Klaus D. Schmidt

AN EXTENSION OF PANJER'S RECURSION

Sundt and Jewell have shown that a nondegenerate claim number distribution Q = {q,}, ~ No satisfies the recursion qn+l=( a b + -h--~)q, for all n _> 0 if and only if Q is a binomial, Poisson or negativebinomial distribution. This recursion is of interest since it yields a recursion for the aggregate claims distribution in the collective model of risk theory when the claim size distribution is integer-valued as well. A similar characterization of claim number distributions satisfying the above recursion for all n >_ 1 has been obtained by Willmot. In the present paper we extend these results and the subsequent recursion for the aggregate claims distribution to the case where the recursion holds for all n >_ k with arbitrary k. Our results are of interest in catastrophe excess-of-loss reinsurance.

Lectures on risk theory

The Claim Arrival Process - The Claim Number Process - The Claim Number Process as a Markov Process - The Mixed Claim Number Process - The Aggregate Claims Process - The Risk Process in Reinsurance - The Reserve Process and the Ruin Problem Appendix: Special Distributions

An Extension of Mack's Model for the Chain Ladder Method

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 nonobservable 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.

Embedding theorems for classes of convex sets

Rådströms embedding theorem states that the nonempty compact convex subsets of a normed vector space can be identified with points of another normed vector space such that the embedding map is additive, positively homogeneous, and isometric. In the present paper, extensions of Rådströms embedding theorem are proven which provide additional information on the embedding space. These results include those of Hörmander who proved a similar embedding theorem for the nonempty closed bounded convex subsets of a Hausdorff locally convex vector space. In contrast to Hörmanders approach via support functionals, all embedding theorems of the present paper are proven by a refinement of Rådströms original method which is constructive and does not rely on Zorns lemma. This paper also includes a brief discussion of some actual or potential applications of embedding theorems for classes of convex sets in probability theory, mathematical economics, interval mathematics, and related areas.

Chain ladder, marginal sum and maximum likelihood estimation

ZusammenfassungDie Chain-Ladder-Methode gehört zu den bekanntesten Methoden der Schadenreservierung: Sie verwendet alle Daten aus dem Abwicklungsdreieck und liefert einfache Schätzer für die erwarteten Endschadenstände. Die einfache Gestalt eines Schätzers ist für seine Verwendung in der Praxis von Bedeutung; seine statistischen Eigenschaften hängen jedoch von den stochastischen Gesetzen ab, die die Erzeugung der Daten bestimmen. In der vorliegenden Arbeit betrachten wir zwei stochastische Modelle, die gewisse elementare Annahmen über den Anfall von Schäden und die Verzögerung bei ihrer Abwicklung widerspiegeln, und zeigen, daß sich in diesen Modellen die Chain-Ladder-Schätzer aus klassischen statistischen Schätzprinzipien ergeben.SummaryThe chain ladder method is one of the most famous methods used in reserving. It exploits all data from the run-off triangle and provides simple estimates of the expected ultimate aggregate claims. Simplicity of an estimator is important for its application in practice, but its performance usually depends upon the stochastic mechanism generating the data. In the present paper we consider two stochastic models which reflect certain elementary ideas on the occurrence and the delay in reporting of claims. We show that in these models the chain ladder estimators of the expected ultimate aggregate claims result from classical statistical estimation principles.

Decompositions of vector measures in Riesz spaces and Banach lattices

The present paper is mainly concerned with decomposition theorems of the Jordan, Yosida-Hewitt, and Lebesgue type for vector measures of bounded variation in a Banach lattice having property (P). The central result is the Jordan decomposition theorem due to which these vector measures may alternately be regarded as order bounded vector measures in an order complete Riesz space or as vector measures of bounded variation in a Banach space. For both classes of vector measures, properties like countable additivity, purely finite additivity, absolute continuity, and singularity can be defined in a natural way and lead to decomposition theorems of the Yosida-Hewitt and Lebesgue type. In the Banach lattice case, these lattice theoretical and topological decomposition theorems can be compared and combined. This paper is organized as follows: In Section 2 we consider order bounded vector measures in an order complete Riesz space. We first note that the class of all order bounded vector measures is an order complete Riesz space itself. From this fact, the Jordan decomposition theorem of Faires and Morrison [5] is immediate, and further decomposition theorems can be deduced by pure Riesz space techniques. We thus obtain the Yosida-Hewitt band decomposition theorem of Congost Iglesias [2] and a new Lebesgue band decomposition theorem. We also generalize the Lebesgue null-set decomposition theorem of Pavlakos [8] and Congost Iglesias [2] for vector measures in a super Dedekind complete Riesz space. These two Lebesgue decompositions usually differ from each other; for order countably additive vector measures they coincide if and only if the dimension of the Riesz space is equal to one. In Section 3 we recall some known results on bounded vector measures and vector measures of bounded variation in a Banach space. In particular, we state the YosidaHewitt decomposition theorem of Uhl [14] and the Lebesgue decomposition theorem of Rickart [9] and Uhl [14] for vector measures of bounded variation. These results are included for reference and in the form they will be needed in Section 4. In Section 4 we study vector measures of bounded variation in a Banach lattice having property (P). As remarked before, the key result of this section is the Jordan decomposition theorem of [11], which generalizes results of Diestel and Faires [3] and Faires and Morrison [5]. The Jordan decomposition theorem is used to prove that the Yosida-Hewitt and Lebesgue decompositions are band decompositions. Furthermore, it is shown that the decomposition theorems of the Yosida-Hewitt type and those of the Lebesgue type differ considerably in the coincidence of their lattice theoretical and

A comparison of models for the chain–ladder method ☆

Abstract The chain–ladder method is the most popular method of loss reserving. In its origin, it is nothing else than a heuristic and appealing algorithm. Because of the stochastic nature of the quantities to which the algorithm is applied, several authors have studied the question whether the chain–ladder method can be justified by a stochastic model and a statistical method related to the model. In the present paper we compare a variety of such models. The comparison results in a flow chart for model selection which may help to decide in a specific situation whether the chain–ladder method should be applied or not.

Bivariate copulas: Transformations, asymmetry and measures of concordance

The present paper introduces a group of transformations on the collection of all bivariate copulas. This group contains an involution which is particularly useful since it provides (1) a criterion under which a given symmetric copula can be transformed into an asymmetric one and (2) a condition under which for a given copula the value of every measure of concordance is equal to zero. The group also contains a subgroup which is of particular interest since its four elements preserve symmetry, the order between two copulas and the value of every measure of concordance.

Bayesian models in actuarial mathematics

Abstract. The present paper provides a unifying survey of Bayesian models in different areas of actuarial mathematics. Bayesian models are discussed with regard to claim number processes, experience rating, and experience reserving. Most models are parametric, but experience rating is also considered in the case of vague prior information and an empirical Bayesian model of experience reserving is studied as well.

Non-optimal prediction by the chain ladder method

Abstract The chain ladder method is one of the most common methods for reserving, and various attempts have been made to justify it in a stochastic model. A particularly interesting model was proposed by Mack (1993, 1994a,b): Under the assumptions of his model, Mack proved that the chain ladder predictors of non-observable aggregate claims are unbiased, and Schmidt and Schnaus (1996) proved that the chain ladder predictor for the first non-observable calendar year is also optimal in the sense that it minimizes expected squared error loss over a wide class of unbiased predictors. In the present paper, it is shown that optimality fails for the chain ladder predictor for the second non-observable calendar year.