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Quantile-based risk sharing with heterogeneous beliefs

  • Paul Embrechts
  • Haiyan Liu
  • Tiantian Mao
  • Ruodu Wang
Full Length Paper Series B
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Abstract

We study risk sharing problems with quantile-based risk measures and heterogeneous beliefs, motivated by the use of internal models in finance and insurance. Explicit forms of Pareto-optimal allocations and competitive equilibria are obtained by solving various optimization problems. For Expected Shortfall (ES) agents, Pareto-optimal allocations are shown to be equivalent to equilibrium allocations, and the equilibrium pricing measure is unique. For Value-at-Risk (VaR) agents or mixed VaR and ES agents, a competitive equilibrium does not exist. Our results generalize existing ones on risk sharing problems with risk measures and belief homogeneity, and draw an interesting connection to early work on optimization properties of ES and VaR.

Keywords

Risk sharing Competitive equilibrium Belief heterogeneity Quantiles Non-convexity Risk measures 

Mathematics Subject Classification

91A06 91B50 46N10 

1 Introduction

The mid to late nineties of the last century were exciting times for Quantitative Risk Management (QRM): Value-at-Risk (VaR) first appeared on Wall Street around 1994, Expected Shortfall (ES)1 was early on considered as a viable (convex) alternative, and mathematicians started looking into optimization problems under VaR and ES objectives or constraints. As one of the early contributors to these developments, Georg Pflug stressed the importance of applications of optimization techniques to QRM problems in finance and insurance; see for instance Pflug [14]. Since then, G. Pflug contributed widely to the broader realm of QRM. Relevant examples are to be found in representation theory [15], portfolio selection [17], stochastic optimization [16], and model ambiguity [20]; see the recent paper [18] for a review, and [19] for a very pedagogic introduction to the field. As a consequence of some of our results, we obtain generalizations of some VaR and ES optimization properties in [14].

The main focus of this paper is risk sharing problems with quantile-based risk measures and heterogeneous beliefs, where various optimization problems naturally appear. Quantile-based risk measures, including VaR and ES, are the standard risk metrics used in current banking and insurance regulation, such as Basel II, III, Solvency II, and the Swiss Solvency Test. Risk sharing problems via VaR or ES are studied in the context of capital optimization; see Embrechts et al. [9] and the references therein.2

In the current regulatory frameworks (e.g. [4]), internal models are extensively used, naturally leading to model heterogeneity, that is, firms use different models for the same future events. See  [8] for a recent discussion on the use of internal models in banking and insurance. Heterogeneous beliefs are typically represented by a collection of probability measures to reflect the divergence of agents’ viewpoints3 on the distributions of risks. In this model landscape, the various agents may not be fully informed on the internal models used by competitors and hence the search for a competitive equilibrium becomes relevant (see Sect. 2 for definitions). For a discussion on heterogeneous beliefs in finance, see e.g. [26] and the references therein. Technically, quantile-based risk sharing problems with heterogeneous beliefs are essentially different from these with homogeneous beliefs or these based on expected utilities. For instance, the risk sharing problem is straightforward for ES agents if all agents use the same probability measure as in [9], but highly non-trivial in the setting of heterogeneous beliefs. Moreover, an expected utility is linear with respect to the underlying probability measure, whereas quantile-based risk measures are not.

In this paper, we concentrate mainly on the mathematical results and provide only brief discussions on their economic relevance. Our main contributions are summarized as follows. Explicit formulas of Pareto-optimal allocations and competitive equilibria are obtained for ES agents, and the Fundamental Theorems of Welfare Economics (see e.g. [25]) are established. For the case of VaR agents and that of mixed VaR, ES and RVaR (see Sect. 6 for a definition) agents, Pareto-optimal allocations share a similar form as in the case of ES agents, but competitive equilibria do not exist. In all cases, we find a Pareto-optimal allocation \((X^*_1, \ldots ,X^*_n)\) of the general (but not unique) form
$$\begin{aligned} X^*_i=(X-x^*)\mathrm {I}_{A_i^*}+\frac{x^*}{n},~~i=1, \ldots ,n, \end{aligned}$$
(1)
where X is the total risk to share, \((A^*_1, \ldots ,A^*_n)\) is a partition of the sample space, and \(x^*\) is a constant. Nevertheless, the determination of \((x^*,A^*_1, \ldots ,A_n^*)\) for ES agents is computationally very different from that for VaR agents. As an interesting consequence of our main results, we obtain a multiple-measure version of the optimization formula of ES of Rockafellar and Uryasev [22, 23] and Pflug [14]. Thanks to the convexity of ES, results in [5] on convex risk measures become helpful in deriving the Pareto-optimal allocations for ES agents; in the case of VaR, which is not convex, optimization problems become more involved. Furthermore, the dependence structure of the Pareto-optimal allocation in (1) can be described as mutual exclusivity (see [21]); this is in sharp contrast to comonotonicity in the classic setting of risk sharing with expected utilities or convex risk measures (see [24]).

2 Preliminaries

2.1 Risk sharing

Let \((\varOmega ,{\mathcal {F}})\) be a measurable space and \({\mathcal {P}}\) be the set of all probability measures on \((\varOmega ,{\mathcal {F}})\). Let \(\mathcal {X}\) be the set of bounded random variables on \((\varOmega ,{\mathcal {F}})\). Given a random variable \(X\in \mathcal {X}\), we define the set of allocations of X as
$$\begin{aligned} \mathbb {A}_n(X)=\left\{ (X_1,\ldots ,X_n)\in \mathcal {X}^n: \sum _{i=1}^nX_i=X\right\} . \end{aligned}$$
(2)
There are n agents in the risk sharing problem. For \(i=1, \ldots ,n\), agent i is equipped with a risk measure \(\rho _i:\mathcal {X}\rightarrow \mathbb {R}\), which is the agent’s objective to minimize. The risk measures \(\rho _1, \ldots ,\rho _n\) used in this paper shall later be specified as VaR and ES under different probability measures.

We consider two classic notions of risk sharing: Pareto optimality and competitive equilibria. First, a Pareto-optimal allocation is one that cannot be strictly improved.

Definition 1

(Pareto-optimal allocations) Fix the risk measures \(\rho _1, \ldots ,\rho _n\) and the total risk \(X\in \mathcal {X}\). An allocation \((X_1, \ldots ,X_n)\in \mathbb {A}_n(X)\) is Pareto-optimal if for any allocation \((Y_1, \ldots ,Y_n)\in {\mathbb {A}}_n(X)\), \(\rho _i(Y_i)\le \rho _i(X_i)\) for all \(i=1, \ldots ,n\) implies \(\rho _i(Y_i)= \rho _i(X_i)\) for all \(i=1, \ldots ,n\).

Next we formulate competitive equilibria for a one-period exchange market in the classic sense of Arrow-Debreu as in [11] and [9]. To reach a competitive equilibrium, agents in the market minimize their own risk measures by trading with each other. Assume that agent i has an initial risk (random loss) \(\xi _i\in \mathcal {X}\) for \(i=1, \ldots ,n\). Let \(X=\sum _{i=1}^n \xi _i\) be the total risk. A probability measure \(Q \in {\mathcal {P}}\) represents the pricing rule (risk-neutral probability measure) for the microeconomic market among the agents, that is, by taking a risk Y in this market,4 one receives a monetary payment of \(\mathbb {E}^{Q}[Y]\).

For each \(i=1, \ldots ,n\), agent i may trade the initial risk \(\xi _i\) for a new position \(X_i\in \mathcal {X}\), and this under the budget constraint \(\mathbb {E}^Q[X_i]\ge \mathbb {E}^Q[\xi _i]\). In general, the budget constraint will be binding (equality is attained) as the admissible set \(\mathcal {X}\) is rich enough. In this setting, each agent’s target is
$$\begin{aligned} \begin{array}{ll} \hbox {to minimize}~~&{} \rho _i(X_i) ~~~\text{ over }~~~X_i\in \mathcal {X}\\ \hbox {subject to}~~&{} \mathbb {E}^Q[X_i]\ge \mathbb {E}^Q[\xi _i], \end{array}~~~~i=1,\ldots ,n. \end{aligned}$$
(3)
To reach an equilibrium, the market clearing equation
$$\begin{aligned} \sum _{i=1}^n X_i^*=X=\sum _{i=1}^n{\xi _i} \end{aligned}$$
needs to be satisfied, where \(X_i^*\) solves (3), \(i=1, \ldots ,n\).

Definition 2

(Competitive equilibria) Fix the risk measures \(\rho _1, \ldots ,\rho _n\), the initial risks \(\xi _1, \ldots ,\xi _n\in \mathcal {X}\) and the total risk \(X=\sum _{i=1}^n \xi _i\). A pair \((Q,(X_1^*, \ldots ,X_n^*))\in {\mathcal {P}} \times \mathbb {A}_n(X)\) is a competitive equilibrium if
$$\begin{aligned} X_i^*\in \mathop {\hbox {arg min}}\limits _{X_i\in \mathcal {X}} \left\{ \rho _i(X_i):\mathbb {E}^{Q}[X_i]\ge \mathbb {E}^Q[\xi _i]\right\} ,~i=1, \ldots ,n. \end{aligned}$$
(4)
The probability measure Q in a competitive equilibrium is called an equilibrium pricing measure, and the allocation \((X_1^*, \ldots ,X_n^*)\) in a competitive equilibrium is called an equilibrium allocation.

It is well known that, through the classic Fundamental Theorems of Welfare Economics (e.g. [25]), Pareto-optimal allocations and equilibrium allocations are closely related. This relationship will become clear in our setting through the main results of the paper.

2.2 VaR, ES, and agents with heterogeneous beliefs

The key feature of this paper is belief heterogeneity among agents. The heterogeneity of probability measures means that the agents hold possibly different beliefs (models) about the future of the market. Following the setup of homogeneous beliefs in [9], we mainly consider two popular risk measures, the Value-at-Risk (VaR) and the Expected Shortfall (ES), both widely used in modern banking and insurance regulation. For a random loss \(X \in \mathcal {X}\) and a given level \(\alpha \in [0, 1)\), its VaR under a probability measure \(Q\in {\mathcal {P}}\) is defined as
$$\begin{aligned} {\mathrm {VaR}}_\alpha ^{Q}(X) =\inf \{x\in \mathbb {R}: Q(X> x)\le \alpha \}. \end{aligned}$$
(5)
Note that \(\mathrm {VaR}_\alpha ^Q(X)\) is the left end-point of the interval of \((1-\alpha )\)-quantiles of X under Q. For \(X\in \mathcal {X}\), the Expected Shortfall (ES) at level \(\alpha \in (0,1)\) under the probability measure \(Q\in {\mathcal {P}}\) is defined as
$$\begin{aligned} \mathrm {ES}_{\alpha }^{Q}(X)= \frac{1}{\alpha }\int _0^\alpha \mathrm {VaR}_u^{Q}(X)\mathrm {d}u. \end{aligned}$$
(6)
A well-known optimization property linking VaR and ES is established in [22] and [14], namely,
$$\begin{aligned} \mathrm {ES}_\alpha ^Q(X)&=\min \left\{ \frac{1}{\alpha }\mathbb {E}^Q[(X-x)_+]+x :x\in \mathbb {R}\right\} , \end{aligned}$$
(7)
and
$$\begin{aligned} \mathrm {VaR}_\alpha ^Q(X)&\in \hbox {arg min}\,\left\{ \frac{1}{\alpha }\mathbb {E}^Q[(X-x)_+]+x :x\in \mathbb {R}\right\} . \end{aligned}$$
(8)
In this paper, we will generalize the above result in a multi-measure setting.

For \(i=1, \ldots ,n\), let agent i be equipped with a probability measure \(Q_i\in {\mathcal {P}}\) representing her belief about the future randomness. This agent’s objective is to minimize a VaR or an ES, and she shall be referred to as a VaR agent or an ES agent, respectively. We will also consider RVaR agents as in [9]; see Sect. 6.

To study risk sharing problems for risk measures, define the inf-convolution of risk measures (see [24]) as
$$\begin{aligned} \mathop { \square } \limits _{i=1}^n \rho _i (X) = \inf \left\{ \sum _{i=1}^n \rho _i(X_i): (X_1, \ldots ,X_n)\in {\mathbb {A}}_n(X) \right\} ,~~X\in \mathcal {X}. \end{aligned}$$
(9)
It is well-known that for monetary risk measures [3] including VaR and ES, Pareto optimality is equivalent to optimality with respect to the sum (see Proposition 1 of [9]). More precisely, \((X_1, \ldots ,X_n)\) is a Pareto-optimal allocation of X if and only if
$$\begin{aligned} \sum _{i=1}^n \rho (X_i) = \mathop { \square } \limits _{i=1}^n \rho _i (X) . \end{aligned}$$
(10)
Therefore, the following two optimization problems are of crucial importance in our study of risk sharing problems, namely,
$$\begin{aligned} \mathop { \square } \limits _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i} (X) = \inf \left\{ \sum _{i=1}^n \mathrm{VaR}_{\alpha _i}^{Q_i}(X_i): (X_1, \ldots ,X_n)\in {\mathbb {A}}_n(X)\right\} , \end{aligned}$$
(11)
and
$$\begin{aligned} \square _{i=1}^n \mathrm {ES}_{\alpha _i}^{Q_i}(X) = \inf \left\{ \sum _{i=1}^n\mathrm {ES}_{\alpha _i}^{Q_i}(X_i): (X_1, \ldots ,X_n)\in {\mathbb {A}}_n(X) \right\} , \end{aligned}$$
(12)
where \(\alpha _i\in (0,1)\), \(i=1, \ldots ,n\).

Notation. Throughout the paper, we use \(\mathrm {I}_A\) to represent the indicator function of the event \(A\in {\mathcal {F}}\), and let \(\pi _n(A)\) be the set of n-partitions of \((A,{\mathcal {F}}|_A).\) For real numbers \(x_1, \ldots ,x_n\), write \(\bigwedge _{i=1}^n x_i=\min \{x_1, \ldots , x_n\}\) and \(\bigvee _{i=1}^nx_i=\max \{x_1, \ldots , x_n\}\).

3 Pareto-optimal allocations for ES agents

In this section, we investigate Pareto-optimal allocations for ES agents. Throughout this section, \(\alpha _1,\ldots ,\alpha _n\in (0,1)\), \(Q_1,\ldots , Q_n\in {\mathcal {P}}\), and the risk measure of agent i is \(\mathrm {ES}_{\alpha _i}^{Q_i}\), \(i=1,\ldots ,n\). We first give a necessary and sufficient condition for the existence of a Pareto-optimal allocation. In the proposition below, \(\sup (\varnothing )\) is set to \(-\infty \) by convention.

Proposition 1

For \(X\in \mathcal {X}\), the following hold.
  1. (i)
    \( \square _{i=1}^n \mathrm {ES}_{\alpha _i}^{Q_i} (X)=\sup \{ \mathbb {E}^Q[X]:Q\in \overline{{\mathcal {Q}}} \}\), where
    $$\begin{aligned} \overline{{\mathcal {Q}}}=\left\{ Q\in {\mathcal {P}}: \frac{\mathrm {d}Q}{\mathrm {d}Q_i} \le \frac{1}{\alpha _i},~i=1,\ldots ,n \right\} .\end{aligned}$$
    (13)
     
  2. (ii)
    A Pareto-optimal allocation of X exists if and only if
    $$\begin{aligned} \sum _{i=1}^n \frac{1}{\alpha _i}Q_i(A_i)\ge 1 ~~\text{ for } \text{ all } (A_1, \ldots ,A_n)\in \pi _n(\varOmega ). \end{aligned}$$
    (14)
     

Proof

Note that each \(\mathrm {ES}_{\alpha _i}^{Q_i}\) is a convex risk measure. The part (i) follows immediately from [5]. Moreover, the “if” part of (ii) follows from Theorem 11.3 of [24]. It suffices to show the “only if” part of (ii). Note that a Pareto-optimal allocation of X exists only if \(\overline{{\mathcal {Q}}}\) is non-empty. We assert that this is in turn equivalent to (14). Indeed, for any \(A\in {\mathcal {F}}\), define
$$\begin{aligned} Q'(A) = \min \left\{ \sum _{i=1}^n\frac{Q_i(A\cap A_i)}{\alpha _i}: (A_1,\ldots ,A_n)\in \pi _n(\varOmega )\right\} . \end{aligned}$$
It can be verified that \(Q'\) satisfies monotonicity and \(\sigma \)-additivity with \(Q'(\varnothing )=0\), that is, \(Q'\) is a measure on \((\varOmega ,{\mathcal {F}})\). On the other hand, for a probability measure \(Q\), \(Q\le Q'\) if and only if \(Q\in \overline{{\mathcal {Q}}}\). To see this, first note that if \(Q\le Q'\), then for \(A\in {\mathcal {F}}\), letting \(A_i=A\) yields \( Q(A) \le Q'(A) \le Q_i(A)/{\alpha _i}\), \(i=1,\ldots ,n\). This implies \({\mathrm {d}Q}/{\mathrm {d}Q_i} \le 1/{\alpha _i}\) and thus, \(Q\in \overline{{\mathcal {Q}}}\). On the other hand, \(Q\in \overline{{\mathcal {Q}}}\) implies
$$\begin{aligned} Q(A) \le \frac{Q_i(A)}{\alpha _i}~~\hbox {for any}~A\in {\mathcal {F}},~~~i=1,\ldots ,n, \end{aligned}$$
and hence, for any \((A_1, \ldots ,A_n)\in \pi _n(\varOmega )\),
$$\begin{aligned} Q(A) = \sum _{i=1}^n Q(A\cap A_i) \le \sum _{i=1}^n \frac{Q_i(A\cap A_i)}{\alpha _i}~~\hbox {for any}~A\in {\mathcal {F}}, \end{aligned}$$
so that \(Q\le Q'\). That is, \(Q\le Q'\) if and only if \(Q\in \overline{{\mathcal {Q}}}\). Hence, \(\overline{{\mathcal {Q}}}\) is non-empty if and only if \(Q'(\varOmega )\ge 1\), that is (14) holds. \(\square \)

Remark 1

From Proposition 1 (ii), the existence of a Pareto-optimal allocation only depends on \((\alpha _1, \ldots ,\alpha _n)\) and \((Q_1, \ldots ,Q_n)\), but not on the total risk X.

Next we explicitly describe Pareto-optimal allocations for the ES agents. First we translate the inf-convolution of ES into another optimization problem. For given \(Q_1, \ldots ,Q_n\in {\mathcal {P}}\), we let Q be a measure dominating \(Q_1, \ldots ,Q_n\), and
$$\begin{aligned} B_j=\left\{ \frac{1}{\alpha _j}\frac{\mathrm {d}Q_j }{\mathrm {d}Q} =\bigwedge _{i=1}^n \frac{1}{\alpha _i}\frac{\mathrm {d}Q_i }{\mathrm {d}Q} \right\} {\Bigr \backslash } \left( \bigcup _{k=1}^{j-1} B_k\right) ,~j=1, \ldots ,n. \end{aligned}$$
(15)
We shall fix \(B=(B_1, \ldots ,B_n)\) as in (15) throughout the rest of Sects. 3 and 4. Apparently, the choice of Q is irrelevant in the definition of \(B_1, \ldots ,B_n\), and one can safely choose \(Q= \sum _{i=1}^n Q_i/n\). Roughly speaking, \(B_j\) is the set of points on which \(\mathrm {d}Q_j/\alpha _j\) is the smallest among \(\mathrm {d}Q_i/\alpha _i\), \(i=1, \ldots ,n\), and we only count once if there is a tie for the minimum. From the definition of \(B_1, \ldots ,B_n\), it is straightforward to verify
$$\begin{aligned} \min \left\{ \sum _{i=1}^n \frac{1}{\alpha _i}Q_i(A_i):(A_1, \ldots ,A_n)\in \pi _n(\varOmega )\right\} = \sum _{i=1}^n \frac{1}{\alpha _i}Q_i(B_i), \end{aligned}$$
and therefore by Proposition 1 (ii), a Pareto-optimal allocation exists if and only if \(\sum _{i=1}^nQ_i(B_i)/{\alpha _i} \ge 1\).

Proposition 2

Assume \( \overline{{\mathcal {Q}}} \) in (13) is non-empty. Then for \(X\in \mathcal {X}\),
$$\begin{aligned}&\mathop { \square } \limits _{i=1}^n \mathrm {ES}_{\alpha _i}^{Q_i} (X) \\&\quad = \min \left\{ \sum _{i=1}^n \frac{1}{\alpha _i} \mathbb {E}^{Q_i}[(X-x)_+\mathrm {I}_{A_i}] +x:(A_1, \ldots ,A_n)\in \pi _n(\varOmega ),~x\in \mathbb {R}\right\} \\&\quad = \min \left\{ \sum _{i=1}^n \frac{1}{\alpha _i} \mathbb {E}^{Q_i}[(X-x)_+\mathrm {I}_{B_i}] +x: x\in \mathbb {R}\right\} . \end{aligned}$$

Proof

Fix \(X\in \mathcal {X}\). As \( \overline{{\mathcal {Q}}} \) is non-empty, or equivalently (14) holds, we have \(\square _{i=1}^n \mathrm {ES}_{\alpha _i}^{Q_i}(X) >-\infty \). Define
$$\begin{aligned} V(X)= \inf \left\{ \sum _{i=1}^n\frac{1}{\alpha _i} \mathbb {E}^{Q_i}[(X-x)_+\mathrm {I}_{A_i}]+x:(A_1, \ldots ,A_n)\in \pi _n(\varOmega ),~x\in \mathbb {R}\right\} . \end{aligned}$$
(16)
We first show
$$\begin{aligned} \mathop { \square } \limits _{i=1}^n \mathrm {ES}_{\alpha _i}^{Q_i} (X)\le V(X). \end{aligned}$$
(17)
For any \((A_1, \ldots ,A_n)\in \pi _n(\varOmega )\) and \(x\in \mathbb {R}\), let \(X_i=(X-x)\mathrm {I}_{A_i}+\frac{x}{n}\), \(i=1,\ldots ,n\). Clearly, \(X_1+\cdots +X_n=X\). Moreover, for \(i=1,\ldots ,n\),
$$\begin{aligned} \mathrm {ES}_{\alpha _i}^{Q_i}(X_i)&=\mathrm {ES}_{\alpha _i}^{Q_i} ((X-x)\mathrm {I}_{A_i})+\frac{x}{n} \\ {}&\le \mathrm {ES}_{\alpha _i}^{Q_i} ((X-x)_+\mathrm {I}_{A_i})+\frac{x}{n}\le \frac{1}{\alpha _i} \mathbb {E}^{Q_i}[(X-x)_+\mathrm {I}_{A_i}]+\frac{x}{n}. \end{aligned}$$
Therefore, for all \(x\in \mathbb {R}\) and \((A_1, \ldots ,A_n)\in \pi _n(\varOmega )\), there exists \((X_1,\ldots ,X_n)\in {\mathbb {A}}_n(X)\) such that
$$\begin{aligned} \sum _{i=1}^n\mathrm {ES}_{\alpha _i}^{Q_i}(X_i)\le \sum _{i=1}^n\frac{1}{\alpha _i} \mathbb {E}^{Q_i}[(X-x)_+\mathrm {I}_{A_i}]+x. \end{aligned}$$
It follows that (17) holds.
Next we need to show \(\square _{i=1}^n \mathrm {ES}_i^{Q_i}(X) \ge V(X)\). For \(A:=(A_1, \ldots ,A_n)\in \pi _n(\varOmega )\), write
$$\begin{aligned} v_A(x)=\sum _{i=1}^n\frac{1}{\alpha _i} \mathbb {E}^{Q_i}[(X-x)_+\mathrm {I}_{A_i}]+x,~~x\in \mathbb {R}. \end{aligned}$$
(18)
Clearly, \(v_A\) is a convex function and has right-derivative at any point in \(\mathbb {R}\). Denote by \(v_A'(x)\) the right-derivative of \(v_A\) at \(x\in \mathbb {R}\) and
$$\begin{aligned} v_A'(x)= -\sum _{i=1}^n\frac{1}{\alpha _i}{Q_{i}} (X>x, A_i) +1. \end{aligned}$$
Therefore, \(v_A'\) is an increasing function of x, with \(v_A'(\infty )=1\) and
$$\begin{aligned} v_A'(-\infty )=1-\sum _{i=1}^n\frac{1}{\alpha _i}{Q_{i}} ( A_i)\le 0 \end{aligned}$$
as a result of the condition  (14). Let \(x_A^*=\inf \{x\in \mathbb {R}: v_A'(x)\ge 0\}\). Obviously \(x_A^*\) minimizes \(v_A\). Moreover, noting that \(Q_i(X>x,A_i)\) is right-continuous in x for \(i=1,\ldots ,n\), \(v_A'\) is a right-continuous function. Therefore, \(v_A'(x_A^*)\ge 0\), and equivalently,
$$\begin{aligned} \sum _{i=1}^n \frac{1}{\alpha _i}{Q_{i}} (X>x^*_A, A_i) \le 1. \end{aligned}$$
(19)
Next, let
$$\begin{aligned} Q_{A}^*(C) = \sum _{i=1}^n \frac{1}{\alpha _i} Q_i(C\cap A_i\cap \{X>x^*_A\}) ,~~C\in {\mathcal {F}}. \end{aligned}$$
Let us verify
  1. 1.

    \(Q_A^*\) is \(\sigma \)-additive, because it is the sum of n measures.

     
  2. 2.

    \(Q_A^*(\varOmega )\le 1\) by (19).

     
Now we make some adjustment to \(Q_A^*\) so that \(Q_A^*(\varOmega )=1\). Note that by a symmetric argument,
$$\begin{aligned} \sum _{i=1}^n \frac{1}{\alpha _i}{Q_{i}} (X\ge x^*_A, A_i) \ge 1. \end{aligned}$$
(20)
Therefore, if \(Q_A^*(\varOmega )< 1\), we can replace \(Q_A^*\) by \(Q_A^{**}\), which is a linear combination of \(Q_A^*\) and \(Q_A'\), defined as
$$\begin{aligned} Q_{A}'(C) = \sum _{i=1}^n \frac{1}{\alpha _i} Q_i(C\cap A_i\cap \{X\ge x^*_A\} ),~~C\in {\mathcal {F}}, \end{aligned}$$
so that \(Q_A^{**}(\varOmega )=1\). In the following we safely assume \(Q_A^*(\varOmega )=1\) (otherwise we just replace it by \(Q_A^{**}\)), that is, \(Q_A^*\) is a probability measure. We can verify
$$\begin{aligned} \mathbb {E}^{Q_A^*}[X]&= \sum _{i=1}^n \frac{1}{\alpha _i}\mathbb {E}^{Q_i}[X\mathrm {I}_{\{X>x_A^*\} }\mathrm {I}_{A_i}]\nonumber \\&=\sum _{i=1}^n \frac{1}{\alpha _i}\mathbb {E}^{Q_i}[(X-x_A^* +x_A^*)\mathrm {I}_{\{X>x_A^*\} }\mathrm {I}_{A_i}]\nonumber \\&=\sum _{i=1}^n \frac{1}{\alpha _i} \mathbb {E}^{Q_i}[(X-x_A^*)_+\mathrm {I}_{A_i}]+ x_A^* Q_A^*(\varOmega )= v_A(x_A^*). \end{aligned}$$
(21)
Therefore,
$$\begin{aligned} V(X)\le \mathbb {E}^{Q_A^*}[X]~~ \text{ for } \text{ all } (A_1, \ldots ,A_n)\in \pi _n(\varOmega ). \end{aligned}$$
(22)
Let \(B_j\), \(j=1,\ldots ,n\) be defined as in (15). Clearly \(B=(B_1,\ldots ,B_n)\in \pi _n(\varOmega )\). It follows that, for \(C\in {\mathcal {F}}\),
$$\begin{aligned} Q_B^*(C)&= \sum _{i=1}^n \frac{1}{\alpha _i} Q_i(C\cap B_i\cap \{X>x^*_B\} ) \\&\le \sum _{i=1}^n \frac{1}{\alpha _j}Q_j(C\cap B_i)\le \frac{1}{\alpha _j} Q_j(C), ~~j=1,\ldots ,n. \end{aligned}$$
As a consequence, we have \(Q_B^*\in \overline{{\mathcal {Q}}}\). It follows that
$$\begin{aligned} \mathbb {E}^{Q_B^*} [X] \le \sup _{Q\in \overline{{\mathcal {Q}}}} \mathbb {E}^Q[X]= \mathop { \square } \limits _{i=1}^n \mathrm {ES}_{\alpha _i}^{Q_i} (X). \end{aligned}$$
Together with (17) and (22), we have
$$\begin{aligned} V(X)\le \mathbb {E}^{Q_B^*}[X]\le \mathop { \square } \limits _{i=1}^n \mathrm {ES}_{\alpha _i}^{Q_i} (X)\le V(X). \end{aligned}$$
(23)
This completes the proof. \(\square \)
With the help of Proposition 2, we are ready to present an explicit form of Pareto-optimal allocations for the ES agents. Define
$$\begin{aligned} x_B ^* =\inf \left\{ x\in \mathbb {R}: \sum _{i=1}^n \frac{1}{\alpha _i}{Q_{i}} (X>x,B_i) \le 1\right\} , \end{aligned}$$
(24)
and
$$\begin{aligned} y_B^*= \inf \left\{ x\in \mathbb {R}: \sum _{i=1}^n \frac{1}{\alpha _i}{Q_{i}} (X>x,B_i)<1\right\} . \end{aligned}$$
(25)
The quantities \(x_B^*\) and \(y_B^*\) will be used repeatedly later in the paper. Note that, if \(Q_1=\dots =Q_n=Q\), then by definition of \((B_1, \ldots ,B_n)\),
$$\begin{aligned} x_B^*= \inf \left\{ x\in \mathbb {R}: \frac{1}{\alpha }{Q} (X>x) \le 1\right\} =\mathrm {VaR}_{\alpha }^{Q}(X), \end{aligned}$$
where \(\alpha =\bigvee _{i=1}^n\alpha _i\). Thus, \(x_B^*\) can be seen as a generalized left-quantile (VaR) of X in the multi-measure framework, whereas \(y_B^*\) is a generalized right-quantile of X. By definition, \(Q_i(x_B^*<X<y_B^*,~B_i)=0\) for \(i=1, \ldots ,n\). Similarly to the left/right-quantiles, \(x_B^*\) and \( y_B^*\) are often identical for practical settings.

Theorem 1

Assume \( \overline{{\mathcal {Q}}} \) in (13) is non-empty. A Pareto-optimal allocation \((X^*_1, \ldots ,X^*_n)\) of \(X\in \mathcal {X}\) is given by
$$\begin{aligned} X^*_i=(X-x^*)\mathrm {I}_{B_i} + \frac{x^*}{n},~~i=1, \ldots ,n, \end{aligned}$$
(26)
for \(x^*\in [x_B^*,y_B^*]\), where \((B_1, \ldots ,B_n)\), \(x_B^*\) and \(y_B^*\) are in (15), (24) and (25).

Proof

For \(i=1,\ldots ,n\), using (7) with \(x={x^*}/{n}\) , we have
$$\begin{aligned} \mathrm {ES}_{\alpha _i}^{Q_i}(X_i^*)&\le \frac{1}{\alpha _i} \mathbb {E}^{Q_i}[(X-x^*)_+\mathrm {I}_{B_i}]+\frac{x^*}{n}. \end{aligned}$$
By taking a derivative of \(v_B\) defined by (18) with \(A_i\) replaced by \(B_i\), \(i=1,\ldots ,n\), we have, for \(x^*\in [x_B^*,y_B^*]\),
$$\begin{aligned} \sum _{i=1}^n\frac{1}{\alpha _i} \mathbb {E}^{Q_i}[(X-x^*)_+\mathrm {I}_{B_i}]+ {x^*} =\min \left\{ \sum _{i=1}^n \frac{1}{\alpha _i} \mathbb {E}^{Q_i}[(X-x)_+\mathrm {I}_{B_i}] +x: x\in \mathbb {R}\right\} . \end{aligned}$$
Therefore, by Proposition 2, we have \(\sum _{i=1}^n \mathrm {ES}_{\alpha _i}^{Q_i}(X_i^*) \le \square _{i=1}^n \mathrm {ES}_{\alpha _i}^{Q_i} (X)\). This implies the Pareto optimality of \((X_1^*, \ldots ,X_n^*)\). \(\square \)

The economic interpretation of the Pareto-optimal allocation in (26) is very simple. For each \(i=1, \ldots ,n\), agent i takes the risk \((X-x^*)\mathrm {I}_{B_i}\) plus a constant (side-payment). Looking at the definition of \(B_i\), it is clear that agent i thinks the event \(B_i\) is the least likely to happen, compared to other agents’ beliefs on the same event. The rest of the risk, which is more likely to happen according to agent i (relative to other agents), is taken by others. This intuitively implies, quoting  [6], “When agents disagree about disaster risk, they will insure each other against the types of disasters they fear most”.

We make some technical observations about Theorem 1.
  1. (i)

    A constant shift (side-payment) among \(X_1^*,\ldots ,X_n^*\) defined in (26) does not compromise the optimality; hence, \((X_1^*+c_1,\ldots ,X_n^*+c_n)\) is also a Pareto-optimal allocation, where \(c_1,\ldots ,c_n\) are constants and \(\sum _{i=1}^nc_i=0\). Later we shall see in Proposition 3 that, under an extra condition, the Pareto-optimal allocation is unique on the set \(\{X>y_B^*\}\) up to constant shifts.

     
  2. (ii)

    The dependence structure of the Pareto-optimal allocation \((X_1^*,\ldots ,X_n^*)\) in (26) is worth noting. On the set \(\{X>x^*\}\), \( X_1^*,\ldots ,X_n^*\) are mutually exclusive, a form of extremal negative dependence (see [21]). This is in sharp contrast to the case of homogeneous beliefs, where a Pareto-optimal allocation for strictly convex functionals is always comonotonic (see [24]), a form of extremal positive dependence.

     
  3. (iii)
    As an immediate consequence of Theorem 1, for \(x^*\in [x_B^*,y_B^*]\),
    $$\begin{aligned} \mathop { \square } \limits _{i=1}^n \mathrm {ES}_{\alpha _i}^{Q_i} (X) = \sum _{i=1}^n \frac{1}{\alpha _i} \mathbb {E}^{Q_i}[(X-x^*)_+\mathrm {I}_{B_i}]+x^*. \end{aligned}$$
    (27)
     
We can easily see that in the case of \(n=1\), Proposition 2 gives, for any \(\alpha \in (0,1)\), \(Q\in {\mathcal {P}}\) and \(X\in \mathcal {X}\),
$$\begin{aligned} \mathrm {ES}_\alpha ^Q(X) =\min \left\{ \frac{1}{\alpha }\mathbb {E}^Q[(X-x)_+]+x :x\in \mathbb {R}\right\} , \end{aligned}$$
(28)
and Theorem 1 (setting \(n=1\)) implies that the above minimum is achieved by \(x^*=\mathrm {VaR}_\alpha ^Q(X)\), a celebrated result [see (7) and (8)] established by [22]. In other words, Theorem 1 can be regarded as a generalization of the result of  [22] in a multiple-measure framework. Using  (28), we obtain the following corollary of Theorem 1, giving a solution to an optimization problem similar to (7) and (8).

Corollary 1

The optimization problem
$$\begin{aligned}&\text{ to } \text{ minimize } ~~ \sum _{i=1}^n \frac{1}{\alpha _i}\mathbb {E}^{Q_i}[(X_i-x_i)_+]+\sum _{i=1}^n x_i \\&~~~ \text{ over } ~~~~x_1, \ldots ,x_n\in \mathbb {R},~(X_1, \ldots ,X_n)\in {\mathbb {A}}_n(X) \end{aligned}$$
admits a solution \((x^*_1, \ldots ,x^*_n,X^*_1, \ldots ,X^*_n)\) where \(x^*_1=\dots =x_n^*=x^*/n\), and \(x^*\) and \(X^*_1, \ldots ,X_n^*\) are given in Theorem 1.

Remark 2

If \(Q_1=\cdots =Q_n={\mathbb {P}}\), Proposition 2 reduces to the classic result
$$\begin{aligned} \mathop { \square } \limits _{i=1}^n \mathrm {ES}_{\alpha _i}^{{\mathbb {P}}} (X)&=\min \left\{ \frac{1}{\bigvee _{i=1}^n\alpha _i} \mathbb {E}^{{\mathbb {P}}}[(X-x)_+]+x:~x\in \mathbb {R}\right\} =\mathrm {ES}^{{\mathbb {P}}}_{{\bigvee _{i=1}^n\alpha _i}}(X). \end{aligned}$$
In this case, according to Theorem 1, the Pareto-optimal allocation is one where all the risk is taken by one agent with the largest \(\alpha _i\) value, and the other agents make side-payments to this agent. This is a special case of Theorem 2 of [9].

Next we study the uniqueness of the form of Pareto-optimal allocations. Since an ES only depends on the tail part of a risk, it is natural that uniqueness can only be established on the set \(\{X>y_B^*\}\). Moreover, it is straightforward to verify that the allocation can be very flexible on the set \(\{\mathrm {d}Q_i/\mathrm {d}Q=0\}\) for each \(i=1, \ldots ,n\), where \( Q=\sum _{i=1}^n Q_i/n\). Hence, we focus our discussion on the case in which \(Q_1, \ldots ,Q_n\) are equivalent.

The following proposition characterizes the form of Pareto-optimal allocations, which requires an intuitive condition
$$\begin{aligned} \text{ the } \text{ sets } \left\{ \frac{1}{\alpha _j}\frac{\mathrm {d}Q_j }{\mathrm {d}Q} =\bigwedge _{i=1}^n \frac{1}{\alpha _i}\frac{\mathrm {d}Q_i }{\mathrm {d}Q} \right\} ,~j=1, \ldots ,n, ~\text{ are } \text{ disjoint },\end{aligned}$$
(29)
so that the sets on which \( ({\mathrm {d}Q_i }/{\mathrm {d}Q})/\alpha _i \) is the smallest, \(i=1, \ldots ,n\), are distinguishable.

Proposition 3

Suppose that \(Q_1, \ldots ,Q_n\) are equivalent to \(Q\in {\mathcal {P}}\) and (29) holds. Any Pareto-optimal allocation \((X_1^*,\ldots ,X_n^*)\) of \(X\in \mathcal {X}\) satisfies, for some constants \(c_1,\ldots ,c_n\in \mathbb {R}\),
$$\begin{aligned} (X_i^*-c_i)_+ = (X - y_B^{*})_+\mathrm {I}_{B_i} ~~Q\text{-a.s. },~~i=1,\ldots ,n, \end{aligned}$$
where \(y_B^*\) is defined in  (25).

Proof

Assume \( \overline{{\mathcal {Q}}} \) in (13) is non-empty so that a Pareto-optimal allocation exists; otherwise there is nothing to show. Let \((X_1^*,\ldots ,X_n^*)\) be a Pareto-optimal allocation and \(y_i=\mathrm {VaR}_{\alpha _i}^{Q_i}(X_i^*)\), \(i=1, \ldots ,n\). Fix \(i=1, \ldots ,n\). We assert that \(X_i^*>y_i\) implies \(X_j^*\ge y_j\) for any \(j\ne i\), \(Q_i\)-almost surely. To see this, assume that there exist i and j such that \(Q_i(X_i^*> y_i,~X_j^*< y_j)>0\). Then there exists \(\delta >0\) such that \(Q_i(X_i^*> y_i,~X_j^*< y_j-\delta )>0\). Let \(A= \{X_i^*> y_i,~X_j^*< y_j-\delta \}\). It follows that \(\mathrm {ES}_{\alpha _i}^{Q_i} (X_i^*-\delta {\mathrm {I}}_{A})< \mathrm {ES}_{\alpha _i}^{Q_i} (X_i^*)\) whereas \(\mathrm {ES}_{\alpha _j}^{Q_j} (X_j^*+\delta {\mathrm {I}}_{A})= \mathrm {ES}_{\alpha _j}^{Q_j} (X_j^*)\) since \(\mathrm{VaR}_{\alpha _i}^{Q_i}(X_i^*)=y_i\) and \(\mathrm{VaR}_{\alpha _j}^{Q_j}(X_j^*)=y_j\). This contradicts the Pareto optimality of \((X_1^*,\ldots ,X_n^*)\). Hence, we have
$$\begin{aligned} Q_i(X_i^*> y_i,~X_j^*< y_j)=0~~~\hbox {for all}~i,j=1, \ldots ,n. \end{aligned}$$
Since \(Q_1, \ldots ,Q_n\) are equivalent, it follows that
$$\begin{aligned} \sum _{i=1}^n ( X_i^*- y_i)_+ =\left( \sum _{i=1}^n X_i^*-\sum _{i=1}^n y_i\right) _+ = \left( X-\sum _{i=1}^n y_i\right) _+~~Q\text{-a.s. } \end{aligned}$$
(30)
Define \(Z_i=\frac{1}{\alpha _i}\frac{\mathrm{d}Q_i}{\mathrm{d}Q}\), \(i=1,\ldots ,n\). By (28), the minimization problem in (12) is equivalent to
$$\begin{aligned} \begin{array}{ll} \text{ to } \text{ minimize } &{}\sum \nolimits _{i=1}^n \mathbb {E}^{Q}[Z_i(X_i- \mathrm {VaR}_{\alpha _i}^{Q_i}(X_i))_+] + \sum \nolimits _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i}(X_i) \\ \text{ over } ~~~&{} (X_1, \ldots ,X_n)\in {\mathbb {A}}_n(X). \end{array} \end{aligned}$$
(31)
From (7) and (30), we know that an optimizer \((X_1^*, \ldots ,X_n^*)\) of (31) satisfies \(\sum _{i=1}^n ( X_i^*- y_i)_+= (X-\sum _{i=1}^n y_i )_+\) Q-almost surely, where \(y_i=\mathrm{VaR}_{\alpha _i}^{Q_i}(X^*_i)\), \(i=1, \ldots ,n\). Consider the optimization problem
$$\begin{aligned} \begin{array}{ll} \text{ to } \text{ minimize } ~~&{} \sum \nolimits _{i=1}^n \mathbb {E}^{Q}[Z_iW_i] + y \\ \text{ over } ~~~&{} (W_1, \ldots ,W_n)\in {\mathbb {A}}_n((X-y)_+),~y\in \mathbb {R}\\ \text{ subject } \text{ to } ~~&{} W_i\ge 0,\,i=1,\ldots ,n. \end{array} \end{aligned}$$
(32)
Note that the constraints in (31) are replaced by weaker constraints, and we allow to choose \(y\in \mathbb {R}\) in (32) which is fixed as \(y=\sum _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i}(X_i) \) in (31). From there, it is clear that the minimum value of (32) is no larger than that of (31). We shall later see that (31) and (32) are indeed equivalent. Recall \(B_j=\{Z_j = \bigwedge _{i=1}^n Z_i \}\), \(j=1,\ldots ,n\), and \(B_1,\ldots ,B_n\) are disjoint. For fixed \(y\in \mathbb {R}\), writing \(W=(X- y )_+\), the optimization problem
$$\begin{aligned} \begin{array}{ll} \text{ to } \text{ minimize } ~~&{} \sum \nolimits _{i=1}^n \mathbb {E}^{Q}[Z_iW_i] + y \\ \text{ over } ~~~&{} (W_1, \ldots ,W_n)\in {\mathbb {A}}_n(W) \\ \text{ subject } \text{ to } ~~&{} W_i\ge 0,\,i=1,\ldots ,n \end{array} \end{aligned}$$
(33)
admits a unique optimizer via point-wise optimization,
$$\begin{aligned} W_i^* = W{\mathrm {I}}_{B_i},~~i=1,\ldots ,n. \end{aligned}$$
Next, we consider the second-step optimization of (32),
$$\begin{aligned} \begin{array}{ll} \text{ to } \text{ minimize } ~~&\sum \nolimits _{i=1}^n \frac{1}{\alpha _i} \mathbb {E}^{Q_i}[ (X-y)_+{\mathrm {I}}_{B_i}] + y ~~~~\text{ over } y\in \mathbb {R}. \end{array} \end{aligned}$$
(34)
By taking a derivative with respect to y, the set of optimizers of the problem (34) is the interval \([x_B^*,y_B^*]\). Let \(y^*\in [x_B^*,y_B^*]\). For \(x_1, \ldots ,x_n\in \mathbb {R}\) with \(\sum _{i=1}^n x_i=y^*\), define \(X_i^*= (X-y^*){\mathrm {I}}_{B_i} + x_i\), \(i=1, \ldots ,n\). We can verify \((X_1^*, \ldots ,X_n^*)\in {\mathbb {A}}_n(X)\) and \((X_i^*-x_i)_+=W_i^*\), \(i=1, \ldots ,n\). Thus, optimization problems  (31) and (32) have the same minimum objective values, and an optimizer \((X_1^*, \ldots ,X_n^*) \) of (31) necessarily satisfies \((X_i^*-x_i)_+= W{\mathrm {I}}_{B_i} = (X-y^*)_+\mathrm {I}_{B_i}\) for some \(x_1, \ldots ,x_n\in \mathbb {R}\) with \(\sum _{i=1}^n x_i=y^*\).
In summary, for any Pareto-optimal allocation \((X_1^*, \ldots ,X_n^*)\), there exist \((x_1, \ldots ,x_n)\) \(\in \mathbb {R}\) and \(y^*\in [x_B^*,y_B^*]\) satisfying \(\sum _{i=1}^n x_i=y^*\), such that for \(i=1, \ldots ,n\),
$$\begin{aligned} (X_i^*-x_i)_+= (X-y^*)_+\mathrm {I}_{B_i}. \end{aligned}$$
Finally, noting that \(\{X>y^*\}=\{X>y_B^*\} \) Q-almost surely and \(y_B^*\ge y^*\), by letting \(c_i=x_i-y^*+y_B^*\), we have
$$\begin{aligned} (X_i^*-c_i)_+= (X-y_B^*)_+\mathrm {I}_{B_i},~~i=1, \ldots ,n. \end{aligned}$$
This completes the proof. \(\square \)

The following corollary of Proposition 3 characterizes all Pareto-optimal allocations for ES agents under the conditions of Proposition 3.

Corollary 2

Suppose that \(Q_1, \ldots ,Q_n\) are equivalent to \(Q\in {\mathcal {P}}\) and (29) holds. A random vector \((X_1^*,\ldots ,X_n^*)\in \mathcal {X}^n\) is a Pareto-optimal allocation of \(X\in \mathcal {X}\) if and only if it has the following form, where \(y_B^*\) is defined by  (25),
$$\begin{aligned} X_i^* = (X - y_B^{*})_+\mathrm {I}_{B_i} + Z_i + c_i ~~Q\text{-a.s. },~~i=1,\ldots ,n, \end{aligned}$$
(35)
for some \((c_1, \ldots ,c_n)\in \mathbb {R}^n\) satisfying \(\sum _{i=1}^n c_i=y_B^*\) and some \((Z_1, \ldots ,Z_n)\in {\mathbb {A}}_n(-(y_B^*-X)_+)\) satisfying \(Z_i\le 0\) and \(Z_i \mathrm {I}_{\{X>y_B^*\} \cap B_i} =0\) for \(i=1, \ldots ,n\).

Proof

By Proposition 3, any Pareto-optimal allocation \((X_1^*, \ldots ,X_n^*)\) satisfies
$$\begin{aligned} (X_i^*-c_i)_+ = (X - y_B^{*})_+\mathrm {I}_{B_i} ~~Q\text{-a.s. },~~i=1,\ldots ,n, \end{aligned}$$
which, together with \(\sum _{i=1}^n X_i^*=X\), gives (35). Next we show that \((X_1^*, \ldots ,X_n^*)\) in (35) is optimal. It is easy to verify that \((X_1^*, \ldots ,X_n^*)\) in (35) satisfies \(\sum _{i=1}^n X_i^*=X\). Note that
$$\begin{aligned} \sum _{i=1}^n \mathrm {ES}_{\alpha _i}^{Q_i} (X^*_i)&\le \sum _{i=1}^n \left( \mathrm {ES}_{\alpha _i}^{Q_i} ((X - y_B^{*})_+\mathrm {I}_{B_i}) + c_i\right) \\&= \sum _{i=1}^n \mathrm {ES}_{\alpha _i}^{Q_i} ((X - y_B^{*})_+\mathrm {I}_{B_i}) + y_B^* \\&\le \sum _{i=1}^n \frac{1}{\alpha _i} \mathbb {E}^{Q_i} [(X - y_B^{*})_+\mathrm {I}_{B_i}] + y_B^* =\mathop { \square } \limits _{i=1}^n \mathrm {ES}_{\alpha _i}^{Q_i} (X), \end{aligned}$$
where the last equality is due to (27). This shows that \((X_1^*, \ldots ,X_n^*)\) is a Pareto-optimal allocation. \(\square \)

4 Competitive equilibria for ES agents

In this section, we study competitive equilibria as in Definition 2 for ES agents. Similarly to Sect. 3, throughout this section, \(\alpha _1, \ldots ,\alpha _n\in (0,1)\), \(Q_1, \ldots , Q_n\in {\mathcal {P}}\), and the risk measure of agent i is \(\mathrm {ES}_{\alpha _i}^{Q_i}\), \(i=1, \ldots ,n\). Each agent’s objective is
$$\begin{aligned} \begin{array}{ll} \text{ to } \text{ minimize }~~ &{} \mathrm {ES}_{\alpha _i}^{Q_i}(X_i)~~~\text{ over }~~~X_i\in \mathcal {X}\\ \hbox {subject to}&{} \mathbb {E}^{Q}[X_i]\ge \mathbb {E}^Q[\xi _i] \end{array} ~~~~~~i=1, \ldots ,n, \end{aligned}$$
(36)
where \((\xi _1, \ldots ,\xi _n)\in {\mathbb {A}}_n(X)\) is the vector of initial risks.
With the Pareto optimality problem solved explicitly in Sect. 3, we establish in this section that for ES agents, a Pareto-optimal allocation is equivalent to an equilibrium allocation. Thus, the two Fundamental Theorems of Welfare Economics (FTWE) hold for ES agents.5 As in the proof of Proposition 2, throughout this section we define the probability measure \(Q_B^*\) via
$$\begin{aligned} Q_{B}^*(C) = \sum _{i=1}^n \frac{1}{\alpha _i} Q_i(C\cap B_i\cap \{X>x^*_B\}) ,~~C\in {\mathcal {F}}, \end{aligned}$$
(37)
where \(B=(B_1,\ldots ,B_n)\) and \(x_B ^*\) are defined by (15) and (24). We can verify that \(Q_B^*\) does not depend on the order of \(Q_1, \ldots ,Q_n\), although the choice of B in (15) does. Later we shall see that the probability measure \(Q_B^*\) turns out to be the unique equilibrium pricing measure for the ES agents. We first present the FTWE for ES agents.

Theorem 2

An allocation of \(X\in \mathcal {X}\) is Pareto-optimal if and only if it is an equilibrium allocation for some initial risks \((\xi _1,\ldots \xi _n)\in {\mathbb {A}}_n(X)\).

Proof

First, an equilibrium allocation is necessarily Pareto-optimal, as the so-called non-satiation condition (see [25]) holds for the ES agents. For the reader who is not familiar with the FTWE, we provide a self-contained simple proof. Suppose that \((Q,(X_1^*, \ldots ,X_n^*))\) is a competitive equilibrium, and \((X_1^*, \ldots ,X_n^*)\) is not Pareto-optimal. Then, there exists \((Y_1, \ldots ,Y_n)\in {\mathbb {A}}_n(X)\) such that \(\mathrm {ES}_{\alpha _i}^{Q_i}(Y_i)\le \mathrm {ES}_{\alpha _i}^{Q_i}(X_i^*)\) for all \(i=1, \ldots ,n\) and there exists \(j\in \{1, \ldots ,n\}\) such that \(\mathrm {ES}_{\alpha _j}^{Q_j}(Y_j)< \mathrm {ES}_{\alpha _j}^{Q_j}(X_j^*)\). If \(\mathbb {E}^Q[Y_j]\ge \mathbb {E}^Q[\xi _j]\), then \(X_j^*\) is not optimal for (36) since it is strictly dominated by \(Y_j\), and thus a contradiction. If \(\mathbb {E}^Q[Y_j]< \mathbb {E}^Q[\xi _j]\), then there exists \(k\in \{1, \ldots ,n\}\) such that \(\mathbb {E}^Q[Y_k]> \mathbb {E}^Q[\xi _k]\). Similarly, \(X_k^*\) is not optimal for (36) since it is strictly dominated by \(Y_k-\mathbb {E}^Q[Y_k]+ \mathbb {E}^Q[\xi _k]\), and thus a contradiction.

Next we show that a Pareto-optimal allocation is necessarily an equilibrium allocation. Let \((X_1^*,\ldots ,X_n^*)\) be a Pareto-optimal allocation. For \(i=1, \ldots ,n\), consider the individual optimization problem in (36) with the initial risks \(\xi _i=X_i^*\), \(i=1, \ldots ,n\), and the pricing measure \(Q_B^*\), namely
$$\begin{aligned} \min _{X_i\in \mathcal {X}}\mathrm{ES}_{\alpha _i}^{Q_i}(X_i) \ \ \hbox {subject to}\ \ \mathbb {E}^{Q_B^*}[X_i]\ge \mathbb {E}^{Q_B^*}[X_i^*]. \end{aligned}$$
(38)
Note that for any \(X_i\in \mathcal {X}\) with \(\mathbb {E}^{Q_B^*}[X_i]\ge \mathbb {E}^{Q_B^*}[X_i^*]\), we have
$$\begin{aligned} \mathrm{ES}_{\alpha _i}^{Q_i}(X_i)=\sup _{Q\in {\mathcal {P}},\, \mathrm{d} Q/\mathrm{d} Q_i\le 1/\alpha _i} \mathbb {E}^Q[X_i]\ge \mathbb {E}^{Q_B^*}[X_i]\ge \mathbb {E}^{Q_B^*}[X_i^*], \end{aligned}$$
where the first inequality follows from \(Q_B^*\in \overline{{\mathcal {Q}}}\) with \(\overline{{\mathcal {Q}}}\) defined in (13). Therefore, the minimum value of the objective in the optimization problem (38) is at least \( \mathbb {E}^{Q_B^*}[X_i^*]\). As a consequence, \(\mathrm{ES}^{Q_i}_{\alpha _i}(X_i^*)\ge \mathbb {E}^{Q_B^*}[X_i^*]\). Noting that \((X_1^*,\ldots ,X_n^*)\) is a Pareto-optimal allocation, from (23) we have
$$\begin{aligned} \sum _{i=1}^n\mathrm{ES}^{Q_i}_{\alpha _i}(X_i^*)=\square _{i=1}^n\mathrm{ES}^{Q_i}_{\alpha _i}(X) = \mathbb {E}^{Q_B^*}[X]. \end{aligned}$$
(39)
Combined with the fact that \(\mathrm{ES}^{Q_i}_{\alpha _i}(X_i^*)\ge \mathbb {E}^{Q_B^*}[X_i^*]\), we have
$$\begin{aligned} \mathrm{ES}^{Q_i}_{\alpha _i}(X_i^*)=\mathbb {E}^{Q_B^*}[X^*_i],~~i=1,\ldots ,n. \end{aligned}$$
That is, \(X_i^*\) is an optimizer of the optimization problem (38). By definition, \((Q_B^*,(\,X_1^*,\ldots ,X_n^*))\) is a competitive equilibrium. \(\square \)

Remark 3

From Proposition 1 and Theorem 2, Pareto-optimal allocations and equilibria may exist for ES agents even if their beliefs are not equivalent. This is in sharp contrast to the classic setting of expected utility agents, where generally no Pareto-optimal allocations or equilibria exist if beliefs are not equivalent.

In the proof of Theorem 2, we have already seen that \(Q_B^*\) is an equilibrium pricing measure for the ES agents. The next theorem verifies that \(Q_B^*\) is indeed the unique equilibrium pricing measure.

Theorem 3

For a given \(X\in \mathcal {X}\), the equilibrium pricing measure is uniquely given by \(Q_B^*\).

Proof

Let \((Q,(X_1^*, \ldots ,X_n^*))\) be a competitive equilibrium. We show the uniqueness of the equilibrium pricing measure in two steps.
  1. (i)
    Assume for the purpose of contradiction that there exists \(A\in {\mathcal {F}}\) with \(A\subset \{X>x_B^*\}\) such that \(Q(A)>Q_B^*(A)\). Since \(Q_B^*(A)= \sum _{i=1}^n \frac{1}{\alpha _i} Q_i(A\cap B_i ),\) we know that \(Q(A \cap B_j )>\frac{1}{\alpha _j}Q_j(A\cap B_j )\) for some \(j\in \{1, \ldots ,n\}\). For a positive constant m, take \(Y_j= X_j^* + m \mathrm {I}_{A\cap B_j }- m Q(A\cap B_j ). \) Obviously, \(\mathbb {E}^Q[Y_j]=\mathbb {E}^Q[X_i^*]\). We can verify that
    $$\begin{aligned} \mathrm {ES}_{\alpha _j}^{Q_j}(Y_j)&= \mathrm {ES}_{\alpha _j}^{Q_j}( X_j^* + m \mathrm {I}_{A\cap B_j }- mQ(A\cap B_j )) \\ {}&= \mathrm {ES}_{\alpha _j}^{Q_j}( X_j^* + m (\mathrm {I}_{A\cap B_j }))- m Q(A\cap B_j ) \\ {}&\le \mathrm {ES}_{\alpha _j}^{Q_j}( X_j^*) + m \mathrm {ES}_{\alpha _j}^{Q_j} (\mathrm {I}_{A\cap B_j })- m Q(A\cap B_j ) \\ {}&\le \mathrm {ES}_{\alpha _j}^{Q_j}( X_j^*) + m \frac{1}{\alpha _j} Q_j(A\cap B_j)- m Q(A\cap B_j ) < \mathrm {ES}_{\alpha _j}^{Q_j}( X_j^*), \end{aligned}$$
    where the first inequality is due to the subadditivity of ES (see e.g. [10]). This contradicts the fact that \((Q,(X_1^*, \ldots ,X_n^*))\) is a competitive equilibrium, since \(Y_j\) strictly dominates \(X_j^*\) in the individual optimization (36). Therefore, we conclude that \(Q(A)\le Q_B^*(A)\) for all \(A\in {\mathcal {F}}\) with \(A\subset \{X>x_B^*\}\).
     
  2. (ii)
    By Theorem 2, \((X_1^*, \ldots ,X_n^*)\) is Pareto-optimal, and hence (39) holds. Since \((Q,(X^*_1, \ldots ,X_n^*))\) is a competitive equilibrium, for \(i=1, \ldots ,n\), we have \(\mathrm {ES}_{\alpha _i}^{Q_i}(X_i^*)\le \mathbb {E}^Q[X_i^*]\), otherwise \(X_i^*\) would have been strictly dominated by \(Y_i=\mathbb {E}^Q[X_i^*]\). By (39), we have
    $$\begin{aligned} \mathbb {E}^{Q_B^*} [X] = \sum _{i=1}^n\mathrm {ES}_{\alpha _i}^{Q_i}(X_i^*) \le \sum _{i=1}^n\mathbb {E}^Q[X_i^*] = \mathbb {E}^Q[X].\end{aligned}$$
    (40)
    From part (i), we know that Q is dominated by \(Q_B^*\) on \(\{X>x_B^*\}\). Assume \(Q(X\le x_B^*)>0\). It follows that
    $$\begin{aligned} \int _{\{X>x_B^*\}} X \mathrm {d}( Q_B^* - Q )&> x_B^*(Q_B^*(X>x_B^*)-Q(X> x_B^*)) \\&= x_B^* Q(X\le x_B^*). \end{aligned}$$
    Therefore,
    $$\begin{aligned} \mathbb {E}^{Q_B^*} [X] - \mathbb {E}^Q[X]= & {} \int _{\{X>x_B^*\}} X \mathrm {d}( Q_B^* - Q ) + \int _{\{X\le x_B^*\}} X \mathrm {d}( Q_B^*- Q)\\> & {} x_B^* Q(X\le x_B^*) -\int _{\{X\le x_B^*\}} X \mathrm {d}Q\\\ge & {} x_B^* Q(X\le x_B^*) - x_B^* Q(X\le x_B^*) =0, \end{aligned}$$
    contradicting (40). From there, we conclude that \(Q(X\le x_B^*)=0\).
     
Combining (i) and (ii), we have \(Q(A)\le Q_B^*(A)\) for all \(A\in {\mathcal {F}}\). Since both Q and \(Q_B^*\) are probability measures, we conclude that \(Q=Q_B^*\). \(\square \)

As a consequence of Theorems 2 and 3, we have the following corollary characterizing all equilibria for given initial risks.

Corollary 3

For any choice of initial risks \((\xi _1,\ldots \xi _n)\in {\mathbb {A}}_n(X)\), a competitive equilibrium is necessarily and sufficiently given by \((Q_B^*,(X_1^*, \ldots ,X_n^*))\), where \((X_1^*, \ldots ,X_n^*)\in {\mathbb {A}}_n(X)\) is a Pareto-optimal allocation such that \(\mathbb {E}^{Q_B^*}[X_i^*]=\mathbb {E}^{Q_B^*}[\xi _i]\), \(i=1, \ldots ,n\).

Recalling Theorem 1, an explicit form of Pareto-optimal allocations is given by
$$\begin{aligned} X^*_i=(X-x^*)\mathrm {I}_{B_i}+c_i,~~i=1,\ldots ,n, \end{aligned}$$
(41)
for any \(x ^*\in [x_B^*,y_B^*]\), where \(\sum _{i=1}^nc_i=x^*\), \(B=(B_1,\ldots ,B_n)\), \(x_B^*\) and \(y_B^*\) are defined in (15), (24) and (25).

Corollary 4

Assume \( \overline{{\mathcal {Q}}} \) in (13) is non-empty. Then \((Q_B^*,(X_1^*, \ldots ,X_n^*))\) given in (37) and (41) is a competitive equilibrium.

5 Risk sharing for VaR agents

In this section, we investigate risk-sharing problems for VaR agents. Throughout this section, \(\alpha _1, \ldots ,\alpha _n\in (0,1)\), \(Q_1, \ldots , Q_n\in {\mathcal {P}}\), and the risk measure of agent i is \(\mathrm {VaR}_{\alpha _i}^{Q_i}\), \(i=1, \ldots ,n\). The main difference between VaR and ES is the non-convexity of VaR, and hence the classic approach based on convex analysis cannot be used.

We introduce a few key quantities in our analysis for VaR agents. For \(X\in \mathcal {X}\) and \(x\in [-\infty ,\infty )\), define the set
$$\begin{aligned} \varGamma (x)= \{(Q_1(X>x,A_1), \ldots ,Q_n(X>x,A_n)): (A_1, \ldots ,A_n)\in \pi _n(\varOmega ) \}+\mathbb {R}_+^n, \end{aligned}$$
(42)
where \(\mathbb {R}_+=[0,\infty )\). Note that for fixed \(A\in {\mathcal {F}}\) and \(i=1, \ldots ,n\), \(Q_i(X>x,A)\) is right-continuous and decreasing in x, with \(Q_i(X>\infty ,A)=0\). Therefore, for each \((\alpha _1, \ldots ,\alpha _n)\in (0,1)^n\), there exists a smallest number \(x^*\in [-\infty ,\infty )\) such that \((\alpha _1, \ldots ,\alpha _n)\in \varGamma (x^*)\). That is,
$$\begin{aligned} x^*= \min \left\{ x\in [-\infty ,\infty ): ( \alpha _1, \ldots ,\alpha _n)\in \varGamma (x) \right\} . \end{aligned}$$
It follows that there exists \((A_1^*, \ldots ,A_n^*)\in \pi _n(\varOmega )\) such that
$$\begin{aligned} (\alpha _1, \ldots ,\alpha _n)\ge (Q_1(X>x^*,A_1^*), \ldots ,Q_n(X>x^*,A_n^*)). \end{aligned}$$
(43)
One can verify that for \(X\in \mathcal {X}\), \(x^*>-\infty \) if and only if
$$\begin{aligned} \bigvee _{i=1}^n \frac{Q_i(A_i)}{\alpha _i} > 1~~~\hbox {for all }~(A_1,\ldots ,A_n)\in \pi _n(\varOmega ). \end{aligned}$$
(44)
To show this, note that \(x^*>-\infty \) if and only if there exists \(x\in \mathbb {R}\) such that for all \((A_1,\ldots ,A_n)\in \pi _n(\varOmega )\), \( Q_i(X>x,A_i)\ge \alpha _i\) for some \(i=1, \ldots ,n\). This is in turn equivalent to (44).

Now we present the Pareto-optimal allocations for VaR agents.

Theorem 4

For \(X\in \mathcal {X}\), the following hold.
  1. (i)
    We have
    $$\begin{aligned} \mathop { \square } \limits _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i} (X) =\min \left\{ x\in [-\infty ,\infty ): ( \alpha _1, \ldots ,\alpha _n)\in \varGamma (x) \right\} =x^*, \end{aligned}$$
    where \(\varGamma \) is given by (42).
     
  2. (ii)
    If (44) holds, that is, \(x^*>-\infty \), a Pareto-optimal allocation \((X^*_1, \ldots ,X^*_n)\) of X is given by
    $$\begin{aligned} X^*_i=(X-x^*)\mathrm {I}_{A_i^*}+\frac{x^*}{n},~~i=1, \ldots ,n, \end{aligned}$$
    where \((A^*_1, \ldots ,A^*_n)\) satisfies (43).
     

Proof

  1. (i)
    First we show \(\square _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i} (X) \ge x^*\). For \((X_1, \ldots ,X_n)\in {\mathbb {A}}_n(X)\), let \(D_i=\{X_i > \mathrm {VaR}_{\alpha _i}^{Q_i}(X_i)\}\), \(i=1, \ldots ,n\). Clearly, \(Q_i(D_i)\le \alpha _i\). Let \(C_i=D_i\cup (\cup _{j=1}^n D_j )^c \) for \(i=1, \ldots ,n\). We have \(\cup _{i=1}^n C_i=\varOmega \), and hence there exists \((A_1, \ldots ,A_n)\in \pi _n(\varOmega )\) such that \(A_i \subset C_i.\) Write \(x=\sum _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i}(X_i)\). We have
    $$\begin{aligned} \{X>x\}&=\left\{ \sum _{i=1}^n X_i>\sum _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i}(X_i)\right\} \subset \bigcup _{i=1}^n D_i.\end{aligned}$$
    Therefore, for \(i=1, \ldots ,n\),
    $$\begin{aligned} Q_i(X>x, A_i)\le Q_i(X>x,C_i)=Q_i(X>x,D_i)\le Q_i(D_i)\le \alpha _i. \end{aligned}$$
    This shows \((\alpha _1, \ldots ,\alpha _n)\in \varGamma (x)\). As a consequence,
    $$\begin{aligned} x^*= \min \left\{ x\in [-\infty ,\infty ): ( \alpha _1, \ldots ,\alpha _n)\in \varGamma (x) \right\} \le x= \sum _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i}(X_i). \end{aligned}$$
    From there we obtain \(\square _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i} (X) \ge x^*\).
    Next we show \(\square _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i} (X) \le x^*\). Take any \(x\in \mathbb {R}\) such that \((\alpha _1, \ldots ,\alpha _n)\in \varGamma (x)\). By definition, there exists \((A_1, \ldots ,A_n)\in \pi _n(\varOmega )\) such that
    $$\begin{aligned} Q_i((X-x)\mathrm {I}_{A_i}>0) =Q_i(X>x, A_i)\le \alpha _i,~~i=1, \ldots ,n.\end{aligned}$$
    (45)
    Note that  (45) implies \(\mathrm {VaR}_{\alpha _i}^{Q_i}((X-x)\mathrm {I}_{A_i})\le 0\) for \(i=1, \ldots ,n\). Let
    $$\begin{aligned} X_i=(X-x)\mathrm {I}_{A_i}+\frac{x}{n},~~i=1, \ldots ,n.\end{aligned}$$
    (46)
    We have \((X_1, \ldots ,X_n)\in {\mathbb {A}}_n(X)\) and
    $$\begin{aligned} \sum _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i}(X_i) = \sum _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i}((X-x)\mathrm {I}_{A_i} ) + x\le x. \end{aligned}$$
    This shows \(\square _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i} (X) \le x\) for all real numbers \(x\ge x^*\). Therefore, \(\square _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i} (X) \le x^*\). In summary, we have \(\square _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i} (X) =x^*\).
     
  2. (ii)
    Suppose \(x^*>-\infty \). Similarly to (45) and (46), from the definition of \(x^*\) and \(A_i^*\), we have
    $$\begin{aligned} \sum _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i}(X_i^*) = \sum _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i}((X-x^*)\mathrm {I}_{A_i^*} ) + x^*\le x^*. \end{aligned}$$
    Together with (i), we conclude that \((X^*_1, \ldots ,X^*_n)\) is a Pareto-optimal allocation of X. \(\square \)
     

As an immediate consequence of Theorem 4 (ii), a Pareto-optimal allocation exists if and only if (44) holds. Similarly to the case of ES agents, the existence of a Pareto-optimal allocation only depends on \((\alpha _1, \ldots ,\alpha _n)\) and \((Q_1, \ldots ,Q_n)\), but not on the total risk X.

The Pareto-optimal allocation for VaR agents in Theorem 4,
$$\begin{aligned} X^{\mathrm {VaR}}_i=(X-x ^*)\mathrm {I}_{A_i^*}+\frac{x^*}{n},~~i=1, \ldots ,n, \end{aligned}$$
and that for ES agents in Theorem 1,
$$\begin{aligned} X^{\mathrm {ES}}_i=(X-y^*)\mathrm {I}_{B_i} + \frac{y^*}{n},~~i=1, \ldots ,n, \end{aligned}$$
share amazing similarity in their forms. Nevertheless, we should clarify that the calculation of \((A_1^*, \ldots ,A_n^*,x^*)\) and \((B_1^*, \ldots ,B_n^*,y^*)\) above are completely different, and these two risk sharing problems have essentially distinct features. We remark two significant differences. First, the optimization problem for VaR agents is a non-convex one, whereas that for ES agents is convex. Second, \((B_1^*, \ldots ,B_n^*,y^*)\) has explicit forms, but \((A_1^*, \ldots ,A_n^*,x^*)\) does not; an efficient way to compute \((A_1^*, \ldots ,A_n^*,x^*)\) seems unavailable at the moment.

Remark 4

If \(Q_1=\cdots =Q_n=\mathbb {P}\), we have
$$\begin{aligned}&\mathop { \square } \limits _{i=1}^n \mathrm {VaR}_{\alpha _i}^{\mathbb {P}} (X)\\&\quad = \inf \left\{ x\in \mathbb {R}~|~ \mathbb {P}(A_i^x)=\alpha _i, A_i^x=A_i\cap \{X>x\}, (A_1,\ldots , A_n)\in \pi _n(\varOmega )\right\} \\&\quad = \inf \left\{ x\in \mathbb {R}~|~ \mathbb {P}(X>x)=\alpha _1+\cdots +\alpha _n\right\} =\mathrm {VaR}^\mathbb {P}_{\sum _{i=1}^n\alpha _i} (X). \end{aligned}$$
This is a special case of Theorem 2 of  [9].
Next, we observe that a competitive equilibrium for VaR agents does not exist. In this setting, each agent’s objective is
$$\begin{aligned} \begin{array}{ll} \text{ to } \text{ minimize }~~ &{} \mathrm {VaR}_{\alpha _i}^{Q_i}(X_i)~~~\text{ over }~~~X_i\in \mathcal {X}\\ \hbox {subject to}&{}\mathbb {E}^{Q}[X_i]\ge \mathbb {E}^Q[\xi _i], \end{array} ~~~~~~i=1, \ldots ,n, \end{aligned}$$
(47)

Proposition 4

For any choice of initial risks, a competitive equilibrium for the VaR agents does not exist.

Proof

First note that there is a correspondence between the optimizer of (47) and that of the following optimization problem
$$\begin{aligned} \text{ to } \text{ minimize }~~ {\mathcal {V}}_i(Y_i)= \mathrm {VaR}_{\alpha _i}^{Q_i}(Y_i) -\mathbb {E}^{Q}[Y_i]~~\text{ over }~~Y_i\in \mathcal {X}, ~~~~~~i=1, \ldots ,n, \end{aligned}$$
(48)
by choosing \(X_i=Y_i-\mathbb {E}^Q[Y_i]+\mathbb {E}^Q[\xi _i].\) Note that for any probability measure Q, one can easily find a random variable \(X_i\) such that \( \mathrm {VaR}_{\alpha _i}^{Q_i}(Y_i) <\mathbb {E}^{Q}[Y_i]\). Then by the positive homogeneity of \({\mathcal {V}}_i\), we have that the infimum of the objective function is \(-\infty \), and hence (47) admits no optimizer. \(\square \)

Although Theorem 4 obtains Pareto-optimal allocations for VaR agents, Proposition 4 shows that there is no competitive equilibrium in this setting. This is a further evidence of the inappropriateness of VaR as a measure of risk in the context of risk sharing; see, for instance [13] and [9] for related discussions on the use of VaR. A possible alternative setting to study competitive equilibria for VaR agents is to restrict the set of admissible positions for each agent and to slightly relax the definition of pricing measures; see [9] for the case allowing only \(0\le X_i\le X\). In the latter paper, for VaR agents with homogeneous beliefs, the equilibrium pricing measure Q is shown to be a zero measure instead of a probability measure, and this is beyond the framework of this paper.

6 Risk sharing for mixed VaR and ES agents

In this section, we consider the risk sharing problem in which some agents are VaR agents and the others are ES agents. The result naturally generalizes to RVaR agents, which shall be defined later. Define the index sets \(I=\{1,\ldots ,m\}\) and \(J=\{m+1,\ldots ,n\}\), \(0\le m< n\). Without loss of generality, assume that for \(i\in I\) and \(j\in J\), the objective of agent i is \(\mathrm {VaR}_{\alpha _i}^{Q_i}\) and that of agent j is \(\mathrm {ES}_{\beta _j}^{Q_j}\), where \(\alpha _i, \beta _j\in (0,1)\) and \(Q_i, Q_j\in {\mathcal {P}}\). Note that here we allow I to be empty but J is assumed non-empty, i.e. there is at least one ES agent. For notional simplicity, in this section we write, for \(X\in \mathcal {X}\),
$$\begin{aligned} V(X)=\inf \left\{ \sum _{i\in I}\mathrm {VaR}_{\alpha _i}^{Q_i}(X_i)+\sum _{j\in J}\mathrm {ES}_{\beta _j}^{Q_j}(X_j) : (X_1,\ldots ,X_n)\in \mathbb {A}_n(X) \right\} . \end{aligned}$$
(49)
We first verify that \({V}(X)>-\infty \) if and only if
$$\begin{aligned} \bigvee _{i=1}^n \frac{Q_i(A_i)}{\alpha _i} > 1~~\hbox {for all}~ (A_{1}, \ldots ,A_m)\in \pi _{m}(\varOmega ), \end{aligned}$$
(50)
and
$$\begin{aligned} \sum _{j\in J}\frac{Q_j(B_j)}{\beta _j}\ge 1~~\hbox {for all}~ (B_{1}, \ldots ,B_n)\in \pi _{n}(\varOmega )~ \mathrm{with}~ Q_i(B_i)\le \alpha _i, i\in I. \end{aligned}$$
(51)
To see this, first note that  (50) is a necessary condition for \(V(X)>-\infty \) by (44). Hence, we only need to show that when (50) holds, \(V(X)>-\infty \) if and only if (51) holds. To show the necessity, assume (51) does not hold. There exists \((B_{1}, \ldots ,B_n)\in \pi _{n}(\varOmega )\) with \(Q_i(B_i)\le \alpha _i\), \(i\in I\), such that \(\sum _{j\in J}{Q_j(B_j)}/{\beta _j}<1\). Define
$$\begin{aligned} X_i =(X+x){\mathrm {I}}_{B_i}- \frac{x}{n},~~i=1,\ldots ,n, \end{aligned}$$
where \(x>0\). Then
$$\begin{aligned} \sum _{i\in I}\mathrm{VaR}_{\alpha _i}^{Q_i}(X_i) + \sum _{j\in J}\mathrm{ES}_{\beta _j}^{Q_j}(X_j)&\le \sum _{j\in J}\mathrm{ES}_{\beta _j}^{Q_j}((X+x){\mathrm {I}}_{B_j}) -x\\&= \sum _{j\in J}\frac{1}{\beta _j}{\mathbb {E}}^{Q_j}[X{\mathrm {I}}_{B_j}]+\sum _{j\in J}\frac{Q_j(B_j)}{\beta _j}x -x. \end{aligned}$$
Letting \(x\rightarrow \infty \), we have that the right hand side of the above equation converges to \(-\infty \), and hence, \(V(X)=-\infty \). The sufficiency is implied by the following theorem.

Theorem 5

Assume that (50) and (51) hold. Then for \(X\in \mathcal {X}\),
$$\begin{aligned} V (X)=\min \left\{ \sum _{j\in J}\frac{1}{\beta _j}\mathbb {E}^{Q_j}[(X-x)_+\mathrm {I}_{B_j}] +x\, \left| \, \begin{array}{l} (B_{1}, \ldots ,B_n)\in \pi _{n}(\varOmega )\\ Q_i(B_i)\le \alpha _i, \,i\in I,~x\in \mathbb {R}\end{array}\right. \right\} . \end{aligned}$$
(52)

Proof

Note that X is bounded and both sides of  (52) are translation invariant.6 Without loss of generality, we assume \(X\ge 0\). Denote by R(X) the right hand side of (52); we will show \(V(X)=R(X)\). For any \((B_{1}, \ldots ,B_n)\in \pi _{n}(\varOmega )\) such that \(Q_i(B_i)\le \alpha _i\) for \(i\in I\), take \(X_i=X\mathrm {I}_{B_i}\), \(i=1,\ldots ,n\). Then, \(\mathrm {VaR}_{\alpha _i}^{Q_i}(X_i)=0\), \(i\in I\), and for \(j\in J\),
$$\begin{aligned} \mathrm {ES}_{\beta _j}^{Q_j}(X_j)&=\min \left\{ \frac{1}{\beta _j}\mathbb {E}^{Q_j}[(X\mathrm {I}_{B_j}-x)_+]+x : x\in \mathbb {R}\right\} \\&=\min \left\{ \frac{1}{\beta _j}\mathbb {E}^{Q_j}[(X\mathrm {I}_{B_j}-x)_+]+x : x\in \mathbb {R}_+\right\} \\&=\min \left\{ \frac{1}{\beta _j}\mathbb {E}^{Q_j}[(X-x)_+\mathrm {I}_{B_j}]+x : x\in \mathbb {R}\right\} . \end{aligned}$$
Therefore, for all \((B_{1}, \ldots ,B_n)\in \pi _{n}(\varOmega )\) such that \(Q_i(B_i)\le \alpha _i\), \(i\in I\), we have
$$\begin{aligned} V(X)&\le \sum _{j\in J}\min \left\{ \frac{1}{\beta _j}\mathbb {E}^{Q_j}[(X-x)_+\mathrm {I}_{B_j}]+x : x\in \mathbb {R}\right\} \\&\le \min \left\{ \sum _{j\in J}\frac{1}{\beta _j}\mathbb {E}^{Q_j}[(X-x)_+\mathrm {I}_{B_j}]+x : x\in \mathbb {R}\right\} . \end{aligned}$$
Hence, \(V(X)\le R(X)\). To show \(V(X)\ge R(X)\), we need to prove that for any \((X_1,\ldots ,X_n)\in \mathbb {A}_n(X) \), there exists \((B_{1}, \ldots ,B_n)\in \pi _{n}(\varOmega )\) with \(Q_i(B_i)\le \alpha _i\), \(i\in I\) such that
$$\begin{aligned} \sum _{i\in I}\mathrm {VaR}_{\alpha _i}^{Q_i} (X_i) + \sum _{j\in J}\mathrm {ES}_{\beta _j}^{Q_j}(X_j) \ge \min _{x\in \mathbb {R}}\left\{ \sum _{j\in J}\frac{1}{\beta _j}\mathbb {E}^{Q_j}[(X-x)_+\mathrm {I}_{B_j}]+x \right\} . \end{aligned}$$
(53)
Because of translation invariance of VaR and ES, without loss of generality, assume \(\mathrm {VaR}_{\alpha _i}^{Q_i} (X_i)=0\), \(i\in I\). As \(Q_i(\{X_i>0\})\le \alpha _i\), \(i\in I\), there exists a set \(B_1\in {\mathcal {F}}\) such that \(\{X_1>0\}\subset B_1\) and \(Q_1(B_1)\le \alpha _1\). Similarly, for \(i\in I\), let \(B_i\) be a set such that \(\{X_i>0\}{\setminus } \cup _{k=1}^{i-1}B_k\subset B_i\) and \(Q_i(B_i)\le \alpha _i\). Let \(B=\cup _{i\in I}B_i\),
$$\begin{aligned} X_i^*=X_i\mathrm {I}_{B}+d\,\mathrm {I}_{B_i},~ i\in I, ~\mathrm{and} ~X_j^*=(X_j-d/(n-m))\mathrm {I}_B+X_j\mathrm {I}_{B^c},~j\in J, \end{aligned}$$
where \(d>0\) is large enough such that
$$\begin{aligned} \sup [X_j-d/(n-m)|B] < \inf [X_j|{B^c}],~~ j\in J. \end{aligned}$$
Clearly, \(X^*_j\le X_j\) for \(j\in J\), and \(\mathrm {VaR}_{\alpha _i}^{Q_i}(X^*_i)=\mathrm {VaR}_{\alpha _i}^{Q_i}(X_i)=0\). By (51), we have \(Q_j(B^c)\!\ge \! \beta _j\), implying \(\mathrm {ES}_{\beta _j}^{Q_j}(X^*_j)\!=\!\mathrm {ES}_{\beta _j}^{Q_j}(X_j\mathrm {I}_{B^c})\). Also note that \( \cup _{i\in I}\{X_i\!>\!0\}\subset B\), implying \(\sum _{j\in J}X_j\mathrm {I}_{B^c}= \left( X-\sum _{i\in I} X_i\right) \mathrm {I}_{B^c}\ge X\mathrm {I}_{B^c}.\) Using the above facts, we have
$$\begin{aligned} \sum _{i\in I}\mathrm {VaR}_{\alpha _i}^{Q_i} (X_i) + \sum _{j\in J}\mathrm {ES}_{\beta _j}^{Q_j}(X_j)&\ge \sum _{i\in I}\mathrm {VaR}_{\alpha _i}^{Q_i} (X^*_i) + \sum _{j\in J}\mathrm {ES}_{\beta _j}^{Q_j}(X^*_j)\\&=\sum _{j\in J}\mathrm {ES}_{\beta _j}^{Q_j}(X_j\mathrm {I}_{B^c}) \\&\ge \mathop { \square } \limits _{j\in J}\mathrm {ES}_{\beta _j}^{Q_j}\left( \sum _{j\in J}X_j\mathrm {I}_{B^c}\right) \ge \mathop { \square } \limits _{j\in J}\mathrm {ES}_{\beta _j}^{Q_j}(X\mathrm {I}_{B^c}). \end{aligned}$$
By Proposition 2, and noting that \((X\mathrm {I}_{B^c}-x)_+\mathrm {I}_A \ge (X-x)_+\mathrm {I}_{B^c\cap A}\) for \(A\in {\mathcal {F}}\) and \(x\in \mathbb {R}\), we have
$$\begin{aligned}&\mathop { \square } \limits _{j\in J}\mathrm {ES}_{\beta _j}^{Q_j}(X\mathrm {I}_{B^c})\\&\quad \ge \min \left\{ \sum _{j\in J}\frac{1}{\beta _j} \mathbb {E}^{Q_j}[(X-x)_+\mathrm {I}_{C_j} ] +x: x\in \mathbb {R}, ~(C_j)_{j\in J} \in \pi _{n-m}(B^c)\right\} \\&\quad \ge { R}(X). \end{aligned}$$
Thus, \(V(X)\ge { R}(X)\), and this completes the proof. \(\square \)
If \(I=\varnothing \), Theorem 5 reduces to Proposition 2. The case of \(J=\varnothing \) (see Theorem 4) is not included in Theorem 5 since the expression in (52) involves a sum of expectations over J. By Theorem 5, there exist \(x^*\in \mathbb {R}\) and \((B_{1}, \ldots ,B_n)\in \pi _{n}(\varOmega )\) such that \( Q_i(B_i)\le \alpha _i, \,i\in I\), and
$$\begin{aligned} V(X)=\sum _{j\in J}\frac{1}{\beta _j}\mathbb {E}^{Q_j}[(X-x^*)_+\mathrm {I}_{B_j}] +x^*. \end{aligned}$$
A Pareto-optimal allocation \((X^*_1, \ldots ,X^*_n)\) of X is given by
$$\begin{aligned} X^*_i=(X-x^*)\mathrm {I}_{B_i} + \frac{x^*}{n},~~i=1, \ldots ,n. \end{aligned}$$
(54)
Nevertheless, analytical formulas of the above \(x^*\in \mathbb {R}\) and \((B_{1}, \ldots ,B_n)\in \pi _{n}(\varOmega )\) are not available. A necessary condition for \((B_{1}, \ldots ,B_n)\in \pi _{n}(\varOmega )\) above is
$$\begin{aligned} B_j\subset \left\{ \frac{1}{\alpha _j}\frac{\mathrm {d}Q_j }{\mathrm {d}Q} =\bigwedge _{i\in J} \frac{1}{\alpha _i}\frac{\mathrm {d}Q_i }{\mathrm {d}Q} \right\} , ~~ j\in J,\end{aligned}$$
but to determine \((B_{1}, \ldots ,B_n)\) seems a very complicated task, even computationally.

Remark 5

When there are mixed VaR and ES agents, as the VaR agents do not care about the risk above a certain quantile, an intuitive idea to find the Pareto-optimal allocation is to first allocate risks for the VaR agents as in Theorem 4, and then allocate risks for the ES agents as in Theorem 1. Such a technical treatment turns out to give an optimal allocation in the setting of homogeneous beliefs in [9]. Unfortunately, it does not necessarily lead to a Pareto-optimal allocation in our setting of heterogeneous beliefs, and hence yields a sharp contrast to the case of homogeneous beliefs treated in [9]; see Example 1 below for a counter-example.

Example 1

Suppose that \(\varOmega =(0,1)\) and \(\mathcal {F}=\mathcal {B}(0,1)\). Let \(Q_1\) be the Lebesgue measure on (0, 1). Take another probability measure \(Q_2\) on \((\varOmega ,{\mathcal {F}})\) such that
$$\begin{aligned} \frac{{\mathrm {d}}Q_2}{{\mathrm {d}} Q_1}(\omega )=\left\{ \begin{array}{ll}2 &{} ~\text{ if }~\omega \in (0,\frac{1}{4}), \\ 1/2 &{} ~\text{ if }~\omega \in [\frac{1}{4},\frac{1}{2})\cup [\frac{3}{4},1),\\ 1&{} ~\text{ if }~\omega \in [\frac{1}{2},\frac{3}{4}). \end{array}\right. \end{aligned}$$
Therefore, \(Q_2\left( (0,1/4)\right) =1/2\), \(Q_2\left( [1/4,1/2)\right) =1/8\), \(Q_2\left( [1/2,3/4)\right) =1/4\), and \(Q_2\left( [3/4,1)\right) =1/8\). Let
$$\begin{aligned} X = \mathrm {I}_{[\frac{1}{4},\frac{1}{2})} +4 \times \mathrm {I}_{[\frac{1}{2},\frac{3}{4})} +5 \times \mathrm {I}_{[\frac{3}{4},1)} \end{aligned}$$
and consider the following optimization problem
$$\begin{aligned} \inf \left\{ \mathrm {VaR}_{1/4}^{Q_1}(X_1)+\mathrm {ES}_{1/4}^{Q_2}(X_2) : X_1+X_2=X, \, X_1,X_2\in \mathcal {X}\right\} . \end{aligned}$$
For any \(A\in \mathcal {F}\), if \(Q_1(A)=1/4\), then \(Q_2(A^c)\ge 1/4\) because \(\frac{{\mathrm {d}}Q_2}{{\mathrm {d}} Q_1}\le 2\) implies that \(Q_2(A)\le 1/2\) and thus \(Q_2(A^c)\ge 1/2>1/4\); therefore (50) and (51) hold and the above infimum is finite by Theorem 5. If we consider to first allocate the worst risk of probability 1/4 to agent 1 (because agent 1 does not care any risk with probability smaller or equal to 1/4), then the resulting allocation is \((X_1,X_2)\) where \(X_1=5 \,\mathrm {I}_{[{3}/{4},1)}\) and \(X_2=X-X_1\). Note that this allocation would have been optimal if \(Q_2=Q_1\); see Theorem 2 of [9]. Consider another allocation \((Y_1,Y_2)\) where \(Y_1=4\times \mathrm {I}_{[1/2,{3}/{4})}\) and \(Y_2=X-Y_1\). One can easily check
$$\begin{aligned} \mathrm {VaR}_{1/4}^{Q_1}(X_1)+\mathrm {ES}_{1/4}^{Q_2}(X_2)=4>3=\mathrm {VaR}_{1/4}^{Q_1}(Y_1)+\mathrm {ES}_{1/4}^{Q_2}(Y_2). \end{aligned}$$
Thus, the Pareto-optimal problem of mixed VaR and ES agents with heterogeneous beliefs cannot be obtained by separately treating VaR agents and ES agents. \(\square \)
Finally, we present the result for a more general class of risk measures, the Range-Value-at-Risk (RVaR), as studied in [9]. Recall that for \(X\in \mathcal {X}\) and \(Q\in {\mathcal {P}}\), the RVaR at level \((\alpha ,\beta )\in [0,1]^2,~\alpha +\beta \le 1\) is defined as
$$\begin{aligned} \mathrm {RVaR}^Q_{\alpha , \beta }(X)=\left\{ \begin{array}{cc} \frac{1}{\beta }\int _\alpha ^{\alpha +\beta }\mathrm {VaR}^Q_\gamma (X)\mathrm {d}\gamma &{} ~\text{ if }~\beta >0, \\ \mathrm {VaR}^Q_\alpha (X) &{} ~\text{ if }~\beta =0. \end{array}\right. \end{aligned}$$
Clearly, the RVaR family includes both VaR and ES as special cases. For more details on RVaR, see [9]. To study risk sharing problems for RVaR agents, the key observation is that RVaR is the inf-convolution of VaR and ES, namely,
$$\begin{aligned} \mathrm {RVaR}^Q_{\alpha , \beta } =\mathrm {VaR}_\alpha ^Q~ \square ~ \mathrm {ES}_\beta ^Q; \end{aligned}$$
see Theorem 2 of [9]. With this result, we can use Theorem 5 to calculate the inf-convolution of RVaR and identify its corresponding Pareto-optimal allocations, by decomposing each RVaR agent into two “imaginary” VaR and ES agents. To guarantee the existence of a Pareto-optimal allocation, or equivalently \(\square _{i=1}^n \mathrm {RVaR}_{\alpha _i,\,\beta _i}^{Q_i} (X)>-\infty \), we require
$$\begin{aligned} \bigvee _{i=1}^n \frac{Q_i(A_i)}{\alpha _i} > 1~~\hbox {for all}~ (A_{1}, \ldots ,A_n)\in \pi _{n}(\varOmega ),\end{aligned}$$
(55)
and
$$\begin{aligned} \sum _{i=1}^n\frac{Q_i(B_{2i})}{\beta _i}\ge 1 ~~~ \begin{array}{c} \text{ for } \text{ all } (B_{11},B_{21}, \ldots ,B_{1n},B_{2n})\in \pi _{2n}(\varOmega )\\ \text{ with } Q_i(B_{1i})\le \alpha _i,~ i=1,\ldots ,n. \end{array} \end{aligned}$$
(56)
Then the following corollary follows directly from Theorem 5.

Corollary 5

Let \(X\in \mathcal {X}\) and \(\alpha _i,\beta _i\in (0,1)\), \(i=1, \ldots ,n\). Assume that (55) and (56) hold. Then
$$\begin{aligned}&\mathop { \square } \limits _{i=1}^n \mathrm {RVaR}_{\alpha _i,\,\beta _i}^{Q_i} (X)\\ {}&\quad =\min \left\{ \sum _{i=1}^n\frac{1}{\beta _i}\mathbb {E}^{Q_i}\left[ (X-x)_+\mathrm {I}_{B_{2i}}\right] +x \, \left| \, \begin{array}{l} (B_{11},B_{21}, \ldots ,B_{1n},B_{2n})\in \pi _{2n}(\varOmega )\\ Q_i(B_{1i})\le \alpha _i, \,i=1,\ldots ,n,~x\in \mathbb {R}\end{array}\right. \right\} . \end{aligned}$$

We conclude this section by observing that, similarly to the case of VaR agents, a competitive equilibrium does not exist for mixed VaR and ES agents or RVaR agents, unless all agents are ES agents.

Remark 6

(A final comment.7) Before ending the paper, we discuss how the results established for bounded random variables can be generalized to the case when the set of risks \({\mathcal {X}}\) is chosen as the set of random variables bounded from below, namely,
$$\begin{aligned} {\mathcal {X}}=\{X\in L^0(\varOmega ,{\mathcal {F}}): \inf X >-\infty \}.\end{aligned}$$
(57)
Note that we now use \(\mathcal {X}\) defined by (57) in all places, including all definitions and all optimization problems.
  1. (i)

    For ES agents, Pareto-optimal allocations do not exist if \(\square _{i=1}^n \mathrm {ES}_{\alpha _i}^{Q_i} (X)=\infty \). Indeed, in this case, for any allocation \((X_1,\ldots ,X_n)\), there exists \(i\in \{1,\ldots ,n\}\) such that \(\mathrm {ES}_{\alpha _i}^{Q_i}(X_i)=\infty \). If \(\mathrm {ES}^{Q_j}_{\alpha _j}(X_j)=\infty \) for all \(j=1, \ldots ,n\), then the allocation \((X,0,\ldots ,0)\) dominates \((X_1,\ldots ,X_n)\). If there exists \(j\ne i\) such that \(\mathrm {ES}_{\alpha _j}^{Q_j}(X_j)<\infty \), then, by letting \(X_i^*=X_i+1\), \(X_j^*=X_j-1\) and \(X_k^*=X_k\) for \(k\ne i,j\), \((X_1^*,\ldots ,X_n^*)\) dominates \((X_1,\ldots ,X_n)\). Therefore one needs to assume \(\square _{i=1}^n \mathrm {ES}_{\alpha _i}^{Q_i} (X)<\infty \) in order for a Pareto-optimal allocation to exist. As one can easily check, assuming \(\square _{i=1}^n \mathrm {ES}_{\alpha _i}^{Q_i} (X)<\infty \), all results and their proofs in Sect. 3 are valid for \({\mathcal {X}}\) in (57).

     
  2. (ii)

    For results in Sect. 4, we add to the definition of a competitive equilibrium that the objective functions at optimum have to be finite. Precisely, a pair \((Q,(X_1^*, \ldots ,X_n^*))\in {\mathcal {P}} \times \mathbb {A}_n(X)\) is a competitive equilibrium if (4) holds and the minimums in (4) are finite. This is to avoid cases where the individual optimization problems give positive or negative infinity. With this natural modification, all results in Sect. 4 hold for \({\mathcal {X}}\) in (57). To guarantee this, we just need to make sure that all optimization problems do no involve infinity. First, it is clear that a competitive equilibrium exists only if \(\square _{i=1}^n \mathrm {ES}_{\alpha _i}^{Q_i} (X)<\infty \). Assuming \(\square _{i=1}^n \mathrm {ES}_{\alpha _i}^{Q_i} (X)<\infty \), there exists an allocation \((X_1^*, \ldots ,X_n^*)\) such that \(\mathrm {ES}_{\alpha _i}^{Q_i}(X^*_i)\) is finite for \(i=1, \ldots ,n\), and, as each of \(\xi _1, \ldots ,\xi _n\) is bounded from below, \(\mathbb {E}^{Q_B^*}[X]<\infty \) implies \(\mathbb {E}^{Q_B^*}[\xi _i]<\infty \), \(i=1,\ldots ,n\). Further, it is straightforward to check that in a competitive equilibrium \((Q,(X_1^*, \ldots ,X_n^*))\), Q has to satisfy \(\mathbb {E}^{Q}[\xi _i]<\infty \), \(i=1,\ldots ,n\), otherwise the objective function for agent i will be \(\infty \) or \(-\infty \). Therefore, there is no issue regarding infinity, and all proofs in Sect. 4 are valid for the set \({\mathcal {X}}\) in (57).

     
  3. (iii)

    For VaR agents, one can easily verify that all results in Sect. 5 are valid for \({\mathcal {X}}\) in (57), due to the fact that whether random variables are unbounded is irrelevant for VaR. In addition, for the risk sharing problem with mixed VaR agents and ES agents, as we only discuss the inf-convolution, the results in Sect. 6 also hold for \({\mathcal {X}}\) in (57) if \(I\not =\varnothing \) (recall that \(I=\varnothing \) corresponds to the case of ES agents).

     

7 Conclusion

By solving various optimization problems, we obtain in explicit forms Pareto-optimal allocations and competitive equilibria for quantile-based risk measures with belief heterogeneity. For ES agents, we show that Pareto-optimal allocations and equilibrium allocations are equivalent, and the equilibrium pricing measure is uniquely determined. In the case of VaR agents, Pareto-optimal allocations are obtained, but competitive equilibria do not exist. Our results and economic interpretations differ significantly from those of [9] where belief homogeneity is assumed. In view of the prominent usage of internal models for various financial institutions, belief heterogeneity seems to be a more reasonable assumption for studying risk sharing problems in the context of regulatory capital calculation and its practical implications.

Footnotes

  1. 1.

    ES is also called CVaR, AVaR or TVaR in various contexts. In particular, CVaR is common in the optimization literature, e.g. [14] and [22, 23]. In this paper, we stick to the term ES following the risk management literature, e.g. [13] and [9].

  2. 2.

    Amongst others [1, 5, 7, 12, 24] and [2] studied risk sharing problems with convex risk measures and expected utilities, different from the setting of quantile-based risk measures in this paper.

  3. 3.

    Following the tradition in the literature of risk sharing, we refer to a participant in the risk sharing problem, such as an investor or a firm, as an agent.

  4. 4.

    In this paper, all random future positions are already discounted, and that is why expectations correspond to prices.

  5. 5.

    Roughly speaking, the first FTWE states that, under some conditions, an equilibrium allocation is Pareto-optimal, and the second FTWE states that, under some conditions, a Pareto-optimal allocation is an equilibrium allocation.

  6. 6.

    Following the risk management literature, a functional \(f:\mathcal {X}\rightarrow \mathbb {R}\) is called translation invariant if \(f(X+c)=f(X)+c\) for all \(X\in \mathcal {X}\) and \(c\in \mathbb {R}\).

  7. 7.

    This remark is based on a very valuable suggestion of an anonymous referee.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  • Paul Embrechts
    • 1
    • 2
  • Haiyan Liu
    • 3
    • 4
  • Tiantian Mao
    • 5
  • Ruodu Wang
    • 6
  1. 1.RiskLab, Department of MathematicsETH ZurichZurichSwitzerland
  2. 2.Swiss Finance InstituteGenevaSwitzerland
  3. 3.Department of MathematicsMichigan State UniversityMichiganUSA
  4. 4.Department of Statistics and ProbabilityMichigan State UniversityMichiganUSA
  5. 5.Department of Statistics and Finance, School of ManagementUniversity of Science and Technology of ChinaHefeiChina
  6. 6.Department of Statistics and Actuarial ScienceUniversity of WaterlooWaterlooCanada

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