Risk Aggregation

Quantitative Risk Management (QRM) standardly concerns a vector of one-period profit-and-loss random variables X = (X1, . . . ,Xd) defined on some probability space (Ω ,F,P). Risk Aggregation concerns the study of the aggregate financial position Ψ (X), for some measurable function Ψ : Rd → R. Under the terms of the New Basel Capital Accord (Basel II), internationally active banks are required to set aside capital to offset various types of risks, i.e. market, credit and operational risk; see [4]. Under the new regulations, the vector X represents the profit-and-loss amounts for particular lines of risk or business, and this over a given period. A risk measure ρ maps the aggregate position Ψ (X) to ρ(Ψ(X)) ∈ R, to be interpreted as the regulatory capital needed to be able to hold the aggregate position Ψ (X) over this predetermined fixed period. The exact calculation of ρ(Ψ(X)) needs the joint distribution function FX of X; when such information is not available, special procedures are called for, typically leading to bounds on ρ(Ψ(X)). Risk Aggregation has often been studied within the framework when only the marginal distributions F1, . . . ,Fd of the individual risks X1, . . . ,Xd are available. A multitude of statistical techniques are available for estimating the univariate (marginal) distributions. It is often more difficult to capture statistically the d-variate structure of dependence of the vector X. Recently, especially in the management of operational risk, cases in which further dependence information is available have become relevant.