The Spillover effect of compulsory insurance

The assumption usually made in the insurance literature that risks are always insurable at the desired level does not hold in the real world: some risks are not—or are only partially—insurable, while others, such as civil liability or health and workers' injuries, must be fully insured or at least covered for a specific amount. We examine in this paper conditions under which a reduction in the constrained level of insurance for one risk increases the demand of insurance for another independent risk. We show that it is necessary to sign the fourth derivative of the utility function to obtain an unambiguous spillover effect. Three different sufficient conditions are derived if the expected value of the exogenous risk is zero. The first condition is that risk aversion be standard—that is, that absolute risk aversion and absolute prudence be decreasing. The second condition is that absolute risk aversion be decreasing and convex. The third condition is that both the third and the fourth derivatives of the utility function be negative. If the expected value of the exogenous risk is positive, a wealth effect is added to the picture, which goes in the opposite direction if absolute risk aversion is decreasing.     

Dynamic Portfolio Choice with Stochastic Wage and Life Insurance

We study optimal insurance, consumption, and portfolio choice in a framework where a family purchases life insurance to protect the loss of the wage earner's human capital. Explicit solutions are obtained by employing constant absolute risk aversion utility functions. We show that the optimal life insurance purchase is not a monotonic function of the correlation between the wage and the financial market. Meanwhile, the life insurance decision is explicitly affected by the family's risk preferences in general. The model also predicts that a family uses life insurance and investment portfolio choice to hedge stochastic wage risk.     

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.     

Portfolio Choice and Life Insurance: The CRRA Case

We solve a portfolio choice problem that includes life insurance and labor income under constant relative risk aversion (CRRA) preferences. We focus on the correlation between the dynamics of human capital and financial capital and model the utility of the family as opposed to separating consumption and bequest. We simplify the underlying Hamilton–Jacobi–Bellman equation using a similarity reduction technique that leads to an efficient numerical solution. Households for whom shocks to human capital are negatively correlated with shocks to financial capital should own more life insurance with greater equity/stock exposure. Life insurance hedges human capital and is insensitive to the family's risk aversion, consistent with practitioner guidance.     

On the Demand for Portfolio Insurance

While insurers manage underwriting risk with various methods including reinsurance, insurers increasingly manage asset risk with options, futures, and other derivatives. Previous research shows that buyers of portfolio insurance pay considerably for downside protection. We add to this literature by providing the first evidence on the cost of portfolio insurance, the payoff to portfolio insurance, and the relative demand for portfolio insurance across VIX levels. We find that the demand for portfolio insurance is relatively high at low levels of VIX, suggesting purchasers demand more downside protection when this protection is cheap on an absolute basis (but expensive on a relative basis). We also provide the first evidence on the hedging behavior of specific investor classes and show that the demand for portfolio insurance is driven by retail investors (individuals) who buy costly insurance from institutional investors. Results are consistent with other types of paradoxical insurance‐buying behavior.     

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.     

Beyond Risk Aversion: Why,How and What's Next?

Risk attitudes other than risk aversion (e.g. prudence and temperance) are becoming important both in theoretical and empirical work. While the literature has mainly focused its attention on the intensity of such risk attitudes (e.g. the concepts of absolute prudence and absolute temperance), I consider here an alternative approach related to the direction of these attitudes (i.e. the sign of the successive derivatives of the utility function).     

Attitudes Towards Income Risk in the Presence of Quantity Constraints

Considering a consumer with standard preferences, I trace out how quantity constraints on markets impact on relative risk aversion and prudence. I first show how this impact decomposes into a local curvature effect and an endogenously changing risk aversion/prudence effect. Next, I calibrate both effects on relative risk aversion and prudence, using estimates on household demand for durables and labour supply. The calibrations show that commitments to durable goods have large effects on attitudes towards risk. And while small wedges between realised and desired levels of labour supply have only moderate effects, becoming full time unemployed on a 60 per cent unemployment benefit significantly raises risk aversion and prudence.     

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.     

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.     

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.     

Decision making with Expected Shortfall and spectral risk measures: The problem of comparative risk aversion

We analyze spectral risk measures with respect to comparative risk aversion following Arrow (1965) and Pratt (1964) for deterministic wealth, and Ross (1981) for stochastic wealth. We argue that the Arrow–Pratt-concept per se well matches with economic intuition in standard financial decision problems, such as willingness to pay for insurance and simple portfolio problems. Different from the literature, we find that the widely-applied spectral Arrow–Pratt-measure is not a consistent measure of Arrow–Pratt-risk aversion. Instead, the difference between the antiderivatives of the corresponding risk spectra is valid. Within the framework of Ross, we show that the popular subclasses of Expected Shortfall, and exponential and power spectral risk measures cannot be completely ordered with respect to Ross-risk aversion. Thus, for all these subclasses, the concept of Ross-risk aversion is not generally compatible with Arrow–Pratt-risk aversion, but induces counter-intuitive comparative statics of its own. Compatibility can be achieved if asset returns are jointly normally distributed. The general lesson is that these restrictions have to be considered before spectral risk measures can be applied for the purpose of optimal decision making and regulatory issues.     

Asset Allocation and the Liquidity Premium for Illiquid Annuities

Academics and practitioners alike have developed numerous techniques for benchmarking investment returns to properly adjust seemingly high numbers for excessive levels of risk. The same, however, cannot be said for liquidity, or the lack thereof. This article develops a model for analyzing the ex ante liquidity premium demanded by the holder of an illiquid annuity. The annuity is an insurance product that is akin to a pension savings account with both an accumulation and decumulation phase. We compute the yield (spread) needed to compensate for the utility welfare loss, which is induced by the inability to rebalance and maintain an optimal portfolio when holding an annuity. Our analysis goes beyond the current literature, by focusing on the interaction between time horizon (both deterministic and stochastic), risk aversion, and preexisting portfolio holdings. More specifically, we derive a negative relationship between a greater level of individual risk aversion and the demanded liquidity premium. We also confirm that, ceteris paribus, the required liquidity premium is an increasing function of the holding period restriction, the subjective return from the market, and is quite sensitive to the individual's endowed (preexisting) portfolio.     

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.     

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.     

Portfolio choice with skewness preference and wealth-dependent risk aversion

This paper studies a dynamic portfolio choice problem for an investor with both wealth-dependent risk aversion and wealth-dependent skewness preferences. In a general economic setting, the solution is characterized in terms of a system of extended Hamilton-Jacobi-Bellman (EHJB) equations and the solution is given in closed form in some special cases. We demonstrate the effects of higher order risk preferences and state-dependent risk aversion on the optimal asset allocation decisions. We find that wealth-dependent risk aversion facilitates risk taking and the skewness preference leads to a more positively skewed portfolio in certain circumstances.     

Risk-aversion concepts in expected- and non-expected-utility models

The non-expected-utility theories of decision under risk have favored the appearance of new notions of increasing risk like monotone increasing risk (based on the notion of comonotonic random variables) or new notions of risk aversion like aversion to monotone increasing risk, in better agreement with these new theories. After a survey of all the possible notions of increasing risk and of risk aversion and their intrinsic definitions, we show that contrary to expected-utility theory where all the notions of risk aversion have the same characterization (u concave), in the framework of rank-dependent expected utility (one of the most well known of the non-expectedutility models), the characterizations of all these notions of risk aversion are different. Moreover, we show that, even in the expected-utility framework, the new notion of monotone increasing risk can give better answers to some problems of comparative statics such as in portfolio choice or in partial insurance. This new notion also can suggest more intuitive approaches to inequalities measurement.     

Investment strategies in the long run with proportional transaction costs and a HARA utility function

We consider an agent who invests in a stock and a money market in order to maximize the asymptotic behaviour of expected utility of the portfolio market price in the presence of proportional transaction costs. The assumption that the portfolio market price is a geometric Brownian motion and the restriction to a utility function with hyperbolic absolute risk aversion (HARA) enable us to evaluate interval investment strategies. It is shown that the optimal interval strategy is also optimal among a wide family of strategies and that it is optimal also in a time changed model in the case of logarithmic utility.     

A Gentle Introduction to Risk Aversion and Utility Theory

While the topics of risk aversion and utility theory have been discussed extensively in the academic literature on risk and insurance, this literature does not include a pedagogical discussion that is widely accessible for classroom use. This article provides a practical introduction to risk aversion that is designed for readers with little prerequisite course work in economics or statistics. We describe a simple model of insurance demand that can be applied to the property, liability, life, and health insurance markets. We also demonstrate how risk aversion affects a variety of real-life insurance decisions made under conditions of uncertainty, including how much the market will bear to pay for insurance administrative expenses and how demand varies for different types of auto insurance coverage. Exercises and practice problems are provided so that readers can test their mastery of the concepts presented in the article. An instructional note on using this article to teach risk aversion in the classroom is also provided.