Rob Kaas

Actuarial theory for dependent risks : measures, orders and models

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.

The Concept of Comonotonicity in Actuarial Science and Finance: Applications

In an insurance context, one is often interested in the distribution function of a sum of random variables (rv’s). Such a sum appears when considering the aggregate claims of an insurance portfolio over a certain reference period. It also appears when considering discounted payments related to a single policy or a portfolio, at different future points in time. The assumption of mutual independence between the components of the sum is very convenient from a computational point of view, but sometimes not a realistic one. In The Concept of Comonotonicity in Actuarial Science and Finance: Theory, we determined approximations for sums of rv’s, when the distributions of the components are known, but the stochastic dependence structure between them is unknown or too cumbersome to work with. Practical applications of this theory will be considered in this paper. Both papers are to a large extent an overview of recent research results obtained by the authors, but also new theoretical and practical results are presented.

Risk Measures and Comonotonicity: A Review

In this paper we examine and summarize properties of several well-known risk measures that can be used in the framework of setting solvency capital requirements for a risky business. Special attention is given to the class of (concave) distortion risk measures. We investigate the relationship between these risk measures and theories of choice under risk. Furthermore we consider the problem of how to evaluate risk measures for sums of non-independent random variables. Approximations for such sums, based on the concept of comonotonicity, are proposed. Several examples are provided to illustrate properties or to prove that certain properties do not hold. Although the paper contains several new results, it is written as an overview and pedagogical introduction to the subject of risk measurement. The paper is an extended version of Dhaene et al. [11] .

Upper and Lower Bounds for Sums of Random Variables.

In this contribution, the upper bounds for sums of dependent random variables X1 + X2 +...+ Xn derived by using comonotonicity are sharpened for the case when there exists a random variable Z such that the distribution functions of the Xi, given Z = z, are known. By a similar technique, lower bounds are derived. A numerical application for the case of lognormal random variables is given.

Economic Capital Allocation Derived from Risk Measures

Abstract We examine properties of risk measures that can be considered to be in line with some “best practice” rules in insurance, based on solvency margins. We give ample motivation that all economic aspects related to an insurance portfolio should be considered in the definition of a risk measure. As a consequence, conditions arise for comparison as well as for addition of risk measures. We demonstrate that imposing properties that are generally valid for risk measures, in all possible dependency structures, based on the difference of the risk and the solvency margin, though providing opportunities to derive nice mathematical results, violates best practice rules. We show that so-called coherent risk measures lead to problems. In particular we consider an exponential risk measure related to a discrete ruin model, depending on the initial surplus, the desired ruin probability, and the risk distribution.

On the Probability and Severity of Ruin

In the usual model of the collective risk theory, we are interested in the severity of ruin, as well as its probability. As a quantitative measure, we propose G(u, y), the probability that for given initial surplus u ruin will occur and that the deficit at the time of ruin will be less than y, and the corresponding density g(u, y). First a general answer in terms of the transform is obtained. Then, assuming that the claim amount distribution is a combination of exponential distributions, we determine g; here the roots of the equation that defines the adjustment coefficient play a central role. An explicit answer is also given in the case in which all claims are of constant size.

The tail probability of discounted sums of Pareto-like losses in insurance

In an insurance context, the discounted sum of losses within a finite or infinite time period can be described as a randomly weighted sum of a sequence of independent random variables. These independent random variables represent the amounts of losses in successive development years, while the weights represent the stochastic discount factors. In this paper, we investigate the problem of approximating the tail probability of this weighted sum in the case when the losses have Pareto-like distributions and the discount factors are mutually dependent. We also give some simulation results.

Risk measurement with equivalent utility principles

Risk measures have been studied for several decades in the actuarial literature, where they appeared under the guise of premium calculation principles. Risk measures and properties that risk measures should satisfy have recently received considerable attention in the financial mathematics literature. Mathematically, a risk measure is a mapping from a class of random variables to the real line. Economically, a risk measure should capture the preferences of the decision-maker. This paper complements the study initiated in Denuit, Dhaene & Van Wouwe (1999) and considers several theories for decision under uncertainty: the classical expected utility paradigm, Yaaris dual approach, maximin expected utility theory, Choquet expected utility theory and Quiggins rank-dependent utility theory. Building on the actuarial equivalent utility pricing principle, broad classes of risk measures are generated, of which most classical risk measures appear to be particular cases. This approach shows that most risk measures studied recently in the financial mathematics literature disregard the utility concept (i.e., correspond to linear utilities), restricting their applicability. Some alternatives proposed in the literature are discussed.

Optimal reinsurance in relation to ordering of risks

Abstract Many criteria for choosing optimal reinsurance treaties have the common property that a treaty is considered preferable to another if it is lower in stop-loss order, which means that it generates uniformly lower stop-loss premiums. Such criteria include maximizing utility of wealth, minimizing ruin probability and many others. If the set of feasible reinsurance treaties contains a minimum element in stop-loss order, this element is the solution to the optimization problem. The retained loss under a stop-loss treaty is such a minimum in the set of all treaties with equal mean, implying that stop-loss reinsurance is optimal with respect to many criteria. This extends the result of Borch (1960) and Kahn (1961) on maximizing the utility of wealth.

A Simple Geometric Proof that Comonotonic Risks have the Convex-largest Sum

In the recent actuarial literature, several proofs have been given for the fact that if a random vector (X1, X2, …, Xn) with given marginals has a comonotonic joint distribution, the sum X1+ X2 + ··· + Xn is the largest possible in convex order. In this note we give a lucid proof of this fact, based on a geometric interpretation of the support of the comonotonic distribution.