Risk Aggregation

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.