Showing papers in "Scandinavian Actuarial Journal in 1992"

Hattendorff's theorem and Thiele's differential equation generalized

Abstract: Hattendorff's classical result on zero means an uncorrelatedness of the losses created in disjoint time intervals by a life insurance policy is an immediate consequence of the very definition of the concept of loss. Thus, the result is formulated and proved here in a setting with quite general payments, discount function, and time intervals, all stochastic. A general representation is given for the variances of the losses. They are easy to compute when sufficient structure is added to the model. The traditional continuous time Markov chain model is given special consideration. A stochastic Thiele's differential equation is obtained in a fairly general counting process framework.

Risk exchange I: A unification of some existing results

Abstract: The paper unifies certain concepts which have arisen within the field of risk exchange. Borch's theorem on Pareto-optimal risk exchanges is shown to be derivable from a Bowley solution when there are only two participants in the risk exchange. This theorem is then extended to an n-party risk exchange by equating this to a sequence of 2-party exchanges between the n participants. Finally, the conditions for constrained Pareto-optimal risk exchanges are derived as extreme cases of Borch's theorem. Thus Borch's theorem and Buhlmann and Jewell's theorem on constrained exchanges are shown to be ultimately derivable from the Bowley solution.

Numerical evaluation of Markov transition probabilities based on the discretized product integral

Abstract: The present paper proposes and investigates a procedure for numerical evaluation of the transition probabilities for a time-inhomogeneous Markov process when the intensities are known (estimated). The procedure is based on Taylor-expansion of the transition probabilities linked with the Chapman-Kolmogorov equations.

Risk exchange II: Optimal reinsurance contracts

Abstract: The paper, a sequel to Taylor (1992), discusses risk exchange (REX) and reinsurance. A REX is global if its recoveries depend on just aggregate claims of the insurer in question; local if it depends on individual claims. A reinsurance is a 2-party REX under which recoveries are payable in only one direction between the two insurers; are dependent on the claims of only the insurer making the recoveries (the cedent); are non-negative; and do not exceed the direct claims on the cedent. Optimality of these forms of REX is discussed. In each case, the optimal REX is characterized (Sections 3 to 6). Optimization here consists of maximization of the expected terminal utility of the one of the two parties who acts as price taker, given the premium principle applied by the other party, who acts as price maker. Section 7 compares global and a particular class of local REXs in which individual recoveries are generated by individual claims and, perhaps surprisingly, find the former always preferable. Comment...

Rates of risk convergence of empirical linear Bayes estimators

Abstract: An empirical linear Bayes estimator is asymptotically optimal in the usual sense if its average risk converges to the risk of the corresponding linear Bayes estimator. The present paper demonstrates that the following result holds for the most commonly used models: If the unknown (structural) parameters are estimated in such a way that their mean square error converges at a certain rate, then the corresponding empirical linear Bayes estimator is asymptotically optimal with the same rate of risk convergence. In particular, this is the case for the random coefficient regression model, and for hierarchical models in the univariate case.