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

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.

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.

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.