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Static marginal expected shortfall: Systemic risk measurement under dependence uncertainty

Jinghui Chen, Edward Furman, X. Sheldon Lin

TL;DR

This work develops sharp bounds for the marginal expected shortfall (MES) under dependence uncertainty, assuming known marginals but unknown joint dependence. It first derives unconstrained bounds and then tightens them using three standard factor-model structures: additive, multiplicative, and minimum-based background risks, illustrating how partial dependence information reduces the MES uncertainty. A complementary approach uses a linear conditional-expectation framework (CAPM/WIPM) to obtain explicit constrained MES bounds, with closed-form results in special cases like nonnegative/nonpositive risks and bivariate normal settings. The authors provide extensive empirical analysis with real market data, introducing the Systemic Risk Criticality Index (SRCI) to identify systemically important firms and demonstrating how dependence-information improves risk assessment in crises such as the 2007–2009 financial collapse and the COVID-19 downturn. The findings offer practically relevant tools for insurers and regulators to allocate capital and monitor systemic risk under dependence uncertainty.

Abstract

Measuring the contribution of a bank or an insurance company to overall systemic risk is a key concern, particularly in the aftermath of the 2007--2009 financial crisis and the 2020 downturn. In this paper, we derive worst-case and best-case bounds for the marginal expected shortfall (MES) -- a key measure of systemic risk contribution -- under the assumption that individual firms' risk distributions are known but their dependence structure is not. We further derive tighter MES bounds when partial information on companies' risk exposures, and thus their dependence, is available. To represent this partial information, we employ three standard factor models: additive, minimum-based, and multiplicative background risk models. Additionally, we propose an alternative set of improved MES bounds based on a linear relationship between firm-specific and market-wide risks, consistent with the Capital Asset Pricing Model in finance and the Weighted Insurance Pricing Model in insurance. Finally, empirical analyses demonstrate the practical relevance of the theoretical bounds for industry practitioners and policymakers.

Static marginal expected shortfall: Systemic risk measurement under dependence uncertainty

TL;DR

This work develops sharp bounds for the marginal expected shortfall (MES) under dependence uncertainty, assuming known marginals but unknown joint dependence. It first derives unconstrained bounds and then tightens them using three standard factor-model structures: additive, multiplicative, and minimum-based background risks, illustrating how partial dependence information reduces the MES uncertainty. A complementary approach uses a linear conditional-expectation framework (CAPM/WIPM) to obtain explicit constrained MES bounds, with closed-form results in special cases like nonnegative/nonpositive risks and bivariate normal settings. The authors provide extensive empirical analysis with real market data, introducing the Systemic Risk Criticality Index (SRCI) to identify systemically important firms and demonstrating how dependence-information improves risk assessment in crises such as the 2007–2009 financial collapse and the COVID-19 downturn. The findings offer practically relevant tools for insurers and regulators to allocate capital and monitor systemic risk under dependence uncertainty.

Abstract

Measuring the contribution of a bank or an insurance company to overall systemic risk is a key concern, particularly in the aftermath of the 2007--2009 financial crisis and the 2020 downturn. In this paper, we derive worst-case and best-case bounds for the marginal expected shortfall (MES) -- a key measure of systemic risk contribution -- under the assumption that individual firms' risk distributions are known but their dependence structure is not. We further derive tighter MES bounds when partial information on companies' risk exposures, and thus their dependence, is available. To represent this partial information, we employ three standard factor models: additive, minimum-based, and multiplicative background risk models. Additionally, we propose an alternative set of improved MES bounds based on a linear relationship between firm-specific and market-wide risks, consistent with the Capital Asset Pricing Model in finance and the Weighted Insurance Pricing Model in insurance. Finally, empirical analyses demonstrate the practical relevance of the theoretical bounds for industry practitioners and policymakers.
Paper Structure (24 sections, 11 theorems, 67 equations, 6 figures, 3 tables)

This paper contains 24 sections, 11 theorems, 67 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. MES bounds with fixed marginal distributions
  3. Preliminaries
  4. Risk bounds
  5. On the sharpness of the lower bounds
  6. The unconstrained lower bound
  7. The constrained lower bound
  8. Dependence uncertainty spread
  9. Additive background risk models
  10. Multiplicative background risk model
  11. Minimum-based background risk model
  12. Alternative constrained risk bounds
  13. An empirical study with real data
  14. Long-term empirical experiment
  15. The financial crisis
  16. ...and 9 more sections

Key Result

Theorem 2.1

Let $X_i=f_i(Y, Z_i)\sim F_i$, where $i\in\mathcal{N},$$Y\sim H$, and $Z_i$ are independent of $Y$, and let $S=\sum_{i=1}^{d}X_i$. Then, for $p\in (0, 1)$ and $j\in\mathcal{N}$, we have

Figures (6)

  • Figure 1: Risk bounds of $\mathrm{MES}_p(X_j,S)$, where $X_i\sim \text{Lognorm}(0,1)$ and $S=\sum_{i=1}^{2}X_i$, as functions of the prudence level $p$. Under the assumptions of factor models, $X_1=\exp(b_1Y+\sqrt{1-b_1^2}Z_1)$ and $X_2=\exp(b_2Y+\sqrt{1-b_2^2}Z_2)$, where $b_1=0.1$, $b_2=0.9$, and $Y\overset{d}{=}Z_i\sim N(0,1)$.
  • Figure 2: $\mathrm{MES}_{0.95}(X_1, S)$ bounds for the standard normal case with $b_1=0.3$ and $b_2\in(-1, 1)$ (left panel), and the degree of improvement (right panel).
  • Figure 3: $\mathrm{MES}_{0.95}(X_1, S)$ bounds for the exponential case with $\lambda=1$ and $\lambda_0\in(0.1, 2.1)$ (left panel), and the degree of improvement (right panel).
  • Figure 4: Unconstrained and constrained risk bounds, together with the empirical values of $\mathrm{MES}_p(X_j, S)$, as functions of the prudence level $p$. $X_i$ denotes the daily loss of the $i$-th S&P 500 constituent that remained continuously in the index from 1 January 2005 to 31 December 2024 $(i=1,2,\dots,213)$. The aggregate loss is given by $S=\sum_{i=1}^{213} X_i$, and $X_j$ corresponds to Apple Inc.
  • Figure 5: Unconstrained and constrained risk bounds, together with the empirical values of $\mathrm{MES}_p(X_j, S)$, as functions of the prudence level $p$. $X_i$ denotes the daily loss of the $i$-th S&P 500 constituent that remained continuously in the index from 18 September 2007 and 17 September 2008 $(i=1,2,\dots,453)$. The aggregate loss is given by $S=\sum_{i=1}^{453} X_i$, and $X_j$ corresponds to Lehman Brothers Holdings Inc.
  • ...and 1 more figures

Theorems & Definitions (26)

  • Definition 2.1: Joint mixability
  • Definition 2.2: Risk concentration
  • Definition 2.3: LES
  • Theorem 2.1: Unconstrained and constrained risk bounds
  • Proposition 2.1
  • Corollary 2.1: Sharpness of the lower bounds
  • Remark 2.1
  • Example 2.1
  • Example 2.2
  • Proposition 2.2: Symmetric marginal
  • ...and 16 more