Capital, aggregate risk, insurance prices and regulation☆
Introduction
Capital requirements and risk constraints are key tools in the regulation of insurance markets. In the presence of incomplete markets and limited commitment, insurers may hold insufficient capital buffers and undiversified, risky asset portfolios that do not optimally protect them against aggregate, non-diversifiable shocks to their assets and liabilities (e.g., de Bandt and Overton (2020)). Insurance regulation also features other policy instruments, such as State Guaranty Funds in the U.S. that provide reinsurance, and bailouts when negative aggregate shocks lead to a large number of insolvent insurers. Previous theoretical studies examine the isolated impacts of individual regulatory tools in partial equilibrium settings. A rigorous analysis of insurance regulation, however, necessitates a general equilibrium framework that endogenizes insurance supply and demand, and incorporates aggregate shocks to insurers' assets and liabilities. Indeed, insurance prices are determined in equilibrium of insurance markets where the demand for insurance equals the supply. Further, to the extent that regulation is intended to improve the efficiency of insurance markets, an analysis of the effectiveness of regulation necessitates an investigation of the sources of inefficiencies in unregulated insurance markets. Because partial equilibrium models by design do not consider the interactions between insurance markets and the broader economy, it is difficult (if not impossible) to characterize efficient allocations in partial equilibrium models. Consequently, such models are not designed for normative analyses that provide insights into how regulatory policies should be optimally designed. Extant literature, therefore, provides limited guidance on the design of insurance regulation and how different regulatory tools interact with each other to mitigate market inefficiencies.
We develop a tractable general equilibrium model of competitive insurance and capital markets to analyze insurance regulation. In the unique equilibrium of the benchmark unregulated economy, insurers maximally leverage their inside equity stakes by raising external capital solely by selling insurance. We derive the Pareto efficient allocation and demonstrate that it can be implemented by multiple regulatory policies that combine risk-based capital requirements, asset risk constraints, reinsurance and bailouts. An efficient regulatory policy with a lower capital requirement must be accompanied by a more stringent asset risk constraint, and one with a capital requirement that is more sensitive to insurer assets should be less sensitive to insurer liabilities. When the aggregate asset and liability risks are below respective thresholds, insurees are fully insured in the optimally regulated economy, and the set of efficient regulatory policies does not vary with aggregate risk. Outside this region, however, insurees must bear aggregate risk with capital and asset risk constraints becoming tighter as either risk increases. The aggregate risks have sharply contrasting effects on the unregulated and regulated economies. Our results highlight the importance of tailoring regulatory policies to aggregate risk levels, and disentangling the impacts of aggregate asset and liability risks in analyses of how regulation influences insurance prices and insurer capitalization.
Our model features continuums of risk-averse insurees, risk neutral investors and insurance firm entrepreneurs (hereafter, insurance firms) each endowed with capital, as well as risk-neutral regular firms with no initial capital. Agents of each type are ex ante identical and can invest in a risk-free asset providing a constant return. Insuree losses are identically distributed, but potentially correlated as the economy is exposed to aggregate losses from a catastrophic event. Our model is applicable to life insurance markets in which case the aggregate loss could stem from a pandemic such as COVID-19, or property and casualty (P&C) insurance markets where aggregate losses could arise from natural events such as hurricanes. In the model, a negative (positive) aggregate shock leads to a higher (lower) proportion of insurees incurring losses; the difference in the proportions is the aggregate loss or liability risk. Regular firms have risky, concave technologies that are identically and binomially distributed, but not independent. There are disjoint sectors of regular firms with all firms in the same sector being exposed to a common sectoral shock that simultaneously affects their success or failure. The economy is also exposed to an aggregate asset shock that leads to correlation among sectoral shocks. A positive (negative) aggregate shock leads to a higher (lower) proportion of sectors experiencing positive shocks; the difference between the proportions is the aggregate asset risk. Agents know the distributions of the aggregate shocks, but not their realizations ex ante or the specific subsets of insurees and sectors that are respectively exposed to the shocks.
In the benchmark unregulated economy, each insuree invests in the risk-free asset, an insurance policy issued by a single insurer, and the diversified market portfolio of regular firms. Investors invest in the risk-free asset as well as equity securities issued by insurance and regular firms. Insurance firms, whose initial capital endowments represent their inside equity capital, can raise external capital via insurance premia paid by insurees and outside equity capital from investors. Insurers, who cannot commit to their investment choices when they raise capital, invest their capital in the risk-free asset and regular firms. Regular firms raise equity capital from investors and insurers. All agents are protected by limited liability.
The unique equilibrium of the benchmark unregulated economy is symmetric. Insurance firms raise external capital entirely via selling insurance policies and default when their assets fail. The result stems from the intuitive observation that unregulated insurance firms maximize their expected payoffs by maximally leveraging their inside equity stakes, but requires rigorous justification in our general equilibrium framework in which capital allocations, prices and returns are endogenous. The intuition for the result is as follows. First, in a competitive equilibrium, regular firms must be indifferent between raising capital from investors and insurers, and investors must be indifferent between investing in insurers and regular firms. Further, insurers too must be indifferent between raising additional capital via outside equity or selling insurance, and must make zero additional profits from raising external capital and investing it. Therefore, the expected marginal productivities of regular firms, the expected equity returns of insurance and regular firms, and insurers' expected marginal costs of raising capital via equity and insurance premia are all equal. Second, the expected return to insurees exceeds the risk-free return in an equilibrium with nonzero insurer default risk because risk-averse insurees must obtain a risk premium. Hence, the (common) expected return on insurer and firm equity exceeds the risk-free return so that risk-neutral insurers and investors do not invest in the risk-free asset. Therefore, investors' capital is ultimately invested in firms either directly or indirectly via their investments in insurance firms. Hence, the total capital invested in regular firms and, therefore, the expected marginal productivity of the representative firm is fully determined by the total capital that insurers raise via insurance premia, which we refer to as their insurance capital for expositional convenience. Finally, in equilibrium, each insurer chooses how much external equity and insurance capital to raise to maximize the expected return on its initial inside equity capital, which equals the expected marginal productivity of each regular firm. The concavity of firms' technology, however, implies that their expected marginal productivity is maximized when the insurance capital raised by insurers is minimized. But the insurance capital is minimized precisely when insurers' equity capital is limited to their inside equity, which maximizes their insolvency risk and minimizes insurance demand by risk-averse insurees.
We examine how the aggregate asset and liability risks affect the equilibrium of the unregulated economy. The equilibrium insurance price declines with the aggregate risks. Interestingly, however, the insurance capital raised by each insurer increases with the aggregate asset risk, but decreases with the aggregate liability risk. The intuition is that the aggregate asset and liability risks have opposing effects on how insurees allocate their capital to the market portfolio and insurance purchases. An increase in the aggregate asset risk does not directly affect the default risk of an individual insurer, but increases the risk of the market portfolio. Hence, insurees invest less capital in the market portfolio and more in purchasing insurance. In contrast, an increase in the aggregate liability risk increases the default risk of insurers, but does not directly impact the market portfolio risk. Hence, insurees invest more capital in the market portfolio and less in the purchase of insurance.1 Because insurers raise external capital entirely via insurance premia in the unregulated equilibrium, the increase (decrease) in insurance capital with the aggregate asset (liability) risk implies that insurer capitalization increases with aggregate asset risk, but decreases with the aggregate liability risk.
The unregulated equilibrium is inefficient, thereby justifying regulatory intervention. First, as insurers raise no outside equity, their high default risk constrains the amount of insurance coverage purchased by insurees. Second, under limited commitment and liability, each insurer maximizes shareholder value by choosing an undiversified portfolio of investments in a single sector. Third, markets for sharing insurer insolvency risk are incomplete. The possibility of insurer default affects the amount of capital that insurers are able to raise by selling insurance and, thereby, influences the total investment in productive firms.
In contrast with partial equilibrium frameworks, our general equilibrium model permits a full characterization of efficiency. We derive the Pareto efficient allocation that maximizes the total expected utility of risk-averse insurees while ensuring that investors, insurers and regular firms receive their expected payoffs in the unregulated equilibrium. The efficient allocation trades off providing protection to insurees against risk and investing in productive regular firms. When the aggregate asset and liability risks are below respective thresholds—that is, the economy is in the normal region— there is sufficient capital to fully protect insurees in all aggregate states even when the investment in regular firms is unconstrained by the need to protect insurees. When the aggregate asset risk and/or the liability risk exceed respective thresholds, however, there is insufficient capital to provide full protection to insurees in all aggregate states without violating the limited liability constraints of investors, insurers and regular firms. Hence, insurees must bear risk in the efficient allocation. Investment declines with the aggregate asset and liability risks.
We show that the efficient allocation can be implemented in a decentralized economy via standard regulatory tools: (i) an ex-post reinsurance scheme akin to the state guaranty fund system in the U.S.; (ii) a constraint that restricts insurers' asset risk; (iii) a risk-based capital requirement contingent on the insurer's risky assets and the capital it raises via insurance premia, which reflects its future liabilities; and (iv) insurer bailouts. There is, in fact, a range of regulatory policies that implement the efficient allocation. As the multiple regulatory policies implement the same allocation, however, the equilibrium of the regulated economy is unique for a given policy, and the regulatory tools that comprise a policy are linked. (a) An equilibrium with a more stringent asset risk constraint is associated with a lower capital requirement, smaller insurer size, and greater bailout subsidies to failed insurers. (b) An equilibrium in which the sensitivity of the capital requirement to an insurer's risky assets is higher must be associated with a lower sensitivity to its liabilities.
The intuition is that the insurance price and total insurance capital raised by insurers depend only on the aggregate asset and liability risks in an efficient regulated equilibrium. Hence, for given aggregate risk levels, the total capital held by insurers is effectively determined by insurers' equity capital. The asset risk constraint specifies the proportion of insurer capital that must be invested in the risk-free asset. To ensure that the level of investment in the risk-free asset by the economy is efficient, a more stringent asset risk constraint must, therefore, be associated with a lower level of capital raised by insurers that, in turn, implies a lower amount of equity capital. Hence, a more stringent asset risk constraint is associated with a lower capital requirement. Because the size of the representative insurer is determined by its equity capital, an equilibrium with a tighter asset risk constraint is associated with smaller insurer size. As an equilibrium with a tighter asset risk constraint features smaller equity capital buffers for insurers, subsidies to failed insurers must be higher. Finally, the sensitivities of the capital requirement to an insurer's assets and liabilities play substitutable roles as they together ensure that the insurer's total equity capital is sufficient relative to its insurance capital. Our results that the same efficient allocation can be implemented by multiple regulatory policies with a more stringent asset risk constraint being accompanied by a more lax capital requirement provide support for proponents as well as opponents of stricter insurance regulation.
We next show how the aggregate risks influence optimal regulatory policies. As investment is unconstrained in the normal region, the efficient level of investment in regular firms is at the first best level where their expected marginal productivity equals the risk-free return. Therefore, insurers are indifferent between regular firms and the risk-free asset, and there is no need for a capital requirement or asset risk constraint. The efficient allocation can be implemented via reinsurance and bailouts to ensure that insurees are fully protected against risk. As the efficient allocation does not vary with aggregate risk, the set of optimal regulatory policies also does not change. Outside the normal region, however, the expected marginal productivity of regular firms exceeds the risk-free return at the efficient allocation so that the efficient level of investment is below the first best level. Hence, insurers are no longer indifferent between regular firms and the risk-free asset so that a binding asset risk constraint and capital requirement are necessary. As the efficient level of investment in the risk-free asset increases with the aggregate asset and liability risks, the interval of asset risk constraints that can implement the efficient allocation shifts to the right. Further, for a given asset risk constraint, the corresponding capital requirement must also increase to ensure that each insurer's capital is sufficient to guarantee that insurers collectively invest efficiently in the risk-free asset.
We contribute to the theoretical literature that examines insurance regulation. Several prior studies examine the effects of specific regulatory tools in partial equilibrium settings. Filipovic et al. (2015) show that capital requirements reduce insurers' insolvency risk stemming from inefficient investments due to limited liability. Rees et al. (1999) argue that regulation does not have beneficial effects when insurers can fully commit to hold sufficient capital. Liu (2019) examines the impact of capital requirements on life insurer's asset risk-taking and underwriting in a partial equilibrium framework in which the cost of capital, insurance price, and capital requirement are exogenous. Koijen and Yogo (2021) develop an equilibrium model of an oligopolistic insurance market. They show that market power and financial frictions stemming from a risk-based capital or economic risk constraint are important determinants of pricing, contract characteristics, and the degree of market incompleteness.
We complement the literature by building a general equilibrium model of competitive insurance and capital markets that incorporates aggregate asset and liability risks. In an unregulated economy with insurer limited commitment and liability, insurers engage in risk-shifting by holding undiversified asset portfolios. Our general equilibrium framework allows for a rigorous characterization of efficiency. We show how the principal tools used to regulate insurance markets can implement the efficient allocation in our normative analysis. We, thereby, endogenize regulatory policies instead of assuming them exogenously as the aforementioned studies do. We demonstrate that reinsurance and bailouts are sufficient for efficiency when the aggregate asset and liability risks are below respective thresholds, but capital and asset risk constraints are also necessary for efficiency outside this region. None of these results can be obtained in partial equilibrium models as they are designed to isolate the impact of specific regulatory tools while abstracting from the rest of the economy. Further, as there is no rigorous characterization of efficiency in partial equilibrium frameworks, they cannot provide a full understanding of why the regulatory tools are there in the first place and how efficient regulatory policies are impacted by aggregate risk.
Ibragimov et al. (2009) develop a model of insurance markets with aggregate liability (catastrophe) risk. They show that limited liability for insurers and heavy-tailed risk distributions combine to create nondiversification traps where insurers choose not to offer catastrophe insurance or participate in reinsurance markets. They show that a centralized agency can play an important coordination role to achieve efficient risk-sharing. They focus solely on aggregate liability risk and examine the role of reinsurance while abstracting away from other regulatory tools. We complement their study by also incorporating aggregate asset risk and show how it interacts with aggregate liability risk to influence optimal regulatory policies that include not just reinsurance, but also insurer bailouts, capital requirements and asset risk constraints. Insurers choose undiversified portfolios in our benchmark model of an unregulated economy due to the convex payoff of shareholders, who are protected by limited liability.
Several studies explore the relation between insurance prices and insurer capitalization employing partial equilibrium models that either focus on insurance supply assuming that insurers are free of insolvency risk (Gron, 1994; Winter, 1994), or insurance demand by incorporating insurer insolvency (Doherty and Garven, 1986; Cummins, 1988; Cummins and Danzon, 1997). In our general equilibrium framework, insurance prices and capitalization are simultaneously and endogenously determined. Taylor (1995) develops a model that extends the framework of Turner (1981) to endogenize the relation between insurance price and capital. In his model, however, insurer capital is determined via the maximization of insurees' expected utility rather than via the capital demand and supply decisions of agents interacting with each other in decentralized markets. Moreover, we study the impact of regulation and derive the novel results that aggregate asset and liability risks could have opposing impacts on insurance prices and insurer capital.
Section snippets
The model
There are two dates 0 and 1 and a single consumption/capital good. There is a continuum of measure 1 of identical risk-averse insurees; a continuum of measure 1 of risk-neutral investors; a continuum of measure M of insurance firms; and a continuum of measure N of regular firms. Agents of each type are ex ante identical. Our results only require that investors are less risk-averse than insurees so that there are gains from sharing insurance risk between insurees and investors via insurance
The equilibrium of the unregulated economy
We now derive the unique equilibrium of the unregulated economy. We make the following standing assumption.
Assumption 1
The first inequality above states that, if the total capital, , held by investors and insurers is invested uniformly in regular firms (so that each regular firm receives ), but no insuree capital is invested in regular firms (either directly by insurees or indirectly via investments by insurers in regular firms), then
Efficient allocation
The equilibrium of the unregulated economy is inefficient for a number of reasons. First, as insurers raise no outside equity, their high default risk constrains the amount of insurance coverage purchased by insurees. Second, due to limited commitment and limited liability, insurers cannot commit to their investment decisions when they raise capital, and each insurer maximizes shareholder value by investing all its capital in an undiversified portfolio of firms in a single sector. Insurers'
Regulation
We now describe how regulatory tools that mimic those observed in insurance markets can be designed to implement the efficient allocation in a decentralized economy. Given our normative objectives, we facilitate our analysis by considering a single “regulator” who embodies the roles of various entities involved in insurance regulation. The decentralized economy is as described in Section 2, but there is now a regulator who has access to intervention tools that influence agents' decisions and,
Conclusions
We develop a general equilibrium model to examine the design and impact of insurance regulation. In the unique unregulated equilibrium, insurers with inside equity stakes raise external capital solely by selling insurance policies. We derive the Pareto efficient allocation that maximizes insurees' expected utility, while ensuring that investors and firms receive their expected payoffs in the unregulated equilibrium. The efficient allocation depends crucially on the relative levels of the
Declaration of Competing Interest
We have no conflicts of interest to disclose.
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We thank two anonymous reviewers and the editor for extensive comments that have significantly improved the paper. We are also grateful to Daniel Bauer, Lixin Huang, Alexander Muermann, Baozhong Yang and seminar audiences at the Risk Theory Society Annual Conference, the Stockholm Institute of Financial Research (SIFR) conference on Insurance Economics, the American Risk and Insurance Association (ARIA) Annual Meeting, the University of Cape Town, the University of Queensland, and the University of Adelaide for valuable comments on earlier versions of the paper. The usual disclaimers apply.