Risk Aggregation

Abstract

Quantitative Risk Management (QRM) often starts with a vector of oneperiod profit-and-loss random variables X = (X 1 , . . . , X d ) ′ 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 : R d → R. A risk measure ρ then maps 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 a predetermined fixed time period. Risk Aggregation has often been studied within the framework when only the marginal distributions F 1 , . . . , F d of the individual risks X 1 , . . . , 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 , . . . , 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 Ψ and risk measures ρ, 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.

The complete mixability and convex minimization problems with monotone marginal densities

“…Embrechts and Puccetti [8] give an example of X 1 , X 2 , X 3 i.i.d. ∼ Pareto(1) (on p. 23), and the distribution function of X 1 + X 2 + X 3 is always less than the distribution function of 3X 1 .…”

“…In particular, this problem is equivalent to the worst Value-at-Risk scenarios in risk management. We refer to [10,19,[6][7][8] for detailed discussions on this topic. Unfortunately, as is mentioned in [7]:…”

Asymptotic Bounds for the Distribution of the Sum of Dependent Random Variables

“…Computing the worst VaR is of great interest in the recent research of quantitative risk management; the reader is referred to Embrechts and Puccetti (2006), Embrechts and Puccetti (2010), Puccetti and Rüschendorf (2013), and Wang et al (2013) for the study of this problem and applications in practice. It is well known that, for a continuous distribution F , m n,F is strictly increasing, invertible, and VaR α (n, F ) = m −1 n,F (α); see, for example, Embrechts and Puccetti (2006) and Wang et al (2013).…”

“…Roughly speaking, finding m ψ,F (s) is equivalent to finding the worst-case value at risk with dependence uncertainty, which plays an important role in the study of risk aggregation. We refer the reader to Embrechts and Puccetti (2010) for an overview on this topic, where the connection between m ψ,F (s) and risk management is 782 R. WANG m n,F (s), and discuss their applications in risk management. In Section 4 we give the dual representation of the complete mixability.…”

Copula Theory: An Introduction

“…X (for example, one knows the lower dimensional marginals of X or some dependence measures among its components), the question is whether one can describe the set of all possible copulas of X, compatible with the given information. As said, this problem has its roots at early works on the Fréchet classes, but its popularity have recently increased due to its connection with several problems arising in risk aggregation (see, e.g., [58]). Copulas and stochastic processes.…”