Christofides (1998) studies the proportional hazards (PH) transform of Wang (1995) and shows that for some parametric families, the PH premium principle reduces to the standard deviation (SD) premium principle. Christofides conjectures that for a parametric family of distributions with constant skewness, the PH premium principle reduces to the SD principle. I will show that this conjecture is false in general but that it is true for location-scale families and for certain other families.

Wang's premium principle has been established as a sound measure of risk in Wang (1995, 1996), Wang, Young, and Panjer (1997), and Wang and Young (1998). Determining when the SD premium principle is a special case of Wang's premium principle is important because it will help identify circumstances under which the more easily applied SD premium principle is a reliable measure of risk.

First, recall that a distortion g is a non-decreasing function from [0, 1] onto itself. Wang's premium principle, with a fixed distortion g, associates the following certainty equivalent with a random variable X, (Wang, 1996) and (Denneberg, 1994):

in which Sx is the decumulative distribution function (ddf) of X, Sx(t) = Pr(X > t), t ∈ R. If g is a power distortion, g(p) = pc, then Hg is the proportional hazards (PH) premium principle (Wang, 1995).

Second, recall that a location-scale family of ddfs is , in which Sz is a fixed ddf. Alternatively, if Z has ddf Sz, then {X = μ + σZ: μ∈ R, σ > 0} forms a location-scale family of random variables, and the ddf of . Examples of location-scale families include the normal, Cauchy, logistic, and uniform families (Lehmann, 1991, pp. 20f). In the next proposition, I show that Wang's premium principle reduces to the SD premium principle on a location-scale family. Christofides (1998) observes this phenomenon in several special cases.

This is a follow-up of a previous paper by the author, where claims reserving in non-life insurance is treated in the framework of a marked Poisson claims process. A key result on decomposition of the process is generalized, and a number of related results are added. Their usefulness is demonstrated by examples and, in particular, the connection to the analogous discrete time model is clarified. The problem of predicting the outstanding part of reported but not settled claims is revisited and, by way of example, solved in a model where the partial payments are governed by a Dirichlet process. The process of reported claims is examined, and its dual relationship to the process of occurred claims is pointed out.

1. In (4.17) a factor W is missing in the numerator of the expression in the middle.

2. The predictor of Xorns proposed in Paragraph 5B is not unbiased in general as a simple counterexample will show. The proposed reserve on an individual orns claim with past history T, U, V > υ′ = τ – T – U, {Y′(υ″); 0 ≤ υ″ ≤ υ′}, is

Suppose that Y = V and Y′(υ″) = Q(υ″/V)V, 0 ≤ υ″ ≤ V, where Q is a non-decreasing, deterministic function on [0,1] such that Q(0) = 0 and Q(1) = 1. In this case R = E[V∣T, U, V > Y′(υ′)] – Y′(υ′). If Q(s) > s for all s ∈ (0, 1), then Y′(υ′) = Q(υ′/V)V > υ′, hence

Numerical evaluation of compound distributions is one of the central numerical tasks in insurance mathematics. Two widely used techniques are Panjer recursion and transform methods. Many authors have pointed out that aliasing errors imply the need to consider the whole distribution if transform methods are used, a potential drawback especially for heavy-tailed distributions. We investigate the magnitude of aliasing errors and show that this problem can be solved by a suitable change of measure.

In this paper, it is shown how to approximate theoretical premium calculation principles in order to make them useful in practice. The method relies on stochastic extrema in moment spaces and is illustrated with the aid of the exponential principle.

In the present paper we generalise Panjer's (1981) recursion for compound distributions to a multivariate situation where each claim event generates a random vector. We discuss situations within insurance where such models could be applicable, and consider some special cases of the general algorithm. Finally we deduce from the algorithm a multivariate extension of De Pril's (1985) recursion for convolutions.

Consider a classical compound Poisson model. The safety loading can be positive, negative or zero. Explicit expressions for the distributions of the surplus prior and at ruin are given in terms of the ruin probability. Moreover, the asymptotic behaviour of these distributions as the initial capital tends to infinity are obtained. In particular, for positive safety loading the Cramer case, the case of subexponential distributions and some intermediate cases are discussed.

Financial time series data are typically found to possess leptokurtic frequency distributions, time varying volatilities, outliers and correlation structures inconsistent with linear generating processes, nonlinear dependence, and dependencies between series that are not stable over time. Regime Switching Vector Autoregressions are of interest because they are capable of explaining the observed features of the data, can capture a variety of interactions between series, appear intuitively reasonable, are vector processes, and are now tractable.

This paper considers a vector autoregression subject to periodic structural changes. The parameters of a vector autoregression are modelled as the outcome of an unobserved discrete Markov process with unknown transition probabilities. The unobserved regimes, one for each time point, together with the regime transition probabilities, are determined in addition to the vector autoregression parameters within each regime.

A Bayesian Markov Chain Monte Carlo estimation procedure is developed which efficiently generates the posterior joint density of the parameters and the regimes. The complete likelihood surface is generated at the same time, enabling estimation of posterior model probabilities for use in non-nested model selection. The procedure can readily be extended to produce joint prediction densities for the variables, incorporating both parameter and model uncertainty.

Results using simulated and real data are provided. A clear separation of the variance between a stable and an unstable regime was observed. Ignoring regime shifts is very likely to produce misleading volatility estimates and is unlikely to be robust to outliers. A comparison with commonly used models suggests that Regime Switching Vector Autoregressions provide a particularly good description of the observed data.

For the construction of bonus-malus systems, we propose to show how to apply, thanks to simple mathematics, a parametric method encompassing those encountered in the literature. We also compare this parametric method with a non-parametric one that has not yet been used in the actuarial literature and that however permits a simple formulation of the stationary and transition probabilities in a portfolio whenever we have the intention to construct a bonus-malus system with finite number of classes.

We derive some decision rules to select best predictive regression models in a credibility context, that is, in a “random effects’ linear regression model with replicates. In contrast to usual model selection techniques on a collective level, our proposal allows to detect individual structures, even if they disappear in the collective.

We give exact, non-asymptotic results for the expected squared error loss for a predictor based on credibility estimation in different models. This involves correct accounting of random model parameters and the study of expected loss for shrinkage estimation. We support the theoretical properties of the new model selectors by a small simulation experiment.

In this paper, we find explicit expressions for the moments of the fund level and the value of the total contribution when arithmetic or geometric rates of return are modeled by a moving average process of order q and when a proportional control is applied to the contributions. Our approach is based on the bilinear Markovian representation.

The three annual 2¼% interest coupons of the Winterthur Insurance convertible bond (face value Chf 4 700) will only be paid out if during their corresponding observation periods no major storm or hail storm on one single day damages at least 6 000 motor vehicles insured with Winterthur Insurance. Data for events, where storm or hail damaged more than 1 000 insured vehicles, are available for the last ten years. Using a constant-parameter model, the estimated discounted value of the three Wincat coupons together is Chf 263.29. A conservative evaluation, which accounts for the standard deviation of the estimate, gives a coupon value of Chf 238.25. However, fitting models which admit a trend or a change-point, leads to substantially higher knock-out probabilities of the coupons. The estimated discounted values of the coupons can drop below the above conservative value; a conservative evaluation as above leads to substantially lower values. Hence, already the model uncertainty is higher than the standard deviations of the used estimators. This shows the dominance of the model risk. Consistency, dispersion, robustness and sensitivity of the models are analysed by a simulation study.

In January 1997, Winterthur Insurance, together with Credit Suisse First Boston (CSFB), issued the first listed CAT bond. The annual “WINCAT0 coupons of this three-year convertible bond are knocked out if any single storm event damages more than 6,000 vehicles insured by Winterthur Insurance in Switzerland.

This was a completely new way of securing insurance risks. The main intention was to test the Swiss capital market for such products and to make investors acquainted with them. Thereby Winterthur, together with CSFB, set new standards in product transparency, fairness of pricing and investor education by making the historical data available via internet and by publishing a special brochure (CSFB (1997)), where the pricing and the mathematical modelling are described in detail. This is also a prerequisite to enable a scientific discussion on pricing aspects of such new financial products. The developers of the bond are therefore grateful to Mr. Schmock for this valuable scientific contribution which can be seen as a thorough and profound statistical analysis on the knock-out probability for the purpose of quantifying the model uncertainty.

In Section 2 we briefly summarize the whole pricing of the bond at the issue date and show that there were several risk premium elements in this pricing where the conservative estimation of the knock-out probability was just one of them. In Section 3 we consider the modelling of low frequency risks from a practitioner's standpoint and formulate some requirements from practice. In Section 4 we make some further comments on the modelling of the Wincat data. Section 5 is a short summary.

Algorithms for the calculation of the distribution of the aggregate claims from a life insurance portfolio have been derived by Kornya (1983), Hipp (1986) and De Pril (1986 and 1989). All these authors considered the distribution of the aggregate claims over a single period. In this paper we derive algorithms for the calculation of the joint distribution of the aggregate claims from a life portfolio over several periods.

In the present discussion we point out the relation of some results in Dickson & Waters (1999) to similar results in Sundt (1998a, b).

We shall need some notation. For a positive integer m let m be the set of all m × 1 vectors with positive integer-valued elements and m+ = m ~ {0}. A vector will be denoted by a bold-face letter and each of its elements by the corresponding italic with a subscript denoting the number of the elements; the subscript · denotes the sum of the elements. Let m0 be the class of probability functions on m with a positive probability at 0 and m+ the class of probability functions on m+. For j = 1,…, m we introduce the m × 1 vector ej where the jth element is one and all the other elements zero. We make the convention that summation over an empty range is equal to zero.

Let g ∈ m0 be the compound probability function with counting distribution with probability function v ∈ 10 and severity distribution with probability function h ∈ m+; we shall denote this compound probability function by v V h. Sundt (1998a) showed that

where φv denotes the De Pril transform of v, given by

Motivated by (2) Sundt (1998a) defined the De Pril transform φg of g by

This defines the De Pril transform for all probability functions in m0. Insertion of (2) in (3) gives

and by solving φg(X) we obtain

Sundt (1998a) studies the De Pril transform defined in this way and found in particular that it is additive for convolutions.

In this paper we consider compound distributions where the counting distribution is a bivariate distribution with the probability function (Pn1,n2)n1,n2≥0 that satisfies a recursion in the form

We present an algorithm for recursive evaluation of the corresponding compound distributions and some examples of distributions in this class.