A note on Mossin’s theorem for deductible insurance given random initial wealth

Mossin’s theorem for deductible insurance given random initial wealth is re-examined. For a fair premium, it is shown that a necessary and sufficient condition, in the spirit of the Generalized Mossin Theorem for coinsurance, is impossible using the notion of expectation dependence. Next, it is established that for a fair premium, full insurance will be optimal for a risk-averse individual if the random loss and the random initial wealth are negative quadrant dependent, improving upon an extant result in the literature. In view of a set of examples given in this paper, such a sufficient condition cannot be obtained using the notion of expectation dependence. Finally, for an unfair premium, it is shown that partial insurance will always be optimal, irrespective of the risk preference of the individual as well as the dependence structure between the random loss and the random initial wealth.     

Mortality Risk and the Value of a Statistical Life: The Dead-Anyway Effect Revis(it)ed

In the expected-utility theory of the monetary value of a statistical life, a well-known result found by Pratt and Zeckhauser [1996] asserts that an individuals’ willingness to pay (WTP) for a marginal reduction in mortality risk increases with the initial level of risk. Their reasoning is based on the so-called “dead-anyway effect” which states that marginal utility of a dollar in the state of death is smaller than in the state of survival. However, this explanation is based on the absence of markets for contingent claims, i.e. annuities and life insurance. This paper reexamines the relationship between WTP and the level of risk under more general circumstances and establishes two main results: first, when insurance markets are perfect, for a risk-averse individual without a bequest motive, marginal WTP for survival does increase with the level of risk but this occurs for a different reason, namely an income effect. Secondly, when the individual has a bequest motive and is endowed with a sufficient amount of wealth from human capital, the effect of initial risk on WTP for survival is reversed: the higher initial risk the lower the value of a statistical life. In the imperfect-markets case we interpret this result as a “constrained-bequest effect”.JEL Classification No.: D8, H43, I18     

Pooling and Separating Equilibria in Insurance Markets with Adverse Selection and Distribution Costs

In the Rothschild-Stiglitz [1976] model of a competitive insurance market with adverse selection, pooling equilibria cannot exist. However in practice, pooling contracts are frequent, notably in health insurance and life insurance. This is due to the fact that distribution costs are nonnegligible and increase rapidly when more contracts are offered. We modify accordingly the Rothschild-Stiglitz model by introducing such distribution costs. We find that, however small these costs may be, they entail possible existence of pooling equilibria. Moreover, in these pooling equilibria, it is the high-risk individuals who are rationed, in the sense that they would be willing to buy more insurance at the current premium/insurance ratio.     

A Voluntary Deductible in Social Health Insurance with Risk Equalization: “Community‐Rated or Risk‐Rated Premium Rebate?”

On January 1, 2006 a new mandatory basic health insurance will be introduced in the Netherlands. One aspect of the new scheme is that the insured can choose to have a deductible. This option should increase the individual responsibility and reduce moral hazard. In the new scheme, a risk equalization system is aimed at avoiding preferred risk selection and insolvency of insurance companies with a relatively high‐risk pool. A crucial issue with respect to a voluntary deductible in this type of social health insurance is whether the premium rebate should be community rated or risk rated. The Dutch government has chosen the former, which means that the premium rebate will be independent of health status and risk. Our analysis shows that, in a situation with “accurate” risk equalization, a community‐rated premium rebate could lead to an adverse selection spiral. Over time, this spiral results in none of the insured taking a deductible and thus no reduction in moral hazard.     

Habit formation and the equity–premium puzzle: a skeptical view

We argue that, ceteris paribus, introducing a habit that resolves the equity–premium puzzle is equivalent to increasing the Arrow-Pratt coefficient of relative risk aversion, AP-RRA. If we constrain the AP-RRA to a constant ‘acceptable’ level, the effect on the equity premium is quantitatively insignificant. In a dynamic setting, the fluctuations of the habit increase the equity premium, slightly, though generates unrealistic fluctuations in the risk-free interest rate. We conclude a habit is observationally equivalent, up to a first-order approximation, to a higher AP-RRA and to a preference shock. These effects cannot resolve the equity–premium puzzle.      

Remarks on the Mossin Theorem

We establish several new results regarding the Mossin Theorem under both nonrandom initial wealth and random initial wealth. For the nonrandom initial wealth case, we show that the Mossin Theorem holds for any constrained class of insurance contracts whose maximum coverage provides full coverage of the potential loss. This result not only settles an open conjecture, but also provides a unified treatment for extant varieties of the Mossin Theorem. For the random initial wealth case, we give a thorough study of the upper-limit insurance. In particular, we show that (1) for a fair premium, the Generalized Mossin Theorem for coinsurance does not hold for upper-limit insurance, and (2) for an unfair premium, partial insurance will always be optimal, regardless of the risk preference of the individual and the dependence structure between the random loss and the random initial wealth.     

偏度风险溢酬及其预测能力研究——基于人民币对外汇期权的实证分析

基于无模型方法,利用在岸人民币对美元期权的市场价格数据,通过提取风险中性三阶矩和已实现三阶矩估算偏度风险溢酬,并考察其时变特征、与波动率风险溢酬的关系和对汇率尾部风险的预测能力。结果表明,外汇期权市场上存在显著为负的时变偏度风险溢酬;偏度风险溢酬与波动率风险溢酬存在共同的驱动因子。此外,偏度风险溢酬在短期内对汇率暴涨的尾部风险事件具备一定的预测能力。当偏度风险溢酬增加时,投资者对偏度风险的厌恶加剧,预期未来人民币大幅贬值的概率上升。     

A Wealth-Based Explanation for Earnings Conservatism

This study extends research on earnings conservatism – the degree to which the accounting system recognizes bad news regarding future cash flows in a more timely manner than good news – by arguing that heterogeneous executives' risk attitudes will influence the degree of conservatism. Prior research has demonstrated that differences in earnings conservatism are mainly the result of differences in institutional factors (Basu (1997) and Ball et al. (2000a)). We hypothesize that more risk-averse managers, who demand a risk premium that offsets the effects of the variance in their compensation, will report more conservative earnings. Earnings conservatism will temper expectations among stakeholders about the future cash flows to be distributed thereby diminishing the likelihood of disappointing outcomes and potential litigation or threats for executives of being fired. The more risk-averse manager would be more inclined to reduce such conflicts, since they will have a destabilizing effect on his future compensation. The empirical results for a sample of Dutch companies over the period of 1983 to 1995 confirm our hypothesis: more risk-averse managers report earnings more conservatively than do less risk-averse managers.     

The pricing of deposit insurance in the presence of systematic risk

Based on the Merton (1977) put option framework, we develop a deposit insurance pricing model that incorporates asset correlations, a measurement for the systematic risk of a bank, to account for the risk of joint bank failures. Estimates from our model suggest that actuarially fair risk-based deposit insurance that considers only individual bank failure risk is underpriced, leaving insurance providers exposed to net losses. Our estimates also capture the size premium where big banks are priced with higher deposit insurance than small banks. This result is particularly relevant to the current regulatory concerns on big banks that are too-big-to-fail. Above all, our approach provides a unifying framework for integrating risk-based deposit insurance with risk-based Basel capital requirements.     

Optimal insurance with background risk: An analysis of general dependence structures

Finance and Stochastics - In this paper, we consider an optimal insurance problem from the perspective of a risk-averse individual who faces an insurable risk as well as some background risk and...     

Zur bedeutung einer Ausfallbedrohtheit von Versicherungskontrakten — ein Beitrag zur Behavioral Insurance

Kahneman / Tversky 1979 introduced the notion of so-called probabilistic insurance contracts. These are insurance policies involving a small probability that the consumer is not reimbursed because of a possible default of the insurance company. Extending the study ofWakker / Thaler / Tversky 1997, the present study contains an experimental analysis of the willingness of potential policyholders to pay for probabilistic insurance in dependency on the rating of the insurance company. It can be shown that people dislike probabilistic insurance and demand a substantial reduction in the premium to compensate for default risk. This reduction is rising with the default risk of the company. In addition, the results show a new phenomenon. The more an insurance company is threatened by default risk the less people are willing to accept contracts of this company at all. Finally the paper discusses implications for the control of insurance companies.     

Incentive Effects of Community Rating in Insurance Markets: Evidence from Massachusetts Automobile Insurance

Rate regulations in insurance markets often impose cross-subsidies in insurance premiums from low-risk consumers to high-risk consumers. This paper develops the hypothesis that premium cross-subsidies affect risk taking by insurance consumers, and tests this hypothesis by examining the marginal impact of premium subsidies and overcharges on future insurance costs. The empirical analysis uses 1990–2003 rating cell-level data from the Massachusetts automobile insurance market, in which regulation produced large cross-subsidies across cells. Consistent with the hypothesized effects, premium subsidies are found to be significantly related to higher future insurance costs, and the opposite effects are found for premium overcharges.     

Optimal proportional reinsurance with a loss-dependent premium principle

ABSTRACT

Empirical studies suggest that many insurance companies recontract with their clients on premiums by extrapolating past losses: a client is offered a decrease in premium if the monetary amounts of his claims do not exceed some prespecified quantities, otherwise, an increase in premium. In this paper, we formulate the empirical studies and investigate optimal reinsurance problems of a risk-averse insurer by introducing a loss-dependent premium principle, which uses a weighted average of history losses and the expectation of future losses to replace the expectation in the expected premium principle. This premium principle satisfies the bonus-malus and smoothes the insurer's wealth. Explicit expressions for the optimal reinsurance strategies and value functions are derived. If the reinsurer applies the loss-dependent premium principle to continuously adjust his premium, we show that the insurer always needs less reinsurance when he also adopts this premium principle than when he adopts the expected premium principle.     

Wealth, Income, and Optimal Insurance

This article considers the decision to purchase insurance against possible losses of a property or wealth. The decision involves a standard economic trade‐off between the benefit of protection against loss and the cost of insurance premium. The premium is paid out of the income and decreases the consumption of other goods and services, rather than out of wealth and decreases the property or wealth. The demand for insurance depends mainly on the income and preferences. As a result, unlike in the standard model, a fair premium is neither necessary nor sufficient for the optimality of full coverage insurance. Rather, the individuals with higher incomes purchase full coverage insurance even at unfair prices of insurance while the individuals with lower income purchase partial coverage insurance at a fair price.     

“Skewness and Stock Option Prices”, Hans Gerber and Bruno Landry,July 1997

We determine the optimal amount of life insurance for a household of two wage earners. We consider the simple case of exponential utility, thereby removing wealth as a factor in buying life insurance, while retaining the relationship among life insurance, income, and the probability of dying and thus losing that income. For insurance purchased via a single premium or premium payable continuously, we explicitly determine the optimal death benefit. We show that if the premium is determined to target a specific probability of loss per policy, then the rates of consumption are identical under single premium or continuously payable premium. Thus, not only is equivalence of consumption achieved for the households under the two premium schemes, it is also obtained for the insurance company in the sense of equivalence of loss probabilities.     

Statistical Implications of the Revenue Transfer Methodology in the Affordable Care Act

The Affordable Care Act (ACA) includes a permanent revenue transfer methodology that provides financial incentives to health insurance plans that have higher than average actuarial risk. In this article, we derive some statistical implications of the revenue transfer methodology in the ACA. We treat as random variables the revenue transfers between individual insurance plans in a given marketplace, where each plan’s revenue transfer amount is measured as a percentage of the plan’s total premium. We analyze the means and variances of those random variables and deduce from the zero-sum nature of the revenue transfers that there is no limit to the magnitude of revenue transfer payments relative to plans’ total premiums. Using data provided by the American Academy of Actuaries and by the Centers for Medicare & Medicaid Services, we obtain an explanation for empirical phenomena that revenue transfers were more variable and can be substantially greater for insurance plans with smaller market shares. We show that it is often the case that an insurer that has decreasing market share will also have increased volatility in its revenue transfers.     

Do Property–Casualty Insurance Firms Locate to Minimize Insurance Premium Taxes?

States levy insurance premium taxes, which are essentially gross receipt taxes on premiums, with insurance companies paying the higher of the tax rate in the state in which the company is domiciled and the state in which the policy is written. Using firm‐level data for the property–casualty (P‐C) insurance industry, we estimate the extra insurance premium tax that P‐C insurance firms pay by not locating in the state that minimizes their insurance premium taxes. We find that only 4.78 percent of P‐C firms are located in the state that minimizes their insurance premium taxes. We explore the relationship between the extra tax paid and other factors that are thought to be associated with firm location choice. We find that P‐C firms appear to trade off higher taxes to locate in a state that is more urban.     

Optimal Loss Mitigation and Contract Design

This work examines the interaction between the premium rates set by an insurer and the incentives of an individual to purchase market insurance and undertake mitigation to reduce the size of a potential loss. A risk‐neutral monopolistic insurer prices insurance according to the price‐elasticity of demand for coverage. The elasticity of demand is affected by the presence of both mitigation and government intervention. The availability of loss reduction activities increases the consumer's elasticity of demand and lowers the optimal rate charged by the monopolist. Government intervention reduces both expenditures on mitigation and the rate charged by the monopolistic insurer.     

A decision theoretic formulation of insurance risk theory

Abstract

In a number of papers Borch has shown how certain insurance problems can be formulated using the concept of utility. (See Borch [3], [4], [5], [6], [7] and [8].) Borch's work is used as a building block in Part I of this report, which presents a Bayesian decision theoretic formulation of some of the main aspects of insurance risk theory. Part I makes use of the concepts of utility and subjective probability. It is admitted that these concepts are more commonly associated with individuals rather than groups of individuals such as insurance companies. However, in this report, we will refer to an insurance company as an individual (albeit a neuter one) and assume that it can quantify its preferences for consequences and its opinions about the occurrence of events. Further, we assume that a company “behaves” according to certain rules of consistent behavior which imply that when presented with several risky courses of action, the company will take the action which has the greatest expected utility. Formal treatments of assumptions that lead to this mode of behavior can be found in Savage [17] and Pratt, Raiffa, and Schlaifer [15].     

Sovereign Debt Portfolios,Bond Risks,and the Credibility of Monetary Policy

We document that governments whose local currency debt provides them with greater hedging benefits actually borrow more in foreign currency. We introduce two features into a government's debt portfolio choice problem to explain this finding: risk-averse lenders and lack of monetary policy commitment. A government without commitment chooses excessively countercyclical inflation ex post, which leads risk-averse lenders to require a risk premium ex ante. This makes local currency debt too expensive from the government's perspective and thereby discourages the government from borrowing in its own currency.