Narrow Results

We study a dynamic mean-variance portfolio selection problem subject to possible limit of market risk. Three measures of market risk are considered: value-at-risk, expected shortfall, and median shortfall. They are all calculated in a dynamic consistent sense. After applying the technique of delta-normal approximation, we can explicitly solve for the optimal solution and calculate the economic loss brought by the risk budget constraint. With the analytical results obtained, influential factors of economic loss are then explored by which some guidelines on trading practice are proposed. The guidelines are independent of risk measures, and are valuable to both institutions and regulators, for they suggest that an institutional investor would spontaneously obey good investment discipline to avoid potential impact of risk constraint. This result meets the purpose of external regulation from the perspective of market discipline.

In a recent consultative document, the Basel Committee on Banking Supervision suggests replacing Value-at-Risk (VaR) by expected shortfall (ES) for setting capital requirements for banks' trading books because ES better captures tail risk than VaR. However, besides ES, another risk measure called median shortfall (MS) also captures tail risk by taking into account both the size and likelihood of losses. We argue that MS is a better alternative than ES as a risk measure for setting capital requirements because: (i) MS is elicitable but ES is not; (ii) MS has distributional robustness with respect to model misspecification but ES does not; (iii) MS is easy to implement but ES is not.