Short courses

3rd Conference in Actuarial Science and Finance in Samos

Risk Measures and Optimal Portfolio Selection (with applications to elliptical distributions)

by J. Dhaene, E. Valdez and T. Hoedemakers

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. As an extension we derive approximations for the scalar product of two random vectors representing the pure actuarial (or technical) risk and the financial (or investment) risk respectively. We study the evaluation of the present value of a series of stochastic payments in the loss reserving context and examine life annuities with stochastic interest rates. We investigate multiperiod portfolio selection problems in a Black & Scholes type market where a basket of 1 riskfree and m risky securities are traded continuously. We look for the optimal allocation of wealth within the class of 'constant mix' portfolios. First, we consider the portfolio selection problem of a decision maker who invests money at predetermined points in time in order to obtain a target capital at the end of the time period under consideration. A second problem concerns a decision maker who invests some amount of money (the initial wealth or provision) in order to be able to fullfil a series of future consumptions or payment obligations. Several optimality criteria and their interpretation within Yaari's dual theory of choice under risk are presented. We examine some risk measures within the class of elliptical distributions, a large class of symmetric distributions which include the familiar Normal distribution. In particular, we focus on the tail conditional expectation as one can derive a nice explicit form within the class of elliptical distributions. As a background, we provide an introduction to definitions, properties and examples of elliptical distributions.

The lectures are based on the following papers that can be downloaded from www.kuleuven.ac.be/insurance (publications):

Bounds for Sums of Non-Independent Log-Elliptical Random Variables, E.A. Valdez, J. Dhaene (2003). Working paper, University of New South Wales. Can a coherent risk measure be too subadditive? [PDF version] J. Dhaene, R. Laeven, S. Vanduffel, G. Darkiewicz & M. Goovaerts (2004). To appear. Comonotonic approximations for optimal portfolio selection problems. [PDF version] J. Dhaene, S. Vanduffel, M. Goovaerts, R. Kaas & D. Vyncke (2004). Journal of Risk and Insurance, to be published. Confidence Bounds for Discounted Loss Reserves. [PDF version] T. Hoedemakers, J. Beirlant, M. Goovaerts & J. Dhaene (2003). Insurance: Mathematics and Economics 33(2), 297-316. Solvency capital, risk measures and comonotonicity: a review. [PDF version] J. Dhaene, S. Vanduffel, Q.H. Tang, M. Goovaerts, R. Kaas & D. Vyncke (2004). Research Report OR 0416, Department of Applied Economics, K.U.Leuven, pp.33. Tail Conditional Expectations for Elliptical Distributions, Z. Landsman, E.A. Valdez (2003). North American Actuarial Journal 7, 55-71. Wang’s Capital Allocation Formula for Elliptically-Contoured Distributions, E.A. Valdez, A. Chernih (2003). Insurance: Mathematics and Economics 33, 517-532.