# Real-Valued Systemic Risk Measures

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Univariate Monetary Risk Measures

## 3. Systemic Risk Measures

#### 3.1. First Aggregate, Then Allocate

#### 3.2. First Allocate, Then Aggregate

#### 3.2.1. Multivariate Shortfall Risk Allocation

#### 3.3. Scenario-Dependent Allocation

#### 3.4. Multidimensional Acceptance Set

## 4. Axiomatic Definition of Systemic Risk Measures

**Definition**

**1.**

**Proposition**

**1.**

**Remark**

**1.**

- $\left\{m\mathbf{1}\in {\mathbb{R}}^{\mathbf{N}}\mid m\in {\mathbb{R}}_{+},\phantom{\rule{4.pt}{0ex}}\mathbf{1}:=[1,\dots ,1]\right\}\subseteq \mathcal{C}$,
- $\Theta (-m\mathbf{1},m\mathbf{1})\in \mathbb{A}$for all$m\in {\mathbb{R}}_{+}$,

**Example**

**1.**

## 5. Risk Measures Associated with Utility Functions and Fairness Concepts

**Assumption**

**1.**

- ${\mathcal{C}}_{0}\subseteq {\mathcal{C}}_{\mathbb{R}}$and$\mathcal{C}={\mathcal{C}}_{0}\cap {M}^{\Phi}$is a convex cone satisfying${\mathbb{R}}^{N}\subseteq \mathcal{C}\subseteq {\mathcal{C}}_{\mathbb{R}}$.
- For all$n=1,\cdots ,N$,${u}_{n}:\mathbb{R}\to \mathbb{R}$is increasing, strictly concave, differentiable, and satisfies the Inada conditions$${u}_{n}^{\prime}(-\infty ):=\underset{x\to -\infty}{lim}{u}_{n}^{\prime}\left(x\right)=+\infty ,\phantom{\rule{4.pt}{0ex}}{u}_{n}^{\prime}(+\infty ):=\underset{x\to +\infty}{lim}{u}_{n}^{\prime}\left(x\right)=0.$$
- $B<\Lambda (+\infty )$, i.e., there exists$\mathbf{M}\in {\mathbb{R}}^{N}$such that${\sum}_{n=1}^{N}{u}_{n}\left({M}^{n}\right)\ge B$.
- For all$n=1,\cdots ,N$, it holds that, for any probability measure$Q\ll \mathbb{P}$$$\mathbb{E}\left[{v}_{n}\left(\frac{dQ}{dP}\right)\right]<\infty \phantom{\rule{1.em}{0ex}}iff\phantom{\rule{1.em}{0ex}}\mathbb{E}\left[{v}_{n}\left(\lambda \frac{dQ}{dP}\right)\right]<\infty ,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4.pt}{0ex}}\forall \lambda >0,$$

**Proposition**

**2**

- (i)
- Suppose that, for some$i,j\in \{1,\cdots ,N\}$,$i\ne j$, we have$\pm ({e}_{i}{1}_{A}-{e}_{j}{1}_{A})\in \mathcal{C}$for all$A\in \mathcal{F}$, where${e}_{1},\cdots ,{e}_{N}$are the elements of the canonical basis of${\mathbb{R}}^{N}$. Then,$$\begin{array}{cc}\hfill \mathcal{D}=dom\left({\alpha}_{B}\right)\cap \{\frac{d\mathbf{Q}}{dP}\in {L}_{+}^{{\Phi}^{*}}\mid & \phantom{\rule{4pt}{0ex}}{Q}^{n}(\Omega )=1\phantom{\rule{4.pt}{0ex}}\forall n,\phantom{\rule{4.pt}{0ex}}{Q}^{i}={Q}^{j}\phantom{\rule{4.pt}{0ex}}and\phantom{\rule{4.pt}{0ex}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \sum _{n=1}^{N}({\mathbb{E}}_{{Q}^{n}}\left[{Y}^{n}\right]-{Y}^{n})\le 0\phantom{\rule{4.pt}{0ex}}for\phantom{\rule{4.pt}{0ex}}all\phantom{\rule{4.pt}{0ex}}\mathbf{Y}\in \mathcal{C}\}.\hfill \end{array}$$
- (ii)
- Suppose that$\pm ({e}_{i}{1}_{A}-{e}_{j}{1}_{A})\in \mathcal{C}$for all$i,j$and all$A\in \mathcal{F}$. Then,$$\mathcal{D}=dom\left({\alpha}_{B}\right)\cap \left\{\frac{d\mathbf{Q}}{dP}\in {L}_{+}^{{\Phi}^{*}}\mid {Q}^{n}(\Omega )=1,\phantom{\rule{4.pt}{0ex}}{Q}^{n}=Q,\phantom{\rule{4.pt}{0ex}}\forall n\right\}.$$

**Definition**

**2.**

**Theorem**

**1**

- (i)
- the optimizer ${\mathbf{Q}}_{\mathbf{X}}=[{Q}_{\mathbf{X}}^{1},\cdots ,{Q}_{\mathbf{X}}^{N}]$ of the dual problem (14) satisfies$${\rho}_{B}\left(\mathbf{X}\right)={\rho}_{B}^{{\mathbf{Q}}_{\mathbf{X}}}\left(\mathbf{X}\right),\phantom{\rule{4.pt}{0ex}}{\pi}_{A}\left(\mathbf{X}\right)={\pi}_{A}^{{\mathbf{Q}}_{\mathbf{X}}}\left(\mathbf{X}\right);$$
- (ii)
- if $A:={\rho}_{B}\left(\mathbf{X}\right)$, the four problems above share the same (unique) solution ${\mathbf{Y}}_{\mathbf{X}}$;
- (iii)
- the dual optimizer ${\mathbf{Q}}_{\mathbf{X}}$ also satisfies$$\sum _{n=1}^{N}{\mathbb{E}}_{{Q}_{{\mathbf{X}}^{n}}}\left[{Y}_{\mathbf{X}}^{n}\right]={\rho}_{B}\left(\mathbf{X}\right)$$
- (iv)
- ${\rho}_{B}\left(\mathbf{X}\right)={max}_{\mathbf{Q}\in \mathcal{D}}{\rho}_{B}^{\mathbf{Q}}\left(\mathbf{X}\right)={\rho}_{B}^{{\mathbf{Q}}_{\mathbf{X}}}\left(\mathbf{X}\right),$ for $\mathcal{D}$ defined in (15). Drawing a parallel between this property and related findings in utility maximization theory, one might say that the domain $\mathcal{D}$ plays the same role here of the set of martingale measures for the underlying stock in the classical theory.

**Definition**

**3.**

#### 5.1. Interpretation and Implementation of $\rho \left(\mathbf{X}\right)$

- (a)
- “the mechanism can be described as a default fund as in the case of a CCP”. Indeed, the properties of ${\rho}_{B}$ inspire the following procedure: at time 0, according to some systemic risk allocation ${\rho}^{n}\left(\mathbf{X}\right)\phantom{\rule{0.166667em}{0ex}}n=1,\cdots ,N,$ satisfying ${\sum}_{n=1}^{N}{\rho}^{n}\left(\mathbf{X}\right)={\rho}_{B}\left(\mathbf{X}\right)$, the amount ${\rho}_{B}\left(\mathbf{X}\right)$ is collected. ${\rho}^{n}$ could be determined consistently using (19). At terminal time, the amount ${\rho}_{B}\left(X\right)$ is distributed among the institutions according to ${Y}_{\mathbf{X}}^{n}$, the optimal scenario-dependent allocations satisfying ${\sum}_{n=1}^{N}{Y}_{\mathbf{X}}^{n}={\rho}_{B}\left(\mathbf{X}\right)$, so that “the fund acts as a clearing house ”.
- (b)
- Alternatively, in terms of capital requirements and risk sharing mechanism, at time 0, ${\rho}^{n}\left(\mathbf{X}\right)$ (a capital requirement) is associated with each institution $n=1,\cdots ,N$ in the system. At terminal time, each bank provides (if negative) or collects (if positive) the amount ${Y}_{\mathbf{X}}^{n}-{\rho}^{n}\left(\mathbf{X}\right)$. This means that, at terminal time, a risk sharing mechanism takes place. Observe that this sharing mechanism is possible given that$$\sum _{n=1}^{N}({Y}_{\mathbf{X}}^{n}-{\rho}^{n}\left(\mathbf{X}\right))=\sum _{n=1}^{N}{Y}_{\mathbf{X}}^{n}-\sum _{n=1}^{N}{\rho}^{n}\left(\mathbf{X}\right)={\rho}_{B}\left(\mathbf{X}\right)-{\rho}_{B}\left(\mathbf{X}\right)=0.$$Remarkably, there is an incentive for a single bank to enter in such a mechanism, based on the principle of choosing a fair risk allocation, as explained below.

#### 5.2. The Exponential Case: Explicit Formulas

**Theorem**

**2**

**Remark**

**2.**

## 6. On Systemic Optimal Risk Transfer Equilibrium

**Proposition**

**3.**

**Proof**

**of**

**Proposition 3.**

#### 6.1. Systemic Optimal (Deterministic) Allocation

#### 6.2. Risk Transfer Equilibrium

#### 6.3. Systemic Optimal Risk Transfer Equilibrium

## 7. Conditional Systemic Risk Measures

- (i)
- the functional ${\rho}_{\mathcal{G}}$ takes values in ${L}^{\infty}(\Omega ,\mathcal{G},P)$ and is both continuous from below and from above;
- (ii)
- the essential infimum in (27) is actually a minimum, attained at the vector of allocations $\mathbf{Y}(\mathcal{G},\mathbf{X})=[{Y}^{1}(\mathcal{G},\mathbf{X}),\dots ,{Y}^{N}(\mathcal{G},\mathbf{X})]\in {\mathcal{C}}_{\mathcal{G}}$;
- (iii)
- ${\rho}_{\mathcal{G}}$ admits a dual representation which specializes the one in (26) with a more explicit formulation of the penalty function $\alpha $ and of the set ${\mathcal{Q}}_{\mathcal{G}}$.
- (iv)
- the supremum in the dual representation (26) of ${\rho}_{\mathcal{G}}$ is actually a maximum, with optimizer $\mathbf{Q}(\mathcal{G},\mathbf{X})=[{Q}^{1}(\mathcal{G},\mathbf{X}),\dots ,{Q}^{N}(\mathcal{G},\mathbf{X})]$ which is a vector of probability measures also satisfying:$$\sum _{n=1}^{N}{\mathbb{E}}_{{Q}^{n}(\mathcal{G},\mathbf{X})}\left[{Y}^{n}(\mathcal{G},\mathbf{X})\mid \mathcal{G}\right]=\sum _{n=1}^{N}{Y}^{n}(\mathcal{G},\mathbf{X})={\rho}_{\mathcal{G}}\left(\mathbf{X}\right)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathbb{P}-\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}.$$

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Doldi, A.; Frittelli, M.
Real-Valued Systemic Risk Measures. *Mathematics* **2021**, *9*, 1016.
https://doi.org/10.3390/math9091016

**AMA Style**

Doldi A, Frittelli M.
Real-Valued Systemic Risk Measures. *Mathematics*. 2021; 9(9):1016.
https://doi.org/10.3390/math9091016

**Chicago/Turabian Style**

Doldi, Alessandro, and Marco Frittelli.
2021. "Real-Valued Systemic Risk Measures" *Mathematics* 9, no. 9: 1016.
https://doi.org/10.3390/math9091016