Vector-valued coherent risk measures

We define (d,n)-coherent risk measures as set-valued maps from into satisfying some axioms. We show that this definition is a convenient extension of the real-valued risk measures introduced by Artzner et al. [2]. We then discuss the aggregation issue, i.e., the passage from valued random portfolio to valued measure of risk. Necessary and sufficient conditions of coherent aggregation are provided.Received: February 2004, Mathematics Subject Classification (2000): 91B30, 46E30JEL Classification: D81, G31     

An axiomatic characterization of capital allocations of coherent risk measures

An axiomatic definition of coherent capital allocations is given. It is shown that coherent capital allocations defined by the proposed axiom system are closely linked to coherent risk measures. More precisely, the associated risk measure of a coherent capital allocation is coherent and, conversely, for every coherent risk measure there exists a coherent capital allocation.     

Operational risk quantified with spectral risk measures: a refined closed-form approximation

The quantification of operational risk has become an important issue as a result of the new capital charges required by the Basel Capital Accord (Basel II) to cover the potential losses of this type of risk. In this paper, we investigate second-order approximation of operational risk quantified with spectral risk measures (OpSRMs) within the theory of second-order regular variation (2RV) and second-order subexponentiality. The result shows that asymptotically two cases (the fast convergence case and the slow convergence) arise depending on the range of the second-order parameter. We also show that the second-order approximation under 2RV is asymptotically equivalent to the slow convergence case. A number of Monte Carlo simulations for a range of empirically relevant frequency and severity distributions are employed to illustrate the performance of our second-order results. The simulation results indicate that our second-order approximations tend to reduce the estimation errors to a great degree, especially for the fast convergence case, and are able to capture the sub-extremal behavior of OpSRMs better than the first-order approximation. Our asymptotic results have implications for the regulation of financial institutions, and may provide further insights into the measurement and management of operational risk.     

Robustness and sensitivity analysis of risk measurement procedures

Measuring the risk of a financial portfolio involves two steps: estimating the loss distribution of the portfolio from available observations and computing a ‘risk measure’ that summarizes the risk of the portfolio. We define the notion of ‘risk measurement procedure’, which includes both of these steps, and introduce a rigorous framework for studying the robustness of risk measurement procedures and their sensitivity to changes in the data set. Our results point to a conflict between the subadditivity and robustness of risk measurement procedures and show that the same risk measure may exhibit quite different sensitivities depending on the estimation procedure used. Our results illustrate, in particular, that using recently proposed risk measures such as CVaR/expected shortfall leads to a less robust risk measurement procedure than historical Value-at-Risk. We also propose alternative risk measurement procedures that possess the robustness property.     

Two-sided coherent risk measures and their application in realistic portfolio optimization

By using a different derivation scheme, a new class of two-sided coherent risk measures is constructed in this paper. Different from existing coherent risk measures, both positive and negative deviations from the expected return are considered in the new measure simultaneously but differently. This innovation makes it easy to reasonably describe and control the asymmetry and fat-tail characteristics of the loss distribution and to properly reflect the investor’s risk attitude. With its easy computation of the new risk measure, a realistic portfolio selection model is established by taking into account typical market frictions such as taxes, transaction costs, and value constraints. Empirical results demonstrate that our new portfolio selection model can not only suitably reflect the impact of different trading constraints, but find more robust optimal portfolios, which are better than the optimal portfolio obtained under the conditional value-at-risk measure in terms of diversification and typical performance ratios.