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Can Nash inform capital requirements? Allocating systemic risk measures

Çağın Ararat, Zachary Feinstein

Abstract

Systemic risk measures aggregate the risks from multiple financial institutions to find system-wide capital requirements. Though much attention has been given to assessing the level of systemic risk, less has been given to allocating that risk to the constituent institutions. Within this work, we propose a Nash allocation rule that is inspired by game theory. Intuitively, to construct these capital allocations, the banks compete in a game to reduce their own capital requirements while, simultaneously, maintaining system-level acceptability. We provide sufficient conditions for the existence and uniqueness of Nash allocation rules, and apply our results to the prominent structures used for systemic risk measures in the literature. We demonstrate the efficacy of Nash allocations with numerical case studies using the Eisenberg-Noe aggregation mechanism.

Can Nash inform capital requirements? Allocating systemic risk measures

Abstract

Systemic risk measures aggregate the risks from multiple financial institutions to find system-wide capital requirements. Though much attention has been given to assessing the level of systemic risk, less has been given to allocating that risk to the constituent institutions. Within this work, we propose a Nash allocation rule that is inspired by game theory. Intuitively, to construct these capital allocations, the banks compete in a game to reduce their own capital requirements while, simultaneously, maintaining system-level acceptability. We provide sufficient conditions for the existence and uniqueness of Nash allocation rules, and apply our results to the prominent structures used for systemic risk measures in the literature. We demonstrate the efficacy of Nash allocations with numerical case studies using the Eisenberg-Noe aggregation mechanism.
Paper Structure (23 sections, 24 theorems, 81 equations, 2 figures, 1 table)

This paper contains 23 sections, 24 theorems, 81 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Literature Review and Systemic Capital Allocations
  3. Background and Notation
  4. Monetary Risk Measures
  5. Systemic Risk Measures
  6. Nash Allocation Rules
  7. Acceptable Allocation Rules
  8. Existence and Uniqueness
  9. Application to Single-Element Aggregation Functions
  10. Insensitive Systemic Risk Measures
  11. Sensitive Systemic Risk Measures
  12. Case Study: Eisenberg-Noe Aggregation
  13. Eisenberg-Noe Aggregation and Decomposition
  14. Nash Allocation
  15. Numerical Examples
  16. ...and 8 more sections

Key Result

Proposition 2.2

fs:sf Let $\rho\colon L^\infty(\mathbb{R})\to\mathbb{R}$ be a risk measure with acceptance set $\mathcal{A}$. Then, for every $Y\in L^\infty(\mathbb{R})$, it holds Moreover, $\rho$ is convex (positively homogeneous, coherent) if and only if $\mathcal{A}$ is a convex set (cone, convex cone); whenever $\rho$ is convex, $\rho$ is continuous from above if and only if $\mathcal{A}$ is closed in the we

Figures (2)

  • Figure 1: Section \ref{['sec:EN-numeric-2']}: Visualizations of the sensitive systemic risk measures and different allocation rules under different risk measures.
  • Figure 2: Section \ref{['sec:EN-numeric-EBA']}: Visualization of the Nash (x-axis) and minimal capital allocations (y-axis). The solid black line indicates the level where the Nash and minimal capital allocations coincide. Banks with larger capital requirements under the minimal capital allocations are blue circles; banks with larger capital requirements under the Nash allocations are red stars.

Theorems & Definitions (75)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • Remark 2.8
  • Example 2.9
  • ...and 65 more