Conférence d'Alexander J. McNeil, University of York
We examine the relationship between multivariate scenarios sets and risk measures. Our interest is motivated by the use of scenario sets in the stress testing of banks and insurance companies whose portfolio values and solvency are dependent on changes in underlying financial risk factors. Although regulators suggest that financial institutions should consider extreme but plausible scenarios, there is no clear guidance on exactly how this should be done. We explain the connection between sets based on the notion of half-space depth (HD) and the Value-at-Risk risk measure. We then introduce general depth concepts related to coherent risk measures and show how these lead to scenario sets based on, for example, the expectile or the expected shortfall risk measure.
We consider the construction of multivariate scenario sets and the implementation of stress tests in practice. In the case of elliptically distributed risk factors, all of the depth-based scenario sets coincide with regions encompassed by the contours of the density function. Our particular interest lies in skewed and/or heavy-tailed multivariate risk factor distributions, where the equivalence of depth contours and density contours does not hold in general; we present a number of example to illustrate the issues that can arise.
