Risk sharing for capital requirements with multidimensional security markets

Abstract

We consider the risk sharing problem for capital requirements induced by capital adequacy tests and security markets. The agents involved in the sharing procedure may be heterogeneous in that they apply varying capital adequacy tests and have access to different security markets. We discuss conditions under which there exists a representative agent. Thereafter, we study two frameworks of capital adequacy more closely, namely polyhedral constraints and distribution-based constraints. We prove existence of optimal risk allocations and equilibria within these frameworks and elaborate on their robustness.

Appendix: Technical supplements

1.1 A.1 The geometry of convex sets

Fix a nonempty convex subset $\mathcal{C}$ of a locally convex Hausdorff topological Riesz space $(\mathcal{X},\preceq ,\tau )$ with dual space $\mathcal{X}^{*}$. The support function of $\mathcal{C}$ is the functional

$$ \sigma _{\mathcal{C}}:\mathcal{X}^{*}\to (-\infty ,\infty ],\qquad \phi \mapsto \sup _{Y\in \mathcal{C}}\phi (Y). $$

The recession cone of $\mathcal{C}$ is the set

$$ 0^{+}\mathcal{C}:=\{U\in \mathcal{X}: Y+kU\in \mathcal{C}, \forall Y \in \mathcal{C}, \forall k\geq 0 \}. $$

A vector $U$ lies in $0^{+}\mathcal{C}$ if and only if $Y+U\in \mathcal{C}$ holds for all $Y\in \mathcal{C}$. $U$ is then called a direction of $\mathcal{C}$. The lineality space of $\mathcal{C}$ is the vector space $\operatorname{lin}(\mathcal{C}):=0^{+} \mathcal{C}\cap (-0^{+}\mathcal{C})$. In the case of an acceptance set $\mathcal{A}$, monotonicity implies $\operatorname {dom}(\sigma _{\mathcal{A}}) \subseteq \mathcal{X}^{*}_{+}$. If $\mathcal{C}$ is closed, the Hahn–Banach separation theorem shows that

$$ \mathcal{C}=\{Y\in \mathcal{X}: \phi (Y)\leq \sigma _{\mathcal{C}}( \phi ), \forall \phi \in \operatorname {dom}(\sigma _{\mathcal{C}})\}. $$

Combining this identity with the definition of the recession cone and the lineality space yields

Lemma A.1

If $\mathcal{C}\subseteq \mathcal{X}$ is closed and convex and $\mathcal{J}\subseteq \operatorname {dom}(\sigma _{\mathcal{C}})$ is such that

$$ \mathcal{C}=\{X\in \mathcal{X}: \phi (X)\leq \sigma _{\mathcal{C}}( \phi ), \forall \phi \in \mathcal{J}\}, $$

then

$$ 0^{+}\mathcal{C}=\bigcap _{\phi \in \mathcal{J}}\mathcal{L}_{0}(\phi )= \{U\in \mathcal{X}: \phi (U)\leq 0, \forall \phi \in \mathcal{J}\} \quad \textit{and}\quad \operatorname{lin}(\mathcal{A})= \bigcap _{\phi \in \mathcal{J}}\ker (\phi ). $$

Finally, we state a decomposition result for closed convex sets specific to finite-dimensional spaces. It follows from arguments in the proofs of [9, Lemmas II.16.2 and II.16.3].

Lemma A.2

Let$\mathcal{C}\subseteq \mathbb{R}^{d}$be convex and closed and$\mathcal{V}:=\operatorname{lin}(\mathcal{C})^{\perp }$. If$\operatorname{ext}( \mathcal{C}\cap \mathcal{V})$denotes the set of extreme points of$\mathcal{C}\cap \mathcal{V}$and$\operatorname{co}(\cdot )$is the convex hull operator, $\mathcal{C}$can be written as

$$ \mathcal{C}=\operatorname{co}\big(\operatorname{ext}(\mathcal{C}\cap \mathcal{V}) \big)+0^{+}\mathcal{C}. $$

1.2 A.2 Infimal convolution

Let $(\mathcal{X},\preceq )$ be a Riesz space and suppose that functions $g_{i}:\mathcal{X}\to (-\infty ,\infty ]$, $i\in [n]$, are given. The infimal convolution or epi-sum of $g_{1},\dots , g_{n}$ is the function $\Box _{i=1}^{n}g_{i}:\mathcal{X}\to [-\infty ,\infty ]$ defined by

$$ \Box _{i=1}^{n}g_{i}(X):=\inf \bigg\{ \sum _{i=1}^{n} g_{i}(X_{i}): X _{1},\dots , X_{n}\in \mathcal{X}, \sum _{i=1}^{n}X_{i}=X\bigg\} ,\quad X \in \mathcal{X}. $$

The convolution is said to be exact at $X\in \mathcal{X}$ if $(\Box _{i=1}^{n}g_{i})(X)\in \mathbb{R}$ and there are $X_{1},\dots , X_{n}\in \mathcal{X}$ with $\sum _{i=1}^{n}X_{i}=X$ such that

$$ \sum _{i=1}^{n}g_{i}(X_{i})=(\Box _{i=1}^{n}g_{i})(X). $$

Lemma A.3

Suppose$\mathcal{X}_{i}\subseteq \mathcal{X}$, $i\in [n]$, are ideals in a Riesz space$(\mathcal{X},\preceq )$such that$\mathcal{X}= \sum _{i=1}^{n}\mathcal{X}_{i}$.

(1) If all$g_{i}:\mathcal{X}\to (-\infty ,\infty ]$are convex, then$\Box _{i=1}^{n}g_{i}$is convex.

(2) If$g_{i}$is monotone on$\mathcal{X}_{i}$with respect to ⪯ for all$i\in [n]$, i.e., $X,Y\in \mathcal{X}_{i}$and$X\preceq Y$implies$g_{i}(X)\leq g_{i}(Y)$, and if$g_{i}|_{ \mathcal{X}\backslash \mathcal{X}_{i}}\equiv \infty$, then$\Box _{i=1}^{n}g_{i}$is monotone on$\mathcal{X}$.

Proof

We only prove (2). Let $X,Y\in \mathcal{X}$ with $X\preceq Y$ and let $\mathbf{X},\mathbf{Y}\in \prod _{i=1}^{n}\mathcal{X}_{i}$ with $\sum _{i=1}^{n}X_{i}=X$ and $\sum _{i=1}^{n}Y_{i}=Y$. We thus have

$$ 0\preceq Y-X=|Y-X|\preceq \sum _{i=1}^{n}|Y_{i}-X_{i}|. $$

By the Riesz space property of $\mathcal{X}$ and the Riesz decomposition property (cf. [3, Sect. 8.5]), there is a vector $\mathbf{Z} \in (\mathcal{X}_{+})^{n}$ such that $Y-X=\sum _{i=1}^{n}Z_{i}$ and such that $Z_{i}=|Z_{i}|\preceq |Y_{i}-X_{i}|$, $i\in [n]$. Because $\mathcal{X}_{i}$ is an ideal, we have in fact $\mathbf{Z}\in \prod _{i=1}^{n}\mathcal{X}_{i}$. By the monotonicity of $g_{i}$ on $\mathcal{X}_{i}$, $i\in [n]$, we obtain

$$ (\Box _{i=1}^{n}g_{i})(X)\leq \sum _{i=1}^{n} g_{i}(Y_{i}-Z_{i})\leq \sum _{i=1}^{n}g_{i}(Y_{i}). $$

Because $(\Box _{i=1}^{n}g_{i})(Y)=\inf \{\sum _{i=1}^{n}g_{i}(Y_{i}): \mathbf{Y}\in \prod _{i=1}^{n}\mathcal{X}_{i}\}$ by the assumption that $g_{i}|_{\mathcal{X}\backslash \mathcal{X}_{i}}\equiv \infty$, taking the infimum over suitable $\mathbf{Y}$ on the right-hand side proves the assertion. □

Note that the risk sharing functional satisfies $\Lambda =\Box _{i=1} ^{n}g_{i}$, where the functions $g_{i}$ are defined by $g_{i}(X)=\rho _{i}(X)$ if $X\in \mathcal{X}_{i}$ and $g_{i}(X)=\infty$ otherwise, $X\in \mathcal{X}$, $i\in [n]$. These functions $g_{i}$ inherit convexity on $\mathcal{X}$ and monotonicity on $\mathcal{X}_{i}$ from $\rho _{i}$.

Lemma A.4

Given a locally convex Hausdorff topological Riesz space$(\mathcal{X},\preceq ,\tau )$and proper functions$g_{i}:\mathcal{X}\to (-\infty ,\infty ]$, $i\in [n]$, the following identities hold:

$$ (\Box _{i=1}^{n}g_{i})^{*}=\sum _{i=1}^{n}g_{i}^{*}\quad \textit{and}\quad \operatorname {dom}\big((\Box _{i=1}^{n}g_{i})^{*}\big)=\bigcap _{i=1}^{n}\operatorname {dom}(g_{i} ^{*}). $$

1.3 A.3 Correspondences

Given two nonempty sets $A$ and $B$, a map $\Gamma : A\to 2^{B}$ mapping elements of $A$ to subsets of $B$ is called a correspondence and denoted by $\Gamma : A\twoheadrightarrow B$. Assume now that $(\mathcal{X},\tau )$ and $(\mathcal{Y},\sigma )$ are topological spaces, and let $\Gamma :\mathcal{X}\twoheadrightarrow \mathcal{Y}$ be a correspondence. A continuous function $\Psi :\mathcal{X}\to \mathcal{Y}$ is a continuous selection for the correspondence $\Gamma$ if $\Psi (x)\in \Gamma (x)$ holds for all $x\in \mathcal{X}$.

If $(\mathcal{X},\sigma )$ is first-countable, $\Gamma$ is upper hemicontinuous at $x\in \mathcal{X}$ if whenever $(x_{k})_{k\in \mathbb{N}}$ is a sequence $\sigma$-converging to $x$ and $(y_{k})_{k \in \mathbb{N}}\subseteq \mathcal{Y}$ is such that $y_{k}\in \Gamma (x _{k})$ for each $k\in \mathbb{N}$, there is a limit point $y\in \Gamma (x)$ of $(y_{k})_{k\in \mathbb{N}}$. If both topological spaces are first-countable, $\Gamma$ is lower hemicontinuous at $x\in \mathcal{X}$ if whenever $(x_{k})_{k\in \mathbb{N}}$ is a sequence $\sigma$-converging to $x$ and $y\in \Gamma (x)$, there are a subsequence $(k_{\lambda })_{\lambda \in \mathbb{N}}$ and $y_{\lambda }\in \Gamma (x_{k_{\lambda }})$, $\lambda \in \mathbb{N}$, such that $y_{\lambda }\to y$ with respect to $\tau$ as $\lambda \to \infty$.¹¹ An example of a lower hemicontinuous correspondence relevant for our investigations is the security allocation map

$$ \mathbb{A}^{s}_{\cdot }:\mathcal{M}\ni Z\mapsto \mathbb{A}_{Z}\cap \prod _{i=1}^{n}\mathcal{S}_{i}. $$

Lemma A.5

The correspondence$\mathbb{A}_{\cdot }^{s}$is lower hemicontinuous on the global security market ℳ and admits a continuous selection$\Psi :\mathcal{M}\to \prod _{i=1}^{n}\mathcal{S}_{i}$with respect to any norm on ℳ.

Proof

Let $\langle \cdot ,\cdot \rangle$ be an inner product on ℳ. Set $\mathcal{S}_{0}:=\{0\}$. We claim that there are natural numbers $0=m_{0}< m_{1}\leq \cdots \leq m_{n}$ and $Z_{1}, \dots ,Z_{m_{n}}\in \bigcup _{i=1}^{n}\mathcal{S}_{i}$ such that for all $i\in [n]$, it holds that $\{Z_{m_{i-1}+1},\dots ,Z_{m_{i}}\}$ is an orthonormal basis of $\{X\in \mathcal{S}_{i}: X\perp \operatorname{span}\{Z_{1},\dots ,Z_{m_{i-1}}\}\}$. Note that every $Z\in \mathcal{M}$ can be expressed as $Z=\sum _{i=1}^{m_{n}}\langle Z_{i},Z \rangle Z_{i}$; hence the mapping $\Psi : Z\mapsto \mathbb{A}_{Z}^{s}$ defined by

$$ \Psi (Z)_{i}:=\sum _{i=m_{i-1}+1}^{m_{i}}\langle Z_{i},Z\rangle Z_{i}, \quad i\in [n], $$

is a selection of $\mathbb{A}_{\cdot }^{s}$ and continuous with respect to the unique locally convex Hausdorff topology on ℳ. Lower hemicontinuity follows immediately. □