Risk Aggregation

Abstract

Quantitative Risk Management (QRM) often starts with a vector of oneperiodprofit-and-loss random variables ${\bf{X}} = (X_1 , \ldots ,X_d )'$ defined on some probability space $(\Omega ,\Im ,\mathbb P)$. Risk Aggregation concerns the study of the aggregate financial position $\psi ({\bf{X}})$, for some measurable function $\psi :\mathbb R^d \to \mathbb R$. A risk measure ρ then maps $\psi ({\bf{X}})$ to $\rho (\psi ({\bf{X}})) \in \mathbb R$, to be interpreted as the regulatory capital needed to be able to hold the aggregate position $\psi ({\bf{X}})$ over a predetermined fixed time period. Risk Aggregation has often been studied within the framework when only the marginal distributions $F_1 , \ldots ,F_d$ of the individual risks $X_1 , \ldots ,X_d$ 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 $(F_1 , \ldots ,F_d )$ 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¬ and risk measures 〉 , 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.