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. In compliance with an approach adopted by Shapley and Shubik (1969), this paper proposes a cooperative market game where agents and institutions play the same role can be developed. We take into account a multiple institutions framework where some of them jointly experience distress events in order to evaluate their individual and collective impact on the remaining institutions in the market. To carry out this analysis, we define a new risk measure (SCoES), generalising the Expected Shortfall of Acerbi (2002) and we characterise the riskiness profile as the outcome of a cost cooperative game played by institutions in distress (a similar approach was adopted by Denault 2001). 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.

Vulnerability-CoVaR: Investigating the Crypto-market

“…, p at time t where j is not part of these CCs. The first considered extension, the SCoVaR, aggregates the variables in the conditional event by taking their sum and was introduced in Bernardi et al (2019). Building on this idea, the SCoVaR in this paper is implicitly defined as follows:…”

“…Cao (2014) introduces the Multi-CoVaR (MCoVaR) with the condition of several CCs being simultaneously in distress. Bernardi et al (2019) propose the System-CoVaR (SCoVaR), in which the conditional variables are aggregated via their sum. Further extensions are detailed in Bernardi et al (2018), Di Bernardino et al (2015), Bernardi et al (2017), andBonaccolto et al (2021).…”

The Skew Normal multivariate risk measurement framework

“…(i) since both integrals (6) and 7contain the same integral, which will be denoted by J 3 , we focus on the explicit calculation of J 3 ;…”

The Sparse Multivariate Method of Simulated Quantiles

“…Several papers try to overcome this problem by introducing systemic risk measures that account for multiple contemporaneous distress events and investigated their theoretical properties, see, e.g., Bernardi et al (2017c), Salvadori et al (2016) and Bernardi et al (2016c). Here, we follow along the same line provided in Bernardi et al (2016a) and we measure how the distress of one institution affects the overall health of all the remaining ones. Formally, let τ ∈ (0, 1) be a confidence level, and let j denote an institution belonging to a given market with d participants, i = 1, 2, .…”

“…While individual risks are assessed using individual Value-at-Risks (VaR), one the most employed systemic risk measure has been becoming the Conditional VaR (CoVaR), introduced by Brunnermeier (2011, 2016). Since then, the assessment of financial risk in a multi-institution framework where some institutions are subject to systemic or non-systemic distress events is one of the hot topics which has received large attention from scholars in Mathematical Finance, Statistics, Management, see, e.g., Acharya et al (2012), Billio et al (2012), Bernardi et al (2017c), Girardi and Ergün (2013), Caporin et al (2013), Engle et al (2014), Hautsch et al (2014, Lucas et al (2014), Bernardi and Catania (2015), , Sordo et al (2015), Bernardi et al (2016b), Bernardi et al (2016c), Bernardi et al (2016a), Brownlees and Engle (2016) and Salvadori et al (2016), just to quote a few of the most relevant approaches. For an extensive and up to date survey on systemic risk measures, see Bisias et al (2012), while the recent literature on systemic risk is reviewed by Benoit et al (2016).…”