We consider an alternative to the usual credibility premium that arises from squared-error loss, namely, a so-called equitable credibility premium (Promislow and Young, 1999). We derive formulas for the credibility weight in certain cases and give sufficient conditions for exact credibility.

Should the pricing of reinsurance catastrophes be related to the price of the default risk embedded in corporate bonds?

If not, why not?

A risk is a risk is a risk, in whatever market it appears. Shouldn't the risk-prices in these different markets be comparable? More basically perhaps, how should reinsurance prices and bond prices be set? How does the market currently set them? These questions are central to the inquiry contained in this paper.

Avoiding unnecessary suspense, our answers are: Yes, cat prices should be related to credit prices because both risks contain a characteristic trade-off between the frequency of and severity of adverse events. We leave the question of how prices should be set to others and focus on the empirical question of how they have been set by the markets. In the process, we develop a fairly robust pricing mechanism and explore its potential uses in many different contexts.

The 1999 Insurance-Linked Securities (ILS) market (a.k.a., Cat Bond market) provides the empirical springboard to the discussion. The ILS market is only 4 years old. As such, it represents a new and unique intersection of reinsurance and financial markets. It provides a wonderful laboratory for exploring risk-pricing.

The ILS market, still in its experimental phase, appears to require more generous (cheap) pricing of insurance risk than does the bond market of default risk. So much so that academics have begun to weigh in on the question of why. Previously, insurance pricing discussions had been confined to practicing insurance professionals, particularly actuaries. For finance professionals, insurance pricing, much less reinsurance pricing, seldom made the index of their financial texts – though even that is beginning to change.

Nielsen (1999) showed the surprising fact that a nonparametric one-dimensional hazard as a function of time can be estimated -consistently if a high quality marker is observed. In this paper we show that the hazard relevant for predicting remaining duration time, given the current status of a high quality marker, can be estimated -consistently if a Markov type property holds for the high quality marker.

Young (1999) discussed the conjecture proposed by Christofides (1998) regarding the premium principle of Wang (1995, 1996). She shows that this conjecture is true for location-scale families and for certain other families, but false in general. In addition Young (1999) states that it remains an open problem to determine under what circumstances Wang's premium principle reduces to the standard deviation (SD) premium principle.

In this paper we will provide further discussion of this problem. We will show that, for a fixed distortion, the natural set on which Wang's premium principle can reduce to the SD premium principle is and only is the union of location-scale families which satisfies some condition. Furthermore, it will be shown that the natural set is and only is a location-scale family if Wang's premium principle can be reduced to the SD premium principle for any distortion.

In this note we give a multivariate extension of the proof of Ospina & Gerber (1987) of the result of Feller (1968) that a univariate distribution on the non-negative integers is infinitely divisible if and only if it can be expressed as a compound Poisson distribution.

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 standard methods for the calculation of total claim size distributions and ruin probabilities, Panjer recursion and algorithms based on transforms, both apply to lattice-type distributions only and therefore require an initial discretization step if continuous distribution functions are of interest. We discuss the associated discretization error and show that it can often be reduced substantially by an extrapolation technique.

This paper proposes a multivariate generalization of the generalized Poisson distribution. Its definition and main properties are given. The parameters are estimated by the method of moments.

Alzheimer's disease (AD) accounts for a significant proportion of long-term care costs. The recent discovery that the ε4 allele of the ApoE gene indicates a predisposition to earlier onset of AD raises questions about the potential for adverse selection in long-term care insurance, about long-term care costs in general, and about the potential effects on costs of gene therapy, or better targetted treatments for AD. This paper describes a simple Markov model for AD, and the estimation of the transition intensities from the medical and epidemiological literature.

A claims reserving method is reviewed which was introduced by Gunnar Benktander in 1976. It is a very intuitive credibility mixture of Bornhuetter/Ferguson and Chain Ladder. In this paper, the mean squared errors of all 3 methods are calculated and compared on the basis of a very simple stochastic model. The Benktander method is found to have almost always a smaller mean squared error than the other two methods and to be almost as precise as an exact Bayesian procedure.

In this paper we discuss the concept of excess of loss reinsurance with reinstatements. The main objective is to provide a methodology to calculate the distribution of total aggregate losses for two or more consecutive layers when there is a limited number of reinstatements. We also compare different premium principles and their properties to price these treaties for any number of free or paid reinstatements.

In the present paper we extend a recursive algorithm developed by Vernic (1999) for compound distributions with bivariate counting distribution and univariate severity distributions to more general multivariate counting distributions.

We extend the Cox-Ingersoll-Ross (1985) model of the short interest rate by assuming a stochastic reversion level, which better reflects the time dependence caused by the cyclical nature of the economy or by expectations concerning the future impact of monetary policies. In this framework, we have studied the convergence of the long-term return by using the theory of generalised Bessel-square processes. We emphasize the applications of the convergence results. A limit theorem proves evidence of the use of a Brownian motion with drift instead of the integral . For practice, however, this approximation turns out to be only appropriate when there are no explicit formulae and calculations are very time-consuming.

We investigate the influence of initial selection (the impact of underwriting during the early years of a policy's life) on individual Permanent Health Insurance claim inceptions. In Gutiérrez-Delgado (1999) a decreasing trend was found. In this paper we include the effect of cause of disability and fit a generalized linear model in order to gain a greater understanding of the phenomenon. Both effects, policy duration and cause of disability, are found to have a significant effect on the number of claims. We describe their influence using factors that collect the information available through the fitted model. Results from both factors suggest that the grouping of diseases selected for the research helps to explain partially our earlier results. In addition there is some evidence of moral hazard in mental disorders and musculoskeletal diseases which also contributes to the understanding of the negative trend found.

In this paper we study some bivariate counting distributions that are obtained by the trivariate reduction method. We work with Poisson compound distributions and we use their good properties in order to derive recursive algorithms for the bivariate distribution and bivariate aggregate claims distribution. A data set is also fitted.

We apply Lemaire's algorithm and a non-parametric mixed Poisson fit to a motor insurance portfolio in order to find the true claim frequency and claim amount distributions. The algorithm we develop accounts for the fact that observed distributions are distorted by bonus hunger, when a bonus-malus system is used by the insurer.

Credibility weighting is helpful in many insurance applications where sparse data crave information from other sources of data. In this paper we aim at estimating a hazard curve using the nonparametric kernel method, where a credibility weighting principle is used locally, so that areas of sparse data for one subgroup can be alleviated by available information from other subgroups. The credibility estimator is found through a Hilbert space projection formulation of Buhlmann-Straub's credibility approach.

This paper explains securitization of insurance risk by describing its essential components and its economic rationale. We use examples and describe recent securitization transactions. We explore the key ideas without abstract mathematics. Insurance-based securitizations improve opportunities for all investors. Relative to traditional reinsurance, securitizations provide larger amounts of coverage and more innovative contract terms.