The Effect of Market Power on Risk-Sharing

# The Effect of Market Power on Risk-Sharing

Michail Anthropelos

anthropel@unipi.gr

Abstract. The paper studies an oligopolistic equilibrium model of financial agents who aim to share their random endowments. The risk-sharing securities and their prices are endogenously determined as the outcome of a strategic game played among all the participating agents. In the complete-market setting, each agent’s set of strategic choices consists of the security payoffs and the pricing kernel that are consistent with the optimal-sharing rules; while in the incomplete setting, agents respond via demand functions on a vector of given tradeable securities. It is shown that at the (Nash) risk-sharing equilibrium, the sharing securities are suboptimal, since agents submit for sharing different risk exposures than their true endowments. On the other hand, the Nash equilibrium prices stay unaffected by the game only in the special case of agents with the same risk aversion. In addition, agents with sufficiently lower risk aversion act as predatory traders, since they absorb utility surplus from the high risk averse agents and reduce the efficiency of sharing. The main results of the paper also hold under the generalized models that allow the presence of noise traders and heterogeneity in agents’ beliefs. I would like to thank Peter Bank, Jakša Cvitanić, Paolo Guasoni, Constantinos Kardaras, Rohit Rahi, George Skiadopoulos, Herakles Polemarchakis, Dimitri Vayanos, Dimitris Voliotis, Gordan Žitković and the seminar and conference participants in Technische Universität in Berlin, London School of Economics, Columbia University, University of Texas at Austin, Leibniz Universität in Hannover and University of Piraeus. I also thank the anonymous referees for their valuable comments and suggestions that have improved this paper.Department of Banking and Financial Management, University of Piraeus, Karaoli and Dimitriou Str 80, Piraeus, 18534, Greece; Tel. +30-2104142551; Fax. +30-2104142331.

Keywords: Optimal risk-sharing, Nash equilibrium in risk-sharing, security designing, predatory trading, thin markets.

JEL Classification: D53, D43, C72.

## 1. Introduction

The concept of risk sharing is central to a large variety of financial fields ranging from investment management and structured finance to insurance and derivative markets. Its importance stems from the fact that investors, financial institutions and insurers find that the sharing of investment payoffs, defaultable incomes and insurance liabilities is often mutually beneficial in terms of reduction of risk exposures. The search for the best way to share risk is connected to financial innovation, in the sense that such sharing is fulfilled by designing and trading new financial securities.

The large majority of the growing body of research in risk-sharing assumes that agents act as perfect competitors and the induced allocation of risk is Pareto optimal. However, when financial agents negotiate the sharing of their (otherwise unhedgeable) risk exposures by designing new or by trading given securities, participation is normally limited (at least at the primary market level) to some institutions and/or some of their clients.Although the large participation in the majority of financial markets does not leave room to large agents for market manipulation, several empirical studies have argued that in financial transactions among institutional investors some form of market power is used (see among others Chan and Lakonishok [16], Keim and Madhavan [31, 32] and Kraus and Stoll [36]). Also, other empirical evidence indicates that in some cases even market makers had acted strategically in placing their orders (see e.g., Christie et al. [17], Christie and Schultz [18], Gibson et al. [25] and the the related discussion by Liu and Wang in [39]). These studies have shown that the market power did exist, was exploited and did cause inefficient allocations and prices. This limited participation implies that at least some of the agents possess the power to affect the equilibrium allocations and prices. As discussed for instance in Rostek and Weretka [48], even in the contemporaneous financial markets, large institutional investors often have the power to dominate the trading and drive the prices according to their own benefit. Moreover, when the market is over-the-counter (OTC), the assumption of perfectly competitive structure is even further away from being realistic. An indicative example is the reinsurance market, where few insurance companies share their portfolios through the trading of new reinsurance contracts. It is apparent that such market can not be considered competitive, since the strategic behavior of each individual company could heavily influence the equilibrium transaction.

Market models that do not take into account agents’ market power lead to equilibria that overestimate the market efficiency (and, as stated by Hellwig in [27] and Kyle in [37], they require agents to behave “schizophrenically” by ignoring their ability to affect the market). If we assume that participating agents do exploit their impact on the way the risk is shared, the equilibrium is different and the efficiency is most likely reduced.

Following the standard setting in the risk-sharing literature, we assume that agents are endowed with some risky portfolios, which are called random endowments. The mutually beneficial sharing of these endowments can be achieved in a complete or in a incomplete-market setting. In the former, agents design and price new securities that Pareto optimally share their risk exposures.The securities that share risk in a Pareto optimal way were introduced by the seminal works of Borch [8, 9] and Wilson [54] (see also Demange and Laroque [20] and the surveys of Allen and Gale in [3] and Duffie and Rahi in [23]). On the other hand, if the securitization of the agents’ endowments is not possible (due to exogenous constrains, such as transaction costs and strict regulation), an incomplete risk-sharing can be attained through the trading of a given vector of securities. Although there is an extended literature dealing with both settings, the effect of agents’ market power on equilibria has not been sufficiently addressed. The main objective of this paper is to fill this gap by (i) modelling the strategic behavior of agents with heterogenous risk aversions when they share risk, (ii) establishing and analyzing the Nash equilibria that occur as the outcome of these strategies and (iii) investigating which are the agents (if any) that benefit from the oligopolistic structure of a risk-sharing transaction.

### 1.1. Model description and main findings

We consider a one-shot, risk-sharing transaction among risk averse agents. If none of them acts strategically, their endowments are shared through the optimal sharing rules; the risk-sharing securities and the pricing kernel are given by specific functions of agents’ endowments, in a way that the aggregate utility after the transaction is maximized.

In our model, we suppose that agents agree that whatever vector of endowments is submitted for sharing, is going to be shared according to the optimal sharing rules. We then take the position of an individual agent and ask whether she has motive to submit risk exposure different than her true one. Since the risk-sharing securities and the pricing kernel are functions of the submitted endowments, the endowment that she chooses to submit directly affects not only the structure of the designed securities, but also their prices. According to the proposed model, she should respond as her to-be-shared endowment the random quantity (called best-endowment response) that maximizes her own utility, when the agreed sharing rules are applied.In contrast to the relevant literature on thin markets with symmetric information (see e.g., Weretka [53] and the references therein), here the strategic set of choices is of infinite dimension; namely it is the set of all random variables that can be considered as agent’s random endowment. This set of choices is in fact equivalent to the set of securities and pricing kernel that are consistent with the optimal sharing rules.The generality of the agent’s random endowment in our model allows its application to a number of financial markets, such as the trading of innovated derivative products, the use of reinsurance contracts or the transactions among inter-dealers (see also the related discussion in Vayanos [51]). Under mean-variance preferences (hereafter M-V preferences), it is shown that it is never optimal for an agent to submit for sharing her true risk exposure. Instead, each agent has motive to declare only a fraction of her true endowment and also to report exposure to the aggregate endowment submitted by the other agents.

A similar best-response strategy is employed when agents negotiate the trading of a given bundle of tradeable securities (incomplete-market setting). Therein, we consider an individual agent who knows the aggregate demand schedule of the rest of the agentsNote that under M-V preferences, the demand schedules are linear and hence a demand function can be inferred by knowing only two of its values (orders). This implies that for the implementation of the proposed strategy (and the induced Nash-game), agents do not need to submit their entire demand functions, but rather only two of its values. This makes the model more practical and applicable. and ask: Which is her best equilibrium price, given the submitted demand from the rest of the agents? Naturally, we define as her “best equilibrium price” the one that clears out the market and at the same time maximizes her own utility. She can then drive the market to this price by submitting a corresponding demand function (called best-demand response). As explained in the sequel, this best-response strategy is appropriate when the transaction involves risk-sharing, since it allows us to identify the hedging needs that are revealed by the submitted demand functions. It is shown that under M-V preferences, the best-demand response is different than the true one in any non-trivial case (i.e., in any case where some risk is to be shared). In fact, the best-demand response coincides with the demand that would be asked by an agent endowed with the best-endowment response.

The aforementioned strategic behavior, when applied by all agents, forms a negotiation scheme on the risk-sharing transactions. The risk-sharing equilibrium is the outcome of a Nash-type game played among the participating agents, where their sets of strategic choices are the risks they choose to submit (or equivalently the securities they are willing to get). In Section 4, it is proved that under M-V preferences the complete-market setting admits a pure-strategy Nash equilibrium. Furthermore, although it is rare to have uniqueness of Nash equilibrium in models with uncertainty (see among others, Grossman [26] and Klemperer and Meyer [33]), the equilibria are indeed unique and fully characterized. In the incomplete market, the game is played on the agents’ reported demand schedules or equivalently on the prices and the allocations of the tradeable assets. The Nash equilibrium is again unique and given in explicit form.

In the Nash risk-sharing, the equilibrium is inefficient in all non-trivial cases, meaning that the aggregate utility is reduced when compared to the optimal one. Interestingly enough though, for agents with sufficiently lower risk aversion the Nash equilibrium yields higher utility. The main message of this theoretical result is that in thin risk-sharing markets, not only there is a loss in the aggregate utility, but also agents with relatively lower risk aversion absorb utility surpluses from their counterparties.The comparison of utility surpluses in competitive and in Nash equilibria is based on the simplified (but indicative) market of two agents with different risk aversions. Another way to interpret this result is through the notion of predatory trading (introduced by Brunnermeier and Pedersen in [11]). More precisely, the market’s thinness gives an opportunity for the relatively lower risk averse agents to act as predatory traders and exploit the intense hedging needs of their counterparties. The strategic behavior is more effective for the more risk tolerant agents, who in a sense “dominate” the transaction. In contrast to the other predatory trading models, this is an endogenous result and holds even if predatory traders are still risk averse.The fact that the lower risk averse agents tend to get more utility in Nash equilibrium is heavily based on the generality of the strategic set and in particular on the agents’ ability to declare as their risk exposure any random variable on the probability space. One may ask whether similar result holds when the set of strategic choices is exogenously restricted. In Appendix B, we consider an example of such restriction where agents choose only the size of their true random endowments that are willing to share. Through this indicative (counter)example, it is shown that the answer is negative, meaning that in complete-market setting the effectiveness of the predatory trading strongly depends on the imposed set of strategic choices. Furthermore, the securities that are designed to share risk are the optimal ones only under the special case of agents with the same risk aversion. Even in this case however, the equilibrium volume and the efficiency are reduced.

In the incomplete-market setting, the induced Nash equilibrium can be considered as an oligopoly variation of the CAPM. In Section 4, it is shown that the Nash equilibrium prices are equal to the perfect-competition ones if and only if agents are homogeneous with respect to their risk preferences. Even in this case however, the allocation in Nash equilibrium is not efficient and the volume is lower (in particular, the volume percentage reduction is , where is the number of participating agents). In the more general situations of heterogeneous agents, we establish an exact measure of the price impact that is caused by the agents’ game. For example, it is shown that an intense upward price pressure occurs not only when the agents with intense hedging demand are also high risk averse, but even when agents with low risk aversion (acting as predatory traders) participate in the trading. As in the complete-market setting, sufficiently lower risk averse agents profit from the market’s inefficiency, for any vector of tradeable securities.

For the definition of market inefficiency, we follow the related literature (see among others Acharya and Bisin [2]) and define the risk-sharing inefficiency as the difference between the aggregate utility surplus at the optimal risk-sharing and the aggregate utility surplus at the Nash equilibrium. As stated above, inefficiency is positive, whereas its size is mainly determined by the number of participating agents and the market’s completion. It is reasonable to expect that the efficiency of the risk-sharing transaction increases as the number of the participating agents increases. Indeed, both in complete and incomplete-market settings, the differences between the Nash and competitive equilibria vanish and the market becomes efficient as the number of agents increases to infinity. In addition, a market that is closer to completion leans to more efficient sharing. It is shown that even when the market is thin and gives rise to market power, each agent gets more utility when the market’s setting is complete. This implies that even in markets with limited participation, financial innovation is mutually beneficial (at least when there are no further transaction costs).

Another possible generalization of the model is regarding agents’ beliefs. As in the majority of the relevant literature, we have assumed that agents’ probability assessments for payoffs of tradeable securities and endowments are common. In Section 6, we withdraw this assumption in the incomplete-market setting and study how this affects the corresponding equilibria. We introduce this generalization following the recently developed models of Kyle et al. [38] (see also Rostek and Weretka [47]) and show that the way agents share the deviations of their beliefs is similar to the way they share their random endowments both in competitive and Nash equilibrium. In particular, except from revealing different hedging needs, agents have motive to declare different beliefs on the tradeable securities too. Homogeneity in agents’ risk aversion keep the equilibrium prices unaffected by the game, but the corresponding volume is again reduced. More interesting is the discussion regarding the utility surpluses. At first glance, one could conjecture that the difference in beliefs results in a more profitable predatory trading, since, apart from the hedging needs, a predatory trader could exploit the deviation on probability assessments. However, this is not necessarily true. In our model, predatory traders are risk averse with specific hedging needs. This means that when the average beliefs of the other agents induce higher (resp. lower) value for the securities a predatory trader buys (resp. sells), her utility surpluses are afflicted. Nevertheless, even under the model that allows different beliefs, sufficiently lower risk averse agents still get more utility at Nash equilibrium than at the competitive one.

### 1.2. Connection with the relevant literature

The paper is related to two main strands of literature: the design of financial securities (related to the complete-market setting) and the trading of standardized securities in thin (OTC) markets under imperfect competition (related to the incomplete-market setting). The contributions of the paper and its connections to these strands of the literature are described below.

The large majority of the literature on financial innovation and more particularly on security design imposes a competitive market structure and does not allow agents to implement any kind of strategic behavior (see among others Acciaio [1], Barrieu and El Karoui [7], Dana [19], and Jouini et al. [30] and the surveys of Allen and Gale [3] and Duffie and Rahi [23]). Relatively little attention has been paid to the more realistic risk-sharing transactions, where agents possess and exploit some sort of market power.

In the existed models of non-competitive risk-sharing, the agents’ market power is asymmetric and stems mostly from asymmetric information or other types of imposed structural differences among players. For example, games on security design under adverse-selection problem (i.e., asymmetric information) have been recently studied (see among others Horst and Moreno-Bromberg [28, 29] and Page and Monteiro [42, 43]), where the market consists of firms (security issuers) that offer a menu of contracts to a continuum of agents. In Section 3, we also consider the case where only one agent has asymmetric information on the aggregate endowment of the rest of the agents. However, the risk-sharing transaction that we study here is not between principal(s) and agent(s), but rather the security design involves a mutually beneficial transfer of all the agents’ risks (agents are not categorized with respect to their types, but with respect to their endowments and risk aversions).

More recently, other types of discriminations among the participating players have been imposed in modelling of security design games. Agents are distinguished as arbitrageurs and investors (see Rahi and Zigrand [46] and Shleifer and Vishny [50]) or security issuers and investors (see Carvajal et al. [14] and the reference therein)Rahi and Zigrand in [46] study a security-design game played among arbitrageurs, where the induced Nash equilibrium is optimal for the arbitrageurs regarding their profits from the mispricings across different market segments. Although the arbitrageurs’ strategic behavior has some common features with the model presented here (e.g., it refers to maximization of quadratic utility functions), the game is played on a different field (arbitrageurs imperfectly compete on the profits that come as a result of segmented markets and they do not share any risk exposure). In Carvajal et al. [14] on the other hand, only some of the participating agents, namely the security issuers (entrepreneurs) design the securities and investors competitively trade them, without securitizing their own endowments.. As already emphasized, the results of this literature do not cover our model, since we do not impose any agents’ discrimination regarding their information or their ability to influence the market. To the best of our knowledge, this is the first attempt to model a symmetric game on risk-sharing transaction where the agents’ bargaining power and security payoffs are endogenously determined in the equilibrium.Another novelty is that the set of strategic choices consists of the all possible endowments that each agent can submit for sharing. This general setting allows, among other things, to conclude that even when agents impose sharing rules that are consistent with the optimal ones, the securities that they design differ from the Pareto-optimal securities, since agents do not submit their true endowments for sharing. In particular, it is an endogenous result that sufficiently lower risk averse agents have higher bargaining power in the game regardless of their true random endowments.

The game on the incomplete market-setting is related to literature on imperfectly competitive financial markets, which is based on the seminal work of Kyle in [37]. According to this theory, agents act strategically on the trading of given securities, and their strategic set of choices consists of the demand functions that they may submit. The related literature follows this approach and studies equilibria under a variety of different sources of imperfection. Mainly (as in Back et al. [6], Kyle [37], Kyle et al. [38] and Vayanos [51]), the source of market’s imperfection is the asymmetric information, where agents are categorized to informed, uniformed and noise ones. Our model departs from this literature, since we assume an oligopolistic market, where each agent is rational and strategically uses her market power even without the presence of noise tradersAs already mentioned, the presence of noise traders and its impact on the equilibrium are addressed in Section 5 as an extension of our basic model.. In other words, as in the complete-market setting, it is the thinness of the market that gives rise to the market power, rather than any type of exogenously imposed asymmetry.

Recently, a number of equilibrium models have been established assuming that market’s imperfection stems from market’s thinness. More precisely, in Carvajal and Weretka [15] (see also Malamud and Rostek [41], Weretka [53] or Carvajal [13] for a broader discussion), a non-competitive market without asymmetric information is considered and (as in our paper) each agent submits demand functions taking into account the impact of her order in the equilibrium. The main substantial difference to our demand-function game is the imposed set of strategic choices. According to the demand games in Carvajal and Weretka [15], Kyle [37] and Weretka [53] (see also Rostek and Weretka [48] for a dynamic version), each agent estimates the price impact of her order, (defined as the slope of the aggregate demand of the rest of the agents) and responds accordingly. Under linear demand functions however, the slope of the demands does not depend on agents’ endowments and hence this demand game is not driven by the agents’ motive to reveal hedging needs different from their true ones. In technical terms, our game is played on the intercept points of the linear demand functions rather than their slopes. As discussed in details in Section 4, it is exactly the strategic behavior on the intercept points that allows us to examine the effective sharing of risks in Nash equilibrium and compare it with the corresponding equilibrium in the complete-market setting.Note also that although in Carvajal and Weretka [15] and Weretka [53] the agents’ preferences are more general, the tractability of their models is based on the assumption of finite probability space. In addition, it should be emphasized at this point that the Nash-game on the intercept points has an equilibrium even in the case of two agents. This comes in sharp contrast to models of Carvajal and Weretka [15], Kyle [37] and Weretka [53], where the two-agent game is in fact ill-posed. This should be highlighted, since many of the real-world risk-sharing transactions are between only two agents.

Also, another branch of the literature on market-power modelling (see e.g., Brunnermeier and Pedersen [11] and Vayanos [52]) distinguishes agents between price-takers and those with market power (or large strategic investors and noisy traders In our model, noise traders do not change the main structure of the equilibrium. As analysed in Section 5, their presence can be summarized through a given submitted endowment or a given order on the tradeable securities (both considered as public information). This eventually means that the net supply, in both complete and incomplete market settings, is not zero.). In particular, in Brunnermeier and Pedersen [11], a scheme of a predatory trading is created by the (exogenously given) need of certain investors to liquidate their positions. The predator investor exploits this need by acting strategically and at first sells the asset and subsequently buys it back. In our model, the predatory trading is given endogenously by the strategic behavior of the agents who respond to the other agents submitted hedging needs. As mentioned above, it is shown that the role of the predator is played by agents with sufficiently lower risk aversion, who exploit the intense hedging needs of the higher risk averse counterparties. Furthermore, if an agent also possesses asymmetric information on the other agents’ endowments, the predatory trading is more excruciating and more profitable.

Finally, we highlight the two main features of this paper, that distinguish it from the related literature. First, the agents’ endowments do not necessarily belong in the span of the tradeable securities (as in Rostek and Weretka [47, 48] and Vayanos [51] for instance). Besides that this generalization is closer to the real-world situations, it allows us to examine how a pre-existing unhedgeable payoff in an agent’s portfolio affects her strategic behavior. Most importantly, assuming general endowments, we are able to designate the link between the equilibria in compete market and incomplete market setting, and in particular that Nash equilibrium demands reflect the hedging needs from the Nash equilibrium endowments.

Secondly, in contrast to the majority of literature in risk-sharing and non-competitive markets that assumes equal agents’ risk aversions, our results emphasize the importance of agents’ heterogeneity on their risk preferences, not only for the allocation and prices, but mainly for the utility surplus that each individual agent gets at the equilibria.

The paper proceeds as follows: Section 2 introduces the market model and sets up the optimal risk-sharing rules in both complete and incomplete market settings. Section 3 establishes the model for an individual agent’s strategy, regarding the risk that she chooses to share and the demand function she chooses to submit, given the other agents’ submitted risks and demands. The Nash risk-sharing equilibria are defined and analyzed in Section 4. The effect on the main findings caused by the presence of noise traders is addressed in Section 5; while Section 6 is dedicated to the market model that allows agents’ with different beliefs. In Appendix B, an indicative example of a restricted strategic set is discussed and the paper’s contributions are summarized in Section 7. For the reader’s convenience, the proofs are omitted from the main body of the paper and provided in Appendix A.

## 2. Risk-Sharing Equilibria without Market Power

We consider a static market model of agents who aim to reduce their risk exposures by trading to each other. Throughout this manuscript, it is assumed that there exists an exogenously priced numéraire in units of which all mentioned payoffs are denominated. Each agent is already exposed to a random endowment that incorporates the net discounted payoffs of all the unhedgeable financial positions that she has taken (the risk that cannot be hedged out by trading in any market outside this setting). These endowments are denoted by random variables , , which are defined in a standard probability space is the so-called “subjective” probability measure, and assumed to be common for each agent. In Section 6, we examine how the the main results of the paper are affected we withdraw the common-beliefs assumption.. The sum of these random endowments is called the aggregate endowment and is denoted by . The preferences of agent are modelled by the M-V utilityThe quadratic utility has be widely used in the risk-sharing literature in a variety of subjects, from adverse-selection problems (see e.g., Carlier et al. [12], Horst and Moreno-Bromberg [28, 29]) to games on financial innovation (as in Rahi and Zigrand [46]). Furthermore, these preferences are consistent with the standard assumptions of exponential utility and normally distributed payoffs (see among others Kyle [37], Vayanos [51] and Liu and Wang [39]). Note also that M-V preferences can generally be considered as a second order approximation of other utility functionals (see for instance Section 4 of Anthropelos and Žitković [5]).

 (2.1) Ui(X)=E[X]−γiVar[X],

for any random payoff that belongs in (hereafter denoted simply by ), where constant is agent ’s risk-aversion coefficient and and stand for the expectation and variance maps under probability measure . With a slight abuse of notation, when is applied to a vector of random variables, it refers to the associated variance-covariance matrix.

In the sequel, we consider two risk-sharing settings: the incomplete-market one, where agents share their risks through the trading of a given vector of securities; and the complete-market setting where risk-sharing involves the designing and trading of new financial securities.

### 2.1. Risk-sharing in incomplete markets

As discussed in the introductory section, the design of new securities that optimally share the agents’ risky endowments and complete the market is rarely possible.Examples of the exogenous constrains that restrict agents from trading the optimal risk-sharing securities are: (i) The stricter regulation on OTC transactions, that imposes restrictions on credit levels, (ii) the indivisibility of some types of random incomes, e.g., real estate investments, revenues of a running business, dividends from illiquid shares etc, (iii) further liquidity concerns and transaction costs that make the optimal risk-sharing trading disadvantageous (an issue addressed among others by Allen and Gale in [3]). In such situations, agents can mutually reduce their risk exposures by trading certain number of standardized securities, whose payoffs are possibly correlated with their endowments. These securities (hereafter called tradeable) could be any structured financial derivatives, such as credit derivatives, asset backed securities, reinsurance contracts etc. Although, trading a given vector of securities is not a Pareto-optimal transaction, an equilibrium allocation could improve the utility of each individual agent.

We assume the existence of tradeable securities, with random vector denoting their payoffs. For any price vector , agent ’s optimization problem is

 (2.2) supa∈Rk{Ui(Ei+a⋅C−a⋅p)}=supa∈Rk{Ui(Ei+a⋅C)−a⋅p}.

The set of vectors that maximize (2.2) for a given price vector is the demand of agent on at price , i.e., Note that this optimization problem imposes no short-selling constrains. One may think that the absence of such constrains implies the possibility of unbounded supply or demand. This can be avoided, for instance, if we impose some regularity requirements on the set of agent’s positions. Unbounded supply (demand) indeed occurs when the prices are sufficiently high (or low). However, the equilibrium arguments, developed in the sequel, will endogenously exclude such extreme situations.. For the set to be a singleton for any price vector , it is sufficient to impose the following assumption.

Standing Assumption. For every considered vector of securities , the matrix is non-singular.

This condition guarantees that is indeed a function from to , which is called the demand function of agent of securities , and under M-V preferences

 (2.3) Zi(p)=(E[C]−p2γi−Cov(C,Ei))⋅Var−1[C],

where stands for the vector , and for any payoff , denotes the vector .

###### Remark 2.1.

Note that the demand has two distinguished sources; the risk premium: , and the correlation between the tradeable securities and agent’s endowment: . The demand for a particular security , for , is a decreasing function of its covariance with agent’s endowment, a fact that supports the use of the M-V criterion for risk management purposes. Indeed, we expect that when is negative, agent is willing to take a long position on as a partial hedging transaction. It is also important to note that only the intercept point (and not the slope) of the linear function depends on agent ’s endowment (see also Demange and Laroque [20] for analogous and more detailed discussion).

In the sequel, we will need some further notation. Let denote the set of matrices that represent allocations of the vector . More precisely, element , and of an allocation , stands for the units of security that agent buys (negative means short position) and the zero-net supply of tradeable securities implies that , for each . Also, for an allocation matrix , denotes its -th row, for each .

When agents do not behave strategically, the risk-sharing achieved by trading is given through a competitive equilibrium (which coincides with the CAPM, see among others Magill and Quinzii [40]).

###### Definition 2.2.

For a given vector of securities , the pair is called a competitive price-allocation equilibrium of if for each .

Taking into account equation (2.3), we get the uniqueness of competitive price-allocation equilibrium and its characterization. Below, stands for the aggregate risk-aversion coefficient, i.e., , and denotes the relative risk tolerance of agent . Also, following the standard notation, we define and , i.e., is the aggregate endowment of the rest of the agents and their aggregate relative risk tolerance.

###### Proposition 2.3 (Capm).

The unique competitive price-allocation equilibrium of a vector of securities is given by

 (2.4) p∗=E[C]−2γCov(C,E)anda∗i=Cov(C,Coi)⋅Var−1[C],

where is the -th row of matrix and

 (2.5) Coi:=λiE−i−(1−λi)Ei,for each i∈{1,...,n}.

Proposition 2.3 states that the equilibrium prices do not depend on the variance-covariance matrix of securities, but only on their expectations and their covariances with the aggregate endowment (this covariance is usually called the covariance value of the securities). Under no market power, the equilibrium price of a security increases as its covariance with the aggregate endowment decreases, a feature that indicates that higher demand of a particular security implies higher equilibrium price.

### 2.2. Optimal risk-sharing

In the case where there are no exogenous constrains or additional transaction costs, agents freely design securities that Pareto-optimally share their risk exposures and complete the market. More precisely, we define the set , which contains all the possible risk-sharing securities, where stands for the payoff of the security that agent receives. The requirement that the sum of these securities is zero implies that risk-sharing transaction is cleared out and there is no further risk added to the market. In Section 5, we withdraw the assumption of zero-net supply, in order to examine how a non-constant risk exposure submitted by noise traders affects the main results of the paper. The formal definition of the optimal risk-sharing follows.

###### Definition 2.4.

A vector of securities is a Pareto-optimal risk-sharing if for all other the following implication holds:
If for some , , then such that

Note that Pareto-optimal risk-sharing equilibrium assumes no market power, and all the agents submit their true risk exposures for the sharing. It is well-known that under M-V preferences, the optimal risk-sharing securities are linear functions of the agents’ endowments (see among others Demange and Laroque [20], Wilson [54]). It follows that we can restrict the problem of finding the optimal sharing to finding the price-allocation of the agents’ vector of endowments . Proposition 2.3 yields that the competitive price-allocation equilibrium of is the pair , where , and the elements of are and for .The exponent in the notation refers to the Pareto-optimal risk-sharing, i.e., when agents are allowed to trade their endowments and do not behave strategically, see also the securities defined in (2.5). One the other hand, for equilibrium prices and allocations in the incomplete market-setting have notation with exponent . The formal statement is given in the following proposition (its proof is placed in Appendix A).

###### Proposition 2.5.

Let be the competitive price-allocation equilibrium of the vector of securities . Then, the vector of securities is the unique Pareto-optimal risk-sharing.

At the optimal risk-sharing transaction, agent gets the total payoff , where denotes the vector . Note that with defined in (2.5). Hence, according to the optimal risk-sharing rules, agent gets security (hereafter called optimal-sharing security) and pays the price

 (2.6) πoi:=aoi⋅po=E[Coi]−2γCov(Coi,E).

In other words, under no use of market power, agent is going to short a part of her true random endowment and long equal parts of the other agents’ aggregate endowment.From Proposition 2.5 it also follows that the pair of the vector of securities and the pricing kernel , defined as , for , is in fact the unique Arrow-Debreu equilibrium among the agents.

### 2.3. Utility surpluses and the inefficiency measure

One of the main goals of this paper is the comparison of each individual agent’s utility surplus at different equilibria. For this, at any price , we denote the utility of agent at her demand as

 vi(p;C)=Ui(Ei+Zi(p)⋅C)−Zi(p)⋅p.

The difference simply measures the utility surplus that agent gets by buying her demand on at price . The advantage of using as a measure of the agent’s utility surplus from a transaction is that it is measured in numéraire units and therefore can be used for comparisons among different equilibria and agents. When the market setting is complete, we use the simplified notation . From Propositions 2.3 and 2.5, we readily get the agent ’s utility at both market settings, provided that none of the agents applies any kind of strategic behavior.

###### Corollary 2.6.

The utility of agent at the Pareto-optimal risk-sharing is given by

 (2.7) voi=γiVar[Coi]+Ui(Ei),

while the utility at the competitive price-allocation equilibrium of a security vector is

 (2.8) vi(p∗;C)=γia∗i⋅Var[C]⋅a∗i+Ui(Ei)=γiCov(C,Coi)⋅Var−1[C]⋅Cov(C,Coi)+Ui(Ei).

It follows that the utility “loss” of each agent when the risk-sharing is in incomplete market and not in complete one, that is the difference , is always non-negative and equal to zero if and only if the optimal security of agent belongs to the span of . We state this fact in the following proposition, the proof of which follows standard arguments.

###### Proposition 2.7.

Each individual agent suffers a loss of utility from market incompleteness. This loss is zero for agent if and only if .

###### Remark 2.8.

The term in (2.8) has a nice interpretation in the case where both and follow multivariate normal distribution. Indeed, an application of the projection theorem (see among others, Chapter 1 of the book by Brunnermeier [10]) yields that the conditional distribution of given , remains normal with conditional variance given by This implies the representation , which connects the utility surplus with the information on that is contained in . Namely, the more information about is revealed from , the higher the agent ’s utility surplus is. Note also that when is measurable with respect to the information generated by , gets its maximum value, and when and are independent, agent is indifferent between trading and not trading at its equilibrium price.

Regarding the aggregate utility, we point out that at the optimal risk-sharing transaction and hence the difference is the maximum aggregate utility surplus that the market gets from the optimal risk-sharing.Note that Pareto optimality and the M-V preferences guarantee that (see also the proof of Proposition 2.5). This allows us to consider the difference between the aggregate utilities in optimal risk-sharing and in any realized (suboptimal) risk-sharing, as a measure of risk-sharing inefficiency, i.e.

 (2.9) Risk-Sharing Inefficiency = Optimal Aggregate Utility − Realized Aggregate Utility.22\lx@notefootnote$22$ThismeasureofaggregatelossofutilityisalsousedinAcharyaandBisin\@@cite[cite][\@@bibrefAchaBin05],whileinVayanos\@@cite[cite][\@@bibrefVay99]theaggregateutilitiesarecomparedintermsofthecertaintyequivalent,whichcoincidewithourcomparison,underM−Vpreferences.

## 3. Best Responses in Risk-Sharing Transactions

As emphasized in the introductory section, financial risk-sharing transactions are normally among relatively few agents. Modelling of such oligopolies should include agents’ best responses to other agents’ actions. This section establishes a novel model for the agents’ strategic behavior in risk-sharing markets, which can be considered also as a (predatory trading) strategy implemented by an agent who has asymmetric information on other agents’ endowments or demand functions.

### 3.1. Best response in complete risk-sharing markets

Proposition 2.5 states that the optimal way to share any vector of submitted random endowments is through the (linear) optimal sharing rules. Given that agents have agreed to adopt this transaction set-up, we ask whether it is preferable for an individual agent to submit her true random endowment or to report a different risk exposure.

More precisely, let’s consider agent , and assume that she knows the aggregate risk exposure of the rest of the agents. According to optimal sharing rule, if she reports endowment , her position at the equilibrium will be (see the payoffs in (2.5) and the prices in (2.6)). However, she may exploit the other agents’ hedging needs that stem from and drive the security-designing to more preferable security and price that the other agents are still willing to offset. The fact that the optimal sharing rules are given linear functions of the submitted endowments means that proposing securities and pricing kernel is in fact equivalent to submitting a corresponding endowment. Therefore, agent can achieve a desirable equilibrium by submitting an appropriate endowment in the sharing scheme. Following the terminology of game theory, we call an optimal choice for submitted endowment best-endowment response.

Intuitively, knowing endowment , agent has two goals: First to hedge as effectively as possible her true endowment (recall that all agents are assumed risk averse), and second to exploit the hedging needs of her counterparties and get a better cash compensation. Regarding the first goal, agent has to submit at least some part of her true endowment and achieve a partial hedging. On the other hand, she may declare exposure to the risk that the other agents possess, since by doing so, she increases the supply (and hence decreases the prices) of the securities she is going to buy.

More formally, given that her counterparties have reported aggregate endowment , if agent reports as her random endowment some random variable , she gets the security with payoff

 (3.1) Coi(B):=λiE−i−λ−iB,

and the accumulated cash she has to pay is

 (3.2) πoi(B):=λi(E[E−i]−2γCov(E−i,E−i+B))−λ−i(E[B]−2γCov(B,E−i+B)).

Hence, her utility after the transaction is

 (3.3) Gi(B;E−i):=Ui(Ei+Coi(B)−πoi(B))=E[Ei+Coi(B)]−γiVar[Ei+Coi(B)]−πoi(B).

Therefore, the best-endowment response of agent is the solution of the following maximization problem, where is the set of her strategic choices

 (3.4) Bri:=argmaxB∈L2{Gi(B;E−i)}.23\lx@notefootnote$23$Notethatlettingthesetofstrategicchoicesbeequalto$L2(Ω,F,P)$impliesthatagent′sreportedendowmentismeasurablewithrespecttotheinformationthatisgeneratedbythetrueendowments,i.e.,allagentsfacethesame$σ$−algebra.

It is important to point out again that (if it exists and is unique) directly determines the payoff of the security that agent gets and the price she pays at the equilibrium. In other words, submitting endowment for sharing is equivalent to proposing the security , determined by equation (3.1), and the induced cash compensation given in (3.2). The solution of problem (3.4) is stated in the following proposition.

###### Proposition 3.1.

For each , the unique (up to constants) best-endowment response of agent , when the rest of the agents have submitted aggregate endowment , is given by

 (3.5) Bri=11+λiEi+λ2i1−λ2iE−i.

By submitting endowment , agent gets security with payoff instead of , where we readily calculate that . Thus, according to best-endowment response, agent gets only a fraction of the Pareto-optimal risk-sharing security, whereas the price she pays is less than , and in fact

 πri=πoi1+λi−2γλiVar[Coi]λ−i(1+λi)2.

A number of further observations are worth some attention. First, the best response is invariant on the probability distribution of the random endowments.This outcome is based on the common-beliefs assumption and the imposed M-V preferences. In particular, under common beliefs, M-V preferences imply that the optimal risk-sharing rules do not depend on the probability distribution of the endowments. This independence is endowed to the optimizer of problem (3.4). In more general preferences’ model however, risk-sharing rules, and hence the corresponding problem (3.4), do depend on the probability distribution of the involved endowments (see for example Jouini et al. [30]). The generalization of the best-endowment response in such models is a very challenging problem and left for future research. Also, given that agent knows the endowments of the rest of the agents, the best-endowment response strategy indicates that she should share only a fraction of her risk exposure and report exposure to the risk that the other agents face. In this way, she increases (resp. decreases) the demand of the securities she sells (resp. buys) at the risk-sharing transaction, which results in a better total cash compensation, . This strategy could well be considered as a predatory trading (in the spirit of Brunnermeier and Pedersen [11]). Agent exploits the hedging needs of her counterparties and acts strategically aiming not only to a good hedging, but also to a higher price of the securities she sells.

The agent’s utility after reporting the best-endowment response is which means that the surplus of utility of agent from applying this strategic behavior is equal to or in other words, compared to utility surplus in (2.7), the percentage increase of utility is (an increasing function of ). Therefore, the application of such predatory trading is more beneficial for agents with relatively low risk aversion; while for homogeneous agents, the percentage increase of utility equals to .

Note also that the difference between becomes equal (up to constants) to if and only if is a constant, which essentially means that agent has nothing to trade with the rest of the agents. Furthermore, and as expected, the difference between and reduces as the number of participating agents increases (the larger the market becomes, the less effective the use of market power is).

###### Remark 3.2.

The case of a speculator, i.e., an agent with constant endowment, is of special interest. In the optimal risk-sharing equilibrium, a speculator gets units of the other agents’ aggregate endowment, in exchange of a risk premium equal to . According to best-endowment response however, she exploits her knowledge on the other agents’ endowment and submits risk exposure to too. In this way, she increases the supply of , which eventually yields to a higher risk premium for those who buy . At resulting equilibrium, not only she gets less units of (which means less risk for her), but also a higher premium.

### 3.2. Best response in incomplete risk-sharing markets

In the case of incomplete market-setting, where agents share their risk only through the transaction of a given vector of securities, each agent reports her demand function and the (competitive) equilibrium is reached at the price vector that sums the demands to zero. In view of the best-endowment response, we conjecture that agents have motive to declare demand functions that do not reflect their true hedging needs. Indeed, it is shown below that when an agent does share some risk with the other agents, it is never optimal to submit her true demand function.

The study of the strategic behaviors under incomplete markets through the submission of demand functions goes back at least to Kyle [37] (see also Rostek and Weretka [48] and the reference therein). Following this literature, we consider agent and suppose that she knows (or can exact) the aggregate demand function of the rest of the agents. Recall from equation (2.3) that under M-V preferences, each agent’s demand function is linear with slope equal to . This implies that the aggregate demand function of the other agents is also linear and given by , where we use the notation . Then, agent submits a demand function that clears out the market at the price that maximizes her own utility. In contrast to the related literatureFor instance, the game developed in Rostek and Weretka [48] or Weretka [53] does not allow agents to submit different hedging needs than those induced by their true endowments. The key difference is that our game is played on the intercept point of the demand functions not on their slopes., we do not assume that agent responds to the slope of demand (which does not contain the agents’ endowments), but rather we ask: Which is the best equilibrium price for agent , given the aggregate demand of the other agents?As explained below, this is a more appropriate way of modelling strategic behavior under endowments that do not belong to the span of the tradeable securities, since it allows us to see how the best responses reflect different hedging needs than the true ones.

Hence, agent ’s problem is written as

 (3.6) pri:=argmaxp∈Rk{ϕi(p;Z−i(p))},

where , for . Below, we state how the competitive price-allocation equilibrium changes when an agent’s market power stems from the asymmetric information on the other agents’ demands.

###### Proposition 3.3.

The price-allocation equilibrium of a vector of securities that maximizes the utility of agent , for each , given the other agents’ aggregate demand function, is provided by

 (3.7) pri=E[C]−2γCov(C,Ei1+λi+E−i1−λ2i),
 (3.8) ari=a∗i1+λiand for all j≠i,arj=a∗j+λiλj1+λiCov(C,λiλ−iE−i−Ei)⋅Var−1[C].

It follows from (3.8) that the units of that agent gets after best-response strategy is reduced when compared to the competitive allocation. In addition, the effective aggregate endowment in the covariance part of the CAPM is equal to instead of . In the effective aggregate endowment, the volume on is reduced by a percentage equal to , whereas the volume on is increased by a percentage equal to . It is implied therefore that agent could drive the market to price if her submitted demand reflects less exposure to her true endowment and some exposure to the other agents’ aggregate endowment (similarly to the best-endowment response). The exact demand function of agent that clears out the market at price is called best-demand response.

To formulate this best-response problem, we have to rewrite problem (3.6) in terms of demand functions. For this, having security vector fixed, we denote by the set of all demand functions that an agent with M-V preferences and risk aversion could submit. Since, all these demand functions have the specific form of (2.3), we can parametrize by the covariance vectors: A function belongs to if and only if there exists a vector , such that

 (3.9) zi(p)=(E[C]−p2γi−c)⋅Var−1[C].

Note that vector reflects the hedging needs, since it takes the position of the covariance between the agent’s endowment and the tradeable securities. Taking the market-clearing into account, the best response of agent is to report the demand , such that or equivalently to choose the vector , such that It follows that where the random variable is the best-endowment response given in (3.5). We summarize this discussion in the following proposition, which states the clear connection between the agent’s best responses in complete and incomplete market settings.

###### Proposition 3.4.

For every vector of securities :

• The best-demand response of an agent coincides with the demand function that is associated with her best-endowment response.

• The best-demand response equals to the true demand if and only if .

• The difference of utilities at the best-response and at competitive equilibrium is

 Ui(Ei+ari⋅(C−pri))−Ui(Ei+a∗i⋅(C−p∗))=γiλ2i1−λ2ia∗i⋅Var[C]⋅a∗i.
###### Remark 3.5.

In view of Corollary 2.6, we have the following relation between the agent ’s utility surpluses

 Utility surplus from best-response=λ2i1−λ2i×Utility surplus in competitive equilibrium.

We conclude that the best-response strategy is more profitable when agent is relatively low risk averse and when her utility surplus from the competitive equilibrium (i.e., without strategic behavior) is large.

Proposition 3.3 states that although the strategy of best-demand response changes the equilibrium prices and reduces the size of the agent’s positions, it does not change the direction of her positions. In particular, by submitting demand , agent increases (resp. decreases) the demand of the securities she is going to short (resp. long) at the equilibrium, and thus creates a beneficial impact on the equilibrium prices, i.e., a predatory trading strategy. A measure of this price impact can simply be given by the difference . For any vector , we readily calculate that this difference equals to . In view of Proposition 2.3, each element of this vector is positive (resp. negative) for the securities that agent sells (resp. buys).

###### Remark 3.6.

The case of a speculator is distinctive (connect it with the discussion in Remark 3.2). In the competitive equilibrium, a speculator satisfies the hedging needs of the other agents by buying (resp. selling) the securities they want to sell (resp. buy). Under this perspective, a speculator could be considered as a (risk averse) market maker. For example, if the payoff of the security, say , is negatively correlated with endowment , the speculator is supposed to take short position on . According to her best-demand response however, knowing the hedging needs of her counterparties, she is going to submit a demand function that reveals some hedging needs (which are in fact of the same direction as the other agents’ aggregate demand). She does so in order to increase the demand on and eventually increases its market value. This is a clear predatory trading. Note that at equilibrium she still takes a short position on (as allocation (3.8) indicates), but her predatory trading strategy increases the risk premium that she is paid by the other agents.

## 4. Nash Risk-Sharing Equilibria

The existence and the uniqueness of best-responses allow us to take the next step and examine whether and how these thin markets equilibrate. Each agent responds to other agents’ choices forming a type of pure-Nash game, where (depending on the market completion) the strategic sets of choices are the set of endowments or demands.

In this section, we study the existence and the uniqueness of the Nash equilibria, and ask whether the agents’ motive to declare different risk exposures than their true ones still holds, when all agents act strategically. We also ask whether there are agents that benefit from the games in thin risk-sharing markets and get higher utility surpluses.

### 4.1. Nash equilibrium in complete risk-sharing market

The way that agents use their market power in the complete risk-sharing market is provided by optimization problem (3.4). Namely, each agent declares the random endowment she chooses to share (the one indicated by solution (3.5)) or equivalently proposes securities that are in line with the optimal sharing rules (see payoff in (3.1) and prices in (3.2)). This procedure sets the conditions of the Nash game in the complete risk-sharing market, and the equilibrium is reached at the security payoffs that all the agents agree on. We call this equilibrium Nash risk-sharing equilibrium.This is a pure-Nash equilibrium where the set of strategic choices is in fact . The definition presupposes that the additional costs and the time of the negotiation are negligible, both of which are standard assumptions in the literature of the Nash equilibria. Before we give the exact definition, we recall from (3.3) that the utility of agent , when she submits endowment and the other agents have submitted endowment is written as

 (4.1) Gi(B;B−i)=Ui(Ei+Coi(B)−πoi(B))=E[Ei+Coi(B)]−γiVar[Ei+Coi(B)]−πoi(B).
###### Definition 4.1.

We call a vector of random variables Nash equilibrium endowments if for each

 (4.2) Gi(B⋄i;B⋄−i)≥Gi(B;B⋄−i),for all B∈L2.

The induced risk-sharing securities given by

 (4.3) C⋄i:=Coi(B⋄i)=λiB⋄−i+λ−iB⋄i,for each i∈{1,...,n},

are called Nash risk-sharing securities.Throughout the paper, exponent refers to Nash equilibrium features.

If a vector of Nash equilibrium endowments exists, the prices of the induced risk-sharing securities are determined by the pricing rule (2.6), where the considered aggregate endowment is the Nash equilibrium one, hereafter denoted by . Hence, for each , the Nash equilibrium price of security is . The following theorem gives the explicit form of the unique Nash risk-sharing equilibrium.

###### Theorem 4.2.

There exists a unique (up to constants) Nash risk-sharing equilibrium characterized as follows: For each , the Nash equilibrium endowment of agent is

 (4.4) B⋄i=λ−iEi+λ2iB⋄,

where

 (4.5) B⋄=1αn∑j=1λ−jEj

and the constant is defined by . The Nash risk-sharing security that agent gets has payoff

 (4.6) C⋄i=λiλ−iα∑j≠iλ−jEj−λ−i(1−λiλ−iα)Ei,

and the utility of agent at the Nash equilibrium is

 (4.7) voi(π⋄i):=Ui(Ei+C⋄i−π⋄i)=γi1+λi1−λiVar[C⋄i]+Ui(Ei)=γiγi+γγi−γVar[C⋄i]+Ui(Ei).

A number of observations are worth some emphasis. First, in consistence with Proposition 3.1, in Nash equilibrium each agent submits only a fraction of her true endowment and reports exposure to the endowments of the other agents. Indeed, the coefficient of in the equilibrium endowment equals to , which is strictly less than 1 for each . As a result, the risk-sharing transaction is significantly departed from the Pareto-optimality. Not only the security that each agent gets at equilibrium is different than the optimal risk-sharing one, but also, in general, the aggregate shared endowment deviates from the true one (they coincide only when agents are homogeneous, see Corollary 4.9 below). In particular, one can readily check that which, in view of (4.6), means that at the Nash equilibrium each agent still sells some of her true endowment, but the size is always smaller than in Pareto-optimal transaction (recall from (2.5) the payoff of security ). Also, for each agent , the quantity of her true risk exposure that she sells at Nash equilibrium is an increasing function of her risk-aversion coefficient (i.e., decreasing function of ), meaning that lower risk averse agents apply the strategic behavior more intensely.

On the other hand, the quantity of that agent buys at Nash equilibrium is not necessarily less than the corresponding quantity in the competitive equilibrium. For this, we compare the coefficients and (recall again (2.5)) and observe that only in the special case of two agents, the size in Nash equilibrium is always less than the Pareto-optimal one (see also the indicative Table 1 below). For , the inequality holds if and are sufficiently largeIt is a matter of simple calculations to get an equivalent inequality of condition . Indeed, for , and , the latter inequality is equivalent to , meaning that agent buys less of endowment if both agents and are sufficiently low risk averse. The latter implies that for relatively low risk averse agents, only a percentage of their true endowments is shared in Nash equilibrium. We summarize the main conclusions of this discussion below.

###### Corollary 4.3.

At Nash risk-sharing equilibrium:

• Each agent reports only a fraction of her true endowment and some exposure to the endowments of the rest of the agents.

• The total quantity of endowment that is shared in Nash equilibrium is less or equal to 1, if and only if

 (4.8) ∑j≠iλ2j∑j≠iλj≤λi.

Note that inequality (4.8) holds strictly when , while for the equality holds when agents have the same risk aversion.

###### Remark 4.4.

Another aspect that highlights the importance of the agents’ risk aversion is the asymptotic interaction of agents’ behaviors. Namely, we observe that -, for every ; while -, for every