distribution function requires the computation of many convolutions of the conditional distribution of the amount of a claim given that a claim has occurred. When the expected number of claims is large, the computation can become unwieldy even with modern large scale electronic computers.

In this paper, a recursive definition of the distribution of total claims is developed for a family of claim number distributions and arbitrary claim amount distributions. When the claim amount is discrete, the recursive definition 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 sufficiently large portfolios.

Results for some specific distributions have been previously obtained using generating functions and Laplace transforms (see Panjer (1980) including discussion). The simple algebraic proof of this paper yields all the previous results as special cases.

Good estimates for the tails of loss severity distributions are essential for pricing or positioning high-excess loss layers in reinsurance. We describe parametric curve-fitting methods for modelling extreme historical losses. These methods revolve around the generalized Pareto distribution and are supported by extreme value theory. We summarize relevant theoretical results and provide an extensive example of their application to Danish data on large fire insurance losses.

A distribution-free formula for the standard error of chain ladder reserve estimates is derived and compared to the results of some parametric methods using a numerical example.

Classical statistics deals with the following standard problem of estimation:

Given: random variables X1, X2 … Xn independent, identically distributed, and

observations x1, X2 … xn,

Estimate: parameter (or function thereof) of the distribution function common to all Xi.

It is not surprising that the “classical actuary” has mostly been involved in solving the actuarial equivalent of this problem in insurance, namely

Given: risks R1R2 … Rn no contagion, homogeneous group,

Find: the proper (common) rate for all risks in the given class.

There have, of course, always been actuaries who have questioned the assumptions of independence (no contagion) and/or identical distribution (homogeneity). As long as ratemaking is considered equivalent to the determination of the mean, there seem to be no additional difficulties if the hypothesis of independence is dropped. But is there a way to drop the condition of homogeneity (identical distribution)?

The authors review various facts about copulas linking discrete distributions. They show how the possibility of ties that results from atoms in the probability distribution invalidates various familiar relations that lie at the root of copula theory in the continuous case. They highlight some of the dangers and limitations of an undiscriminating transposition of modeling and inference practices from the continuous setting into the discrete one.

(a) The notion of premium calculation principle has become fairly generally accepted in the risk theory literature. For completeness we repeat its definition:

A premium calculation principle is a functional assigning to a random variable X (or its distribution function Fx(x)) a real number P. In symbols

The interpretation is rather obvious. The random variable X stands for the possible claims of a risk whereas P is the premium charged for assuming this risk.

This is of course formalizing the way actuaries think about premiums. In actuarial terms, the premium is a property of the risk (and nothing else), e.g.

(b) Of course, in economics premiums are not only depending on the risk but also on market conditions. Let us assume for a moment that we can describe the risk by a random variable X (as under a)), describe the market conditions by a random variable Z.

Then we want to show how an economic premium principle

can be constructed. During the development of the paper we will also give a clear meaning to the random variable Z:

In the market we are considering agents i = 1, 2, …, n. They constitute buyers of insurance, insurance companies, reinsurance companies.

Each agent i is characterized by his

utility function ui(x) [as usual: ]

initial wealth wi.

In this section, the risk aspect is modelled by a finite (for simplicity) probability space with states s = 1, 2, …, S and probabilities πs of state s happening.

This paper presents a universal framework for pricing financial and insurance risks. Examples are given for pricing contingent payoffs, where the underlying asset or liability can be either traded or not traded. The paper also outlines an application of the framework to prescribe capital allocations within insurance companies, and to determine fair values of insurance liabilities.

The correlation order, which is defined as a partial order between bivariate distributions with equal marginals, is shown to be a helpfull tool for deriving results concerning the riskiness of portfolios with pairwise dependencies. Given the distribution functions of the individual risks, it is investigated how changing the dependency assumption influences the stop-loss premiums of such portfolios.

It is now widely accepted that stochastic mortality – the risk that aggregate mortality might differ from that anticipated – is an important risk factor in both life insurance and pensions. As such it affects how fair values, premium rates, and risk reserves are calculated.

This paper makes use of the similarities between the force of mortality and interest rates to examine how we might model mortality risks and price mortality-related instruments using adaptations of the arbitrage-free pricing frameworks that have been developed for interest-rate derivatives. In so doing, the paper pulls together a range of arbitrage-free (or risk-neutral) frameworks for pricing and hedging mortality risk that allow for both interest and mortality factors to be stochastic. The different frameworks that we describe – short-rate models, forward-mortality models, positive-mortality models and mortality market models – are all based on positive-interest-rate modelling frameworks since the force of mortality can be treated in a similar way to the short-term risk-free rate of interest. While much of this paper is a review of the possible frameworks, the key new development is the introduction of mortality market models equivalent to the LIBOR and swap market models in the interest-rate literature.

These frameworks can be applied to a great variety of mortality-related instruments, from vanilla longevity bonds to exotic mortality derivatives.

A recent result by Panjer provides a recursive algorithm for the compound distribution of aggregate claims when the counting law belongs to a special recursive family. In the present paper we first give a characterization of this recursive family, then describe some generalizations of Panjer's result.

In the usual model of the collective risk theory, we are interested in the severity of ruin, as well as its probability. As a quantitative measure, we propose G(u, y), the probability that for given initial surplus u ruin will occur and that the deficit at the time of ruin will be less than y, and the corresponding density g(u, y). First a general answer in terms of the transform is obtained. Then, assuming that the claim amount distribution is a combination of exponential distributions, we determine g; here the roots of the equation that defines the adjustment coefficient play a central role. An explicit answer is also given in the case in which all claims are of constant size.

We reconsider the problem of producing fair and accurate tariffs based on aggregated insurance data giving numbers of claims and total costs for the claims. Jørgensen and de Souza (Scand Actuarial J., 1994) assumed Poisson arrival of claims and gamma distributed costs for individual claims. Jørgensen and de Souza (1994) directly modelled the risk or expected cost of claims per insured unit, μ say. They observed that the dependence of the likelihood function on μ is as for a linear exponential family, so that modelling similar to that of generalized linear models is possible. In this paper we observe that, when modelling the cost of insurance claims, it is generally necessary to model the dispersion of the costs as well as their mean. In order to model the dispersion we use the framework of double generalized linear models. Modelling the dispersion increases the precision of the estimated tariffs. The use of double generalized linear models also allows us to handle the case where only the total cost of claims and not the number of claims has been recorded.

We propose practical solutions for the determination of optimal retentions in a stop-loss reinsurance. We develop two new optimization criteria for deriving the optimal retentions by, respectively, minimizing the value-at-risk (VaR) and the conditional tail expectation (CTE) of the total risks of an insurer. We establish necessary and sufficient conditions for the existence of the optimal retentions for two risk models: individual risk model and collective risk model. The resulting optimal solution of our optimization criterion has several important characteristics: (i) the optimal retention has a very simple analytic form; (ii) the optimal retention depends only on the assumed loss distribution and the reinsurer’s safety loading factor; (iii) the CTE criterion is more applicable than the VaR criterion in the sense that the optimal condition for the former is less restrictive than the latter; (iv) if optimal solutions exist, then both VaR- and CTE-based optimization criteria yield the same optimal retentions. In terms of applications, we extend the results to the individual risk models with dependent risks and use multivariate phase type distribution, multivariate Pareto distribution and multivariate Bernoulli distribution to illustrate the effect of dependence on optimal retentions. We also use the compound Poisson distribution and the compound negative binomial distribution to illustrate the optimal retentions in a collective risk model.

The compound binomial model is a discrete time analogue (or approximation) of the compound Poisson model of classical risk theory. In this paper, several results are derived for the probability of ruin as well as for the joint distribution of the surpluses immediately before and at ruin. The starting point of the probabilistic arguments are two series of random variables with a surprisingly simple expectation (Theorem 1) and a more classical result of the theory of random walks (Theorem 2) that is best proved by a martingale argument.

We give an extension of the Economic Premium Principle treated in Astin Bulletin, Volume 11 where only exponential utility functions were admitted. The case of arbitrary risk averse utility functions leads to similar quantitative results. The role of risk aversion in the treatment is essential. It also permits an easy proof for the existence of equilibrium.

We consider a situation originally discussed by De Finetti (1957) in which a surplus process is modified by the introduction of a constant dividend barrier. We extend some known results relating to the distribution of the present value of dividend payments until ruin in the classical risk model and show how a discrete time risk model can be used to provide approximations when analytic results are unavailable. We extend the analysis by allowing the process to continue after ruin.

This paper discusses the modelling and control of pension funds.

A continuous-time stochastic pension fund model is proposed in which there are n risky assets plus the risk-free asset as well as randomness in the level of benefit outgo. We consider Markov control strategies which optimise over the contribution rate and over the range of possible asset-allocation strategies.

For a general (not necessarily quadratic) loss function it is shown that the optimal proportions of the fund invested in each of the risky assets remain constant relative to one another. Furthermore, the asset allocation strategy always lies on the capital market line familiar from modern portfolio theory.

A general quadratic loss function is proposed which provides an explicit solution for the optimal contribution and asset-allocation strategies. It is noted that these solutions are not dependent on the level of uncertainty in the level of benefit outgo, suggesting that small schemes should operate in the same way as large ones. The optimal asset-allocation strategy, however, is found to be counterintuitive leading to some discussion of the form of the loss function. Power and exponential loss functions are then investigated and related problems discussed.

The stationary distribution of the process is considered and optimal strategies compared with dynamic control strategies.

Finally there is some discussion of the effects of constraints on contribution and asset-allocation strategies.

The objective of this paper is to provide an extension of well-known models of tarification in automobile insurance. The analysis begins by introducing a regression component in the Poisson model in order to use all available information in the estimation of the distribution. In a second step, a random variable is included in the regression component of the Poisson model and a negative binomial model with a regression component is derived. We then present our main contribution by proposing a bonus-malus system which integrates a priori and a posteriori information on an individual basis. We show how net premium tables can be derived from the model. Examples of tables are presented.

In this paper, we reexamine the two optimal reinsurance problems studied in Cai et al. (2008), in which the objectives are to find the optimal reinsurance contracts that minimize the value-at-risk (VaR) and the conditional tail expectation (CTE) of the total risk exposure under the expectation premium principle. We provide a simpler and more transparent approach to solve these problems by using intuitive geometric arguments. The usefulness of this approach is further demonstrated by solving the VaR-minimization problem when the expectation premium principle is replaced by Wang's premium principle.