A Value at Risk Approach to Background Risk

This paper studies the effects of an uninsurable background risk (BR) on the demand for insurance (proportional and with deductible). We study both the case of BR uncorrelated with the insurable one and the perfectly correlated one, in a Gaussian world. In order to perform our study, we exploit the new risk measure known as Value at Risk (VaR) and consider insurance contracts which are Mean-VaR efficient. We obtain results which depend on the parameters (moments) of both risks and on the magnitude of loadings charged by the insurance company, instead of depending on the risk attitudes of the insured, such as risk aversion and prudence.We demonstrate that, if loadings are not too high, the demand for insurance increases with positively correlated BR; it decreases with BR negatively correlated if the latter is less risky than the insurable one (in this case it can even go to zero, if loadings are too high); it goes to zero with BR which is negatively correlated and more risky than the insurable one.     

Background Uncertainty and the Demand for Insurance Against Insurable Risks

Theory suggests that people facing higher uninsurable background risk buy more insurance against other risks that are insurable. This proposition is supported by Italian cross-sectional data. It is shown that the probability of purchasing casualty insurance increases with earnings uncertainty. This finding is consistent with consumer preferences being characterized by decreasing absolute prudence.     

Variance Vulnerability, Background Risks, and Mean-Variance Preferences

An agent with two-parameter, mean-variance preferences is called variance vulnerable if an increase in the variance of an exogenous, independent background risk induces the agent to choose a lower level of risky activities. Variance vulnerability resembles the notion of risk vulnerability in the expected utility (EU) framework. First, we characterize variance vulnerability in terms of two-parameter utility functions. Second, we identify the multivariate normal as the only distribution such that EU- and two-parameter approach are compatible when independent background risks prevail. Third, presupposing normality, we show that—analogously to risk vulnerability—temperance is a necessary, and standardness and convex risk aversion are sufficient conditions for variance vulnerability.     

Insurance demand in the presence of loss-dependent background risk

We analyze insurance demand when insurable losses come with an uninsurable zero-mean background risk that increases in the loss size. If the individual is risk vulnerable, loss-dependent background risk triggers a precautionary insurance motive and increases optimal insurance demand. Prudence alone is sufficient for insurance demand to increase in two cases: the case of fair insurance and the case where the smallest possible loss exceeds a certain threshold value (referred to as the large loss case). We derive conditions under which insurance demand increases or decreases in initial wealth. In the large loss case, prudence determines whether changes in the background risk lead to more insurance demand. We generalize this result to arbitrary loss distributions and find conditions based on decreasing third-degree Ross risk aversion, Arrow–Pratt risk aversion, and Arrow–Pratt temperance.     

Background Risk, Demand for Insurance, and Choquet Expected Utility Preferences

This article deals with demand for insurance with a background risk in a nonprobabilized uncertainty framework, where preferences are represented by a nonadditive model of decision making. The Choquet expected utility model that we use generalizes expected utility and allows for a separation of the attitude towards uncertainty and the attitude towards wealth. When the insurable and the background risk are comonotone, the impact of the background risk on the demand for insurance is related to the attitude towards wealth. In contrast, when the two risks are anticomonotone, the attitude towards uncertainty is determinant. In this case, some of the resulting behaviors cannot be explained by the standard expected utility model.     

Optimal Insurance Under Random Auditing

We provide a characterization of an optimal insurance contract (coverage schedule and audit policy) when the monitoring procedure is random. When the policyholder exhibits constant absolute risk aversion, the optimal contract involves a positive indemnity payment with a deductible when the magnitude of damages exceeds a threshold. In such a case, marginal damages are fully covered if the claim is verified. Otherwise, there is an additional deductible that disappears when the damages become infinitely large. Under decreasing absolute risk aversion, providing a positive indemnity payment for small claims with a nonmonotonic coverage schedule may be optimal.     

When Is a Coinsurance‐Type Insurance Policy Inferior or Even Giffen?

This article derives the necessary and sufficient conditions for a coinsurance‐type insurance policy covering a particular risk to be inferior and to be Giffen. Mossin's decreasing absolute risk aversion assumption for insurance to be inferior is avoided. The result generalizes Hoy and Robson and Briys, Dionne, and Eeckhoudt's results to the case with a continuum of states and relaxes their assumption of constant relative risk aversion. It is shown that knowledge about the distribution of risk can be used to relax assumptions on an utility function for a coinsurance‐type insurance policy to be inferior and to be Giffen.     

Is consumption risk priced in the stock market?

The paper examines whether the risk in the consumption of stockholders caused by incomplete consumption insurance is priced in the cross-section of average stock returns. Using Taylor series expansion of the average marginal utility of consumption, we show that the risk in the consumption of stock market participants can be decomposed into two components, insurable (hedgeable using financial assets) and uninsurable (caused by incomplete consumption insurance) consumption risks. We argue that the growth rate of average consumption may be viewed as a proxy for the insurable component of consumption risk, while the growth rates of the rescaled higher-order cross-sectional consumption distribution moments may be regarded as a multivariate proxy for uninsurable risk in consumption. Exploiting microlevel household quarterly consumption data from the US Consumer Expenditure Survey, we find that both components of consumption risk are significantly priced when the limited stock market participation is taken into account. Neither the insurable and uninsurable components of consumption risk nor the Fama–French risk factors are rejected as capturing important components of systematic risk when tested against each other in an integrated multifactor asset pricing model.     

A Note on Deductible Insurance and Production

This paper examines the optimal production decision of a firm facing revenue risk. We show that the purchase of actuarially fair deductible insurance unambiguously induces the firm to produce more if the firm is not only risk averse but also prudent. If the firm's perferences satisfy constant absolute risk aversion, buying actuarially unfair deductible insurance unambiguously enhances production should the positive loading factor be sufficiently small. When there are moral hazard problems in that the firm's output cannot be contracted upon, we show that the purchase of actuarially fair deductible insurance unambiguously induces the firm to produce more if the firm's utility function is quadratic.     

Public Versus Private Insurance with Dual Theory: A Political Economy Argument

This paper analyzes the political support for public insurance in the presence of a private insurance alternative. The public insurance is compulsory and offers a uniform insurance policy. The private insurance is voluntary and can offer different insurance policies. Adopting Yaari's [Econometrica, 55, 95–115, 1987] dual theory to expected utility (i.e., risk aversion without diminishing marginal utility of income), we show that adverse selection on the private insurance market may lead a majority of individuals to prefer public insurance over private insurance, even if the median risk is below the average risk (so that the median actually subsidizes high-risk individuals). We also show that risk aversion makes public insurance more attractive and that the dual theory is less favourable to a mixed insurance system than the expected utility framework. Lastly, we demonstrate how the use of genetic tests may threaten the political viability of public insurance.     

More on Properness: The Case of Mean-Variance Preferences

This paper focuses on the situations where individuals with mean-variance preferences add independent risks to an already risky situation. Pratt and Zeckhauser (Econometrica, 55, 143–154, 1987) define a concept called proper risk aversion in the expected utility framework to describe the situation where an undesirable risk can never be made desirable by the presence of an independent undesirable risk. The assumption of mean-variance preferences allows us to study proper risk aversion in an intuitive manner. The paper presents an economic interpretation for the quasi-concavity of a utility function derived over mean and variance. The main result of the paper says that quasi-concavity plus decreasing risk aversion is equivalent to proper risk aversion.     

Precautionary Saving and Consumption Smoothing across Time and Possibilities

This paper examines how aversion to risk and aversion to intertemporal substitution determine the strength of the precautionary saving motive in a two-period model with Selden/Kreps–Porteus preferences. For small risks, we derive a measure of the strength of the precautionary saving motive that generalizes the concept of "prudence" introduced by Kimball (1990b) . For large risks, we show that decreasing absolute risk aversion guarantees that the precautionary saving motive is stronger than risk aversion, regardless of the elasticity of intertemporal substitution. Holding risk preferences fixed, the extent to which the precautionary saving motive is stronger than risk aversion increases with the elasticity of intertemporal substitution. We derive sufficient conditions for a change in risk preferences alone to increase the strength of the precautionary saving motive and for the strength of the precautionary saving motive to decline with wealth. Within the class of constant elasticity of intertemporal substitution, constant-relative risk aversion utility functions, these conditions are also necessary.     

To Insure or Not to Insure?: An Insurance Puzzle

The selection of a deductible level in insurance is governed by the willingness to limit the risk borne by risk-averse agents at an acceptable cost, given the deadweight insurance loading. We examine the demand for insurance in a simple lifecycle model with a liquidity constraint and no serial correlation in the insurable risk. This allows for consumers to follow a time-diversification (self-insurance) strategy by accumulating buffer stock wealth. We conclude that insurance would only be demanded for catastrophic risks, or by people that are currently liquidity constrained. The added value of the insurance sector is thus surprisingly low in such an economy.     

我国巨灾风险可保性的理性思考

巨灾保险虽然一直未能真正踏上破冰之旅,但近年频发加剧的巨灾,使巨灾保险再次成为保险业的聚焦领域。巨灾风险可保性争论不休,业界徘徊不前,厘清巨灾风险的保险属性,可否引导市场参与巨灾风险的有效管理,特别是金融市场、保险市场和资本市场的联动推进,更是尚需解决的问题。基于巨灾风险管理的理论分析认为,我国巨灾风险具备可保性有其理...     

创新运用保险技术分解社会风险探究

处于经济社会转型期的当代中国所面临的社会风险比以往任何时候更具复杂性和威胁性,社会风险管理面临严重挑战。通过阐述国际社会风险管理的新理念和趋势、分析现实中国的主要社会风险及其风险源,提出创新运用保险技术分解社会风险的必要性、可行性和现实路径及方法。重点提出通过商业保险对社会风险源的定性、定量分析,转化部分社会风险成可保风险,达到相应社会风险的被分解和预控。     

The Demand for Insurance With an Upper Limit on Coverage

The demand for insurance is examined when the indemnity schedule is subject to an upper limit. The optimal contract is shown to display full insurance above a deductible up to the cap. Some results derived in the standard model with no upper limit on coverage turn out to be invalid; the optimal deductible of an actuarially fair policy is positive and insurance may be a normal good under decreasing absolute risk aversion. An increase in the upper limit would induce the policyholder with constant absolute risk aversion to reduce his or her optimal deductible and therefore this would increase the demand for insurance against small losses.     

Precautionary Insurance Demand With State-Dependent Background Risk

This article considers a zero‐mean background risk that is uncorrelated with insurable losses, but is not necessarily statistically independent. In particular, the size of the background risk can vary in different insurable‐loss states. We show how a prudent individual will buy either more insurance or less insurance than with no background risk, depending on the relative size of the background risk in the loss states vis‐á‐vis the no‐loss states. If we consider two individuals, with one more risk averse than the other, we need to compare the intensities of their precautionary motives, in addition to their measures of risk aversion, before we can determine who buys more insurance coverage in the presence of the state dependent background risk.     

背景风险对保险需求的影响——基于中国健康保险市场的实证研究

保险需求理论研究表明,不可保的背景风险会增加投保人对可保风险的保险需求。本文把2003年发生的SARS看作一次自然实验,来检验背景风险对保险需求的影响。本文用我国健康保险市场的时间序列数据,通过邹-检验和时间序列结构突变检验方法,对存在背景风险时的保险需求理论进行实证研究。研究发现,背景风险会增加投保人对可保风险的保险需求,但这种影响只具有短期效应。     

Wealth effects and committed cost allocation: An agency theory perspective

In this article we examine the effects of committed costs (CC) on compensation and effort (production) decisions in a principal-agent (P-A) setting. In the case where moral hazard is present, the compensation and effort (production) decisions are independent of CC whenever P has constant absolute risk aversion. When P has decreasing absolute risk aversion, he demands as increased risk premium, therefore increases the spread of the compensation schedule, and induces A to increase effort (production) and vice versa. The optimal compensation scheme can be decomposed to conform to incentive schemes that are generally observed in practice. In particular, we decompose the optimal compensation scheme that depends on CC into two parts—a part that is based on cash flows and a part that is based on income (after allocating committed costs). In the case where effort is observable, CC does not effect the compensation scheme and effort decisions, when P has constant absolute risk aversion. In contrast to earlier studies that examine the owner-manager case, when P has decreasing absolute risk aversion, effort (production) could either increase or decrease. The presence of moral hazard affects the effort (production) decision differently than when risk-sharing considerations alone exist. The reduction in A's compensation induced by increased CC never exceeds the amount of CC.     

Flexible Spending Accounts as Insurance

We model flexible spending accounts (FSAs) as a special type of insurance policy. We prove the following results given losses drawn from a continuous distribution: (1) the optimal election amount, F^*, is increasing in the consumer's level of risk aversion; (2) F^* is increasing in the level of the maximum loss; If utility is decreasing in absolute risk aversion (DARA), then F^* is (3) decreasing in income and (4) increasing in the marginal tax rate.