A new approach to assessing model risk in high dimensions ¬リニ

Background and Research Objectives The risk assessment of high dimensional portfolios (X1, X2, . . . , Xd) is a core issue in the regulation of financial institutions and in quantitative risk management. In this regard, one usually attempts to measure the risk of the aggregate portfolio (defined as the sum of individual risks Xi) using a risk measure (such as the variance or the Value-at-Risk (VaR)). Solving this problem is a numerical issue once the joint distribution of (X1, X2, . . . , Xd) is completely specified. Unfortunately, its estimation is a difficult task, prone to model misspecification. At present, there is no generally accepted framework for quantifying model risk. A natural way to do so consists in finding the extreme possible values of a chosen risk measure in a family of candidate models. In a recent paper, Embrechts, Puccetti, and Rüschendorf (2013) find bounds on the VaR of high dimensional portfolios, assuming that marginal distributions of the individual risks are known (or prone to negligible model risk) and that the dependence structure (copula) among the risks is not specified. This assumption is natural, as fitting the marginal distribution of a single risk Xi (i = 1, 2, ..., d) can often be performed in a relatively accurate manner, whereas fitting a multivariate model for (X1, X2, . . . , Xd) is challenging, even when the number of observations is large. The bounds derived by Embrechts, Puccetti, and Rüschendorf (2013) are wide, as they neglect all information on the interaction among the individual risks. In this paper, we propose to integrate in a natural way the information from a fitted multivariate model. While a multivariate Gaussian or Student distribution can be suitable as a fit to a dataset “on the whole”, it is usually a poor choice if one seeks accurate estimates of the probability of simultaneous extreme (“tail”) events, or, equivalently, if one wants to estimate the VaR of the aggregate portfolio S = ∑d i=1Xi at a given high confidence interval. Our objective is to assess model risk when aggregating a large number of risks and to extend the work of Embrechts, Puccetti, and Rüschendorf (2013) and Barrieu and Scandolo (2013) by including potential information from a well fitted multivariate model where there is enough data.