Faculty of Actuarial Science and Insurance Seminar with Francesca Centrone

Capital Allocation for convex and quasi-convex non differentiable risk measures

Francesca Centrone and Emanuela Rosazza Gianin

Abstract:

To address future uncertainty about their net worth, firms, insurance companies, and portfolio managers often have to satisfy capital requirements, that is, to hold an amount of riskless assets to hedge themselves. This fact then raises the issue of how to share all this immobilized capital in an a priori fair way among the different lines or business units (see, for example [4], [6]). As risk capital is commonly accepted in the literature to be modeled through the use of risk measures ([1], [4], [5], [8]), capital allocation problems in risk management and the theory of risk measures are naturally linked. Starting from Deprez and Gerber's ([7]) work on convex risk premiums, Tsanakas ([9]) defines a Capital Allocation Rule (C.A.R) for Gateaux differentiable convex risk measures inspired to the game theoretic concept of Aumann and Shapley value ([2]), and studies its properties for some widely used classes of convex risk measures (such as the distortion exponential), also providing explicit formulas. Anyway, the case of general non Gateaux-differentiable risk measures is left substantially open, although there exist meaningful examples of convex and quasi-convex ([3], [8]) non Gateaux differentiable risk measures also borrowed from the insurance literature, such as the mean value premium principle. The purpose of this work is to try to fill, though not in full generality, these gaps. To this aim, we propose a family of C.A.R. based on the dual representation theorems for risk measures, study their properties and show that they reduce to Tsanakas' one, when we assume Gateaux differentiability. In the meantime, we discuss the suitability of the use of quasi-convex risk measures for capital allocation purposes.