Attributing Systemic Risk to Individual Institutions

Abstract

Tools that attribute system-wide risk to individual institutions are key elements of an operational macroprudential approach to financial stability. We propose to measure institutions' systemic importance via an attribution methodology that is based on a game theoretic construct: the Shapley value. This methodology has a number of appealing features. First, owing to weak underlying assumptions, it can be used in conjunction with all popular risk measures. Second, it provides an exact allocation of risk that satisfies a concrete fairness criterion. Third, it accommodates easily uncertainty about model parameters and, more generally, uncertainty about the validity of the risk model.We apply this methodology to a number of stylised banking systems and two risk metrics, value-at-risk and expected shortfall. This allows us to study the interaction of different drivers of systemic importance: size, risk profile and common-risk-factor exposures of individual institutions. We also prove a theorem that, under mild conditions, the systemic importance of an institution increases more than proportionately with its size. An implication of this finding is that a policy intervention calibrated to be proportional to the size of an institution would understate the systemic importance of large firms relative to that of small ones. In addition, we demonstrate how the Shapley value attribution methodology can be used for the calibration of macroprudential capital requirements that have an objective defined at the level of the overall system but are applied at the level of individual institutions. Finally, we argue that different policy tools call for using different applications of the general Shapley value methodology.

Measuring the systemic importance of interconnected banks

“…The measure corresponding to this approach can be interpreted as the actuarially fair premium that each institution should pay to a (hypothetical) provider of insurance covering system-wide losses in systemic events. Tarashev et al (2010) propose a measure that captures the contribution of institutions to system-wide risk. This measure is based on Shapley values, a game theoretical concept developed by Shapley (1953) for allocating the value created in cooperative games across individual players.…”

“…For example, Tarashev et al (2010) highlight that bank size, institution-specific probabilities of default and exposures to common risk factors interact in a non-linear fashion to determine the contribution of financial institutions to system-wide risk. In a related paper, Staum (2010) uses Shapley values to design a deposit insurance scheme in the presence of fire-sale externalities and mergers.…”

Network Structure and Systemic Risk in Banking Systems

“…These studies give insights on the global level of systemic risk in the entire network, but do not allow to measure the systemic importance of a given financial institution, which is our focus here. Rather than compute a global measure of systemic risk then allocating it to individual institutions as in Tarashev et al (2010); Zhou et al (2009);Liu and Staum (2011), we use a direct metric of systemic importance, the Contagion Index (Cont, 2009;Cont and Moussa, 2010), which allows to rank institutions in terms of the risk they pose to the system by quantifying the expected loss in capital generated by an institutions default in a macroeconomic stress scenario. Recent studies such as Acharya et al (2010); Zhou et al (2009) have also proposed measures of systemic importance based on market data such as CDS spreads or equity volatility.…”

Modeling systematic risk and point-in-time probability of default under the Vasicek asymptotic single-risk-factor model framework

“…It is a tool complementing the existing internal models for capital allocation ( [2], [5], [8], [10], [13], [15], [23]). In practice, stress testing focuses on systematic risk, with shocks from the external market or macroeconomic factors ( [7], [13], [24]). With the dynamic model (1.4) and model (2.3), stress testing can be conducted through shocking the systematic risk factors in the model (2.3), then propagate to entity default risk by model (1.4).…”