Risk aggregation

Quantitative Risk Management (QRM) often starts with a vector of oneperiodprofit-and-loss random variables X=(X1,\u2026,Xd)\u2032 defined on some probability space (\u3a9,\u2111,\u2119). Risk Aggregation concerns the study of the aggregate financial position \u3c8(X), for some measurable function \u3c8:\u211dd\u2192\u211d. A risk measure \u3c1 then maps \u3c8(X) to \u3c1(\u3c8(X)) 08\u211d, to be interpreted as the regulatory capital needed to be able to hold the aggregate position \u3c8(X) over a predetermined fixed time period. Risk Aggregation has often been studied within the framework when only the marginal distributions F1,\u2026,Fd of the individual risks X1,\u2026,Xd are available. Recently, especially in the management of operational risk, cases in which further dependence information is available have become relevant. We introduce a general mathematical framework which interpolates between marginal knowledge (F1,\u2026,Fd) and full knowledge of F X, the distribution of X. We illustrate the basic issues through some pedagogic examples of actuarial and financial interest. In particular, we study Risk Aggregation under different mathematical set-ups, for different aggregating functionals\uac and risk measures \u3009 , focusing on Value-at-Risk. We show how the theory of Mass Transportations and tools originally developed to solve so-called Monge-Kantorovich problems turn out to be useful in this context. Finally, we introduce some new numerical integration techniques which solve some open aggregation problems and raise new interesting research issues