Counterparty risk allocation

We address the problem of minimizing the risk of an exposure to a small number of defaultable counterparties based on spectral risk measures, in particular expected shortfall. A typical application is the allocation of cash holdings to a few banks. The resulting risk-minimal allocation turns out to be economically odd or implausible in a number of ways. When the loss distribution is discrete, only a sparse finite set of solutions (including corner solutions) can be optimal. The optimization problem is ill-posed, as the risk-minimal allocation does not depend continuously on the input parameters. With two counterparties, only a total allocation to one counterparty or a fifty-fifty solution can be optimal. In general, the risk-minimal allocation is not monotonic in the confidence level used for calculating the expected shortfall. This nonmonotonicity also holds for continuous loss distributions. These results strengthen the doubts over the appropriateness of spectral risk measures in the target function for economic decision-making.