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

We propose practical solutions for the determination of optimal retentions in a stop-loss reinsurance. We develop two new optimization criteria for deriving the optimal retentions by, respectively, minimizing the value-at-risk (VaR) and the conditional tail expectation (CTE) of the total risks of an insurer. We establish necessary and sufficient conditions for the existence of the optimal retentions for two risk models: individual risk model and collective risk model. The resulting optimal solution of our optimization criterion has several important characteristics: (i) the optimal retention has a very simple analytic form; (ii) the optimal retention depends only on the assumed loss distribution and the reinsurer’s safety loading factor; (iii) the CTE criterion is more applicable than the VaR criterion in the sense that the optimal condition for the former is less restrictive than the latter; (iv) if optimal solutions exist, then both VaR- and CTE-based optimization criteria yield the same optimal retentions. In terms of applications, we extend the results to the individual risk models with dependent risks and use multivariate phase type distribution, multivariate Pareto distribution and multivariate Bernoulli distribution to illustrate the effect of dependence on optimal retentions. We also use the compound Poisson distribution and the compound negative binomial distribution to illustrate the optimal retentions in a collective risk model.

/pdf/optimal-retention-for-a-stop-loss-reinsurance-under-the-var-10izncfplj.pdf

Optimal Retention for a Stop-loss Reinsurance Under the VaR and CTE Risk Measures

Estimation of the Pareto tail index from extreme order statistics is an important problem in many settings such as income distributions (for inequality measurement), finance (for the evaluation of the value at risk), and insurance (determination of loss probabilities) among others. The upper tail of the distribution in which the data are sparse is typically fitted with a model such as the Pareto model from which quantities such as probabilities associated with extreme events are deduced. The success of this procedure relies heavily not only on the choice of the estimator for the Pareto tail index but also on the procedure used to determine the number k of extreme order statistics that are used for the estimation. For the choice of k most of the known procedures are based on the minimization of (an estimate of) the asymptotic mean square error of the maximum likelihood (or Hill) estimator (MLE) which is the traditional choice for the estimator of the Pareto tail index. In this paper we question the choice of the estimator and the resulting procedure for the determination of k, because we believe that the model chosen to describe the behaviour of the tail distribution can only be considered as approximate. If the data in the tail are not exactly but only approximately Pareto, then the MLE can be biased, i.e. it is not robust, and consequently the choice of k is also biased. We propose instead a weighted MLE for the Pareto tail index that downweights data “far” from the model, where “far” will be measured by the size of standardized residuals constructed by viewing the Pareto model as a regression model. The data that are downweighted this way do not systematically correspond to the largest quantiles. Based on this estimator and proceeding as in Ronchetti and Staudte (1994), we develop a robust prediction error criterion, called RC-criterion, to choose k. In simulation studies, we will compare our estimator and criterion to classical ones with exact and/or approximate Pareto data. Moreover, the analysis of real data sets will show that a robust procedure for selection, and not just for estimation, is needed.

/pdf/a-robust-prediction-error-criterion-for-pareto-modelling-of-obuhbmc30s.pdf

A robust prediction error criterion for Pareto modelling of upper tails.

In this paper, we consider a portfolio of n dependent risks X 1 , … , X n and we study the stochastic behavior of the aggregate claim amount S = X 1 + ⋯ + X n . Our objective is to determine the amount of economic capital needed for the whole portfolio and to compute the amount of capital to be allocated to each risk X 1 , … , X n . To do so, we use a top–down approach. For ( X 1 , … , X n ) , we consider risk models based on multivariate compound distributions defined with a multivariate counting distribution. We use the TVaR to evaluate the total capital requirement of the portfolio based on the distribution of S , and we use the TVaR-based capital allocation method to quantify the contribution of each risk. To simplify the presentation, the claim amounts are assumed to be continuously distributed. For multivariate compound distributions with continuous claim amounts, we provide general formulas for the cumulative distribution function of S , for the TVaR of S and the contribution to each risk. We obtain closed-form expressions for those quantities for multivariate compound distributions with gamma and mixed Erlang claim amounts. Finally, we treat in detail the multivariate compound Poisson distribution case. Numerical examples are provided in order to examine the impact of the dependence relation on the TVaR of S , the contribution to each risk of the portfolio, and the benefit of the aggregation of several risks.

TVaR-based capital allocation for multivariate compound distributions with positive continuous claim amounts

In parametric modeling of loss distributions in actuarial science, a versatile choice with intermediate tail weight is the lognormal distribution. Surprisingly, however, the fitting of this model using estimators that are at once efficient and robust has not been seriously addressed in the extensive literature. Consequently, typical estimators of the lognormal mean and variance fail to be both efficient and robust. In particular, the highly efficient maximum likelihood estimators lack robustness because of their limited sensitivity to outliers in the sample. For the two-parameter lognormal estimation problem, the author considers the problem of efficient and robust joint estimation of the mean and variance of a normal model. He introduces generalized-median-type estimators that yield efficient and robust estimators of various parameters of interest in the lognormal model. The paper provides detailed treatment of the lognormal mean and discusses extension of the approach to the much more complicat...

/pdf/efficient-and-robust-fitting-of-lognormal-distributions-5ayyn7nkdv.pdf

Efficient and Robust Fitting of Lognormal Distributions

In this paper, we propose two constructions which allow dependence between the risks of an insurance portfolio in the individual risk model. In the first construction, each risk’s experience is influenced by an individual and a collective risk factor, as well as a class factor if the portfolio is divided into different classes. The second construction uses copulas. The impact on the cumulative distribution function of the aggregate claim amount and on the stop-loss premium is presented via numerical examples.

On two dependent individual risk models

In this paper, we consider the robustness properties of some estimators in the context of a few natural problems in risk theory. They are the calculation of excess of loss premiums, stop-loss premiums in both individual and collective risk models, and the probability of ruin. For that purpose, we introduce the influence function (and its empirical equivalent, the sensitivity function) and we apply it to the non-parametric plug-in estimators of those quantities. We find that they all have unbounded influence function. In order to obtain estimators with bounded influence function, we consider parametric estimators. We note that the shape of the influence function for any function of the parameters is fixed by the method used for estimating them. We propose the use of minimum distance methods in order to obtain robust estimators. We compare one of them, the minimum Cramer–von Mises estimator, with the maximum likelihood estimator and the non-parametric plug-in estimator with various illustrations.

On robustness in risk theory

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On two dependent individual risk models

On robustness in risk theory