The increasing complexity of insurance and reinsurance products has seen a growing interest amongst actuaries in the modelling of dependent risks. For efficient risk management, actuaries need to be able to answer fundamental questions such as: Is the correlation structure dangerous? And, if yes, to what extent? Therefore tools to quantify, compare, and model the strength of dependence between different risks are vital. Combining coverage of stochastic order and risk measure theories with the basics of risk management and stochastic dependence, this book provides an essential guide to managing modern financial risk. * Describes how to model risks in incomplete markets, emphasising insurance risks. * Explains how to measure and compare the danger of risks, model their interactions, and measure the strength of their association. * Examines the type of dependence induced by GLM-based credibility models, the bounds on functions of dependent risks, and probabilistic distances between actuarial models. * Detailed presentation of risk measures, stochastic orderings, copula models, dependence concepts and dependence orderings. * Includes numerous exercises allowing a cementing of the concepts by all levels of readers. * Solutions to tasks as well as further examples and exercises can be found on a supporting website.

Actuarial Theory for Dependent Risks: Measures, Orders and Models

Most credibility formulas that have been developed till today refer to the net premium, and the problem of an adequate loading has not yet been solved satisfactorily. In the present paper an approach to premium calculation based on loss functions is introduced, and credibility formulas for the resulting premium principles (net premium principle, exponential principle, Esscher principle, e.g.) are derived. These formulas turn out to be Bayes rules in the sense of statistical decision theory. This gives rise to considering minimax rules as well.

Decision theoretic foundations of credibility theory

This paper presents a new axiomatic characterization of risk measures that are additive for independent random variables. In contrast to previous work, we include an axiom that guarantees monotonicity of the risk measure. Furthermore, the axiom of additivity for independent random variables is related to an axiom of additivity for comonotonic random variables. The risk measure characterized can be regarded as a mixed exponential premium.

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A comonotonic image of independence for additive risk measures

We discuss some properties of a class of multivariate mixed Erlang distributions with different scale parameters and describes various distributional properties related to applications in insurance risk theory. Some representations involving scale mixtures, generalized Esscher transformations, higher-order equilibrium distributions, and residual lifetime distributions are derived. These results allows for the study of stop-loss moments, premium calculation, and the risk allocation problem. Finally, some results concerning minimum and maximum variables are derived and applied to pricing joint life and last survivor policies.

http://hub.hku.hk/bitstream/10722/211945/1/content.pdf

On some properties of a class of multivariate erlang mixtures with insurance applications

Order preserving and order inducing properties of the Escher premium calculation principle are investigated. It is found that using this principle higher premiums might be asked for smaller risks, and also for less variable risks.

H.U. Gerber

Papers

Credibility for Esscher premiums