The object of our study is

where each Sn is a m-dimensional stochastic (real valued) vector, i.e.

denned on a probability space (Ω, , P) and adapted to a filtration (n)0≤n≤N with 0 being the σ-algebra consisting of all null sets and their complements. In this paper we interpret as the value of some financial asset k at time n.

Remark: If the asset generates dividends or coupon payments, think of as to include these payments (cum dividend process). Think of dividends as being reinvested immediately at the ex-dividend price.

Definition 1

(a) A sequence of random vectors

where

is called a trading strategy. Since our time horizon ends at time N we must always have ϑN ≡ 0.

The interpretation is obvious: stands for the number of shares of asset k you hold in the time interval [n,n + 1). You must choose ϑn at time n.

(b) The sequence of random variables

where Sn stands for the payment stream generated by ϑ (set ϑ−1 ≡ 0).

In performing Bayesian analysis of 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. Young (1997) applies techniques from nonparametric density estimation to estimate the prior and uses the estimated model to calculate the predictive mean of future claims given past claims. A shortcoming of this method is that, in estimating the prior, one assumes the average claim amount equals the conditional claim. In this paper, we consider a class of priors obtained by perturbing the one determined nonparametrically, as in Young (1997). We thereby reflect the uncertainty in the prior that arises from the randomness in the claim data. We, then, calculate intervals for the corresponding predictive means. We illustrate our method with data from Dannenburg et al. (1996) and compare the intervals of the predictive means with nonparametric confidence intervals.

This paper provides bonus-malus systems which rest on different types of claims. Consistent estimators are given for some moments of the mixing distribution of a multi equation Poisson model with random effects. Bonus-malus coefficients are then obtained with the expected value principle, and from linear credibility predictors. Empirical results are presented for two types of claims, namely claims at fault and not at fault with respect to a third party.

In this paper we compare, from the point of view of reinsurance, the several risk adjusted premium calculation principles considered in Wang (1996b). We conclude that, with the exception of the proportional hazard (PH) premium calculation principle, all the others behave in a way similar to the expected value principle. We prove that the stop loss reinsurance premium when calculated using the PH premium principle gives a higher premium than any of the other transforms, provided that the priority is big enough. We observe a similar behaviour with respect to excess of loss reinsurance in all the examples given.

We also study the behaviour of the adjustment coefficient, both from the insurer's and the reinsurer's point of view as functions of the priority, when the PH principle is used as opposed to the expected value principle.

The Kolmogorov distance is used to transform arithmetic severities into equispaced arithmetic severities in order to reduce the number of calculations when using algorithms like Panjer's formulae for compound distributions. An upper bound is given for the Kolmogorov distance between the true compound distribution and the transformed one. Advantages of the Kolmogorov distance and disadvantages of the total variation distance are discussed. When the bounds are too big, a Berry-Esseen result can be used. Then almost every case can be handled by the techniques described in this paper. Numerical examples show the interest of the methods.

Largest claims reinsurance covers are reconsidered. Allowing the original claims sizes to be not necessarily independent, a new, upper premium bound is derived and explored.

Bühlmann-Straub credibility is used to find an estimate of the mortality loss ratio for a company, relative to a standard table, for use in the statutory valuation of life insurance business. A method for calculating the margin for adverse deviation to be added to the mortality rate (in accordance with the general principle of Canadian statutory valuation) is derived. Applying credibility further to the variance of the mortality loss ratio gives a methodology for calculating the amount of the surplus (i.e. capital) required to cover annual fluctuations in mortality experience. The necessary structural parameters are calculated from industry statistics; the methodology is illustrated using Canadian life insurance data.