# Quantile-based risk sharing with heterogeneous beliefs

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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’ viewpoints^{3} 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.

*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

*allocations*of

*X*as

*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 \(\mathbb {E}^{Q}[Y]\).

*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

### 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

*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

*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

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.

*inf-convolution*of risk measures (see [24]) as

*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

**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

- (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)
- (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

*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

### 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*.

*Q*be a measure dominating \(Q_1, \ldots ,Q_n\), and

*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

### Proposition 2

### Proof

*x*, with \(v_A'(\infty )=1\) and

*x*for \(i=1,\ldots ,n\), \(v_A'\) is a right-continuous function. Therefore, \(v_A'(x_A^*)\ge 0\), and equivalently,

- 1.
\(Q_A^*\) is \(\sigma \)-additive, because it is the sum of

*n*measures. - 2.
\(Q_A^*(\varOmega )\le 1\) by (19).

*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

### Proof

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”.

- (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.

- (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. - (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)

### Corollary 1

### Remark 2

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.

### Proposition 3

### Proof

*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

*Q*-almost surely, where \(y_i=\mathrm{VaR}_{\alpha _i}^{Q_i}(X^*_i)\), \(i=1, \ldots ,n\). Consider the optimization problem

*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 \(\sum _{i=1}^n x_i=y^*\).

*Q*-almost surely and \(y_B^*\ge y^*\), by letting \(c_i=x_i-y^*+y_B^*\), we have

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

### Corollary 2

### Proof

## 4 Competitive equilibria for ES agents

*i*is \(\mathrm {ES}_{\alpha _i}^{Q_i}\), \(i=1, \ldots ,n\). Each agent’s objective is

^{5}As in the proof of Proposition 2, throughout this section we define the probability measure \(Q_B^*\) via

*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.

### 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

- (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 thatwhere 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 \(A\subset \{X>x_B^*\}\).$$\begin{aligned} \mathrm {ES}_{\alpha _j}^{Q_j}(Y_j)&= \mathrm {ES}_{\alpha _j}^{Q_j}( X_j^* + m \mathrm {I}_{A\cap B_j }- mQ(A\cap B_j )) \\ {}&= \mathrm {ES}_{\alpha _j}^{Q_j}( X_j^* + m (\mathrm {I}_{A\cap B_j }))- m Q(A\cap B_j ) \\ {}&\le \mathrm {ES}_{\alpha _j}^{Q_j}( X_j^*) + m \mathrm {ES}_{\alpha _j}^{Q_j} (\mathrm {I}_{A\cap B_j })- m Q(A\cap B_j ) \\ {}&\le \mathrm {ES}_{\alpha _j}^{Q_j}( X_j^*) + m \frac{1}{\alpha _j} Q_j(A\cap B_j)- m Q(A\cap B_j ) < \mathrm {ES}_{\alpha _j}^{Q_j}( X_j^*), \end{aligned}$$ - (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 haveFrom part (i), we know that$$\begin{aligned} \mathbb {E}^{Q_B^*} [X] = \sum _{i=1}^n\mathrm {ES}_{\alpha _i}^{Q_i}(X_i^*) \le \sum _{i=1}^n\mathbb {E}^Q[X_i^*] = \mathbb {E}^Q[X].\end{aligned}$$(40)
*Q*is dominated by \(Q_B^*\) on \(\{X>x_B^*\}\). Assume \(Q(X\le x_B^*)>0\). It follows thatTherefore,$$\begin{aligned} \int _{\{X>x_B^*\}} X \mathrm {d}( Q_B^* - Q )&> x_B^*(Q_B^*(X>x_B^*)-Q(X> x_B^*)) \\&= x_B^* Q(X\le x_B^*). \end{aligned}$$contradicting (40). From there, we conclude that \(Q(X\le x_B^*)=0\).$$\begin{aligned} \mathbb {E}^{Q_B^*} [X] - \mathbb {E}^Q[X]= & {} \int _{\{X>x_B^*\}} X \mathrm {d}( Q_B^* - Q ) + \int _{\{X\le x_B^*\}} X \mathrm {d}( Q_B^*- Q)\\> & {} x_B^* Q(X\le x_B^*) -\int _{\{X\le x_B^*\}} X \mathrm {d}Q\\\ge & {} x_B^* Q(X\le x_B^*) - x_B^* Q(X\le x_B^*) =0, \end{aligned}$$

*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 \(\mathbb {E}^{Q_B^*}[X_i^*]=\mathbb {E}^{Q_B^*}[\xi _i]\), \(i=1, \ldots ,n\).

## 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.

*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,

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

### Theorem 4

- (i)We havewhere \(\varGamma \) is given by (42).$$\begin{aligned} \mathop { \square } \limits _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i} (X) =\min \left\{ x\in [-\infty ,\infty ): ( \alpha _1, \ldots ,\alpha _n)\in \varGamma (x) \right\} =x^*, \end{aligned}$$
- (ii)

### Proof

- (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 haveTherefore, for \(i=1, \ldots ,n\),$$\begin{aligned} \{X>x\}&=\left\{ \sum _{i=1}^n X_i>\sum _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i}(X_i)\right\} \subset \bigcup _{i=1}^n D_i.\end{aligned}$$This shows \((\alpha _1, \ldots ,\alpha _n)\in \varGamma (x)\). As a consequence,$$\begin{aligned} Q_i(X>x, A_i)\le Q_i(X>x,C_i)=Q_i(X>x,D_i)\le Q_i(D_i)\le \alpha _i. \end{aligned}$$From there we obtain \(\square _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i} (X) \ge x^*\).$$\begin{aligned} x^*= \min \left\{ x\in [-\infty ,\infty ): ( \alpha _1, \ldots ,\alpha _n)\in \varGamma (x) \right\} \le x= \sum _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i}(X_i). \end{aligned}$$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 thatNote that (45) implies \(\mathrm {VaR}_{\alpha _i}^{Q_i}((X-x)\mathrm {I}_{A_i})\le 0\) for \(i=1, \ldots ,n\). Let$$\begin{aligned} Q_i((X-x)\mathrm {I}_{A_i}>0) =Q_i(X>x, A_i)\le \alpha _i,~~i=1, \ldots ,n.\end{aligned}$$(45)We have \((X_1, \ldots ,X_n)\in {\mathbb {A}}_n(X)\) and$$\begin{aligned} X_i=(X-x)\mathrm {I}_{A_i}+\frac{x}{n},~~i=1, \ldots ,n.\end{aligned}$$(46)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 \(\square _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i} (X) =x^*\).$$\begin{aligned} \sum _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i}(X_i) = \sum _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i}((X-x)\mathrm {I}_{A_i} ) + x\le x. \end{aligned}$$
- (ii)Suppose \(x^*>-\infty \). Similarly to (45) and (46), from the definition of \(x^*\) and \(A_i^*\), we haveTogether with (i), we conclude that \((X^*_1, \ldots ,X^*_n)\) is a Pareto-optimal allocation of$$\begin{aligned} \sum _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i}(X_i^*) = \sum _{i=1}^n \mathrm {VaR}_{\alpha _i}^{Q_i}((X-x^*)\mathrm {I}_{A_i^*} ) + x^*\le x^*. \end{aligned}$$
*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*.

### Remark 4

### Proposition 4

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

### Proof

*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

*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 \(X\in \mathcal {X}\),

### Theorem 5

### Proof

*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 \(j\in J\),

*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

*X*is given by

### 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

### Corollary 5

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,

- (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).

- (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). - (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.
- 2.
- 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.
In this paper, all random future positions are already discounted, and that is why expectations correspond to prices.

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

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