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Mainly due to new capital adequacy standards for banking and insurance, an increased interest exists in the aggregation properties of risk measures like Value-at-Risk (VaR). We show how VaR can change from sub-to superadditivity depending on the properties of the underlying model. Mainly, the switch from a finite to an infinite mean model gives a completely… (More)

- Paul Embrechts, Dominik D. Lambrigger, Mario V. Wüthrich
- 2008

Properties of risk measures for extreme risks have become an important topic of research. In the present paper we discuss sub-and superadditivity of quantile based risk measures and show how multivariate extreme value theory yields the ideal modeling environment. Numerous examples and counterexamples highlight the applicability of the main results obtained.

We assume that the claims liability process satisfies the distribution-free chain-ladder model assumptions. For claims reserving at time I we predict the total ultimate claim with the information available at time I and, similarly, at time I + 1 we predict the same total ultimate claim with the (updated) information available at time I + 1. The claims… (More)

We prove a two-dimensional version of the famous Pickands–Balkema–de Haan theorem of extreme value theory. The bivariate random variables are generated using the copula language. This representation of dependence structures allows to derive asymptotic results for bivariate excess distributions. Résumé Une version en dimension 2 du célèbre théorème de… (More)

In recent Solvency II considerations much effort has been put into the development of appropriate models for the study of the one-year loss reserving uncertainty in non-life insurance. In this article we derive formulas for the conditional mean square error of prediction of the one-year claims development result in the context of the Bayes chain ladder… (More)

Often in non-life insurance, claims reserves are the largest position on the liability side of the balance sheet. Therefore, the prediction of adequate claims reserves for a portfolio consisting of several runoff subportfolios from dependent lines of business is of big importance for every non-life insurance company. In the present paper we consider the… (More)

We present a novel stochastic model for claims reserving that allows to combine claims payments and incurred losses information. The main idea is to combine two claims reserving models (Hertig's model [11] and Gogol's model [8]) leading to a log-normal paid-incurred chain (PIC) model. Using a Bayesian point of view for the parameter modelling we derive in… (More)

Often in non-life insurance, claims reserves are the largest position on the liability side of the balance sheet. Therefore, the estimation of adequate claims reserves for a portfolio consisting of several runoff subportfolios is relevant for every non-life insurance company. In the present paper we provide a framework in which we unify the multivariate… (More)

We study utility indifference pricing of claim streams with in-tertemporal consumption and power (CRRA) utilities. We derive explicit formulas for the derivatives of the utility indifference price with respect to claims and wealth. The simple structure of these formulas is a reflection of surprising operator identities for the derivatives of the optimal… (More)

We consider the chain ladder reserving method in a Bayesian set up, which allows for combining individual claims development data with portfolio information as for instance development patterns from industry-wide data. We derive the Bayes estimators and the credibility estimators within this Bayesian framework. We show that the credibility esti-mators are… (More)