Insurance Research Paper

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There is insurance when one party agrees to pay an indemnity to another party in the event of the occurrence of a predefined random event generating a damage for the latter. The most standard example is the insurance contract where the first party—the insurer— is compensated by being paid a premium by the second party, the policyholder. But many other contracts involve some forms of insurance. For example, in share-cropping contracts, the landlord agrees to reduce the rent of land in case of a bad crop. In many instances, employers agree to pay their employees more than the competitive wage in bad years. In cost-plus contracts, the buyer accepts to pay a larger price if the producer incurred an unexpected increase in cost. In the case of income taxes, the state partially insures the losses of taxpayers by reducing the tax payment when incomes are low.

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The shifting of risk is of considerable importance for the functioning of our modern economies. Insurance is desirable for risk-averse agents as a risk-reduction device through the use of the Law of Large Numbers. Moreover, insurance allows for disentangling investment decisions from risk-taking decisions. Without it, there would certainly not have been the experience of the historical economic growth of the twentieth century. Ford, Solvay, Rockefeller, and the others would not have taken the investment risks that they actually took without the possibility to share the risk with shareholders and insurers. Similarly, most consumers would not purchase new expensive cars or houses if they would not be able to insure them.

However, various informational problems (observation of the risk, solvency issues, of the prevention efforts and/or of the loss by the insurer) can make competitive insurance markets inefficient by not providing enough coverage at an acceptable price.

1. The Social Value Of Insurance

There is an added value to insurance only because policyholders are risk-averse, that is they dislike zeromean risk on wealth. Consider an agent facing a random loss X to their wealth. An insurance contract stipulates a premium P and an indemnity schedule I(.) that determines the indemnity I(X) for each possible loss X. There is full coverage if I(.) is the identity function. The actuarial value of the contract is the expected indemnity EI(X). The insurance premium is said to be actuarially fair if it is equal to the actuarial value of the contract: P =EI(X). Suppose that it is the case. Then, the purchase of a full insurance contract at an actuarially fair premium has the effect of replacing a random loss X by its expectation P =EX. The private value of such a contract is equal to the risk premium attached to the initial risk by the policyholder. It is increasing with the policyholder’s degree of risk aversion and with the riskiness of the loss. As a first approximation, the riskiness of the loss can be measured by its variance.

In order to determine the social value of insurance, the effect of the transfer of the risk on the insurer’s welfare should also be measured. Consider an insurer selling n fair full insurance contracts to n policy- holders, each of them bearing a random loss Xi, with X1, … ,Xn being identically and independently distributed. The common wisdom is that the insurer does not bear any risk in the aggregate if it is able to sell enough such insurance contracts to cover independent risks, i.e., if n is large enough. This is not true, as explained by Samuelson (1963). It is a fallacious interpretation of the Law of Large Numbers which leads people to believe that accumulating several independent risks generates diversification. Indeed, the aggregate indemnity to be paid by the insurer is the sum o f the Xi. It s riskiness measured by its variance equals nσ2, where σ2 is the variance of each individual risk Xi. If the insurer is risk-averse, the risk transfer is not Pareto- improving since it makes the insurer worse off. The classical view of modern finance on this problem is to recognize that insurance companies are not owned by a single person, but rather by a large set of shareholders. If insurable losses are uncorrelated with financial markets returns, the riskiness of insurance companies has no adverse effect on shareholders. In such a situation, the risk transfer to insurance companies is Pareto-improving, and the social value of insurance companies can be measured by the sum of the risk premia associated by policyholders to their random loss.

An alternative insurance scheme is to organize a mutual arrangement among the n risk-averse agents. Consider an arrangement in which the pool guarantees to each of its member a complete coverage of losses against an ex-post contribution equaling the average loss in the pool. In such a scheme, the random loss Xi is replaced by (X1+… +Xn)/n, whose variance is 1/n the variance of Xi. In consequence, this mutual arrangement is Pareto-improving. At the limit, when n tends to infinity, individual risks are completely washed out by diversification.

2. Efficient Risk Sharing With Complete Markets

An important question is to characterize socially efficient allocations of risks in an economy where individuals may have very different initial risk exposures and risk attitudes. Wilson (1968) solved this question when individuals are expected—utility maximizers. Any efficient allocation of risk must satisfy two conditions. The first one is often called the mutuality principle. It states each ex-post individual’s consumption only depends upon the ex-post average wealth in the economy. This means that all members raise their consumption in good states, and reduce it in bad states. There is no loser in an improving environment. Notice that this condition is satisfied in the case of the mutual arrangement examined above. The intuition for the mutuality principle is the need for diversification: all individual risks must be gathered in the pool. The second characteristic of a socially efficient risk-sharing rule deals with the sharing of the aggregate risk in the pool. As suggested by the intuition, a socially efficient sharing of the aggregate risk is such that the risk borne by a member of the pool must be inversely proportional to their Arrow-Pratt degree of absolute risk aversion. At the limit, if there is a risk-neutral agent in the economy, this agent should bear all risks, i.e., he should insure everyone else in the economy.

Whether decentralized economies can allocate risks in an efficient way is another question that has been solved by the pioneering works of Arrow (1953), Debreu (1959), and Borch (1962). Suppose that the economic system is such that it is possible to freely insure against any bet, at fixed odds, any amount wished on the occurrence of any event which will not affect the welfare of the insurers in any way. Markets are said to be complete in this case. Suppose also that there is no transaction cost on these markets, and no asymmetric information of any kind. The standard first theorem of welfare economics can then be applied to obtain that competition on these markets yields a Pareto-efficient allocation of risks in this economy. The competitive allocation must in particular satisfy the mutuality principle. At the competitive equilibrium, all households should sell their entire human and physical capital in exchange for a stake of the aggregate economy. A part of this story can be found in the Capital Asset Pricing Model in which all investors have the same composition of the portfolio of risky assets, the ‘market portfolio.’

Townsend (1994) tested the mutuality principle in small rural villages in India. He did not reject the hypothesis of efficient risk sharing within each village. But he rejected the hypothesis of efficient risk sharing across villages. More generally, it is quite obvious that the mutuality principle does not hold in large economies. Some workers may see their income reduced in a booming economy. In case of a laborsaving macroeconomic shock, workers may lose against capital owners. Some classes of retirees may enjoy an increase in their pension benefit even when there is a recession. Some countries may be in recession in a growing world economy. Shiller (1993) recommends that new institutions be created to share these society’s large economic risks that are currently inefficiently shared. The welfare impacts of imperfect risk sharing are likely to be very large. In the remainder of this research paper, the reasons for these market imperfections are examined.

3. Optimal Insurance With Transaction Costs

Insurance entails transaction costs. In many lines of casualty insurance, transaction costs may be as large as 30 percent of the premium. Consider an economy with a risk-neutral insurer and consumers with idiosyncratic risks X. suppose that each euro of indemnity yields k euros of deadweight cost. In a competitive insurance market, the premium associated with insurance contract I(.) must be equal to (1+k)EI(X). Mossin (1968) showed that the optimal insurance contract for the policyholder is no longer the full insurance contract. It is always optimal for him to retain part of the risk in order to reduce the insurance cost. This is because risk is a second-order effect relative to transaction costs.

Arrow (1965) examined the optimal form of the risk retention. There are various ways for policyholders to retain a share of the risk. The most standard is to accept a straight deductible, in which the indemnity is either zero if the loss is less than the prespecified deductible, or the loss minus the deductible otherwise. Alternatively, the insurance contract can contain a coinsurance rule in which case the indemnity is a prespecified percentage of the loss. Other clauses can be considered, as upper caps on indemnities, disappearing deductibles, etc. Arrow showed that the optimal insurance contract takes the form of a straight deductible. Any nondeductible insurance contract is dominated by a straight deductible contract with the same actuarial value, but yielding a smaller retained risk for the policyholder. Straight deductibles provide the best compromise between the willingness to cover the risk and the limitation of the insurance deadweight cost.

One can link transaction costs to undiversifiable risks. Obviously, many natural, environmental, or technological risks are in the class of large risks that are difficult to eliminate by using the mutuality principle. The insurer’s risk-neutrality for these risks may be questioned. Insurance companies will not provide fair insurance premiums for them. Indeed, shareholders will not be able to diversify the risk associated to the dividends paid by insurance companies that cover these large risks. They will ask for a risk premium, which will increase the cost of capital of these companies. This cost will be passed on to policyholders through a larger premium rate for the component of individual risks that is systematic. It will induce them to retain part of their individual risk. In short, the fact that the risk is systematic induces insurance premiums to contain a positive loading that has an effect equivalent to a transaction cost. This is the logic behind larger deductibles for systematic risks.

4. Adverse Selection

Since the seminal paper by Rothschild and Stiglitz (1976), it is recognized that the fact that insurers face a heterogeneous population of consumers is a source of inefficiency on insurance markets. The classical model presented above allows for a heterogeneous population as long as the characteristics of the risk borne by each agent is common knowledge. For example, the fact that women are safer drivers than men is compatible with full insurance of every driver at the competitive equilibrium with a risk-neutral insurance industry. The premium rate for every category of risk will be fair, thereby inducing each individual to purchase full insurance at the optimum.

A problem arises when the population is heterogeneous, but the observable characteristics of the agents are not perfectly correlated to the intensity of their risk. The adverse selection problem originates from the observation that if insurance companies calculate the premium rate on the basis of the average probability distribution in the population, the less risky agents will purchase less insurance than riskier agents. In the extreme case, the low-risk agent will find the premium rate too large with respect to their actual probability of loss. They will prefer not to insure their risk. Insurers will anticipate this reaction, and they will increase the premium rate to break even, only on the population of high-risk policyholders. The literature on adverse selection is devoted to characterizing an equilibrium. Insurers will use the fact that low-risk agents and large-risk agents behave differently in the face of a large set of insurance contracts. In particular, low-risk agents could credibly signal their type by selecting a contract with a large deductible, something that high-risk agents dislike.

The presence of high-risk agents generates a negative externality to lower-risk agents who are unable to find an insurance premium at an acceptable premium rate. To illustrate, this is probably why the proportion of households that purchase life insurance is so small, despite the potential severity of the risk. People have private information about their health status that cannot be observed by insurance companies. Then, only those with the lowest life expectancy purchase life insurance.

The standard policy recommendation for improving risk-sharing efficiency under an adverse selection problem is to make public all relevant information about risks. For example, insurers should be allowed to know whether the potential policyholder has some severe illness. They should also be allowed to use genetic testing. Obviously, redistributional reasons may go against such a policy recommendation if the State is not in a position to compensate poor high-risk agents.

5. Moral Hazard

The population of risks can be heterogeneous not only because agents bear intrinsically different risks, but also because they do not invest the same amount of their energy, wealth, or time in risk prevention. In particular, it has long been recognized that individuals that are better covered by insurance invest less in risk prevention if the link between the premium rate and the size of these investments is weak. It will be the case if insurers are not in a position to observe the investment in risk prevention by the policyholder. In that case, the premium rate is not sensitive to the effort made by the policyholder to prevent losses. Obviously, there will be an inverse relationship between risk prevention and insurance coverage: policyholders will not internalize the benefits of their efforts. The level of risk prevention will be inefficient. This is ex ante moral hazard, common to all principal-agent problems. Anticipating this low degree of prevention and the higher frequency of losses that it entails, insurers will raise their premium rate. Full insurance will not be optimal for agents. At the limit, no insurance can be an equilibrium. Holmstrom (1979) characterized the equilibrium insurance contract with ex ante moral hazard.

To illustrate, this is why it is not possible to insure against the absence of promotion on the workplace, about failure at school or university, about the lack of demand for a new product, or about divorce. To some extent, this is also why it is hard to insure against unemployment, or against environmental and technological risks.

The policy recommendation to fight against ex ante moral hazard is the enforcement of norms for risk prevention. This is the case for environmental risks in which ships transporting chemical products have to satisfy various safety requirements that are imposed by regulatory agencies. Automobile driving norms are also standard, as speed limits, alcohol-free driving, etc. Why these norms are mostly organized by a regulatory agency rather than by insurers is not completely clear. One reason is due to the combination of negative externalities and limited liability. If there is more than one principal supervising the implementation of norms, the information among the different principals should be pooled to save on monitoring costs. For example, auto insurers should be allowed to get the information about driver fines by the police.

Another policy recommendation is to allow insurers to discriminate prices among different policyholders that exercise various preventive efforts. Allowing for discrimination is a way to provide incentive to policyholders to invest in risk-reducing activities. In France again, insurers are not allowed to discriminate homeowner premium rates on the basis of natural risks such as earthquake, flood, and storms. The consequences are by now notorious: many households built their houses in areas that were secularly known to be flooded periodically.

6. Insurance Fraud

The classical model assumed that the size of the loss is observable. There are many instances in which this is at best a crude approximation of the real world. Contracts can be made contingent only upon observable events. The problem here is to give the good incentives to the policyholder to report their actual loss. The difficulty for insurers to verify claims is at the origin of why it is not possible to insure against loss of happiness, or against some forms of suffering that cannot be measured by physicians.

There exist other types of risk for which outcomes can be observed by the insurer only at a relatively high auditing cost. Townsend (1979), Mookherjee and Png (1989), and others analyzed the optimal risk-sharing scheme in this case. If there is no limit on the penalty that can be imposed to policyholders that do not declare the actual level of their loss, the first-best solution can be attained. Indeed, insurers should announce that they will audit claims with some probability p that is very low. If the policyholder made a fraudulent claim, an unbounded penalty (‘death penalty’) would be imposed on him. This is enough to give the incentive not to defraud the insurance contract, even if p is very small. In this case, the fact that there is costly claim verification is not detrimental to welfare, and the risk is insurable in full.

But there are several reasons to believe that an unbounded penalty in case of a fraudulent claim is not a realistic assumption: ethics, limited liability, risk of legal errors, etc. Ex ante, it is then Pareto-efficient to limit the size of the penalty. In order to report their loss correctly, the insurer will have to audit claims at a high frequency. This entails additional costs on the insurance contract. If the auditing cost is high, or if the frequency of audit necessary to give the good incentive for the policyholder to reveal the truth is too high, consumers would be better off by not insuring the risk. Notice that another way to reduce the willingness to submit a fraudulent claim is to limit the indemnity. The maximal indemnity that is compatible with truth telling is an increasing function of the penalty and of the probability of audit. Consumers would like to announce ex ante that they will not submit fraudulent claims ex post. That would allow insurers to save the audit cost, thereby reducing the equilibrium premium rate, but the announcement is not credible.

Is ex post moral hazard an important problem? It is often suggested that the cost of fraudulent claims may well amount up to 10 percent of premiums paid for such insurance lines as automobile insurance or homeowner insurance. This estimation is the cost of unjustified indemnities to policyholders, not the auditing cost to fight against fraud. This percentage is comparable to the rate of transaction costs, whose effects on insurability have been previously examined.

The policy recommendation is clear from the discussion above: one should impose a larger penalty on policyholders that have been convicted of a fraudulent claim. Several countries in Africa, and to a smaller extent in Europe, have been weak in this area, recognizing fraud as a ‘national sport’ that should be forgiven. By doing so, the legal system imposes a possibly large cost to Society in terms of a loss of insurability. This weakness has been particularly clear for insurance lines where the indemnity payer does not have the good incentives to be tough on fraud. For example, one may question whether social security organizations are fighting fraudulent claims efficiently. This yields a general distrust of the system, which is detrimental to the unemployed themselves.

7. Limited Liability

An individual can cause damage to others, either in the course of his/her profession (medicine, surgery, house building, etc.) or because of other activities (e.g., driving a car). The same kind of external random effect occurs for firms. In most countries, the agent found liable for damage to others must indemnify them accordingly. This is done to force decision makers to internalize all costs generated by their choice. But indemnification is only possible up to the decision maker’s financial capacity. Limited liability is a way to protect risk-takers against an excessive financial distress. But it has long been recognized that limited liability distorts the decision of the risk-taker in a way that is socially inefficient. The US Saving and Loans crisis is often explained by the fact that ‘zombie’ S and Ls adopted in the early 1980s a very risky attitude in an attempt to ‘bet for resurrection’ after some blows were delivered to their portfolio of (real estate) assets. This is because limited liability gives the agent the equivalent of a free put option. Put it in simpler terms, under limited liability, an insolvent agent can only benefit from taking more risk, because he does not bear the burden of losses. Therefore, if the agent is risk-neutral, he will seek to maximize the expectation of a convex function of his wealth. As a result, he will systematically exhibit a risk-loving behavior, and adopt a very risky attitude. This is a kind of moral hazard problem. Risk aversion mitigates this result, but only for agents who are well capitalized.

The effect of limited liability of the policyholder on their demand for insurance is thus unambiguous: if he is risk-neutral, it is never optimal to cover a risk of loss, even in the most favorable case where the premium rate is fair. Insuring the risk would yield a sure reduction in wealth equaling the expected loss. Not insuring the risk would yield an expected reduction of wealth that is less than it, since the agent bears only part of the risk of loss. Another way of looking at this problem is that the insurance contract creates a ‘deep pocket’ where victims can find compensation for their losses. This kind of problem is particularly crucial when examining the demand for insurance by firms for catastrophic environmental risks. Limited liability on the part of the insurance also reduces the demand of insurance, since it makes the indemnity dependent on solvency.

Limited liability thus raises several important questions. How to organize compensation for those who bear the negative externalities? How to build an incentive-compatible mechanism that increases loss prevention by decision makers with limited liability? How to solve the market failure of liability insurance markets? How to force firms not to under-capitalize their subsidiaries which are in charge of managing the riskiest part of the business? Two routes have been used. The first one is compulsory insurance. This solves the misallocation of risk in the economy and the organization of a system to compensate the victims. But, most of the time, compulsory insurance has been funded by a flat, non-discriminatory, non-incentive compatible insurance tariff. The policyholder’s investment in loss prevention is not observed by the fund, either because it is difficult to get information on it, or because the fund did not organize an incentive compatible system.

The second route has been to organize a ‘deep pocket’ for decision-makers. For example, the hospital that employs an uninsured physician may be made liable in case of the physician’s insolvency. Under the US CERCLA, when a bank has been relatively closely involved in the monitoring of a firm’s activities, it may be considered by the courts as liable for cleaning up the environmental damages generated by the insolvent firm. The objective of this strategy is to force risk-takers to internalize the full cost of potential losses: the hospital will reduce the income of the careless physician, and banks will increase the loan rate of riskier firms. If there is no asymmetric information between the principal (the hospital, the bank) and the agent (the physician, the firm), the agent will select the socially efficient level of care and insurance. There would then be no insurability problem. But, there is no reason to believe that the principal can monitor the agent at no cost. The CERCLA legislation, for example, introduces more asymmetric information on credit markets. Consequently, there will be more credit rationing, the cost of capital will be larger, and the structure of banking contracts for firms will be affected. Is insurability worth these costs?

8. Regulation Of The Insurance Sector

Insurance markets are among the most heavily regulated ones around the world. In the Middle Ages, the Catholic Church was reluctant to recognize insurance as an ethically acceptable activity. The French revolution prohibited insurance companies in 1793. States have been very active to control insurance tariffs, in particular in life insurance. They also imposed compulsory insurance. More recently, they established solvency rules similar to those of the banking sector.

There is a European tradition to use insurance for a redistributive purpose. Insurance implements wealth redistribution if the expected profit of the insurer depends upon the observable characteristics of the policyholders. The expected profit on the low risk class of policyholders is used to cover the expected loss on the high risk class. This scheme is standard in unemployment, health, and disability insurance, but it can also be found in other insurance sectors like automobile (for young and old drivers), homeowner (for natural catastrophes), and life (for the prohibition of using health factors to price life insurance). However, the solidarity among different risk classes is difficult to organize with decentralized insurance markets without strong regulation. The absence of discrimination on the basis of policyholders’ observable characteristics that are correlated with the risk introduces adverse selection. It also generates moral hazards if these characteristics can be affected by the preventive effort of the policyholder. In order to alleviate the adverse selection problem, regulators often impose compulsory insurance, as in social security systems. Pricing regulation is also necessary to react to the natural tendency of competitive markets to discriminate. Finally, insurers should be forced to supply coverage to all customers at the undiscriminating premium. Various reforms of the health insurance sector in OECD countries have been proposed to deal with the problems inherent in such heavy regulation.

There is a form of saving in an insurance contract from the policyholder’s point of view. Indeed, the indemnity, if any, may be paid a long time after the premium. This is particularly true for life insurance. The lender, i.e., the policyholder, should check the solvency of the borrower-insurer. In perfect competitive markets, insurers facing a higher risk of insolvency should be penalized by collecting smaller premia, to compensate the risk borne by the policyholders. This is enough to provide good incentives for insurance companies to manage their financial reserves efficiently. However, policyholders may not be in a position to evaluate correctly the probability of failure of the insurers. This asymmetric information calls for centralized solvency regulation. It takes two forms. The first type of regulation is about solvency ratios, similar to those imposed on banks. The second type of regulation is about the kind of risks that can be taken by insurers on the investments they make with their financial reserves. Typically, there is an upper limit on the share of reserves invested in stocks and real estate. Many countries prohibit the use of derivative assets, in spite of their potential benefits for risk reduction. These constraints must be strong enough to fight against the moral hazard problem generated by asymmetric information and limited liability. It should not be too strong in order to allow for efficient Asset-Liability Management (ALM) by insurers.


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