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Occurrences and developments of claims are modelled as a marked point process. The individual claim consists of an occurrence time, two covariates, a reporting delay, and a process describing partial payments and settlement of the claim. Under certain likelihood assumptions the distribution of the process is described by 14 one-dimensional components. The modelling is nonparametric Bayesian. The posterior distribution of the components and the posterior distribution of the outstanding IBNR and RBNS liabilities are found simultaneously. The method is applied to a portfolio of accident insurances.
We give 2 explicit formulae for the hedging portfolio of Asian options. One is based on the usual Lognormal approximation, and the other on an Inverse Gaussian approximation. Both give excellent results as replicating strategies when the parameters of the model are in a reasonable range.
Simple analytical lower and upper bounds are obtained for stop-loss premiums and ruin probabilities of compound Poisson risks in case the mean, variance and range of the claim size distribution are known. They are based on stop-loss extremal distributions and improve the bounds derived earlier from dangerous extremal distributions. The special bounds obtained in case the relative variance of the claim size is unknown, but its maximal value is known, are related to other actuarial results.
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.
Recursions are derived for a class of compound distributions having a claim frequency distribution of the well known (a,b)-type. The probability mass function on which the recursions are usually based is replaced by the distribution function in order to obtain increasing iterates. A monotone transformation is suggested to avoid an underflow in the initial stages of the iteration. The faster increase of the transformed iterates is diminished by use of a scaling function. Further, an adaptive weighting depending on the initial value and the increase of the iterates is derived. It enables us to manage an arbitrary large portfolio. Some numerical results are displayed demonstrating the efficiency of the different methods. The computation of the stop-loss premiums using these methods are indicated. Finally, related iteration schemes based on the cumulative distribution function are outlined.
In the present note we deduce a class of bounds for the difference between the stop-loss transforms of two compound distributions with the same severity distribution. The class contains bounds of any degree of accuracy in the sense that the bounds can be chosen as close to the exact value as desired; the time required to compute the bounds increases with the accuracy.
This paper shows how a multivariate Bayes estimator can be adjusted to satisfy a set of linear constraints. In the direct approach, the constraint is enforced by a restriction on the class of admissible estimators. In an alternative approach, the constraint is merely encouraged by a mixed risk function which penalises misbalance between the estimator and the constraint. The adjustment to the optimal unconstrained estimator is shown to depend on the risk function and the linear constraints only, not on the probability model underlying the Bayes estimator. Two practical examples are given, one of which involves reconciliation of independently assessed share values with current market values.
The chain ladder method is a simple and suggestive tool in claims reserving, and various attempts have been made aiming at its justification in a stochastic model. Remarkable progress has been achieved by Schnieper and Mack who considered models involving assumptions on conditional distributions. The present paper extends the model of Mack and proposes a basic model in a decision theoretic setting. The model allows to characterize optimality of the chain ladder factors as predictors of non-observable development factors and hence optimality of the chain ladder predictors of aggregate claims at the end of the first non-observable calendar year. We also present a model in which the chain ladder predictor of ultimate aggregate claims turns out to be unbiased.