1 Introduction
We introduce the concept of Systemic Optimal Risk Transfer Equilibrium, denoted by SORTE, that conjugates the classical Bühlmann’s notion of an equilibrium risk exchange with capital allocation based on systemic expected utility optimization.
The capital allocation and risk sharing equilibrium that we consider can be applied to many contexts, such as: equilibrium among financial institutions, agents, or countries; insurance and reinsurance markets; capital allocation among business units of a single firm; wealth allocation among investors.
In this paper we will refer to a participant in these problems (financial institution or firms or countries) as an agent; the class consisting of these N agents as the system; the individual risk of the agents (or the random endowment or future profit and loss) as the risk vector \(\mathbf {X}:=[X^{1},\ldots ,X^{N}]\); the amount \(\mathbf {Y} :=[Y^{1},\ldots ,Y^{N}]\) that can be exchanged among the agents as random allocation. We will generically refer to a central regulator authority, or CCP, or executive manager as a central bank (CB).
We now present the main concepts of our approach and leave the details and the mathematical rigorous presentation to the next sections. In a one period framework, we consider
N agents, each one characterized by a concave, strictly monotone utility function
\(u_{n}:\mathbb {R\rightarrow R}\) and by the original risk
\(X^{n}\in L^{0}(\Omega ,{\mathcal {F}},P),\) for
\(n=1,\ldots ,N\). Here,
\((\Omega ,{\mathcal {F}},P\mathbf {)}\) is a probability space and
\( L^{0}(\Omega ,{\mathcal {F}},P)\) is the vector space of real valued
\({\mathcal {F}} \)-measurable random variables. The sigma-algebra
\({\mathcal {F}}\) represents all possible measurable events at the final time
T.
\({\mathbb {E}}\left[ \cdot \right] \) denotes the expectation under
P. Given another probability measure
Q,
\(E_{Q}\left[ \cdot \right] \) denotes the expectation under
Q. For the sake of simplicity and w.l.o.g., we are assuming zero interest rate. We will use the bold notation to denote vectors.
1.
Bühlmann’s risk exchange equilibrium
We recall Bühlmann’s definition of a risk exchange equilibrium in a pure exchange economy (or in a reinsurance market). The initial wealth of agent
n is denoted by
\(x^{n}\in {\mathbb {R}}\) and the variable
\(X^{n}\) represents the original risk of this agent. In this economy each agent is allowed to exchange risk with the other agents. Each agent has to agree to receive (if positive) or to provide (if negative) the amount
\({\widetilde{Y}}^{n}(\omega )\) at the final time in exchange of the amount
\(E_{Q}[{\widetilde{Y}}^{n}]\) paid (if positive) or received (if negative) at the initial time, where
Q is some pricing probability measure. Hence
\({\widetilde{Y}}^{n}\) is a time
T measurable random variable. In order that at the final time this risk sharing procedure is indeed possible, the exchange variables
\({\widetilde{Y}} ^{n}\) have to satisfy the
clearing condition$$\begin{aligned} \sum _{n=1}^{N}{\widetilde{Y}}^{n}=0 \ P{\text {-}}a.s.\,\,\,. \end{aligned}$$
As in Bühlmann [
12,
13], we say that a pair (
\(\widetilde{\mathbf {Y}}_{\mathbf {X}},Q_{\mathbf {X}})\) is an risk exchange equilibrium if:
(a)
for each n, \({\widetilde{Y}}_{\mathbf {X}}^{n}\) maximizes: \({\mathbb {E}} \left[ u_{n}(x^{n}+X^{n}+{\widetilde{Y}}^{n}-E_{Q_{\mathbf {X}}}[{\widetilde{Y}} ^{n}])\right] \) among all variables \({\widetilde{Y}}^{n}\);
(b)
\(\sum _{n=1}^{N}{\widetilde{Y}}_{\mathbf {X}}^{n}=0\) P-a.s. .
It is clear that only for some particular choice of the equilibrium pricing measure \(Q_{\mathbf {X}}\), the optimal solutions \({\widetilde{Y}}_{\mathbf {X} }^{n}\) to the problems in (a) will also satisfy the condition in (b).
In addition it is evident that the clearing condition in (b) requires that all agents accept to exchange the amount \({\widetilde{Y}}_{ \mathbf {X}}^{n}(\omega )\) at the final time T.
Define
$$\begin{aligned} {\mathcal {C}}_{{\mathbb {R}}}:=\left\{ \mathbf {Y}\in (L^{0}(\Omega ,{\mathcal {F}} ,P))^{N}\mid \sum _{n=1}^{N}Y^{n}\in {\mathbb {R}}\right\} \end{aligned}$$
(1)
that is,
\({\mathcal {C}}_{{\mathbb {R}}}\) is the set of random vectors such that the sum of the components is
P-a.s. a deterministic number.
Observe that with the change of notations
\(Y^{n}:=x^{n}+{\widetilde{Y}} ^{n}-E_{Q_{\mathbf {X}}}[{\widetilde{Y}}^{n}]\), we obtain variables with
\(E_{Q_{ \mathbf {X}}}[Y^{n}]=x^{n}\) for each
n, and an optimal solution
\(Y_{\mathbf { X}}^{n}\) still belonging to
\({\mathcal {C}}_{{\mathbb {R}}}\) and satisfying
$$\begin{aligned} \sum _{n=1}^{N}Y_{\mathbf {X}}^{n}=\sum _{n=1}^{N}x^{n}\quad P{\text {-}}a.s.\,\,. \end{aligned}$$
(2)
As can be easily checked
$$\begin{aligned} \sup _{{\widetilde{Y}}^{n}}{\mathbb {E}}\left[ u_{n}(x^{n}+X^{n}+{\widetilde{Y}} ^{n}-E_{Q_{\mathbf {X}}}[{\widetilde{Y}}^{n}])\right] =\sup _{Y^{n}}\left\{ {\mathbb {E}}\left[ u_{n}(X^{n}+Y^{n})\right] \mid E_{Q_{\mathbf {X} }}[Y^{n}]\le x^{n}\right\} . \end{aligned}$$
Hence the two above conditions in the definition of a risk exchange equilibrium may be equivalently reformulated as
(a’)
for each n, \(Y_{\mathbf {X}}^{n}\) maximizes: \({\mathbb {E}}\left[ u_{n}(X^{n}+Y^{n})\right] \) among all variables satisfying \(E_{Q_{\mathbf {X} }}[Y^{n}]\le x^{n}\);
(b’)
\(\mathbf {Y}_{\mathbf {X}}\in {\mathcal {C}}_{{\mathbb {R}}}\) and \( \sum _{n=1}^{N}Y_{\mathbf {X}}^{n}=\sum _{n=1}^{N}x^{n}\) P-a.s.
We remark that here the quantity \(x^{n}\in {\mathbb {R}}\) is preassigned to each agent.
2.
Systemic optimal (deterministic) allocation
To simplify the presentation, we now suppose that the initial wealth of each agent is already absorbed in the notation
\(X^{n}\), so that
\(X^{n}\) represents the initial wealth plus the original risk of agent
n. We assume that the system has at disposal a total amount of capital
\(A\in {\mathbb {R}}\) to be used at a later time in case of necessity. This amount could have been assigned by the Central Bank, or could have been the result of the previous trading in the system, or could have been collected ad hoc by the agents. The amount A could represent an insurance pot or a fund collected (as guarantee for future investments) in a community of homeowners. For further interpretation of
A, see also the related discussion in Sect. 5.2 of Biagini et al. [
7]. In any case, we consider the quantity
A as exogenously determined. This amount is allocated among the agents in order to optimize the overall systemic satisfaction. If we denote with
\(a^{n}\in {\mathbb {R}}\) the cash received (if positive) or provided (if negative) by agent
n, then the time
T wealth at disposal of agent
n will be
\((X^{n}+a^{n})\). The optimal vector
\(\mathbf {a_{\mathbf {X}}\in } {\mathbb {R}}^{N}\) could be determined according to the following aggregate time-
T criterion
$$\begin{aligned} \sup \left\{ \sum _{n=1}^{N}{\mathbb {E}}\left[ u_{n}(X^{n}+a^{n})\right] \mid \mathbf {a\in }{\mathbb {R}}^{N}\text { s.t. }\sum _{n=1}^{N}a^{n}=A\right\} . \end{aligned}$$
(3)
Note that each agent is not optimizing his own utility function. As the vector
\( \mathbf {a\in }{\mathbb {R}}^{N}\) is deterministic, it is known at time
\(t=0\) and therefore the agents have to agree to provide or receive money only at such initial time.
However, under the assumption that also at the final time the agents have confidence in the overall reliability of the other agents, one can combine the two approaches outlined in Items 1 and 2 above to further increase the optimal total expected systemic utility and simultaneously guarantee that each agent will optimize his/her own single expected utility, taking into consideration an aggregated budget constraint assigned by the system. Of course an alternative assumption to trustworthiness could be that the rules are enforced by the CB.
We denote with
\({\mathcal {L}}^{n}\subseteq L^{0}(\Omega ,{\mathcal {F}},P)\) a space of admissible random variables and assume that
\({\mathcal {L}}^{n}+ \mathbb {R=}{\mathcal {L}}^{n}\). We will consider maps
\(p^{n}:{\mathcal {L}} ^{n}\rightarrow {\mathbb {R}}\) that represent the pricing or cost functionals, one for each agent
n. As we shall see, in some relevant cases, all agents will adopt the same functional
\(p^{1}=\cdots =p^{N}\), which will then be interpreted as the equilibrium pricing functional, as in Bühlmann’s setting above, where
\(p^{n}(\cdot ):=E_{Q}[\cdot ]\) for all
n. However, we do not have to assume this a priori. Instead we require that the maps
\(p^n\) satisfy for all
\(n=1,\ldots ,N\):
(i)
\(p^{n}\) is monotone increasing;
(iii)
\(p^{n}(Y+c)=p^{n}(Y)+c\) for all \(c\in {\mathbb {R}}\) and \(Y\in {\mathcal {L}} ^{n}\).
Such assumptions in particular imply
\(p^{n}(c)=c\) for all constants
\(c\in {\mathbb {R}}\). A relevant example of such functionals are
$$\begin{aligned} p^{n}(\cdot ):=E_{Q^{n}}[\cdot ]\,, \end{aligned}$$
(4)
where
\(Q^n\) are probability measures for
\(n=1,\ldots ,N\). Another example could be
\(p^{n}=-\rho ^{n}\), for convex risk measures
\(\rho ^{n}\).
Now we will apply both approaches, outlined in Items 1 and 2 above, to describe the concept of a Systemic Optimal Risk Transfer Equilibrium.
3.
Systemic optimal risk transfer equilibrium
As explained in Item 1, given some amount
\(a^{n}\) assigned to agent
n, this agent may buy
\({\widetilde{Y}}^{n}\) at the price
\(p^{n}({\widetilde{Y}} ^{n}) \) in order to optimize
$$\begin{aligned} {\mathbb {E}}\left[ u_{n}(a^{n}+X^{n}+{\widetilde{Y}}^{n}-p^{n}({\widetilde{Y}} ^{n}))\right] . \end{aligned}$$
The pricing functionals
\(p^{n}\),
\(n=1,\ldots ,N\) have to be selected so that the optimal solution verifies the clearing condition
$$\begin{aligned} \sum _{n=1}^{N}{\widetilde{Y}}^{n}=0\quad P-\text {a.s.} . \end{aligned}$$
However, as in Item 2,
\(a^{n}\) is not exogenously assigned to each agent, but only the total amount
A is at disposal of the whole system. Thus the optimal way to allocate
A among the agents is given by the solution
\(( {\widetilde{Y}}_{\mathbf {X}}^{n},p_{\mathbf {X}}^{n},a_{\mathbf {X}}^{n})\) of the following problem:
From (
5) and (
6) it easily follows that an optimal solution
\(({\widetilde{Y}}_{ \mathbf {X}}^{n},p_{\mathbf {X}}^{n},a_{\mathbf {X}}^{n})\) fulfills
$$\begin{aligned} \sum _{n=1}^{N}p_{\mathbf {X}}^{n}({\widetilde{Y}}_{\mathbf {X}}^{n})=0. \end{aligned}$$
(7)
Further, letting
\(Y^{n}:=a^{n}+{\widetilde{Y}}^{n}-p_{\mathbf {X}}^{n}( {\widetilde{Y}}^{n})\), from the cash additivity of
\(p_{\mathbf {X}}^{n}\) we deduce
\(p_{\mathbf {X}}^{n}(Y^{n})=a^{n}+p_{\mathbf {X}}^{n}({\widetilde{Y}} ^{n})-p_{\mathbf {X}}^{n}({\widetilde{Y}}^{n})=a^{n}\) and
\(\sum _{n=1}^{N}Y_{ \mathbf {X}}^{n}=\sum _{n=1}^{N}a^{n}+\sum _{n=1}^{N}{\widetilde{Y}}_{\mathbf {X} }^{n}-\sum _{n=1}^{N}p_{\mathbf {X}}^{n}({\widetilde{Y}}_{\mathbf {X} }^{n})=\sum _{n=1}^{N}a^{n}\) and, as before, the above optimization problem can be reformulated as
where analogously to (
7) we have that a solution
\((Y_{\mathbf {X }}^{n},p_{\mathbf {X}}^{n},a_{\mathbf {X}}^{n})\) satisfies
\(\sum _{n=1}^{N}p_{ \mathbf {X}}^{n}(Y_{\mathbf {X}}^{n})=A\), by (
8) and (
9).
The two optimal values in (
5) and (
8) coincide. We see that while each agent is behaving optimally according to his preferences, the budget constraint
\(p_{\mathbf {X}}^{n}(Y^{n})\le a^{n}\) are not a priori assigned, but are endogenously determined through an aggregate optimization problem. The optimal value
\(a_{\mathbf {X}}^{n}\) determines the optimal risk allocation of each agent. It will turn out that
\(a_{\mathbf {X} }^{n}=p_{\mathbf {X}}^{n}(Y_{\mathbf {X}}^{n})\). Obviously, the optimal value in (
5) is greater than (or equal to) the optimal value in (
3), which can be economically translated into the statement that
allowing for exchanges also at terminal time increases the systemic performance.
In addition to the condition in (
9), we introduce further possible constraints on the optimal solution, by requiring that
$$\begin{aligned} \mathbf {Y}_{\mathbf {X}}\in {\mathcal {B}}, \end{aligned}$$
(10)
where
\({\mathcal {B}}\subseteq {\mathcal {C}}_{{\mathbb {R}}}\).
In the paper, see Sect.
3.4, we formalize the above discussion and show the existence of the solution
\((Y_{\mathbf {X}}^{n}\),
\(p_{\mathbf {X} }^{n},a_{\mathbf {X}}^{n})\) to (
8), (
9) and (
10), which we call Systemic Optimal Risk Transfer Equilibrium (SORTE). We show that
\(p_{\mathbf {X}}^{n}\) can be chosen to be of the particular form
\(p_{ \mathbf {X}}^{n}(\cdot ):=E_{Q_{\mathbf {X}}^{n}}[\cdot ]\), for a probability vector
\(\mathbf {Q}_{\mathbf {X}}=[Q_{\mathbf {X}}^{1},\ldots ,Q_{\mathbf {X}}^{N}]\) . The crucial step, Theorem
4.5, is the proof of the dual representation and the existence of the optimizer of the associated problem (
29). The optimizer of the dual formulation provides the optimal probability vector
\(\mathbf {Q}_{\mathbf {X}}\) that determines the functional
\( p_{\mathbf {X}}^{n}(\cdot ):=E_{Q_{\mathbf {X}}^{n}}[\cdot ]\). The characteristics of the optimal
\(\mathbf {Q}_{\mathbf {X}}\) depend on the feasible allocation set
\({\mathcal {B}}\). When no constraints are enforced, i.e., when
\({\mathcal {B}}={\mathcal {C}}_{{\mathbb {R}}}\), then all the components of
\(\mathbf {Q}_{\mathbf {X}}\) turn out to be equal. Hence we find that the implicit assumption of one single equilibrium pricing measure, made in the Bühlmann’s framework, is in our theory a consequence of the particular selection
\({\mathcal {B}}={\mathcal {C}}_{{\mathbb {R}}}\), but for general
\(\mathcal {B }\) this in not always the case. At this point it might be convenient for the reader to have at hand the example of the exponential utility function that is described in Sects.
3.5, and
5, where we obtain an explicit formulation of the optimal solution
\(\mathbf {Y}_{\mathbf {X}}\), of the equilibrium pricing measure
\(\mathbf {Q}_{\mathbf {X}}\) and of the optimal vector
\(\mathbf {a}_{\mathbf {X}}\).
Example 1.2
In order to ignore all integrability issues, in this example we assume that
\( \Omega \) is a finite set, endowed with the sigma algebra of all its subsets and the uniform probability measure. Consider
\(N=4,\) \(u_{n}(x):=(1-e^{- \alpha _{n}x}),\) \(\alpha _{n}>0,\) \(\,n=1,\dots ,4\), and some vectors
\( \mathbf {x}\in {{\mathbb {R}}}^{4}\), and
\(\mathbf {X}\in (L^{\infty })^{4}\). Moreover take
$$\begin{aligned} {\mathcal {B}}=\left\{ \mathbf {Y}\in {\mathcal {C}}_{{{\mathbb {R}}}}\mid Y^{1}+Y^{2}=0,Y^{3}+Y^{4}=0\right\} . \end{aligned}$$
Thus
\(\mathbf {X}\) and
\({\mathcal {B}}\) model a single system of 4 agents which can exchange the risk only in a restricted way (agent 1 with agent 2, and agent 3 with agent 4), so that in effect the system consists of two isolated clusters of agents. Then a constrained risk exchange equilibrium in general does not exists. By contradiction, suppose that (
\( \widetilde{\mathbf {Y}}_{\mathbf {X}},Q_{\mathbf {X}})\) is a constrained risk exchange equilibrium. It is easy to verify that
\(([\widetilde{{Y}}_{\mathbf {X }}^{1},\widetilde{{Y}}_{\mathbf {X}}^{2}],Q_{\mathbf {X}})\) is a (unconstrained) risk exchange equilibrium with respect to
\([X^{1},X^{2}]\) and
\([x^{1},x^{2}]\) (i.e. it satisfies (a) and (b) for
\(N=2\)). Similarly,
\(([ \widetilde{{Y}}_{\mathbf {X}}^{3},\widetilde{{Y}}_{\mathbf {X}}^{4}],Q_{ \mathbf {X}})\) is a (unconstrained) risk exchange equilibrium with respect to
\([X^{3},X^{4}]\) and
\([x^{3},x^{4}]\). This implies using Eq. (2) in Bühlmann [
13] that
$$\begin{aligned} \frac{\exp {\left( \eta (X^{1}+X^{2})\right) }}{{\mathbb {E}}\left[ \exp { \left( \eta (X^{1}+X^{2})\right) }\right] }=\frac{\mathrm {d}Q_{\mathbf {X}}}{ \mathrm {d}P}=\frac{\exp {\left( \theta (X^{3}+X^{4})\right) }}{{\mathbb {E}} \left[ \exp {\left( \theta (X^{3}+X^{4})\right) }\right] },\,\,\,\,\,\eta = \frac{1}{\alpha _{1}}+\frac{1}{\alpha _{2}}, \theta =\frac{1}{\alpha _{3}}+\frac{1}{\alpha _{4}}, \end{aligned}$$
which clearly gives a contradiction, since
\(\mathbf {X}\) is arbitrary.
Observe, however, that in this example a constrained equilibrium exists if we allow for possibly different pricing measures, namely if we may replace the measure \(Q_{\mathbf {X}}\) with a vector \(\mathbf {Q}_{\mathbf {X }}\). This would amount to replacing (a2) with (a3) below, namely to require that:
(a3) for each n, \({\widetilde{Y}}_{\mathbf {X}}^{n}\) maximizes: \({\mathbb {E}} \left[ u_{n}(x^{n}+X^{n}+{\widetilde{Y}}^{n}-E_{Q_{\mathbf {X}}^{n}}[\widetilde{ Y}^{n}])\right] \) among all variables \({\widetilde{Y}}^{n}\);
(b2) \(\widetilde{\mathbf {Y}}_{\mathbf {X}}\in {\mathcal {B}}\) and \(\sum _{n=1}^{N} {\widetilde{Y}}_{\mathbf {X}}^{n}=0\) P-a.s. .
Then such an equilibrium exists. Indeed, by the results in Bühlmann [
13], we can guarantee the existence of the risk exchange equilibrium
\( ([\widetilde{{Y}}_{\mathbf {X}}^{1},\widetilde{{Y}}_{\mathbf {X}}^{2}],Q_{ \mathbf {X}}^{12})\) with respect to
\([X^{1},X^{2}]\) and
\([x^{1},x^{2}]\), and the risk exchange equilibrium
\(([\widetilde{{Y}}_{\mathbf {X}}^{3},\widetilde{ {Y}}_{\mathbf {X}}^{4}],Q_{\mathbf {X}}^{34})\) with respect to
\([X^{3},X^{4}]\) and
\([x^{3},x^{4}]\). Then
\(([\widetilde{{Y}}_{\mathbf {X}}^{1},\widetilde{{Y}}_{\mathbf {X}}^{2}, \widetilde{{Y}}_{\mathbf {X}}^{3},\widetilde{{Y}}_{\mathbf {X}}^{4}],[Q_{ \mathbf {X}}^{12},Q_{\mathbf {X}}^{12},Q_{\mathbf {X}}^{34},Q_{\mathbf {X} }^{34}])\) satisfies (a3) and (b2). The conclusion is that, even in the Bühlmann case, the presence of constraints implies multiple equilibrium pricing measures.
From the mathematical point of view, this fact is very easy to understand in our setup, described in Assumption
3.10. More constraints implies a smaller set
\({\mathcal {B}}_{0}\) of feasible vectors
\(\widetilde{\mathbf {Y}} \in {\mathcal {B}}\) such that
\(\sum _{n=1}^{N}{\widetilde{Y}}_{\mathbf {X}}^{n}=0\) and this in turn implies a larger polar set of
\({\mathcal {B}}_{0}\) (which we will denote with
\({\mathcal {Q}}\), see the definition in Sect. item 4). The equilibrium exists only if we are allowed to pick the pricing vector
\(\mathbf {Q}_{\mathbf {X}}\) in this larger set
\({\mathcal {Q}}\) , but the elements in
\({\mathcal {Q}}\) don’t need to have all equal components. Economically, multiple pricing measures may arise because the risk exchange mechanism may be restricted to clusters of agents, as in this example, and agents from different clusters may well adopt a different equilibrium pricing measure. For further details on clustering, see the Examples
3.17 and
4.20.
Bühlmann’s equilibrium (
\(\mathbf {Y}_{\mathbf {X}}\)) satisfies two relevant properties:
Pareto optimality (there are no feasible allocation
\(\mathbf {Y}\) such that all agents are equal or better off - compared with
\(\mathbf {Y}_{\mathbf {X}}\) - and at least one of them is better off) and
Individual Rationality (each agent is better off with
\(Y_{ \mathbf {X}}^{n}\) than without it). Any feasible allocation satisfying these two properties is called an
optimal risk sharing rule, see Barrieu and El Karoui [
4] or Jouini et al. [
30].
We show that a SORTE is unique (once the class of pricing functionals is restricted to those in the form
\(p^{n}(\cdot )=E_{Q^{n}}[\cdot ]\)). We also prove Pareto optimality, see the Definition
3.1 and the exact formulation in Theorem
4.17.
However, a SORTE lacks Individual Rationality. This is shown in the toy example of Sect.
5.2, but it is also evident from the expression in Eq. (
8). As already mentioned, each agent is performing rationally, maximizing her expected utility, but under a budget constraint
\( p_{\mathbf {X}}^{n}(Y^{n})\le a_{\mathbf {X}}^{n}\) that is determined globally via an additional systemic maximization problem (
\(\sup _{\mathbf { a\in }{\mathbb {R}}^{N}}\{\ldots \mid \sum _{n=1}^{N}a^{n}=A\}\)) that assigns priority to the systemic performance, rather than to each individual agent. In the SORTE we replace individual rationality with such a
systemic induced individual rationality, which also shows the difference between the concepts of SORTE and of an optimal risk sharing rule. We also point out that the participation in the risk sharing mechanism may be appropriately mitigated or enforced by the use of adequate sets
\({\mathcal {B}}\), see e.g. Example
4.20 for risk sharing restricted to subsystems. From the technical point of view, we will not rely on any of the methods and results related to the notion of inf-convolution, which is a common tool to prove existence of optimal risk sharing rules (see for example [
4] or [
30]) in the case of monetary utility functions, as we do not require the utility functions to be cash additive. Our proofs are based on the dual approach to (systemic) utility maximization. This is summarized in Sect.
4.1. Furthermore, the exponential case is treated in detail in Sect.
5.
Review of literature This paper originates from the systemic risk approach developed in Biagini et al. [
6,
7]. In [
7] the main focus was the analysis of the systemic risk measure
$$\begin{aligned} \rho (\mathbf {X}):=\inf _{\mathbf {Y}\in {\mathcal {B}}\subset {\mathcal {C}}_{ {\mathbb {R}}}}\left\{ \sum _{n=1}^{N}Y^{n}\mid {\mathbb {E}}\left[ \sum _{n=1}^{N}u_{n}(X^{n}+Y^{n})\right] \ge B\right\} , B\in {\mathbb {R}}, \end{aligned}$$
(14)
which computes systemic risk as the minimal capital
\(\sum _{n=1}^{N}Y^{n}\in {\mathbb {R}}\) that secures the aggregated system
\(({\mathbb {E}}\left[ \sum _{n=1}^{N}u_{n}(X^{n}+Y^{n})\right] \ge B)\) by injecting the random allocation
\(Y^{n}\) into the single institution
\(X^{n}\).
The notion of a SORTE is inspired by the following utility maximization problem, associated to the risk minimization problem (
14),
$$\begin{aligned} \sup _{\mathbf {Y}\in {\mathcal {B}}\subset {\mathcal {C}}_{{\mathbb {R}}}}\left\{ {\mathbb {E}}\left[ \sum _{n=1}^{N}u_{n}(X^{n}+Y^{n})\right] \mid \sum _{n=1}^{N}Y^{n}\le A\right\} , A\in {\mathbb {R}}, \end{aligned}$$
(15)
that was also introduced in [
7]. Related papers on systemic risk measures are Feinstein et al. [
23], Acharya et al. [
2], Armenti et al. [
3], Chen et al. [
17], Kromer et al. [
32]. For an exhaustive overview on the literature on systemic risk, see Hurd [
29] and Fouque and Langsam [
27].
For a review on Arrow–Debreu Equilibrium (see Debreu [
20]; Mas Colell and Zame [
34] for the infinite dimensional case) we refer to Sect. 3.6 of Föllmer and Schied [
26], which is close to our setup. In the spirit of the Arrow–Debreu Equilibrium, Bühlmann [
12,
13] proved the existence of risk exchange equilibria in a pure exchange economy. Such risk sharing equilibria had been studied in different forms starting from the seminal papers of Borch [
11], where Pareto-optimal allocations were proved to be comonotonic for concave utility functions, and Bühlmann and Jewell [
14]. The differences with Bühlmann’s setup and our approach have been highlighted before in detail.
In Barrieu and El Karoui [
4] inf-convolution of convex risk measures has been introduced as a fundamental tool for studying risk sharing. Existence of optimal risk sharing for law-determined monetary utility functions is obtained in Jouini et al. [
30] and then generalized to the case of non-monotone risk measures by Acciaio [
1] and Filipovi ć and Svindland [
25], to multivariate risks by Carlier and Dana [
15] and Carlier et al. [
16], to cash-subadditive and quasi-convex measures by Mastrogiacomo and Rosazza Gianin [
35]. Further works on risk sharing are also Dana and Le Van [
19], Heath and Ku [
28], Tsanakas [
39], Weber [
40]. Risk sharing problems with quantile-based risk measures are studied in Embrechts et al. [
22] by explicit construction, and in [
21] for heterogeneous beliefs. In Filipovi ć and Kupper [
24] Capital and Risk Transfer is modelled as (deterministically determined) redistribution of capital and risk by means of a finite set of non deterministic financial instruments. Existence issues are studied and related concepts of equilibrium are introduced. Recent further extensions have been obtained in Liebrich and Svindland [
33].
2 Notations
Let
\((\Omega ,{\mathcal {F}},P\mathbf {)}\) be a probability space and consider the following set of probability vectors on
\((\Omega ,{\mathcal {F}})\)$$\begin{aligned} {\mathcal {P}}^{N}:=\left\{ \mathbf {Q=[}Q^{1},\ldots ,Q^{N}\mathbf {]}\mid \text { such that }Q^{j}\ll P\text { for all }j=1,\ldots ,N\right\} . \end{aligned}$$
For a vector of probability measures
\(\mathbf {Q}\) we write
\(\mathbf {Q}\ll P\) to denote
\(Q^{1}{\ll P},\dots ,Q^{N}\ll P\). Similarly for
\(\mathbf {Q}\sim P \) . Set
\(L^{0}(\Omega ,{\mathcal {F}},P;{\mathbb {R}}^{N}\mathbf {)}=(L^{0}(P))^{N}\). For
\(Q\in {\mathcal {P}}^{1}\) let
\(L^{1}(Q\mathbf {):=}L^{1}(\Omega ,{\mathcal {F}} ,Q;{\mathbb {R}}\mathbf {)}\) be the vector space of
\(Q-\) integrable random variables and
\(L^\infty (Q):=L^{\infty }(\Omega ,{\mathcal {F}},Q;{\mathbb {R}} \mathbf {)}\) be the space of
\(Q-\) essentially bounded random variables. Set
and
. For
\(\mathbf {Q}\in {\mathcal {P}}^{N}\) let
$$\begin{aligned}&L^{1}(\mathbf {Q):=}L^1(Q^{1}\mathbf {)\times \cdots \times }L^1(Q^{N})\,,\,\,\,\, \,\,\,\,L^{1}_+(\mathbf {Q):=}L^1_+(Q^{1}\mathbf {)\times \cdots \times } L^1_+(Q^{N})\,,\\&L^\infty (\mathbf {Q}):=L^\infty (Q^1)\times \dots \times L^\infty (Q^N)\,,\,\,\,\,\,\,\,\,L^\infty _+(\mathbf {Q}):=L^\infty _+(Q^1) \times \dots \times L^\infty _+(Q^N). \end{aligned}$$
We also set
\(\mathbb {R}_{+}=[0,+\infty )\) and
\(\mathbb {R}_{++}=(0,+\infty )\).
For each
\(j=1,\ldots ,N\) consider a vector subspace
\({\mathcal {L}}^{j}\) with
\({ {\mathbb {R}}}\subseteq {\mathcal {L}}^{j}\subseteq L^{0}(\Omega ,{\mathcal {F}},P; {\mathbb {R}}\mathbf {)}\) and set
$$\begin{aligned} {\mathcal {L}}\mathbf {:=}{\mathcal {L}}^{1}\times \cdots \times {\mathcal {L}}^{N}\mathbf { \subseteq }(L^{0}(P))^{N}. \end{aligned}$$
Consider now a subset
\(\mathscr {Q}\subseteq {\mathcal {P}}^{N}\) and assume that the pair
\(({\mathcal {L}},\mathscr {Q})\) satisfies that for every
\(\mathbf {Q }\in \mathscr {Q}\)$$\begin{aligned} {\mathcal {L}}\subseteq L^{1}(\mathbf {Q}). \end{aligned}$$
One could take as
\({\mathcal {L}}^{j}\), for example,
\(L^{\infty }\) or some Orlicz space. Our optimization problems will be defined on the vector space
\( {\mathcal {L}}\) to be specified later.
For each \(n=1,\ldots ,N\), let \(u_{n}:\mathbb {R\rightarrow R}\) be concave and strictly increasing. Fix \(\mathbf {X=[}X^{1},\ldots ,X^{N}\mathbf {]\in }{\mathcal {L}}\).
For
\((\mathbf {Q},\,\mathbf {a,}\,A)\in \mathscr {Q}\mathbf {\times }{\mathbb {R}} ^{N}\mathbf {\times }{\mathbb {R}}\) define
$$\begin{aligned} U_{n}^{Q^{n}}(a^{n})&:&=\sup \left\{ {\mathbb {E}}\left[ u_{n}(X^{n}+Y)\right] \mid Y\in {\mathcal {L}}^{n},E_{Q^{n}}[Y]\le a^{n}\right\} \,, \end{aligned}$$
(16)
$$\begin{aligned} S^{\mathbf {Q}}(A)&:&=\sup \left\{ \sum _{n=1}^{N}U_{n}^{Q^{n}}(a^{n})\mid \mathbf {a\in }{\mathbb {R}}^{N}\text { s.t. }\sum _{n=1}^{N}a^{n}\le A\right\} \,, \end{aligned}$$
(17)
$$\begin{aligned} \Pi ^{\mathbf {Q}}(A)&:&=\sup \left\{ {\mathbb {E}}\left[ \sum _{n=1}^{N}u_{n}(X^{n}+Y^{n})\right] \mid \mathbf {Y}\in {\mathcal {L}} ,\,\sum _{n=1}^{N}E_{Q^{n}}[Y^{n}]\le A\right\} \,. \end{aligned}$$
(18)
Obviously, such quantities depend also on
\(\mathbf {X}\), but as
\(\mathbf {X}\) will be kept fixed throughout most of the analysis, we may avoid to explicitly specify this dependence in the notations. As
\(u_{n}\) is increasing we can replace, in the definitions of
\(U_{n}^{Q^{n}}(a^{n}),\) \(S^{ \mathbf {Q}}(A)\) and
\(\Pi ^{\mathbf {Q}}(A)\) the inequality in the budget constraint with an equality.
When a vector
\(\mathbf {Q}\in \mathscr {Q}\) is assigned, we can consider two problems. First, for each
n,
\(U_{n}^{Q^{n}}(a^{n})\) is the optimal value of the classical one dimensional expected utility maximization problem with random endowment
\(X^{n}\) under the budget constraint
\(E_{Q^{n}}[Y]\le a^{n}\) , determined by the real number
\(a^{n}\) and the valuation operator
\( E_{Q^{n}}[\cdot ]\) associated to
\(Q^{n}\). Second, if we interpret the quantity
\(\sum _{n=1}^{N}u_{n}(\cdot )\) as the aggregated utility of the system, then
\(\Pi ^{\mathbf {Q}}(A)\) is the maximal expected utility of the whole system
\(\mathbf {X,}\) among all
\(\mathbf {Y}\in {{\mathcal {L}}}\) satisfying the overall budget constraint
\(\sum _{n=1}^{N}E_{Q^{n}}\left[ Y^{n} \right] \le A\). Notice that in these problems the vector
\(\mathbf {Y}\) is not required to belong to
\({\mathcal {C}}_{{\mathbb {R}}}\), but only to the vector space
\({\mathcal {L}}\). We will show in Lemma
4.11 the quite obvious equality
\(S^{\mathbf {Q}}(A)=\Pi ^{\mathbf {Q}}(A).\)