# Measuring and Allocating Systemic Risk

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## Abstract

**:**

## 1. Introduction

- It views financial institutions as parts of the financial system. For instance, a bank that behaves as part of a herd is allocated more systemic risk than a bank that acts independently of the rest of the financial sector.
- It relates the financial industry to the real economy. As a consequence, costs of externalities grow faster than proportionally if they become large compared to a country’s GDP. Moreover, negative externalities that happen in states of the world where the overall economy is strong cost less than those that happen when the economy is weak.
- It satisfies the so-called clone property. That is, if a financial institution is split into n equal parts, the total systemic risk does not change and the risk attributed to the institution also splits into n equal parts.
- It can detect risks that might not be fully reflected in market quotes of traded securities since it considers scenarios in which taxpayers provide support to financial institutions, resulting in distorted market prices.
- It can detect systemic risk in low volatility environments in which the risk of large economic shocks is low but the financial system is prone to negative feedback spirals in case a crisis erupts.
- It is based on a tolerance parameter that can be adjusted over time so as to implement countercyclical regulation.

## 2. Measuring Total Systemic Risk

- $X\ge {X}^{\prime}$ if $X\left(\omega \right)\ge {X}^{\prime}\left(\omega \right)$ for all $\omega \in \Omega $,
- $X>{X}^{\prime}$ if $X\ge {X}^{\prime}$ and $X\left(\omega \right)>{X}^{\prime}\left(\omega \right)$ for at least one $\omega \in \Omega $,
- $X\gg {X}^{\prime}$ if $X\left(\omega \right)>{X}^{\prime}\left(\omega \right)$ for all $\omega \in \Omega $,
- ${L}_{+}:=\left\{X\in L:X\ge 0\right\}$, ${L}_{++}:=\left\{X\in L:X\gg 0\right\}$,
- ${\u2225X\u2225}_{\infty}:={max}_{\omega \in \Omega}\left|X\left(\omega \right)\right|$, ${\u2225X\u2225}_{1}:={\sum}_{\omega \in \Omega}\left|X\left(\omega \right)\right|$,
- ${B}_{\infty}^{\epsilon}\left(X\right):=\left\{{X}^{\prime}\in L:{\u2225X-{X}^{\prime}\u2225}_{\infty}\le \epsilon \right\}$,
- $\mathcal{P}:=$ the set of all probability measures on $\Omega $,
- ${\mathcal{P}}^{f}:=$ the set of all probability measures on $\Omega $ with full support.

#### 2.1. The Regulator

**Differentiability**$\mathrm{int}\phantom{\rule{0.166667em}{0ex}}\mathrm{dom}\phantom{\rule{0.166667em}{0ex}}U$ is non-empty and U is differentiable on $\mathrm{int}\phantom{\rule{0.166667em}{0ex}}\mathrm{dom}\phantom{\rule{0.166667em}{0ex}}U$,

**Strict monotonicity**$U\left(X\right)>U\left({X}^{\prime}\right)$ for all $X,{X}^{\prime}\in \mathrm{dom}\phantom{\rule{0.166667em}{0ex}}U$ such that $X>{X}^{\prime}$.

#### 2.2. The Financial System

- ${V}_{i}\in L$ is the net worth of institution i at the end of the measuring period, calculated e.g., as market value of assets minus book value of liabilities. ${V}_{i}$ is assumed to already include nonlinearities generated by negative feedback effects occurring during a crisis. A reduced form stochastic model for ${V}_{i}$ is proposed in Section 4 below. ${V}_{i}^{-}$ denotes the negative part $max\left\{0,-{V}_{i}\right\}$ and ${({V}_{i}-{v}_{i})}^{+}$ the positive part $max\left\{{V}_{i}-{v}_{i},0\right\}$.
- ${\alpha}_{i}\ge 0$ is a constant, and the term ${\alpha}_{i}{V}_{i}^{-}$ describes a potential cost to society if institution i’s net worth falls below zero. We assume ${\alpha}_{i}$ to be a number between 0 and 1. To avoid moral hazard, it is important that there be no bailout guarantee. However, we assume that, in case of a crisis, the government decides on a case by case basis which institutions have to be supported to prevent negative effects on the real economy. Companies could receive a mix of bailout payments and government loans at preferred conditions, or they could be nationalized. In all of these cases, some of the losses have to be borne by the taxpayer. We allow ${\alpha}_{i}$ to depend on the type of institution i since the cost caused by a bankruptcy depends on the structure of a financial company. For instance, a financial institution with several different business lines is more difficult to save or liquidate than one which concentrates only on a few activities. Furthermore, a typical bank relies heavily on short-term debt financing. Thus, to protect the economy from an impending collapse, a relatively high fraction of its losses will have to be covered. Other components of the financial system, such as insurance companies or pension plans have more long-term liabilities and can continue operating even if their net worth falls below zero. As a consequence, it might be possible to help them through a crisis with a minimal amount of funding.
- ${\beta}_{i},{v}_{i}\ge 0$ are constants, and ${\beta}_{i}{({V}_{i}-{v}_{i})}^{+}$ models a possible positive externality. For instance, if institution i’s net worth exceeds the level ${v}_{i}$, the government may benefit from additional tax revenues of the form ${\beta}_{i}{({V}_{i}-{v}_{i})}^{+}$.

#### 2.3. Total Systemic Risk

**Definition**

**1.**

**Lemma**

**1.**

**(N) Normalization at the tolerance level:**$\rho \left(e\right)=0,$**(M) Monotonicity:**$\rho \left(E\right)\ge \rho \left({E}^{\prime}\right)$ for all $E,{E}^{\prime}\in L$ such that $E\le {E}^{\prime},$**(T) Translation property:**$\rho (E+m)=\rho \left(E\right)-m$ for all $E\in L$ and $m\in \mathbb{R}$,**(L) Lipschitz-continuity:**$|\rho \left(E\right)-\rho \left({E}^{\prime}\right)|\le {\u2225E-{E}^{\prime}\u2225}_{\infty}$ for all $E,{E}^{\prime}\in L$.

- (I)
- $Y+E+\rho \left(E\right)\in \mathrm{int}\phantom{\rule{0.166667em}{0ex}}\mathrm{dom}\phantom{\rule{0.166667em}{0ex}}U$.

**Theorem**

**1.**

**Proposition**

**1.**

**Corollary**

**1.**

## 3. Systemic Risk Allocation

**(FA)****Full allocation:**${\sum}_{i=1}^{I}{k}_{i}=\rho \left(E\right)$.

**(RA)****Riskless allocation:**${k}_{i}=-{E}_{i}$ if ${E}_{i}$ is deterministic,**(CR)****Causal responsibility:**If the i-th externality changes from ${E}_{i}$ to ${E}_{i}^{\prime}={E}_{i}+\Delta {E}_{i}$ for some $\Delta {E}_{i}\in L$ and ${E}_{j}^{\prime}={E}_{j}$ for $j\ne i$, then the adjusted allocation ${k}^{\prime}$ satisfies ${k}_{i}^{\prime}-{k}_{i}=\rho (E+\Delta {E}_{i})-\rho \left(E\right)$.

**(AD)****Additivity:**If firms i and j are merged in such a way that the externality of the combined unit becomes ${E}_{i+j}={E}_{i}+{E}_{j}$ and the externalities of the other firms stay the same, then the new allocation ${k}^{\prime}$ satisfies ${k}_{i+j}^{\prime}={k}_{i}+{k}_{j}$ as well as ${k}_{l}^{\prime}={k}_{l}$ for $l\ne i,j$.

**Proposition**

**2.**

**Definition**

**2.**

## 4. Modeling Losses in the Financial Sector

#### 4.1. Initial Losses

#### 4.2. Feedback Mechanisms

- Direct spillovers are caused by contractual connections between firms.
- Indirect spillover effects that played an important role in the subprime mortgage crisis starting in 2007 were asset fire sales and the dry up of funding liquidity. Asset fire sales occur when firms sell illiquid assets. This creates downward pressure on their prices and affects other institutions that are holding them. Funding liquidity dries up when, due to distress in some parts of the financial sector, lenders become more risk averse and suddenly demand higher interest. This is especially problematic for firms with a lot of short term debt since they have to refinance more frequently.

#### 4.3. The Regulator

#### 4.4. Mergers and Spinoffs

- Amplification mechanisms are nonlinear. Thus, $\widehat{V}$ is not necessarily equal to the sum ${\sum}_{j=1}^{m}{V}_{{i}_{j}}$.
- Externalities are nonlinear in final net worths. Thus, even if $\widehat{V}$ equals ${\sum}_{j=1}^{m}{V}_{{i}_{j}}$, $\widehat{E}$ is in general different from ${\sum}_{j=1}^{m}{E}_{{i}_{j}}$.

#### 4.4.1. Clone Property

**Clone property:**

#### 4.4.2. Reducing Systemic Risk through Spinoffs

#### 4.4.3. Cases Where Spinoffs Increase Systemic Risk

## 5. Managing Systemic Risk: A Discussion

#### 5.1. Systemic Risk Limits

#### 5.2. Systemic Risk Charges

#### 5.3. Cap and Trade

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proofs

**Proof**

**of**

**Lemma**

**1.**

**Proof**

**of**

**Theorem**

**1.**

**Proof**

**of**

**Proposition**

**1.**

**Proof**

**of**

**Corollary**

**1.**

**Proof**

**of**

**Proposition**

**2.**

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1 | See Brunnermeier et al. (2014) and Bai et al. (2018) for a discussion of different types of liquidity and a definition of the liquidity mismatch index. |

2 | We refer to Brunnermeier and Oehmke (2012) for a survey of financial crises and feedback effects that played an important role. |

3 | Risk Assessment Model of Systemic Institutions |

4 | Constant Relative Risk Aversion |

5 | The shocks ${F}_{1},\cdots ,{F}_{n},{W}_{1},\cdots ,{W}_{I}$ can be modeled to be dependent. In particular, firm-specific shocks can be correlated with the macroeconomic variables as well as other firm-specific shocks. |

6 |

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**MDPI and ACS Style**

Brunnermeier, M.K.; Cheridito, P. Measuring and Allocating Systemic Risk. *Risks* **2019**, *7*, 46.
https://doi.org/10.3390/risks7020046

**AMA Style**

Brunnermeier MK, Cheridito P. Measuring and Allocating Systemic Risk. *Risks*. 2019; 7(2):46.
https://doi.org/10.3390/risks7020046

**Chicago/Turabian Style**

Brunnermeier, Markus K., and Patrick Cheridito. 2019. "Measuring and Allocating Systemic Risk" *Risks* 7, no. 2: 46.
https://doi.org/10.3390/risks7020046