JSTOR

I. Introduction

As Hansmann (1996) and Hart and Moore (1998) note, it is often thought that, in the absence of government intervention, the only efficient ownership structure in a market economy is one in which firms are investor owned. Yet, as Hansmann's book indicates, this perspective is clearly too narrow. Employee‐owned firms are, and have long been, prevalent in the service professions such as law, accounting, and medicine. Producer cooperatives are predominant in the markets for many agricultural commodities. Nonprofit firms, which effectively have no owners at all, are highly important in the provision of hospital and educational services. Consumer owned firms are important in providing utility services and financial services.

The mutual form of organization, where the consumer and owner interests are merged, has proven to be an effective financial institution, particularly for the provision of insurance. During the late 1980s the formation of new mutual insurers was widely noted in the markets for medical malpractice insurance and municipal liability insurance. In these and other lines of insurance, there has also been a movement toward group captive insurance companies, reciprocals, and industry pools.1 These organizational forms are similar to mutuals in that economic agents with exposure to similar risks share these risks through the organization and these economic agents retain an equity position in the organization. The fact that mutual insurers are owned by their policyholders implies that mutual policyholders necessarily retain some risk. Many of the newly formed mutuals are relatively small, and many small mutuals continue to exist over long periods of time. Policyholders in these small mutuals may actually retain substantial amounts of risk, and it seems unlikely that small mutuals offer risk‐sharing advantages over conventional insurance. These organizations insure risks that are presumably insurable through conventional investor‐owned stock companies. Given that stock insurance companies can spread unsystematic risk across the entire capital market while mutuals can spread this risk only across their membership, it is clear that these mutuals must offer their members some other advantage in order to be viable institutions.

A more general question is why both stock and mutual organizational forms exist and often coexist in the same insurance market. One explanation for the existence of mutual insurance focuses on the conflicts of interest among policyholders, owners, and managers and is based on the ownership structure of mutuals. Policyholders' ability to withdraw assets on demand provides a device for disciplining managers (Fama and Jensen 1983a, 1983b). Since withdrawal of assets is a blunt instrument of control, this suggests mutuals are more likely to form in lines where there is less scope for managerial discretion. Since policyholders are the owners of a mutual insurer, agency problems between owners and policyholders are internalized (Mayers and Smith 1981, 1988). The analysis based on agency problems suggests that the market for any given line of insurance is dominated by one organizational form and does not explain the coexistence of stock and mutual insurers in a given line of business.

A second explanation for the existence of mutual insurance focuses on the participating nature of mutual insurance policies. Under mutual insurance the risk of the insurance pool, including any aggregate risk, is born by those insured rather than transferred to outside stakeholders. Marshall (1974), extending the earlier work of Borch (1962), argues that mutual insurance is more efficient unless there are sufficiently many independent risks that the law of large numbers can be applied. Doherty and Dionne (1993) show that the composite risk transfer implicit in mutual insurance weakly dominates the simple risk transfer implicit in stock insurance. Doherty (1991) argues that changes in policy terms have shifted aggregate risk to policyholders, presumably because changes in loss distributions have made it more difficult for the law of large numbers to reduce the risk in insurance pools. The different risk‐sharing attributes of the mutual form extend to depository financial institutions. Schrand and Unal (1998) show that, when mutual thrifts convert to stock institutions, there is an increase in the total risk of the institution because of the increased ability and incentives for risk taking.

More generally, increasing interest has centered on the question of the coexistence of alternative mechanisms for the delivery of financial services. For example, Boot and Thakor (1997a, 1997b); Repullo and Suarez (1998); and Rajan (1992) developed models in which banks and capital markets coexist as sources of capital to the firms in the economy. In each of these models, coexistence occurs because of heterogeneity in the population of firms consuming the financial services. In a somewhat similar vein, Gorton and Pennacchi (1990) suggest bank deposits coexist with capital market debt because the former solve an adverse selection problem in the capital market. The model developed here is consistent with these studies in that coexistence is driven by heterogeneity, unobservable in our case, in the consuming population. Smith and Stutzer (1990a), in a paper also broadly consistent with the above approaches, argue that, in a Rothschild‐Stiglitz (1976) adverse selection economy with exogenous aggregate uncertainty, participating policies serve as a self‐selection device. High‐risk individuals fully insure against both individual and aggregate risk, while low‐risk individuals partially insure against both individual and aggregate risk.2 The Smith and Stutzer argument is based on the important insight that, in the presence of aggregate risk, their participating nature makes mutual, or mutuallike, insurance policies an efficient risk‐sharing mechanism.

This study extends this prior research by recognizing that, in pure mutual insurance, risk‐averse policyholders participate in the aggregate risk of the insurance pool as the result of their equity position in the risk‐sharing arrangement, even in the absence of economywide aggregate risk. The risk associated with the pool may be significant if the pool size is small. This suggests that organizational size may be an important component of the institutional structure. In fact, we observe the existence of relatively small mutual insurers. Neither of the existing explanations of mutual insurance fully accounts for this difference in the size distributions of stock and mutual insurers. Since there appears to be less risk sharing, the small mutuals must offer their policyholders other advantages, possibly in solving problems of adverse selection. The purpose of this study is to formalize this intuition and state more precisely the conditions under which it is valid. The analysis provides an alternative explanation for the existence of mutual insurance, for the coexistence of stock and mutual insurers, and for differences in the size distribution of the two insurer types.

This study also extends the literature on the economic theory of organizations. A central issue in this paper can be viewed as a question of whether insurance transactions should take place between agents within the firm (mutual insurers) or between agents and firms in an external market (stock insurers). For the most part, the literature has treated this as an “either/or” issue. We consider the question of whether similar transactions can take place both within and between firms simultaneously within an industry. The “property rights” approach defines the firm by the ownership of assets (e.g., Grossman and Hart 1986; Hart and Moore 1990; Hart 1995) and the corresponding residual control rights. In this approach, insurers' assets are the claims on wealth used to indemnify losses in the event losses exceed premium revenue. Mutual insurers have a claim on members' wealth, while stock insurers have a claim on shareholders' wealth. This paper is also relevant to the property rights approach in that it asks the question whether the production of insurance should take place where the rights of consumers and investors are merged (mutual insurers) or separate (stock insurers). For our purposes, it seems most natural to view the firm as a nexus of contracts (Alchain and Demsetz 1972). We distinguish between mutual and stock insurers by the different types of risk‐sharing contracts they produce. A mutual is an agreement among its members to share all loses equally, while a stock insurer provides fixed premium indemnity contracts.

This study is also closely related to the work of Hansmann (1996) and Hart and Moore (1998) on cooperatives. Hart and Moore (1998) examine the relative efficiency of cooperative versus outside ownership. In their model, individuals are ex ante identical but differ in their ex post willingness to pay. The ex post informational asymmetry is the source of inefficiency in their model. In contrast, ex ante informational asymmetry is the source of inefficiency in our model. Hansmann points out that cooperatives are widespread in the U.S. economy. An important theme of Hansmann's book is that cooperatives are more successful when their membership is homogeneous. In insurance markets, this implies that mutuals are more successful when their members have similar risk characteristics. Discussing the development of property‐liability insurance in the mid‐1800s, Hansmann argues that an important reason mutual insurance arose was to solve adverse selection problems.3 As he puts it: “mutual companies evidently arise in part because insurance companies cannot easily distinguish between prospective insureds that differ in the risks they represent” (p. 277). Our analysis provides some insight into the formation of mutuals in the property‐liability insurance industry.4

We develop a model of mutual insurance under adverse selection. We define a mutual as a risk pool, that is, an agreement to share risks among the members of the pool. Since a mutual is an agreement to share risks, separation of high‐ and low‐risk consumers requires separate risk pools. We derive conditions under which there is an equilibrium with high‐risk and low‐risk individuals in separate mutuals. These include a condition on the population size and conditions analogous to those required for the existence of a separating equilibrium in the Rothschild‐Stiglitz (1976) model. We then introduce risk‐neutral (stock) insurers into the model. A single stock insurance company, whether a monopoly or a firm in a competitive industry, can offer a menu of contracts that separate high‐ and low‐risk consumers. We derive the conditions under which stock and mutual insurers can coexist for the monopoly and competitive cases. High‐risk consumers buy insurance from stock insurers and low‐risk consumers form mutuals. While the adverse selection problem limits their size, the mutual can offer higher expected indemnity to low‐risk consumers than the stock insurance policy without attracting high‐risk consumers. The menu of contracts offered by stock insurers is dominated by the menu of institutions offered by the different forms of organization.

We present empirical evidence showing that the size distributions of stock and mutual property‐liability insurers differ in a manner consistent with our theoretical model. Specifically, the proportion of small mutual insurers is much greater than the proportion of small stock insurers. In addition, consistent with the predictions of our theoretical model, the average size of mutual property‐liability insurers is smaller, the variance in the size of mutuals is much greater, and the coefficient of variation of mutual insurer size is much larger than that of stock insurers.

The paper is organized as follows. The next section considers whether it is possible to have distinct mutual insurance companies dedicated to low‐risk and high‐risk consumers. Section III introduces stock insurance companies and considers whether stock and mutual insurance companies can coexist. Some relevant empirical results are presented in section IV, and section V concludes the paper.

II. Risk Pooling and Adverse Selection

In this section we consider the problem of forming a risk pool (i.e., a mutual insurer) in an environment with adverse selection. We show first that adverse selection can create incentives for the formation of distinct mutual insurers. We also show that adverse selection limits the size of these low‐risk mutuals.

The model is a variation of the familiar Rothschild‐Stiglitz (1976) adverse selection model. We assume a finite number, N, of individuals who are indistinguishable in all relevant observable respects. Each individual has a von Neumann‐Morgenstern utility function u(·) exhibiting standard risk aversion (Kimball 1993) and exogenous nonrandom initial wealth, w. Individuals are subject to a risk of loss in the fixed amount l, where . Of these individuals, are “low risk,” having loss probability p, and are “high risk,” having loss probability q, where . Let and . We assume that p and q are fixed, so there is no moral hazard. Each individual is assumed to know his or her own loss probability and that probability is private information.5 The remaining parameters are assumed to be common knowledge.

Since we are interested in the formation of mutual insurers, we restrict the nature of the contracts that individuals can write in forming such risk pools. A mutual insurer (risk pool) is an agreement to divide all incurred losses equally among a group of n members.6 Note that, while there is full stated coverage of losses in such an arrangement, all pool members are assessed for their share of the total loss. If there are k losses, then each individual effectively retains a portion of the aggregate loss equal to kl/n and, if he or she suffers a loss, faces an effective deductible of l/n. All risk pooling contracts are binding and enforceable and are agreed to before losses are realized. An individual wants to join a particular risk pool if doing so would increase the individual's expected utility. The individual is admitted to the risk pool only if doing so increases the expected utility of the members of the pool.

Since individuals can make binding contracts, the problem here is one of cooperative game theory. The solution concept is the modified concept of the core suggested by Boyd, Prescott, and Smith (1988).7 The core is the set of coalitions (mutuals) such that no individual or group of individuals can be made better off by an alternative partition of the population. Here, following Boyd, Prescott, and Smith, the private information possessed by individuals is incorporated in the individual rationality constraints for forming a coalition to block possible core configurations. We refer to this as the information constrained core. The information constrained core is therefore second best because the informational constraints prohibit a first‐ best allocation.8 The problem considered here is to show that the information constrained core is not empty.

All of the members of a risk pool face an identical fraction of the same aggregate loss distribution and therefore have the same expected utility from any given risk pool regardless of their individual characteristics. In general, the loss distribution for any given risk pool depends on the composition of the risk pool. Since individuals are assumed to be identical up to their type, the loss distribution, and thus the expected utility of the members, depends on the number of individuals of each type in the risk pool. An individual's expected utility from membership in a risk pool with low risk and high risk members can be written as . This formulation is conceptually similar to the price‐quantity contract found in the Rothschild‐Stiglitz analysis. Adding members, irrespective of type, subdivides the risk further, which decreases the effective deductible and increases the quantity of insurance. The relative number of high‐ and low‐risk members in the pool influences the price. Since it cannot raise expected losses and increases risk sharing, we assume that adding low‐risk individuals to any risk pool is always desirable (i.e., , for all ) and adding high‐risk members to a pool composed of high‐risk individuals is also always desirable, . Also, adding low‐risk individuals to any risk pool is more desirable than adding high‐risk individuals, .

While it may be more desirable to add low‐risk members than high‐risk members to a pool, high‐risk individuals cannot be expressly excluded, since it cannot be determined a priori who they are. Therefore, a risk pool cannot control its composition directly. However, it can control its size, and although the composition of a risk pool cannot be observed, it may be possible to infer the pool's composition from its size. Therefore, the critical problem becomes determining the optimal sizes for the risk pools that will form.

We show that, under certain conditions, a separating equilibrium exists in which high risks form a large mutual and low risks form small mutuals. The conditions under which this separating equilibrium exists are analogous to those under which a separating equilibrium exists in the Rothschild‐Stiglitz model. The first condition, where is essentially a self‐selection constraint. It requires that high‐risk individuals prefer a pool composed of a large number of high‐risk members to a pool composed of a smaller number of individuals represented in their population proportions. Since the pool of mixed risks offers a lower expected loss, eq. (1) holds only if it implies significantly greater risk retention, that is, if . Depending on the relative numbers of high‐ and low‐risk individuals in the population, eq. (1) may also imply .

The second condition, is the analog of the condition for existence of a separating equilibrium in the Rothschild‐Stiglitz (RS) model. The RS condition is that low‐risk individuals prefer a low‐coverage/low‐price contract (which earns zero profits when purchased only by low‐risk individuals and high‐risk individuals avoid because of the limited coverage) over a contract offering any level of coverage at the pooled (population average) price (which earns zero profits when purchased by all types). Note, the expected utility‐maximizing risk pool that combines low‐ and high‐risk members in their population proportions contains the entire population and yields . Condition (2) guarantees that low‐risk individuals' expected utility at the low‐risk pool is greater than the maximum expected utility they would receive from any pool in which both high‐ and low‐risk individuals participate proportionately. It implies that low‐risk individuals prefer to join a mutual containing a relatively small number of low‐risk individuals. Condition (2) holds if additional risk‐sharing gains that result from adding high‐risk members are not sufficient to compensate for the increase in pool members' expected losses that also result. This is essentially a requirement that the loss probabilities are sufficiently different, the proportion of high‐risk individuals in the population is sufficiently large, or individuals are not overly risk averse.

We assume throughout the rest of the paper that eqq. (1) and (2) hold. We let n* denote the largest value of n for which both eqq. (1) and (2) hold.

Proposition 1. If conditions (1) and (2) hold and is an integer, then (a) the information constrained core is nonempty and (b) high‐risk individuals form a single mutual of size and low‐risk individuals subdivide themselves into small mutuals of size n*. That is, if the high‐risk self‐selection and no pooling constraints hold, then there is a separating equilibrium in which high‐risk individuals form a single large mutual and low‐risk individuals form small mutuals.

Proof. If the composition of a risk pool cannot be controlled then the expected utility of membership in a pool of size n is at most It is straightforward that this utility is maximized at , since the addition of members does not change the expected relative proportion of high‐ and low‐risk members and does increase risk sharing. That is, for a mixed composition pool, the expected utility‐maximizing coalition is the grand coalition In the absence of a utility improving alternative, the grand coalition would form.

Now consider whether coalitions comprising low‐risk members can be formed.9 Let the action of joining such a coalition be designated “join.” Let the alternative to joining one of these coalitions be to remain in a pool composed of all nondeviating individuals. Let this action be designated “stay.” The maximum expected utility from join is , a pool composed entirely of low‐risk individuals. The minimum expected utility from join is U(0, n*), a pool composed entirely of high‐risk individuals. Because risk type is not observable, consumers cannot use it in reaching a cooperative solution. Hence, the choice over the proposed coalitions can be viewed as a normal form noncooperative game with the following payoffs and actions:

Conditions (1) and (2) are the analogs of the conditions needed for separating equilibrium in the Rothschild‐Stiglitz model. Proposition 1 can be viewed as the analog of their result for mutual insurance. If conditions (1) and (2) hold, we would see a market equilibrium in which the high‐risk individuals form a single large mutual and the low‐risk individuals subdivide themselves into a number of mutuals of size n*. Since these pools contain exclusively low‐risk members, n* represents the largest coalition size consistent with a separating equilibrium.11

As Dionne and Doherty (1994) show, the possibility of renegotiation can be fatal to separating equilibria in adverse selection economies. If renegotiation is anticipated, then the expected payoffs from particular strategies are changed in ways that undermine the conditions for separation. The separating equilibrium are renegotiation‐proof if either or holds, where n_H is an arbitrary number of high risks with 1 ≤ n_H ≤ N_H. If eq. (5) holds, a high‐risk individual has no interest in being the first high‐risk member in an otherwise low‐risk mutual of size n* + 1. If eq. (6) holds, the low‐risk members in the mutual have no incentive to admit even one high‐risk member. Conditions (5) and (6) are considerably stricter than conditions (1) and (2). However, only one of these conditions needs to hold to prevent renegotiation. In simulated loss distributions satisfying conditions (1) and (2), we find that condition (5) is difficult to satisfy. However, condition (6) is satisfied quite frequently for reasonable parameter values. Losses must be large in relation to wealth, pool size quite small, and loss probabilities of high‐ and low‐risk individuals similar for condition (6) to be violated. Thus, in general, the separating equilibrium is renegotiationproof, but this arises because low risk members have an interest in ex ante limiting the size of the mutual to prevent entry by high‐risk individuals rather than because high‐risk individuals do not wish to join. We note that, in practice, many small mutuals make efforts to limit membership size by restricting membership to a particular organization, geographical area, profession/industry, or some combination of these. Henceforth, the discussion assumes that either condition (5), condition (6), or both also is met or that the consumers can make binding commitments to a particular pool size at the time of formation. The assumption that N_L/n_L* is an integer is also important to the issue of renegotiation. This assumption avoids the introduction of an integer problem where individuals in non‐optimally sized mutuals have incentives to disturb equilibrium.

If parameter shifts change a preexisting pooling equilibrium, low‐risk individuals who may have preferred the grand coalition may gain from mutual formation. For example, consider the effect of an increase in q, the loss probability for high‐risk individuals, or an increase in θ_H, the proportion of high‐risk individuals. In either case, the utility of the grand coalition falls and both conditions (1) and (2) are more likely to hold. Changes in legal standards could produce these results.12 One consequence is that formation of small mutual insurers is likely to occur in waves following shocks to the market. Other theories do not predict time variation in mutual formation. This analysis may provide some possible insights into the formation of small mutual insurers in some lines of insurance in the late 1980s.

III. The Introduction of Stock Insurance Companies

We now consider whether the equilibrium proposed in the preceding section leaves a role for stock insurance companies. To see that it does, assume there is an (N + 1)st agent, the “insurance company,” that is risk neutral and has initial wealth W ≥ Nl.13 The insurance company can offer price‐quantity contracts, C = (α, β), where α is the fixed premium and β is the gross indemnity. We let U_t(C) denote the expected utility of a type t individual under the insurance contract C. The insurance company designs contracts that maximize its expected profits subject to self‐selection and participation constraints. The insurance company's problem is similar to the problem considered in Stiglitz (1977). The ability to form mutuals implies that the reservation expected utility levels defining the participation constraints for a type are given by the expected utility attained by membership in the optimal mutual for that type, rather than the expected utility of no insurance. The insurance company chooses its contract offerings, then consumers form coalitions as in the preceding section. The insurance company and the consumers it attracts represent a separate coalition within the economy.

The insurance company can make a strictly positive expected profit. To see this, observe that, since N_H is finite, individuals in the large mutual necessarily retain some risk. The insurance company can offer full coverage, β = l, at a premium equal to the certainty equivalent of the risk associated with the large risk pool. Let C_H′ = (z_H, l) denote this contract, where z_H is the certainty equivalent implicitly defined by u(w − z_H) = U(0, N_H). Since individuals are strictly risk averse, z_H > ql, and the contract earns positive expected profits. While the insurance company is a monopoly, the ability of the high‐risk individuals to form a mutual limits the insurer's profits.

Can the insurance company also profitably attract low‐risk policyholders? Analysis of this question requires an understanding of the comparison of a default‐free policy offered by the stock insurer and a mutual policy with the same expected premium and coverage. If the mutual pool size is small, the law of large numbers does not apply and the purchase of a mutual policy involves participation in the risk of the pool. For example, an individual in a low‐risk pool with n members has expected utility where k represents the number of losses and the expectation is over a binomial distribution with parameters p and n − 1. Purchasing a mutual policy is equivalent to purchasing a stock insurance policy with the same expected premium and coverage plus a background risk arising from the equity position in the mutual. That is, the expected utility in eq. (7) can be rewritten as where β_n = (n − 1)l/n is the gross indemnity, α_n = pβ_n = p(n − 1)l/n is the premium and x_n = [p(n − 1) − k]l/n is the zero‐mean background risk. Kimball (1993) shows that, for an individual whose utility function follows standard risk aversion, the introduction of background risk makes the individual worse off and makes the individual behave as if he or she has become more risk averse. Agarwal and Ligon (2002) show that this implies that the indifference curves of consumers become more concave in premium‐indemnity space (or, equivalently, more convex in state space) and that, at equivalently priced full‐coverage policies, individuals with a background risk have lower utility.14

The consequences of this can most easily be seen by referring to figure 1. The lines 0H, 0L, and 0P are the zero expected profit constraints for policies bought exclusively by high‐risk individuals, low‐risk individuals, and a pool with proportional representation. High‐risk individuals are indifferent between membership in the large mutual and membership in mutual pools with expected premium and expected benefit levels lying along the indifference curve U_H², which yields U(0, N_H). Since the purchase of the policy C_H′ effectively removes the background risk from the high‐risk individuals' utility assessment, its premium can be higher at full coverage and still generate the same level of utility. The high‐risk indifference curve for policies offered by the stock insurer and yielding the same utility as C_H′ lie along indifference curve U_H¹. Since U_H(C_H′) = U(0, N_H), the indifference curves U_H¹ and U_H² represent the same level of expected utility, but they differ because the background risk associated with the mutual pool represents a source of disutility and affects the willingness to trade premium for indemnity. Since the insurance company cannot observe individual risk type, it must offer C_H′ along with a policy C_L′ intended to attract low risk individuals. The policy C_L′ is actuarially fairly priced to low risk individuals and satisfies the self‐selection constraint U_H(C_L′) = U_H(C_H′). This is shown in figure 1 at the intersection of the indifference curve U_H¹ and the low risk fair odds line 0L.

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Fig. 1.— Analysis of Equilibrium

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As the size of the low‐risk mutual increases, the premium approaches pl, the coverage approaches l, and the background risk collapses around zero.15 This implies that for n sufficiently large, Low‐risk individuals will join the mutual if it offers higher expected utility than the stock insurance policy, so eq. (9) can be interpreted as a participation constraint for the low‐risk mutual. Let denote the smallest n for which eq. (9) holds. Similarly, in a competitive market, insurers offer a menu of policies intended to separate high‐ and low‐risk individuals. We let C_H* = (ql, l) denote the policy that provides full coverage at an actuarially fair price for high‐risk individuals. Stock insurance policies yielding the same expected utility to high‐risk individuals as C_H* lie along the indifference curve U_H³ in figure 1. Mutual pools yielding the same utility as C_H* lie along the indifference curve U_H⁴ in figure 1. The indifference curves U_H³ and U_H⁴ represent the same level of expected utility, again, they differ because of the background risk of the mutual. We let n** denote the largest value of n for which We let C_L* denote the policy that is fairly priced to low‐risk individuals and satisfies the self‐selection constraint U_H(C_H*) ≥ U_H(C_L*). This is shown in figure 1 at the intersection of the indifference curve U_H³ and the fair odds line 0L. The pair of policies (C_H*, C_L*) are the policies that comprise the Rothschild‐Stiglitz separating equilibrium; these are also shown in figure 1. Again, for n sufficiently large, we have This can also be interpreted as a participation constraint for the low‐risk mutual, and we let denote the smallest n for which this inequality holds.

Proposition 2. Assume there is a monopoly insurance company, conditions (1) and (2) hold and N_L/n* is an integer. If , and , then (a) the information constrained core is not empty; and (b) in equilibrium, high‐risk individuals purchase C_H′ from the insurer and low‐risk individuals form mutuals of size n*. That is, if the high‐risk self‐selection, no pooling, and mutual participation constraints hold, there is a separating equilibrium in which high‐risk individuals buy full‐coverage policies from the monopoly insurer and low‐risk individuals form small mutuals.

Proof. The insurance company offers the policies C_H′ and C_L′, high‐risk individuals choose C_H′. But, since , we have U(n*, 0) > U( ) > U_L(C_L′), and the policy C_L′ fails to attract low‐risk individuals.

Suppose the insurance company attempts to attract low‐risk individuals by decreasing the profits earned from policies sold to high‐risk consumers to increase the coverage levels available to low‐risk consumers. Suppose the insurer offers C_H* = (ql, l) to high‐risk individuals, which attracts only them and yields zero expected profit. As before, high‐risk individuals compare other insurance company offerings along the relatively less risk averse indifference curve U_H³. Possible mutual pools are compared to C_H* along the indifference curve U_H⁴ and U_H³ = U_H⁴. The insurer can offer a policy such as C_L^′′ that is unattractive to high‐risk consumers, would earn positive profits if purchased by low‐risk consumers, and has a higher utility level than U(n*, 0). However, because the reservation utility of high‐risk individuals has increased with their ability to buy the reduced premium policy C_H*, the proposed mutual size can be expanded to yield U(n**, 0) > U( ) > U_L(C_L*) > U_L(C_L^′′). The insurer can attempt to attract low‐risk consumers by reducing the expected profit it attempts to earn, ultimately offering C_L*, which earns zero expected profit. But, since , as just shown, U(n**, 0) > U( ) > U_L(C_L*). Thus, there is no policy that attracts low‐risk consumers, does not attract high‐risk consumers, and earns nonnegative expected profit for the insurer. In this event, the most profitable strategy for the insurance company is to offer C_H′, attracting the high‐risk individuals and earning positive profits. The insurer makes expected profits of N_H(z_H − ql). Low‐risk consumers form mutuals of size n*. If N_L/n* is an integer, then since all individual are in an optimal coalition, the information constrained core is not empty. Q.E.D.

Proposition 2 shows that a monopoly stock insurance company and mutual insurance companies can coexist. High‐risk individuals prefer the stock insurance policies to a single large high‐risk mutual. Low‐risk individuals prefer to join small mutuals rather than purchase partial‐coverage stock insurance policies.

Since the high‐risk individuals behave as if they are less risk averse when assessing the stock insurance company's risk‐free policies, C_L′ implies a lower expected benefit level than that implied by U(n*, 0). This is so because n* is defined as the largest pool size for which the selection constraint (1) holds. The expected wealth levels implied by the utility level of the pool on the right‐hand side of (1) when (1) is binding corresponds to point a in figure 1. The indemnity level consistent with the expected indemnity level of a pool of size n* is the vertical line through a. The low‐risk indifference curve through the intersection of this line and the low‐risk fair odds line gives the expected utility, U(n*, 0), associated with the low‐risk mutual of size n*. The problem then is to determine whether U(n*, 0) is greater or less than U_L(C_L′). If , then the mutual is large enough that the increased expected indemnity available through the mutual is sufficient compensation for the background risk this introduces without attracting high‐risk individuals. That is, both the participation constraint (9) and the selectivity constraint (2) are satisfied.

Suppose, instead of a single monopoly insurance company, a sufficient number of insurance companies exists to enforce competition.16 As before, insurance companies choose contract offerings and the consumers form coalitions. Suppose also that the RS condition holds, and let C_H* and C_L* denote the RS equilibrium policies.

First, observe that stock insurers earn zero expected profit. Any coalition containing stock insurers earning strictly positive expected profit is blocked by a coalition containing a stock insurer extracting a smaller profit. Second, observe that any level of cross‐subsidization is blocked by exclusively low‐risk mutuals, where the optimal size of such mutuals depends on the level of subsidy. Then, the high‐risk individuals obtain an actuarially fairly priced full‐coverage policy, C_H* = (ql, l). Since the number of high‐risk individuals is finite, forming a risk pool yields strictly lower expected utility to high‐risk individuals, u(w − ql) > U(0, N_H). The high‐risk individuals purchase the conventional fixed‐premium insurance policy C_H*. The low‐risk individuals strictly prefer the policy C_L* to C_H*. However, in a mutual, expected utility is an increasing function of the number of low‐risk individuals, and as suggested, since high‐risk reservation utility has increased, the low‐risk mutual can be expanded to n** members without attracting high‐risk consumers.

In terms of figure 1, high‐risk individuals evaluate stock insurance policies along U_H³ and mutuals along U_H⁴. The point b is at the intersection of the indifference curve U_H⁴ and the pooled fair odds line 0P. The indemnity level consistent with the expected utility of a low‐risk pool of size n** is given by the vertical line through b. The low‐risk indifference curve through the intersection of this vertical and the low risk fair odds line gives the expected utility associated with the low risk pool of size n**. The problem then is to determine if U(n**, 0) is greater or less than U_L(U_L*).

Proposition 3. Assume there is a competitive insurance industry, condition (10) holds, and N_L/n** is an integer. If , then (a) the information constrained core is not empty; and (b) in equilibrium, high‐risk consumers purchase C_H* from insurers and low‐risk consumers form mutuals of size n**. That is, if the high‐risk stock participation constraint and the low‐risk mutual participation constraint both hold for size n**, then there is a separating equilibrium in which high‐risk consumers buy full‐coverage policies from competitive insurers and low‐risk consumers form small mutuals.

Proof. Since insurance firms cannot observe individuals' risk types, they offer a menu of policies consisting of C_H* and C_L*, and the high‐risk consumers choose the policy C_H*. Since , we have U(n**, 0) > U( ) > U_L(C_L*), and low‐risk consumers are better off forming mutuals of size n**. If N_L/n** is an integer, then the information constrained core is not empty. Q.E.D.

Proposition 3 shows that mutual insurance companies can coexist with a competitive stock insurance industry. High‐risk individuals prefer the fairly priced full‐coverage stock insurance policies to membership in a single large high‐risk mutual. Low‐risk individuals prefer to join small mutuals rather than purchase partial‐coverage stock insurance contracts.

In both the monopoly and competitive cases, provided N_L is large enough, the exclusively low‐risk mutual pools can always be made large enough to provide greater expected utility to low‐risk individuals than the partial‐coverage policies offered by the insurance companies. The question is whether this makes them so large that they attract the high‐risk individuals.

Proposition 4.(a) Under the assumptions of proposition 2, if , then the information constrained core is empty. (b) Under the assumptions of Proposition 3, if , then the information constrained core is empty. That is, if the participation constraints for the low‐risk mutuals do not hold, then separating equilibrium does not exist.

Proof. (a) If an equilibrium exists, the monopoly insurer offers the pair of policies C_H′ and C_L′ and low‐risk mutuals have at least members. Since , U_L(C_L′) > U(n*, 0), and instead of conditions (1) and (2) we have and That is, condition (2) continues to hold at , but condition (1) does not. Because risk type is nonobservable, insurance companies and consumers cannot use it in reaching a cooperative solution. Hence, the choice between stock insurance policies and joining a mutual of size can be viewed as a normal form noncooperative game with the following payoffs and actions:

		Low Risk
		Stock	Mutual
High Risk	Stock	UH(CH′), UL(CL′)	UH(CH′), U( )
	Mutual	U(0, ), UL(CL′)	U(θL , θH ),
			U(θL , θH )

If in the monopoly case or if in the competitive case, then stock and mutual insurance companies cannot coexist. The problem in both cases is that the selection constraint is violated. The signaling cost born by the low‐risk individuals consists of partial coverage and the risk of their equity position in the mutual. As the size of the mutual increases, the coverage level increases and the risk of the equity position decreases. The critical values and are the sizes at which signaling costs are lower through the mutual mechanism than through the partial‐coverage stock policy in the monopoly and competitive cases, respectively. The critical values n* and n** are the corresponding monopoly and competitive case maximum sizes where the high‐risk self‐selection constraint still holds. When or , then exclusively low‐risk mutual insurance companies are not viable and there is no market equilibrium where stock insurers and mutuals can coexist.

Assuming stock and mutual insurers can coexist, then parameter shifts affect the size and number of mutuals. Again consider the effect of an increase in q, the loss probability for high‐risk individuals. This increases the amount of risk that low‐risk individuals bear as a signaling cost to separate themselves from the high‐risk individuals. In the RS model, this takes the form of a lower indemnity. In the present model, this takes the form of smaller, and therefore riskier, mutuals. Since N_L is fixed, this implies that the number of mutuals increases. Now consider the effect of an increase in θ_H, the proportion of high‐risk individuals. In the RS model, assuming equilibrium exists, the increase in θ_H has no effect; this is not true in the present model. Here, the increase in θ_H reduces the expected utility that high‐risk individuals would obtain by switching from C_H* to a mixed mutual of a given size. Put differently, mutuals can become larger without creating an incentive for high‐risk individuals to switch. Thus, the increase in the proportion of high‐risk individuals leads to a smaller number of larger low‐risk mutuals and, if C_L′ were originally preferred by low‐risk individuals, can lead to formation of mutuals where stock companies were previously dominant.

We conclude that, for appropriate parameter values, both stock insurers and mutuals can coexist in the market. If so, then policyholders of stock insurance companies are exclusively high‐risk individuals and the members of the risk pools are exclusively low‐risk individuals. The self‐selection constraints limit the size of the mutuals. The mutuals must be small enough, hence risky enough, to remain unattractive to high‐risk individuals. The results of this section also suggest that it is possible for a menu of institutions to dominate a menu of contracts. The intuition underlying this result is relatively straightforward. In the case of a menu of contracts the cost of a high‐risk deviation (to the low‐risk contract) is borne entirely by third parties (i.e., the shareholders), because the premium cost is fixed ex ante. Under conventional insurance, stockholders bear the cost of high‐risk deviations and the stock company must restrict effective coverage levels relatively more to prevent high‐risk individuals from purchasing C_L*. In the case of a menu of institutions (where the low‐risk alternative is a small mutual), the cost of a deviation by high‐risk individuals is always borne, in part, by the high‐risk consumers themselves because the cost of mutual membership depends on the pool composition. Thus, the mutual pool can have higher expected benefit levels than the stock insurer's standard deductible contract without attracting high‐risk individuals. If the increase in expected benefit levels is great enough, a small mutual pool can be constructed that dominates the stock insurer's low‐risk deductible contract for the low‐risk individuals, making them better off than they would be in an all stock insurer world. High‐risk individuals are at least as well off with a monopolistic stock insurer and strictly better off with competitive insurers, because of the elimination of the background risk, than they would be in an all mutual world. Hence, the coexistence of both mutual and competitive conventional insurers allows a Pareto improvement in the information constrained core allocations.

IV. Empirical Evidence Related to the Model

Our analysis suggests that, for the mutual organizational form to solve problems of adverse selection, the size of the mutual must be limited. In the absence of these asymmetric information problems, the optimal mutual size would be the grand coalition of all insured individuals facing a particular risk. Since, in the presence of asymmetric information, mutuals have an incentive to remain small or grow very large, the size distribution of mutual insurers may differ from that of stock insurance companies that do not face similar constraints. Specifically, if provision of insurance is dominated by single‐line companies17 and a separating equilibrium obtains in some property‐liability insurance lines, we would expect to see relatively more small mutuals than small stock insurers, since there is a constraint on the size of mutuals serving better‐risk individuals with no corresponding constraint on stock insurer size in these lines. Also, if single‐line companies predominate and in some property‐liability lines a pooling equilibrium obtains (the grand mutual coalition) while in others a separating equilibrium obtains (small mutuals), the standard deviation and coefficient of variation of the mutual distribution should be larger than that of the stock distribution, since the overall mutual distribution reflects the very large firms in some lines and very small firms in others while no size constraints are imposed on the stock insurer distribution. The mean of the mutual distribution may be smaller than that of the stock, if asymmetric information is a common problem (i.e., occurs in most lines) in property‐liability insurance markets. Larger mutuals may predominate if asymmetric information is a less‐common problem (i.e., occurs in relatively fewer lines) in property‐liability insurance markets. So long as the size of some mutuals is limited by asymmetric information, we would expect the standard deviation of size to be larger for mutuals.

To test these implications, data were collected on the total admitted assets and direct premiums written of all stock and mutual insurance companies listed in the 1995 edition of Best's Key Rating Guide: Property‐Casualty Edition, including the firms not rated by Best's. This data source contains 999 mutual insurance companies and 1927 stock insurance companies. The statistics for the mean, standard deviation, and coefficient of variation of assets and premiums for both company types appear in table 1.

TABLE 1 Mean, Standard Deviation, and Coefficient of Variation by Insurer Type (in millions of dollars)

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Table 1 indicates that the standard deviation and coefficient of variation of the mutual distribution are larger than that of the stock distribution with respect to both assets and premiums. The standard deviation of asset size was $1,740.2 million and the coefficient of variation 9.97 for the mutuals and $1,217.1 million and 4.38, respectively, for the stocks. The standard deviation of premiums was $738.1 million and the coefficient of variation 10.14 for mutuals and $454.5 million and 4.28, respectively, for stocks. In the case of both assets and premiums, mutuals have higher standard deviations and higher coefficients of variation than stocks, suggesting greater variation in size. In addition, on average, mutuals are smaller than stocks with a mean asset size of $174.6 million and premium size of $72.8 million compared to $278.0 million and $106.2 million, respectively, for stocks. The question remains whether the greater variation in the size distribution for the mutuals is driven by the presence of a greater number of small mutuals or simply by greater overall variation in size. That question is answered by the frequency distributions for the two types of firms presented in tables 2 and 3.

TABLE 2 Frequency Distribution of Asset Size by Insurer Type (in millions of dollars)

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TABLE 3 Frequency Distribution of Premiums by Insurer Type (in millions of dollars)

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Table 2 presents the frequency distribution of asset size and table 3 presents the frequency distribution of premiums. Both cases show a relatively larger number of small mutual firms. In the case of assets, 68.4% of mutual firms have assets of less than $10 million. For stocks, the percentage of firms with less than $10 million in assets is 27.8%. However, above the $10 million asset level, the size distributions are quite similar. For example, the conditional probability that a mutual has more than $200 million in assets given that it has more than $10 million is 26%, while for stocks this conditional probability is 26.4%. The primary difference between the size distributions is the greater likelihood of very small mutuals.

The premium data in table 3 tell a similar story. For mutuals, 50.4% have premiums of less than $1 million while only 24.9% of stocks have premiums of less than $1 million. As before, the conditional size distribution above the $10 million premium level is virtually identical. The conditional probability of premiums above $200 million given that premiums are above $10 million is 17% for mutuals and 17.6% for stocks.

Moral hazard and the regulatory environment might account for the differences in the size distributions of stock and mutual insurers. Lee and Ligon (2001) show that moral hazard can provide incentives to limit the size of mutual insurers. Policyholders in small mutuals retain substantial amounts of risk, and this provides the incentive to undertake loss‐prevention and loss‐reduction efforts. However, nothing in their results guarantees stock‐mutual coexistence. Ligon and Thistle (2000) show that moral hazard alone does not account for the coexistence of stock and mutual insurers. They show that, in an environment with both moral hazard and adverse selection, the incentives for a separating equilibrium are similar to those developed here.

The regulatory environment is another possible explanation for the differences in the size distributions.18 Insurance companies face state‐level regulation, and differential regulation could account for the difference in the size of stocks and mutuals. The most‐important regulations are statutory minimum capital requirements, risk‐based capital requirements, and solvency monitoring. An examination of states' statutory minimum capital requirements show that almost all states have the same minimum capital requirement for both stock and mutual insurers.19 In many states, the minimum capital requirements are quite small.20 These minima seem to be too small and, more importantly, too uniform across states to account for our empirical results.

The fixed minimum capital requirements are more appropriate for startup companies than established insurers. Risk‐based capital (RBC) requirements vary with the amounts and types of an insurer's premiums, assets, and liabilities; insurers with riskier activities have higher RBC requirements.21 An insurer's actual capital is compared to its RBC and regulatory action taken if actual capital is too low relative to RBC. However, the RBC calculations and regulatory intervention rules are the same for both stock and mutual insurers. Solvency monitoring includes the Insurance Regulatory Information System (IRIS) and the Financial Analysis and Solvency Tracking (FAST) model.22 These impose certain requirements on insurers' financial ratios, for example, on the ratio of premiums to surplus. However, these apply equally to stock and mutual insurance companies. Overall, it seems unlikely that differences in regulation account for our empirical results.

The data clearly indicate that the existence of small mutuals results in significant differences in the size distributions of stock and mutual insurers. The existence of these small mutuals is consistent with the theoretical impact of asymmetric information on mutual formation discussed previously. Other explanations of stock‐mutual coexistence do not predict this difference in the size distributions.

V. Conclusions

In this paper, we analyzed the conditions under which both stock and mutual insurance companies can coexist in the market for the same line of insurance. Our basic argument is that mutuals, and especially the combination of stock and mutual insurers, arise as a solution to adverse selection problems. While existing theories of stock‐mutual coexistence based on agency costs and aggregate risk clearly contribute to our understanding of insurance markets, the current analysis provides some insight into the development of small mutual insurance companies in medical malpractice and municipal liability, the formation of risk retention groups, group captives, and similar risk sharing arrangements and the fact that numerous mutual insurance companies remain very small.

Stock insurance companies solve adverse selection problems by allowing consumers to choose from a menu of contracts. The issue we address is whether mutual insurance companies, or the combination of stock and mutual insurance companies can solve adverse selection problems. We derive the conditions under which mutuals solve adverse selection problems by separating high‐ and low‐risk individuals into different risk pools; these conditions are analogous to those for a Rothschild‐Stiglitz separating equilibrium. We also derive the conditions under which stock and mutual insurers coexist and the combination of both organizational forms solves the adverse selection problem. These conditions include a participation constraint and a self‐selection constraint that mutual insurers must satisfy. Adverse selection, in particular, the need to satisfy the selection constraint, limits the size of the mutual. In this case, high‐risk individuals buy conventional fixed‐premium policies from stock insurers and low‐risk individuals form mutuals. High‐risk individuals are no worse off (under monopoly) or strictly better off (under competition) buying coverage from the stock insurer than joining a mutual. Low‐risk individuals are strictly better off forming mutuals than buying stock insurance policies. The menu of institutions provided by the different organizational forms is a Pareto improvement over the menu of contracts provided by stock insurers.

The empirical evidence suggests that asymmetric information may play an important role in the coexistence of stock and mutual insurers. One empirical implication of our theoretical analysis is that asymmetric information may create incentives for some mutuals to be small. There is no corresponding incentive for stock insurers to be small. We find that the empirical distribution of insurer size by type corresponds precisely with what our theoretical analysis predicts. There exists a proportionately larger number of very small mutuals. This characteristic of the size distribution is not predicted by agency cost or aggregate risk theories of stock‐mutual coexistence. Also, the formation of small mutuals was widely noted in medical malpractice and municipal liability insurance in the late 1980s. Such activity is little noted today. Our theory offers an explanation of time variation in formation of mutual insurers that is not predicted by other theories of stock‐mutual coexistence. Mutual formation would be more likely in periods when parameter shifts have destabilized an existing pooling equilibrium or stock dominated equilibrium in certain lines of insurance. Finally, the evidence here is also consistent with the wider trend in the literature that recognizes that considering the roles played by asymmetric information and heterogeneity improves our understanding of the existence of competing financial institutions and markets in general and the coexistence of stock‐mutual insurers in particular. Heterogeneity in institutions and markets suggests heterogeneity in the populations they serve.