Optimal Scenario-Dependent Multivariate Shortfall Risk Measure and its Application in Capital Allocation

In this paper, we consider a multivariate shortfall risk measure with scenario-dependent allocation weights and examine its properties such as convexity and quasi-convexity. For fixed allocation weights, we show that the resulting risk measure is a convex systemic risk measure in which case the property of translation invariance is dependent on the allocation weights. However, if the allocation weights are chosen optimally on a feasible set, then the resulting risk measure is a quasi-convex systemic risk measure. To evaluate the systemic risk of a financial system with the proposed risk measure computationally, we reformulate it as a two-stage stochastic program which is a finite convex program when the underlying uncertainty is discretely distributed. In the case when the uncertainty is continuously distributed, we propose a polynomial decision rule to tackle the semi-infinite two-stage stochastic program which restricts the scenario-dependent allocation weights to be a class of polynomials of the underlying uncertainty parameters and subsequently reformulate the evaluation problem as a tractable optimization problem. Some convergence results are established for the approximation scheme. Moreover, we apply the proposed risk measure to capital allocation problem and introduce scenario-dependent allocation strategy and deterministic allocation strategy. Finally, we carry out some preliminary tests on the proposed computational schemes for a continuous system, a discrete system and a capital allocation example for a life insurance company.