Multivariate Shortfall Risk Allocation and Systemic Risk

Abstract

The ongoing concern about systemic risk since the outburst of the global financial crisis has highlighted the need for risk measures at the level of sets of interconnected financial components, such as portfolios, institutions or members of clearing houses. The two main issues in systemic risk measurement are the computation of an overall reserve level and its allocation to the different components according to their systemic relevance. We develop here a pragmatic approach to systemic risk measurement and allocation based on multivariate shortfall risk measures, where acceptable allocations are first computed and then aggregated so as to minimize costs. We analyze the sensitivity of the risk allocations to various factors and highlight its relevance as an indicator of systemic risk. In particular, we study the interplay between the loss function and the dependence structure of the components. Moreover, we address the computational aspects of risk allocation. Finally, we apply this methodology to the allocation of the default fund of a CCP on real data.

On fairness of systemic risk measures

“…In particular, ρ(X+εV) = ρ(X) − ε N n=1 V n for εV ∈W C and (5.2) follows. The marginal risk contribution d dε ρ(X+εV)| ε=0 was also considered in [13] and [3] and is an important quantity which describes the sensitivity of the risk of X with respect to the impact V ∈L 0 (R N ). The property (5.2) cannot be immediately generalized to the case of random vectors V as N n=1 V n / ∈ R in general.…”

“…(a) In practice, the mechanism can be described as a default fund as in the case of a CCP (see [3]). The amount ρ(X) would be collected at time 0 according to some systemic risk allocation ρ n (X) , n = 1, · · · , N, satisfying N n=1 ρ n (X) = ρ(X).…”

“…It might be more relevant to measure systemic risk as the minimal cash that secures Here, the amount m i is added to the financial position X i of institution i ∈ {1, · · · , N } before the corresponding total loss Λ(X + m) is computed (we refer to [3], [7] and [27]).…”

Magic Points in Finance: Empirical Integration for Parametric Option Pricing

“…Third, a variety of other relevant financial quantities that need to be repeatedly evaluated for different parameter constellations can be expressed via Fourier transforms analogously to the Fourier representation of option prices. For one example we refer the reader to Armenti et al [1], who propose a method to quantify risk allocation. At the heart of their algorithm lies an optimization routine that requires the repeated evaluation of parametric expectations that can be expressed as Fourier integrals.…”

“…The resulting error is determined by the decay of the integrand and will be further analyzed in Appendix C. From now on we set Ω := Ω 1 ×· · ·×Ω d with bounded open intervals Ω 1 ⊂ R + +iη 1 and Ω j ⊂ R + iη j for j = 2, . .…”