Capital, Systemic Risk, Insurance Prices and Regulation

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1 Caital, Systemic Risk, Insurance Prices and Regulation Ajay Subramanian J. Mack Robinson College of Business Georgia State University Jinjing Wang J. Mack Robinson College of Business Georgia State University Aril 1, 2015 Abstract We develo a unied equilibrium model of cometitive insurance markets that incororates the demand and suly of insurance as well as insurers' asset and liability risks. Insurers' assets may be exosed to both idiosyncratic and systemic shocks. We obtain new insights into the relationshi between insurance remia and insurers' internal caital that otentially reconcile the conicting redictions of revious theories that investigate the relation using artial equilibrium frameworks. Equilibrium eects lead to a non-monotonic U-shaed relation between insurance rice and internal caital. We study the eects of systemic risk on the otimal asset and liquidity management by insurers as well as risk-sharing between insurers and insurees. In the rst-best benchmark scenario, we show that, when systemic risk is low, both insurees and insurers hold no liquidity reserves, insurees are fully insured, and insurers bear all the systemic risk. When systemic risk takes intermediate values, both insurees and insurers still hold no liquidity reserves, but insurees artially share systemic risk with insurers. When systemic risk is high, however, both insurees and insurers hold some liquidity reserves, and insurees artially share systemic risk with insurers. The rst best asset and liquidity management olicies as well as the systemic risk allocation can be imlemented through a regulatory intervention olicy that combines a minimum liquidity requirement, ex ost taxation contingent on the aggregate state, and comrehensive insurance olicies. We also rovide imlications for the solvency regulation of insurers facing systemic risk. 1

2 1 Introduction Financial insitutions such as insurers and banks are usually required to hold sucient equity caital on the liability side of their balance sheets and liquid reserves on the asset side as a buer against the risk of insolvency, esecially when their loss ortfolios are imerfectly diversied and/or returns on their assets shrink dramatically. The nancial crisis of was reciitated by the resence of insucient liquidity buers and excessive debt levels in the nancial system that made nancial institutions vulnerable to large aggregate negative shocks. In the context of insurers, the imerfect incororation of the externality created by systemic risk on their investment decisions may lead them to hold insucient liquidity buers. The resulting increase in insurer insolvency risk has an imact on the amount of insurance they can suly to insurees and, therefore, the degree of risk-sharing in the insurance market. Indeed, emirical evidence shows that, in resonse to Risk Based Caital (RBC) requirements, under-caitalized insurers not only increase their caital holdings to meet minimum caital requirements, but also take more risks to reach higher returns in both roertyliability and life insurance industries (Cummins and Sommer, 1996; Shim, 2010; Sager, 2002). Insurers' roensity to reach for yield contributes to their overall insolvency risk. 1 Systemic risk may, therefore, lead to a misallocation of caital and subotimal risk sharing among insurees and insurers. To the best of our knowledge, however, the above arguments have yet to be theoretically formalized in an equilibrium framework that endogenizes the demand and suly of insurance as well as insurers' asset and liability risks. We contribute to the literature by develoing an equilibrium model of cometitive insurance markets where insurers' assets may be exosed to idiosyncratic and aggregate/systemic shocks. In the unregulated economy, we show that the equilibrium insurance rice varies non-monotonically in a U-shaed manner with the level of internal caital held by insurers. In other words, the insurance rice decreases with a ositive shock to internal caital when the internal caital is below a threshold, but increases when the internal caital is above the threshold. We thereby reconcile conicting redictions in revious literature on the relation between insurance remia and internal caital that are obtained in artial equilibrium frameworks that focus on either demand-side or suly-side forces. We also obtain additional testable imlications for the eects of insurers' idiosyncratic asset 1 Cox(1967) describes bank's tendency to invest in high risk loans with higher returns. Becker and Ivashina (2013)) suort insurers' reaching for yield behavior by examining insurers' bond investment decisions 2

3 risks on remia and the level of insurance coverage. An increase in asset risk, that raises the default robability, raises insurance remia and reduces coverage. In the benchmark rst best economy, where systemic risk is fully internalized, we analyze the eects of systemic risk on the allocation of insurer caital to liquidity reserves and risky assets as well as risk sharing among insurees and insurers. We show that, when systemic risk is below a threshold, it is otimal for insurers and insurees to hold zero liquidity reserves, insurees are fully insured, and insurers bear all systemic risk. When systemic risk takes intermediate values, both insurees and insurers still hold no liquidity reserves, but insurees artially share systemic risk with insurers. When the systemic risk is high, however, both insurees and insurers hold nonzero liquidity reserves, and insurees artially share systemic risk with insurers. We demonstrate that the rst best allocations can be imlemented through regulatory intervention that combines comrehensive insurance olicies sold by insurers that combine insurance and investment, reinsurance for insurers, a minimum liquidity requirement, and ex ost budget-neutral taxation contingent on the realized aggregate state. We use our results to shed some light on the solvency regulation of insurers. Our model features two tyes of agents: a continuum of ex ante identical, risk averse insurees with logarithmic utility each facing a risk of incurring a loss in their endowment of caital, and a continuum of ex ante identical risk neutral insurers each endowed with a certain amount of internal equity caital. There is a storage technology/safe asset that rovides a constant risk free return and a continuum of risky assets that generate higher exected returns than the risk free asset. Although both insurees and insurers can directly invest in the safe asset, only insurers have access to the risky assets. In addition to their risk-sharing function, insurance rms, therefore, also serve as intermediaries to channel individual caital into roductive risky assets. Insuree losses are indeendently and identically distributed, but insurers' assets are exosed to systemic risk. Secically, a certain roortion of insurers is exosed to a common asset shock, while the remaining insurers' asset risks are idiosyncratic. A riori, it is unkown whether a articular insurer is exosed to the common or idiosyncratic shock. The roortion of insurers who are exosed to the common shock is, therefore, the natural measure of the systemic risk in the economy. Insurees invest a ortion of their caital in the risk-free asset and use the remaining caital to urchase insurance. Insurers invest their internal caital and the external caital raised from selling insurance claims in the risk-free and risky assets. 3

4 We rst derive the market equilibrium of the unregulated economy. In the unregulated economy, insurees make their insurance urchase decisions rationally anticiating insurers' investment strategy and default risk given their observations of insurers' internal caital, the size of the insurance ool, and the insurance rice (the remium er unit of insurance). Ceteris aribus, an increase in insurers' internal caital or a decrease in asset risk increases the demand for insurance due to the lower likelihood of insurer insolvency. An increase in the risk of insuree losses leads to a decrease in insurance demand because it increases the roortion of insurees who suer losses and, therefore, decreases the amount that each insuree recovers if he incurs a loss, but the insurance comany is insolvent. Insurers, in turn, choose how much insurance to sell taking as given the insurance rice and the loss role of their insuree ools. Cometition among insurers ensures that, in equilibrium, each insurer earns zero exected economic rots that incororate the oortunity costs of internal caital that is used to make loss ayments when insurers are insolvent. An increase in the insurance rice, therefore, lowers the amount of insurance that each insurer sells in equilibrium leading to a downward sloing cometitive insurer suly curve. An increase in the internal caital or an increase in asset risk, ceteris aribus, increases the oortunity costs of roviding insurance, thereby increasing the amount of insurance that rovides zero economic rots to insurers in cometitive markets. An increase in the loss roortion increases the cost of claims, thereby ushing u the cometitive suly level. In cometitive equilibrium, the insurance rice is determined by market clearingthe demand for insurance must equal the sulyand zero economic rots for insurers. The insurance demand curve and the cometitive suly curve are both downward sloing with the demand curve being steeer due to the risk aversion of insurees. Consequently, any factor that increases the insurance demand curve, ceteris aribus, decreases the equilibrium rice, while a factor that increases the cometitive suly curve has a ositive eect. We analytically characterize the cometitive equilibrium of the economy and exlore its comarative statics. We demonstrate that there is a U-shaed relation between the insurance rice and insurers' internal caital. Secically, the insurance rice decreases with a ositive shock to internal caital when the internal caital is below a threshold, but increases when the internal caital is above the threshold. The intuition for the non-monotonic U-shaed relation hinges on the inuence of both demand-side and suly-side factors. An increase in insurers' internal caital increases 4

5 the cometitive suly of insurance coverage because of the increased oortunity costs of internal caital. Because insurers are risk-neutral, however, the change in the cometitive suly of insurance coverage is linear in the internal caital. On the demand side, an increase in insurers' internal caital increases insurers' insolvency buer, thereby increasing the demand for insurance coverage. An increase in internal caital also increases the funds available for investment that further has a ositive imact on the demand for insurance. The demand, however, is concave in the internal caital due to insurees' risk aversion. Because the insurance suly varies linearly with caital, while the insurance demand is concave, there exists a threshold level of caital at which the demand eect equals the suly eect. Consequently, the demand eect dominates the suly eect so that the equilibrium remium rate goes down when the internal caital level is lower than the threshold. When the caital is above the threshold, the suly eect dominates so that the remium rate increases. As suggested by the above discussion, equilibrium eects that integrate both demand side and suly side forces lay a central role in driving the U-shaed relation between the insurance rice and insurer caital. Our results, therefore, reconcile and further rene the oosing redictions for the relation in the literature that stem from a focus on only demand or only suly eects in artial equilibrium frameworks. Secically, the caacity constraints theory, which focuses on the suly of insurance, redicts a negative relationshi between insurance rice and caital by assuming that insurers are free of insolvency risk (Gron, 1994; Winter, 1994). In contrast, the risky debt theory incororates the default risk of insurers, but redicts a ositive relationshi between insurance rice and caital (Doherty and Garven, 1986; Cummins, 1988, Cummins and Danzon, 1997). Emirical evidence on the relationshi is also mixed. We make the simle, but fundamental oint that the insurance rice reects the eects of caital on both the demand for insurance and the suly of insurance in equilibrium. We show that the relative dominance of demand-side and suly-side forces deends on the level of internal caital, thereby generating a U-shaed relation between rice and internal caital. Next, we show that an increase in insurers' asset risk, which increases their insolvency robability, increases the insurance rice and reduces the insurance coverage in equilibrium. The intuition for the results again hinges on a subtle interlay between the eects of an increase in asset risk on insurance suly and demand. A ositive shock to insurers' asset risk, ceteris aribus, has the 5

6 direct eect of increasing the cometitive suly of insurance coverage, that is, the level of insurance suly at which insurers earn zero economic rots. Consequently, the amount of funds available to ay loss claims in distress increases, thereby having the indirect eect of increasing the demand for insurance. On the other hand, an increase in the asset risk increases the insurers' insolvency robability that has a negative eect on the demand for insurance. We show that, under reasonable conditions, the direct eect outweighs the indirect eect. Consequently, an increase in asset risk reduces insurance demand, but increases the zero-economic-rot suly level, thereby increasing the insurance rice and decreasing the coverage level in equilibrium. Our results imly that the resonse to the increased asset risk of insurance rms is the shift of insuree's caital accumulation from indirect investment in risky assets to direct storage in safe assets. We roceed to analyze the benchmark rst best economy in which systemic risk is fully internalized. We derive the otimal allocation of insurer caital between the safe asset (liquidity reserves) and risky assets as well as the sharing of risk between insurers and insurees. When the systemic risk is low, there is sucient aggregate caital in the economy to rovide full insurance to insurees so that insurers bear all the aggregate risk. Further, because the exected return from risky assets exceeds the risk-free return, it is otimal to allocate all caital to risky assets so that neither insurers nor insurees have holdings in the risk-free asset. When systemic risk takes intermediate values, insurees cannot be rovided with full insurance because of the limited liability of insurers in the bad aggregate state. Consequently, insurers and insurees share aggregate risk, but it is still otimal to exloit the higher exected surlus generated by the risky assets so that all the caital in the economy is invested in the risky assets. When systemic risk is very high, however, risk-averse insurees would bear excessively high losses in the bad aggregate state if all caital were invested in risky assets. Consequently, both insurees and insurers hold ositive liquidity reserves, and share aggregate risk. We demonstrate that a regulator/social lanner can imlement the rst-best allocation olicies through a combination of comrehensive insurance olicies sold by insurers that combine insurance with investment, reinsurance for insurers, a minimum liquidity requirement, and ex ost taxation that is contingent on the aggregate state. The comrehensive insurance olicies rovide direct access to the risky assets for insurees. Reinsurance achieves risk-sharing among insurers, while ex ost taxation transfers funds from solvent to insolvent insurers. The minimum liquidity requrement, 6

7 which is imosed when systemic risk exceeds a threshold, forces insurers to maintain the rst best level of liquidity reserves. The lan for the aer is as follows. We further discuss related literature in Section 2. We resent the model in Section 3 and derive the equilibrium of the unregulated economy in Section 3.1. We analyze the imact of systemic risk and regulation in Section 4. Section 5 and we relegate all roofs to the Aendix. 2 Related Literature Our aer is related to two lines of literature that investigate the relation between caital and rice. The rst branch rooses the caacity constraint" theory, which assumes that insurers are free from insolvency risk. The rediction of an inverse relation between insurance rice and caitalization crucially hinges on the assumtion that insurers are limited by regulations or by innitely risk averse olicyholders so that they can only sell an amount of insurance that is consistent with zero insolvency risk (e.g.,gron, 1994; Winter, 1994). Winter(1994) exlains the variation in insurance remia over the insurance cycle" using a dynamic model. Emirical tests using industry-level data rior to 1980 suort the redicted inverse relation between insurance caital and rice, but data from the 1980s do not suort the rediction. Gron (1994) nds suort for the result using data on short-tail lines of business. Cagle and Harrington (1995) redict that the insurance rice increases by less than the amount needed to shift the cost of the shock to caital given inelastic industry demand with resect to rice and caital. Another signicant stream of literaturethe risky cororate debt theoryincororates the ossibility of insurer insolvency and redicts a ositive relation between insurance rice and caitalization (e.g., Doherty and Garven, 1986; Cummins, 1988). The studies in this strand of the literature emhasize that, because insurers are not free of insolvency risk in reality, the ricing of insurance should incororate the ossibility of insurers' nancial distress. Higher caitalization levels reduce the chance of insurer default, thereby leading to a higher rice of insurance associated with a higher amount of caital. Cummins and Danzon (1997) show evidence that the insurance rice declines in resonse to the loss shocks in the mid-1980s that deleted insurer's caital using data from 1976 to While the caacity constraint" theory concentrates on the suly of in- 7

8 surance, the ricing of risky debt" theory focuses on caital's inuence on the quality of insurance rms and, therefore, the demand for insurance. The emirical studies suort the mixed results for dierent eriods and business lines. We comlement the above streams of the literature by integrating demand-side and suly-side forces in an equilibrium framework. We show that there is a U-shaed relation between rice and internal caital. In contrast with the literature on risk debt ricing", which assumes an exogenous rocess for asset value, we endogenize the asset value which deends on the total invested caital including both internal caital and caital raised through the selling of insurance olicies. Insurers' assets and total liabilities are, therefore, simultaneously determined in equilibrium in our analysis. Our aer is also related to the studies that examine the relation between caital holdings and risk taking of insurance comanies. Cummins and Sommer (1996) emirically show that insurers hold more caital and choose higher ortfolio risks to achieve their desired overall insolvency risk using data from 1979 to It is argued that insurers resonse to the adotion of RBC requirements in both roerty-liability and life insurance industry by increasing caital holdings to avoid regulation costs, and by investing in riskier assets to obtain high yields (e.g.,barano and Sager, 2002; Shim, 2010). Insurers are hyothesized to choose risk levels and caitalization to achieve target solvency levels in resonse to buyers' demands for safety. Our aer ts into the literature by studying the resonse of the market rice to exogenous shocks to internal caital and assets in an equilibrium framework. Our results shed some light on insurance investment regulation. Relaxing liquidity constraints on insurers' assets induces insurees to choose relatively greater self-insurance, thereby shrinking the insurance markets and, therefore, the channeling of insurance caital to value-creating assets. Our aer also contributes to the literature on caital allocation and insurance ricing. Zanjani (2002) argues that rice dierences across markets are driven by dierent caital requirements to maintain solvency assuming that caital is costly to hold. Our aer endogenizes the cost of caital in terms of the oortunity cost of holding internal caital, which is used to ay for loss claims when insurers default. We highight insurees' and insurers' resonses to internal caital shocks. Consequently, insurance rices reect insurees' demand for safety and insurers' abilities to rovide insurance with imerfect rotection. 8

9 3 The Model We consider a single-eriod economy with two dates 0 and 1. There is a single consumtion/caital good. There are two tyes of agents: a continuum of measure 1 of risk-averse insurees or olicy holders and a continuum of measure 1 of risk-neutral insurance rms. Each insuree is endowed with 1 unit of caital at date 0 and has a logarithmic utility function. Each insurance rm is endowed with K units of internal caital. There is a storage technology/safe asset that is in suciently large suly that it rovides a constant return of R f er unit of caital invested. At date 1, an insuree i can incur a loss l 1 so that a ortion of each insuree's endowment is at risk. Losses are indeendently and identically distributed across insurees. Each insuree's loss robability is. At date 0, each insuree invests a ortion of her caital in the safe asset and the remainder in buying an insurance olicy at a remium κ er unit of loss, where κ is endogenously determined in equilibrium. Insurance markets are cometitive so that insurees and insurers act as rice takers by taking the insurance remium rate as given in making their decisions. Insurers have internal caital K and raise external caital by selling insurance olicies. Each insurance rm j has access to a risky technology that generates a return of R H er unit of invested caital with robability 1 q when it succeeds but R L < R H with robability q when it fails. A roortion 1 τ of insurance rms are exosed to idiosyncratic technology shocks, that is, the technology shocks are indeendently and identically distributed for this grou of insurance rms. The remaining roortion τ of insurers are, however, exosed to a common shock, that is, the technology shock described above is the same for these insurers. Although insurers know that a roortion τ of them is exosed to a common shock, an individual insurer does not know whether it is exosed to an idiosyncratic or common shock a riori. τ is a measure of the systemic risk in the economy. We assume that (1 q)r H + qr L R f. (1) The above condition ensures that the exected return on the risky roject is at least as great as the risk-free rate. While olicy holders can directly invest in the safe asset, only insurance rms have access to the roduction technology. Consequently, in addition to the rovision of insurance to olicy holders, insurance rms also lay imortant roles as nancial intermediaries who channel 9

10 the caital sulied by olicy holders to roductive assets. If the insurance remium er unit of loss is κ, the total external caital raised by an insurer j is κc j, where C j is the face value of insurance claims sold by the insurer. Each insurer can invest its total caital, K + κc j in a ortfolio comrising of the risk-free storage technology and the risky roject. In an autarkic economy with no regulation, it follows from condition (1) that it is otimal for risk-neutral insurance rms to invest their entire caital in the risky technology. By our earlier discussion, the total liability of an insurer is C j because a roortion of its ool of insurees incur losses. Insurers default if their total liability cannot be covered by the total investment returns when the risky technology fails, that is when C j > (K + κc j )R L. (2) In the event of default, the total available caital of an insurer is slit u among insurees in roortion to their resective indemnities. The internal caital lays the role of a buer that increases an insurer's caacity to meet its liabilities and, thereby, the amount of insurance it can sell. The cost of holding internal caital in our model is an oortunity cost, which refers to the returns from the invested internal caital that are deleted to ay out liabilities when insurers default. Insurees observe the total caital, K + κc j, held by each insurer j. In making the decision on the level of insurance coverage to urchase, insurees rationally anticiate the ossibility of default, and the amount they will be aid for a loss when insurers' asset returns are insucient to ay out the aggregate loss claims as shown by (2). 3.1 The Equilibrium We now derive the equilibrium of the insurance markets by analyzing the demand and suly of insurance by insurees and insurers, resectively. In equilibrium, the demand for insurance equals the suly. 10

11 3.1.1 Insurance Demand Each insuree chooses its ortfolio, which comrises of his investment in the safe asset (self-insurance) and his choice of insurance coverage, to maximize his exected utility. Without insurance, each insuree i's exected utility is given by the autarkic utility level, Autarkic Utility = ln(r f l) + (1 ) ln(r f ). (3) Insurees are cometitive and take the insurance rice as given in making their urchase decisions. They observe the total caital held by insurers and, therefore, rationally anticiate the ossibility that they may not be fully indemnied in the scenario where they incur losses, but the insurer is insolvent. Insurees also rationally incororate insurers' investment ortfolio choices in making their insurance demand decisions. As reviously stated, insurers invest all their caital in the risky technology, thereby causing insurers to be likely to default in the bad state where the technology fails. The likelihood that insurees' loss claims may not be fully indemnied is then aected by the risk in the investment ortfolio of insurance rms and the total liabilities insured by them. In general, the loss ayment obtained by each insuree is determined by three factors: the roortion of insurees in the insurer's ool who incur losses, the total amount of caital held by the insurer, and the return of the insurer's investment roject. Given that insurees and insurers are ex ante identical, we focus on symmetric equilibria where insurees make identical ortfolio choices and insurers have ex ante identical ools of insurees. Without loss of generality, therefore, we focus on a reresentative insurer. Suose that the amount of coverage urchased by a reresentative insuree is C d. If κ is the insurance remium er unit of coverage and C s is the total face value of the insurance olicies sold by the insurer, its total caital is K + κc s. The insurer's available caital if its roject fails is, therefore, (K + κc s )R L. Consequently, the ayment received by each insuree who incurs a loss when the insurer's roject fails is min(c d, (K+κCs)R L ). It is clear from our subsequent results that it is subotimal for the insurer to sell so much coverage that it is unable to meet losses in the good state where its roject succeeds. In the following, therefore, we assume this result to avoid unnecessarily comlicating the exosition. The insuree's otimal demand for insurance coverage maximizes its exected utility subject to 11

12 its budget constraint, that is, it solves { }} { max(1 q) ln [(1 κc d )R f l + C d ]+ C d insuree incurs loss in insurer's good state insuree incurs loss in insurer's bad state { [ }} { q ln (1 κc d )R f l + min(c d, (K + κc ] s)r L ) insuree does not incur loss { }} { + (1 ) ln [(1 κc d )R f ] (4) such that κc d 1 (5) As is clear from the above, an atomistic insuree makes his insurance urchase decision based on his robability of a loss and the robability that the insurer's roject fails. Because he observes the insurer's total caital when he makes his decision, the insuree's decision rationally incororates the exected roortion of the oulation of insurees that will incur losses. The roerties of the logarithmic utility function guarantee that it is subotimal for insurees to invest all their caital in risky insurance so that the budget constraint, (5) is not binding. The necessary and sucient rst order condition for the insuree's otimal choice of insurance coverage, C d, is (1 q)(1 κr f ) qκr f (1 κcd )R f l + Cd (1 κcd )R f l + (K+κC s)r L { (1 κr f ) + (1 κcd )R f l + Cd (1 )κr f (1 κcd )R f (1 )κr f (1 κc d )R f 1 {C (K+κC s )R L } } 1 {Cs <(K+κC s )R L } = 0 (6) The solution to the above equation can be exressed as a function, Cd (K, C s, κ), where we suress the deendence of the otimal demand on the liability and asset risk arameters, and q, and the safe asset return, R f, to simlify the notation. The following lemma characterize the insuree's otimal demand for insurance coverage for a given insurance remium rate, κ. Lemma 1 Suose insurees assume that insurers will defaut in the bad state where the technology fails. For a given insurance remium rate, κ, the otimal insurance demand C d is 12

13 given by C d = C d (K, C s, κ), (7) where C d (K, C s, κ) satises equation (1 q)(1 κr f ) (1 κcd )R f l + Cd qκr f (1 κc d )R f l + (K+κC s )R L (1 )κr f (1 κc d )R f = 0 (8) Suose insurees assume that insurers do not default in the bad state where the technology fails. For a given insurance remium rate, κ, the otimal insurance demand Cd is given by Cd = C d (κ), (9) where Cd (κ) satises (1 κr f ) (1 κc d )R f l + C d (1 )κr f (1 κc d )R f = 0 (10) By (8) and (10), we note that the insurer's internal caital, K, total suly, C s, and asset risk arameter, q, inuence the otimal demand for insurance coverage only when insurees foresee insurer insolvency in the bad state, where its assets fail. For generality, we allow for the case that the market insurance remium rate might lead to over insurance, i.e., C d > l. The following lemma shows how the otimal demand for insurance coverage varies with the fundamental arameters of the model that will be useful when we derive the equilibrium of the insurance market. Lemma 2 (Variation of Insurance Demand) The otimal demand for insurance, Cd, (i) decreases with the remium rate, κ; (ii) decreases with the return, R f, on the safe asset; (iii) increases with insurers' internal caital, K; (iv) increases with the total face value of olicies sold by the insurer, C s ; increases with the insurer's asset return in the low state, R L ; and (v) decreases with the insurer's exected robability of failure; q. The otimal demand for insurance claims reects the tradeo between self-insurance through investments in the safe asset and the urchase of insurance coverage with otential default risk for 13

14 the insurer and, therefore, imerfect insurance for the insuree. Caital allocated in safe assets lays an alternative role in buering the losses that cannot be indemnied by insurers when their assets fail. The insurance demand decreases with the insurance remium rate, that is, the demand curve is downward-sloing, since the utility function of insurees satises the roerties highlighted by Hoy and Robson (1981) for insurance to be a normal good. An increase in the risk-free return raises the autarkic utility level, thereby diminishing the demand for insurance coverage. In addition to functioning as a risk warehouse, which absorbs and diversies each insuree's idiosyncratic loss, insurance rms also serve as nancial intermediaries who channel external caital sulied by olicyholders to roductive assets. In our model, the overall insolvency risk faced by insurance rms are simultaneously determined by the asset and liability sides of insurer's balance sheets. An increase in the aggregate loss roortion of the insuree ool; a decrease in the internal caital held by insurers; a decrease in the amount of external caital raised by the insurer from selling insurance; and a decrease in the asset return in the low state all lower the insurance coverage of an insuree when the insurer is insolvent so that the otimal insurance demand declines Cometitive Insurance Suly Each insurer chooses its otimal suly of insurance coverage to maximize its total net exected ayos from roviding insurance for insurees and investing the caital it raises. As discussed earlier, in the absence of regulatory intervention, it is otimal for each insurer to invest its entire caital in the risky roject due to its risk neutrality and the asset return condition (1). Insurers are cometitive and take the insurance remium rate as given in making their insurance suly decisions. An insurer chooses to suly insurance if and only if its exected net rots are at least as great as its autarkic exected ayo, that is, its exected ayo from not selling insurance and investing its internal caital. An insurer's autarkic exected ayo is Autarkic Exected Payo = K ((1 q) R H + qr L ). (11) Each insurer makes its suly decision knowing the exected roortion of insurees,, who will incur losses. In the bad state where its technology fails, if its available caital is lower than the total loss ayments to insurees, then the caital is divided equally among the insurees. The otimal 14

15 suly of insurance coverage level, therefore, solves max C s {(1 q) ((K + κc s ) R H C s )} + {q ((K + κc s ) R L C s )} 1 {Cs (K+κC s)r L } (12) such that {(1 q) ((K + κc s ) R H C s )} + {q ((K + κc s ) R L C s )} 1 {Cs (K+κC s)r L } K ((1 q) R H + qr L ) (P.C ) (13) The articiation constraint, (13), ensures that the insurer chooses to sell a nonzero amount of coverage if and only if its exected net rot exceeds its exected ayo in autarky. From (12) and (13), it is clear that it is otimal for the insurer to suly no coverage if the remium rate, κ < R H and innite coverage if κ > R L. In equilibrium, therefore, we must have κ [ R H, R L ]. It also follows from the linearity of the objective function and the articiation constraint that the articiation constraint, (13), must bind in equilibrium, that is, insurers make zero exected economic rots. Consequently, the cometitive suly of insurance coverage for any remium rate κ [ R H, R L ] is C s (K, κ) = qkr L (1 q)(κr H ) (14) Lemma 3 (Cometitive Insurance Suly) For κ ( R H, R L ), the cometitive insurance suly level, C s (K, κ), (i) decreases with the insurance remium rate, κ; (ii) increases with insurers' internal caital, K; (iii) increases with insurers' exected default robability, q; (iv) increases with the asset return, R L, in the bad state; and (v) increases with the loss robability of insurees,. An increase in the remium rate increases the exected return from sulying insurance and, therefore, decreases the coverage level at which each insurer's articiation constraint, (13), is binding. For given κ ( R H, R L ), an increase in the insurer's internal caital, asset risk, or the aggregate risk of the ool of insurees lowers the exected returns from roviding insurance and, therefore, increases the cometititive insurance suly level. 15

16 3.1.3 Cometitive Equilibria of Unregulated Insurance Market We now derive the insurance market equilibrium that is characterized by the insurance rice (er unit of coverage) κ. The equilibrium satises the following conditions. 1. Given the equilibrium rice κ, the face value of coverage sulied by each insurer is C s (K, κ ) given by (14) and insurers make zero exected economic rots. 2. Given the equilibrium rice κ and the cometitive suly level C s (K, κ ), the coverage urchased by each insuree is C d (K, C s (κ ), κ ) given by (7) and (9). 3. The equilibrium rice κ clears the market, that is, C d (K, C s (κ ), κ ) = C s (K, κ ) = C. The following roosition characterizes the equilibria of the insurance market. We begin with some necessary denitions. Dene the exected return from the insurer's risky technology, ER = (1 q)r H + qr L. (15) Dene the excess demand function F (K, κ) = C d (K, C s (κ), κ) C s (K, κ), (16) where C d (K, C s (κ), κ) is the demand function described by Lemma (1) and C s (K, κ) is given by ((14)). Proosition 1 (Cometitive Equilibria) Suose K K 1, where K 1 is given by F (K = K 1, κ = ER ) = 0 (17) In equilibrium, insurers default in the bad state when their assets fail. The equilibrium rice, κ, satises: F (K, κ ) = 0. (18) Suose K > K 1. In equilibrium, insurers do not default in the bad state where their assets fail. The equilibrium insurance rice is κ = 16 ER and the equilibrium coverage level, C, is

17 given by C = κ (1 )(R f l) 1 κ R f > l. The above roosition shows that there are two ossible equilibria that are determined by the internal caital of insurers. When the internal caital is lower than the threshold level K 1, the reresentative insurer defaults in the bad state that is rationally foreseen by all agents. When the internal caital is higher than the threshold K 1, the insurer faces no insolvency risk and this is rationally anticiated by all agents. In the equilibrium, therefore, the suly of insurance is comletely elastic, and the rice is determined by the aggregate loss roortion of insurees adjusted by the exected return from the risky technology, (13) is binding. ER, at which the insurer's articiation constraint We focus on the more interesting rst scenario in which insurers with insucient internal caital default after their technologies fail. Figure 1 shows the equilibrium. The equilibrium rice κ and coverage level C, satisfy the system of equations (33),(34) and(35). In addition, the condition (36) ensures that the insurer, indeed, defaults in the bad state. The equilibrium remium rate must be greater than bad state, the equilibrium insurance rate must be less than R H. To ensure that the insurer defaults in its ER. The demand curve for insurance coverage and the cometitive insurance suly curve are both downward sloing, but the demand curve is steeer than the suly curve. Thus the existence condition for a solution κ to the system of equations is F (κ) κ ER = lim κ ER C d (κ, C s (κ)) lim κ ER C s (κ) > 0. (19) Any solution, κ, to the system of equations must also satisfy the following constraint: qkr (1 q)(κ R H ) < 1 κ so that κ is indeed the equilibrium remium rate. As there may be multile ossible equilibria corresonding to remium rates that satisfy the above conditions, we choose the smallest equilibrium remium that maximizes the exected utility of insurees and, therefore, the social welfare, that is the smallest κ, at which F (κ) κ κ > 0. 17

18 We next identify the eects of shocks in the economy on the equilibrium insurance rice, coverage level and social welfare. 3.2 The Eects of Caital and Risk Internal Caital Internal caital inuences the equilibrium insurance rice through the demand for and sulyof insurance. By (14), an increase in internal caital increases the cometitive insurance suly level. There are both direct and indirect eects of an increase in internal caital on the demand for insurance. An increase in internal caital has the direct eect of increasing the demand for insurance because of the higher available caital to meet insurance claims. The demand for insurance coverage is further enlarged by the insurees' anticiation of the increase in the cometitive suly of insurance with the increase in internal caital. demand for insurance is also ositive. Consequently, the overall eect of internal caital on the The net eects of an increase in internal caital on the equilibrium remium rate deend on the relative dominance of demand-side and suly-side eects. The smallest equilibrium rice κ satises F (K,κ) κ κ=κ > 0. The marginal eects of internal caital on the insurance rice can be understood through its eects on the excess demand function. F (K, κ ) K = C d (K, C s, κ ) + C d(k, Cs, κ ) Cs (K, κ ) Cs (K, κ ) } K {{ } Cs. K } {{ } } K {{ } direct eect on demand indirect eect on demand direct eect on cometitive suly rice. The following roosition describes the eects of internal caital on the equilibrium insurance 18

19 Proosition 2 (The Eects of Internal Caital) Suose F (K,κ ) K K 0 > 0. There exist a threshold K such that the equilibrium remium rate κ decreases with internal caital when K < K, and increases when K > K. Suose F (κ,k) K K 0 < 0, the equilibrium remium rate increases with the amount of internal caital. Let κ be the equilibrium remium rate corresonding to the internal caital level K. K and are jointly determined by the following two equations: F (K, κ ) K K= K,κ = κ; = 0 (20) ( Cd K, C s ( κ, K), ) κ, = Cs ( K, κ). (21) The above roosition shows that the insurance remium decreases with insurers' internal caital when the internal caital level is relatively low, while it increases with insurers' internal caital when its level is relatively high. This result reconciles the conicting redictions on the relation between insurance rice and caital in revious literature. The caacity constraint" theory relies on the assumtion that insurance rms are free of insolvency. Winter(1990) argues that insurance rms can only write the volume of business consistent with zero insolvency due to regulation. The total caital amount determines the caacity of the insurance market. A signicant negative shock to insurer caitalshrinks the suly of insurance in imerfect caital markets. It follows that the insurance rice increases and insurance coverage declines while the demand for insurance is not aected in the absence of insurer insolvency. The ricing of risky debt" theory incororates the insolvency risk of insurance rms. Cummins and Sommer (1996) theoretically show both a ostive and negative relation between rice and a retroactive loss shock based on an otimal endogenous caitalization structure of insurance rms. As mentioned earlier, an increase in internal caital increases the insurance demand and suly so the net imact deends on which of the two eects is dominant. By (14), the cometitive suly of insurance is linear in the internal caital level. Because insurees are risk-averse, the demand for insurance is concave in the insurer's internal caital. Consequently, the excess demand function, F (K, κ ), is concave in the internal caital, that is, 19

20 If F (κ,k) K K 0 > 0, then there exists, in general, a threshold level of internal caital, K, at which the marginal eect of internal caital on the excess demand is zero. It follows from the concavity of the excess demand that the marginal eect of internal caital on the excess demand is ositive for K < K and negative for K > K. In other words, the risk aversion of insurees causes the demand eect of an increase in internal caital on the insurance rice to dominate the suly eect for K < K and vice versa for K > K. Hence, the equilibrium insurance remium varies in a U-shaed manner with the level of internal caital. If F (κ,k) K K 0 0, then the marginal eect of internal caital on the excess demand is always non-ositive so that the equilibrium insurance remium increases with internal caital The Eects of Asset Risk We now address the imacts of the reresentative insurer's asset risk on the equilibrium remium rate and insurance coverage. The resence of asset induced insolvency comlicates the decisions on both the demand and suly sides. The imact of asset risk on insurance suly indirectly inuences insurance demand by aecting the total caital available to the insurer to meet liabilities in insolvency. Secically, it follows from (14) that an increase in asset risk increases the cometitive insurance suly level. Because insurees rationally foresee the likelihood that their losses will not be fully indemnied by insurers, the direct eect of an increase in asset risk on insurance demandis negative. The increase in the cometitive suly level with asset risk, however, increases the amount each insuree is able to recover if it incurs a loss, but the insurer is insolvent. The indirect imact of an increase in asset risk on insurancey demand is, therefore, ositive.thenet imact of asset risk on the equilibrium remium rateis determined via its eect on the excess demand function, F (κ, q) q C d (κ, q) = + C d(κ, q) q C } {{ } s direct eect on demand<0 C s q } {{ } indirect eect on suly>0 Cs (κ, q), q } {{ } direct eect on zero-economic-rot suly>0 (22) where we exlicitly indicate the deendence of the demand and suly functions on the asset risk arameter, q. The following roosition characterizes the eect of asset risk on the equilibrium insurance rice and coverage. 20

21 Proosition 3 (The Eects of Asset Risk) Suose R L increases with the asset risk, q, while the coverage level declines. If R L risk on the insurance rice is ambiguous. < R f. The equilibrium insurance rice R f, then the eect of asset The intuition for the condition R L < R f is as follows. R L catures the marginal contribution of an increase in the suly of insurance claims to the marginal utility of each insuree in the default state is, while R f measures the marginal contribution of an increase in insurance demand to marginal utility of each insuree in the default state. Consequently, the condition R L < R f imlies that the marginal contribution of insurance suly to the marginal utility is less than that of insurance demand. In other words, one unit increase in insurance suly will induce less than one unit increase in insurance demand. It follows that the indirect eect of an increase in asset risk on insurance demand through the increase in the cometitive insurance suly level is less than the direct eect on the cometitive suly level. Consequently, the excess demand decreases with asset risk so that the equilibrium rice increases. 4 Regulation There are three sources of ineciencies in the unregulated economy as analyzed in the revious section. First, each insurer makes its insurance suly decisions and investment decisions incororating its individual asset return distribution without fully internalizing the otential correlation of asset returns across insurers arising from the fact that a roortion τ of insurers is exosed to a common shock. Without considering systemic risk, insurers may hold insucient liquidity reserves and over-invest their caital in risky assets. Second, insurees' idiosyncratic losses may not be fully insured by insurers when each insurer's internal caital is relatively low. Insurees bear insurers' default risk driven by the asset side of their balance sheets when there is no eective risk sharing mechanisms among insurers to insure their idiosyncratic asset risk. Third, each insuree has no access to the risky assets, and insurance rms serve as the only intermediaries that channels the insuree's caital into more roductive risky assets. Insurers, however, cannot eectively share the investment risk with insurees through the insurance olicies that only rotect insuree's losses without combining investment returns to insurees. The equilibrium rice and insurance coverage level in the unregulated economy, therefore, do not internalize the externalities created by systemic 21

22 risk of insurers' assets and the lack of instruments that achieve full risk-sharing. Consequently, we otentially have a misallocation of insuree caital to the urchase of insurance and misallocation of insurer caital to safe and risky assets. Regulatory intervention could imrove allocative eciency by internalizing the externalities created by systemic risk, imosing necessary liquidity reserves requirement, and also roviding risk sharing mechanisms through ex ost taxation transfers among insurers. We now consider a regulated economy in which the regulator ossesses the same information as the agents in the economy including the roortion of insurees who incur losses as well as the roortion of insurance rms whose assets fail. We rst study a hyothetical benchmark scenario in which both insurees and insurers have access to risky technology, insurers fully insure insuree's idiosyncratic risk of incurring losses and fully/artially share the asset returns risk deending on the roortion τ of insurers exosed to a common shock. We then derive how regulatory tools can be combined to achieve otimal asset allocation and risk allocation in the hyothetical benchmark scenario. 4.1 Benchmark First Best Scenario We begin by studying a hyothetical benchmark scenario that full internalizes the ineciencies in the unregulated economy due to systemic risk and ineective risk sharing mechanisms among insurees and insurers. In this economy, we assume that there are eective risk sharing mechanisms among insurers so that insuree's idiosyncratic losses can be fully insured by insurers. Without loss of generality, we can assume that there is a single reresentative risk averse insuree with 1 unit of caital good and a single reresentative risk neutral insurer with K units of caital goods. We assume that both the insuree and the insurer have access to risky assets that may be subject to aggregate shocks. With robability q, the economy oerates in the bad aggregate state where a roortion τ of risky investments earn a low return R L. Thus the return er unit caital invested is M L, where M L = (1 q)(1 τ)r H + q(1 τ)r L + τr L. With robability 1 q, the economy oerates in the good aggregate state where a roortion τ of the risky investments earn a high rate of return R H. It follows that the return er unit caital invested is M H, where M H = (1 q)(1 τ)r H + q(1 τ)r L + τr H. The Insurer rovides insurance to cover the insuree;s idosyncratic losse, but also share the aggregate risk associated with investments in the risky assets. Let C H and C L be the insurer's combined returns from investing caital in risky technology and 22

23 selling insurance in the good and bad aggregate states, resectively. Insurers decide the roortion α of safe assets as liquidity reserves buer. In the equilibrium, insurer's ayo from investing in risky assets and selling insurance is as great as its autarkic ayo, that is αkr f + [(1 q)c H + qc L ] = K[(1 q)r H + qr L ] The insuree invests a roortion β of its caital in safe assets and the rest in urchasing risky insurance. Let D H and D L be the bundled insurance claims received by insurees including both individual losses and returns from risky assets and/or insurance. The otimal returns from investment in insurance olicies and the otimal investment in safe assets are solved as follows: max q ln(βr f + D L ) β,α,d L,D H } {{ } bad aggregate shock + (1 q) ln(βr f + D H ) } {{ } good aggregate shock (23) subject to αkr f + [(1 q)c H + qc L ] = K[(1 q)r H + qr L ] (24) D L + C L = [(1 β) + (1 α)k]m L l (25) D H + C H = [(1 β) + (1 α)k]m H l (26) αkr f + C L 0 (27) αkr f + C H 0 The following roosition shows the otimal asset allocation between risky and safe assets, and the otimal risk allocation among the reresentative insuree and insurer. Proosition 4 (Benchmark Asset allocation and Risk Sharing among Insurees and Insurers) Suose (i) q < 0.5, and K < (ER l)(er R f ) ER R f 1 q 1 2q (ii)(er + R f R H R L )R f + l(er R f ) < 0, then when τ τ 1, where τ 1 = buer, such that β = 0, K ER (1+K)(ER R L ), or when q > 0.5, and for any K;, both insuree and insurer hold zero liquidity reserves α = 0. The otimal return er unit caital invested in insurance 23

24 olicies in good and bad aggregate states are the same, and insurees are fully insured, that is D H = D L = D = ER l when τ 1 < τ < τ 2, where τ 2 = (1 q)(1 + K)(R H + R L 2ER)ER + q ER K(ER R L ) 2(1 q)(1 + K)(R H ER)(ER R L ) ( (1 q)(1 + K)(R H + R L 2ER)ER + q ER K(ER R L ) ( (qk 4 (1 q)(1 + K)(R H ER)(ER R L ))( (1 q)(1 + K) ) ) ER + l(1 q) (ER R f ) + 2(1 q)(1 + K)(R H ER)(ER R L ) ) 2 both insuree and insurer still holds zero liquidity reserves buer such that β = 0, and α = 0. Insurees are imerfectly insured, and the otimal returns er unit caital invested in insurance roducts in good and bad aggregate states are resectively: D H = (1 + K)M H l ER 1 q K, DL = (1 + K)M L l Insurer's limited liability constraint 27 binds, and its otimal returns er unit caital of investing in risky assets and roviding insurance in good and bad aggregate states are resectively C H = ER 1 q K CL = 0 In this case, insurers and insurees share the systemic risk. when τ > τ 2, both insuree and insurer in total have to hold some roortion of safe assets such that β + α K = (1 + K)(ER R f M H M L ) + l(er R f ) qer K (R f M L ) 1 q (M H R f )(R f M L ) Insurees are imerfectly insured, and the oitmal return er unit caital invested in insurance 24

25 roducts in good and bad aggregate states are resectively: D H = ( 1 + K (β + α K) ) M H l ER 1 q K + α KR f D L = ( 1 + K (β + α K) ) M L l + α KR f Insurer's limited liability constraint 27 binds, and its otimal returns er unit caital of investing in risky assets and roviding insurance in good and bad aggregate states are resectively C H = ER 1 q K α K R f C L = α K R f In this case, insurers and insurees share the systemic risk. The above roosition shows the eects of systemic risk on the otimal asset allocation and risk sharing arrangement when the systemic risk are hyothetically internalized. When the systemic risk measure τ is relatively low, insuree's idosyncratic losses and returns from investment in risky assets can be fully insured by insurers. Thus it is otimal to invest all social caital in risky assets to roduce the highest level of allocative social caital. When the systemic risk measure τ is in the intermediate level, insurers may not have enough caital to ayo its insurance claims in the bad aggregate so that insurer defaults and its limited liability constraint in that state is binding. It may still oitmal for both insurers and insurees to invest all in risky technology to achieve the highest level of total allocative caital. 2 When the systemic risk measure τ is above a threshold, however, the marginal increase in the total exceted allocative caital returns from risky assets is insucient to comensate from the disutility to the reresentative insuree arising from the imerfect insurance ayos due to aggregate shocks. It is, therefore, to hold certain amount of safe assets. Figure 4.1 summarizes the relationshi between systemic risk measure τ and the otimal investment in safe assets. It reects the tradeos between total allocative returns from investments and the risk sharing among insurees and insurers. When the systemic risk is low, the total allocative caital reaches the maximum level, and insurees are fully insured, and insurers take all systemic risk. When 2 When asset default robability is suciently high and insurer's internal caital is relatively low, the marginal increase in total exected allocative caital returns from risky assets may be insucient to comenate the disutility arising from the imerfect insurance ayos due to aggregate shocks. It, thus may be oitmal to hold some safe assets as in the third case. 25

26 Figure 1: Systemic Risk and Liquidity Reserves Buer the systemic risk is in the intermediate level, the total allocative caital also reaches the highest level, and insurees are imerfectly insured, and insurees and insurers share the systemic asset risk. When the systemic risk is high, the marginal decrease in the total allocative caital due to some investment in safe assets trades o the wedge between insurance claims received by insurees in good and bad aggregate states. We next analyze how the benchmark level of investment ortfolios and risk sharing can be imlemented through regulatory intervention. 4.2 Regulatory intervention The inecient investment allocation and risk sharing ga between the autarkic economy and the hyothetical benchmark arises from three factors: the lack of eective idiosyncratic asset risk sharing among insurees, lack of the eective systemic asset risk sharing among both insurees and insurers, and lack of the eective mechanism internalizing eect of systemic risk on insurer's investment ortfolios. The above three factors rovide regulators the room to reduce the market ineciency using comrehensive tools. Liquidity Requirement and Systemic Risk Proosition 4and Figure 4.1 imly the oitmal investment in safe assets. Without internalizing systemic risk, insurers always invest all available caital into risky assets. This rovides regulators the room to imosing the minimum amount of liquidity reserves when systemic risk is high enough. 26

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