Allocation of risk capital in a cost cooperative game induced by a modified expected shortfall

Abstract

The standard theory of coherent risk measures fails to consider individual institutions as part of a system which might itself experience instability and spread new sources of risk to the market participants. This paper fills this gap and proposes a cooperative market game where agents and institutions play the same role. We take into account a multiple institutions framework where some institutions jointly experience distress, and evaluate their individual and collective impact on the remaining institutions in the market. To carry out the analysis, we define a new risk measure (SCoES) which is a generalization of the Expected Shortfall of and we characterize the riskiness profile as the outcome of a cost cooperative game played by institutions in distress. Each institution’s marginal contribution to the spread of riskiness towards the safe institutions in then evaluated by calculating suitable solution concepts of the game such as the Banzhaf–Coleman and the Shapley–Shubik values.

Keywords:

• Risk measures
• systemic risk
• cooperative market game
• Shapley value
• value–at–risk
• expected shortfall

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 A mechanism for allocation of risk capital to subportfolios of pooled liabilities has been proposed by Tsanakas and Barnett in 2003 (see Tsanakas and Barnett, 2003) and by Tsanakas in 2009 (see Tsanakas, 2009), based on the Aumann–Shapley value.

2 This property relies on the acceptance sets, which are axiomatized in Artzner et al. (1999).

3 Property TI is sometimes denoted as Risk Free Condition (RFC).

4 Also note that the estimates on δ are as many as the possible coalitions S except the empty set, i.e. $2^{\left|\mathcal{D}\right|}-1,$ hence there are at most $2^{\left|\mathcal{D}\right|}-1$ levels of δ that must be exceeded. Because the choice of δ in the definition of (2)(2) $$E{S}_{\tau }(X)=-\frac{\delta }{\tau }\sum _{j=1}^{\tau }{\tilde{X}}_j.$$(2) is arbitrary, taking the maximum level among such values implies that such condition in (7)(7) $$\mathit{SCoE}{S}_{i|S}^{{\tau }_1|{\tau }_2}\equiv -\mathbb{E}\left[X_i|X_i\le -\mathit{SCoVa}{R}_{i|S}^{{\tau }_1|{\tau }_2},\sum _{j_k\in S}{X}_{j_k}\le -E{S}_{{\tau }_2}\left(\sum _{j_k\in S}{X}_{j_k}\right)\right].$$(7) is always satisfied, consequently $\mathit{SCoE}{S}_{i|S}^{{\tau }_1|{\tau }_2}\equiv \mathbb{E}\left[X_i|X_i\le \mathit{SCoVa}{R}_{i|S}^{{\tau }_1|{\tau }_2}\right]$ for all $i\in \mathcal{P}\setminus \mathcal{D}.$

5 There exists a large number of contributions on axiomatizations of values in literature, see for example Feltkamp (1995) and van den Brink and van der Laan (1998).

6 Nonetheless, some recent contributions have been published on the Shapley value without the efficiency axiom, see Einy and Haimanko (2011) for simple voting games and Casajus (2014) for different classes of games.

Additional information

Funding

This research is supported by the Italian Ministry of Research PRIN 2013–2015, “Multivariate Statistical Methods for Risk Assessment” (MISURA), and by the “Carlo Giannini Research Fellowship”, the “Centro Interuniversitario di Econometria” (CIdE) and “UniCredit Foundation”. We would like to thank Rosella Castellano, Umberto Cherubini, Rita D’Ecclesia, Fabrizio Durante, Viviana Fanelli, Gianfranco Gambarelli, Piotr Jaworski, Sabrina Mulinacci, Roland Seydel, Marco Teodori, the audience at CFE–ERCIM 2014 in Pisa (Italy), and the audience at EURO 2016 in Poznan (Poland) for their valuable comments and suggestions. The usual disclaimer applies.