Risk aversion under preference uncertainty

We show that if an agent is uncertain about the precise form of his utility function, his actual relative risk aversion may depend on wealth even if he knows his utility function lies in the class of constant relative risk aversion (CRRA) utility functions. We illustrate the consequences of this result for optimal asset allocation: poor agents that are uncertain about their risk aversion parameter invest less in risky assets than wealthy investors with identical risk aversion uncertainty.     

Cross-hedging with futures and options: The effects of disappointment aversion

This article examines the effect of disappointment aversion on cross-hedging decisions. We show that, when both futures and options markets are unbiased, disappointment aversion has no effect on the optimal hedge positions. In case that either market is biased, disappointment aversion induces the hedger to behave more conservatively. In addition, as the hedger becomes more disappointment averse, his action is more reserved. It is also found that disappointment aversion tends to depress the importance of the put options whereas the effect of risk aversion is not uniform. Analytical predictions are supplemented by numerical exercises.     

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.     

Saving and investing for early retirement: A theoretical analysis

We study optimal consumption and portfolio choice in a framework where investors adjust their labor supply through an irreversible choice of their retirement time. We show that investing for early retirement tends to increase savings and reduce an agent's effective relative risk aversion, thus increasing her stock market exposure. Contrary to common intuition, an investor might find it optimal to increase the proportion of financial wealth held in stocks as she ages and accumulates assets, even when her income and the investment opportunity set are constant. The model predicts a decrease in risk aversion following strong market gains like those observed in the nineties.     

Pricing bounds and approximations for discrete arithmetic Asian options under time-changed Lévy processes

We derive efficient and accurate analytical pricing bounds and approximations for discrete arithmetic Asian options under time-changed Lévy processes. By extending the conditioning variable approach, we derive the lower bound on the Asian option price and construct an upper bound based on the sharp lower bound. We also consider the general partially exact and bounded (PEB) approximations, which include the sharp lower bound and partially conditional moment matching approximation as special cases. The PEB approximations are known to lie between a sharp lower bound and an upper bound. Our numerical tests show that the PEB approximations to discrete arithmetic Asian option prices can produce highly accurate approximations when compared to other approximation methods. Our proposed approximation methods can be readily applied to pricing Asian options under most common types of underlying asset price processes, like the Heston stochastic volatility model nested in the class of time-changed Lévy processes with the leverage effect.     

Asset pricing and energy consumption risk

This paper proposes energy consumption in the US as a new measure for the consumption capital asset pricing model. We find that (i) industrial energy growth produces reasonable values for the relative risk aversion coefficient and the implied risk-free rate; (ii) compared to alternative consumption measures, industrial energy performs well in explaining the cross-sectional variation in stock returns with the lowest implied risk aversion and pricing errors; (iii) the industrial energy consumption risk model performs equally well as the Fama–French three-factor model in the cross-sectional asset pricing tests; and (iv) total energy consumption risk is priced in the presence of the Fama–French factor risks.     

A Generalisation of the Mean-Variance Analysis

In this paper we consider a decision maker whose utility function has a kink at the reference point with different functions below and above this reference point. We also suppose that the decision maker generally distorts the objective probabilities. First we show that the expected utility function of this decision maker can be approximated by a function of mean and partial moments of distribution. This 'mean-partial moments' utility generalises not only mean-variance utility of Tobin and Markowitz, but also mean-semivariance utility of Markowitz. Then, in the spirit of Arrow and Pratt, we derive an expression for a risk premium when risk is small. Our analysis shows that a decision maker in this framework exhibits three types of aversions: aversion to loss, aversion to uncertainty in gains, and aversion to uncertainty in losses. Finally we present a solution to the optimal capital allocation problem and derive an expression for a portfolio performance measure which generalises the Sharpe and Sortino ratios. We demonstrate that in this framework the decision maker's skewness preferences have first-order impact on risk measurement even when the risk is small.     

Sampling distribution of the relative risk aversion estimator: Theory and applications

Brown and Gibbons (1985) developed a theory of relative risk aversion estimation in terms of average market rates of return and the variance of market rates of return. However, the exact sampling distributions of the relative risk aversion estimators have not been derived. The main purpose of this paper is to derive the exact sampling distribution of an appropriate relative risk aversion estimator. First, we have derived theoretically the density of Brown and Gibbons' maximum likelihood estimator. It is shown that the centralt is not appropriate for testing the significance of estimated relative risk aversion distribution. Then we derived the minimum variance unbiased estimator by a linear transformation of the Brown and Gibbons' maximum likelihood estimator. The density function is neither a central nor a noncentralt distribution. The density function of this new distribution has been tabulated. There is an empirical example to illustrate the application of this new sampling distribution.     

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.     

Robust portfolio choice with uncertainty about jump and diffusion risk

We analyze the portfolio planning problem of an ambiguity averse investor. The stock follows a jump-diffusion process. We find that there are pronounced differences between ambiguity aversion with respect to diffusion risk and jump risk. Ignoring ambiguity with respect to jump risk causes larger losses in an incomplete market, whereas ignoring ambiguity with respect to diffusion risk is more severe in a complete market. For a deterministic jump size we show that the loss from market incompleteness is always increasing in the level of ambiguity aversion with respect to one risk factor and decreasing in the level of ambiguity aversion with respect to the other risk factor.     

Does knowledge of finance mitigate the gender difference in financial risk-aversion?

We investigate the gender difference in financial risk aversion using a survey of finance professors from universities across the United States. We compare their actual portfolio allocations to that of respondents in the Federal Reserve's Survey of Consumer Finances (SCF). We find that among highly educated individuals, women are significantly more risk averse than men. However, we find that when men and women have both attained a high level of financial education, they are equally likely to invest a significant portion of their portfolio in risky assets, suggesting that financial education mitigates the gender difference in financial risk aversion.     

Markets versus governments

We provide a simple framework for comparing market allocations with government-regulated allocations. Governments can collect information about individuals’ types and enforce transfers across individuals. Markets (without significant government intervention) have to rely on transactions that are ex post beneficial for individuals. Consequently, governments achieve better risk sharing and consumption smoothing than markets. However, politicians in charge of collective decisions can use the centralized information and the enforcement power of government for their own benefits. This leads to political economy distortions and rents for politicians, making government-operated allocation mechanisms potentially worse than markets. We provide conditions under which it is ex ante beneficial for the society to tolerate the political economy distortions in exchange for the improvement in risk sharing. For example, more effective controls on politicians or higher discount factors of politicians make governments more attractive relative to markets. Moreover, when markets cannot engage in self-enforcing risk-sharing arrangements and income effects are limited, greater risk aversion and greater uncertainty make governments more attractive relative to markets. Nevertheless, we also show theoretically and numerically that the effect of risk aversion on the desirability of markets may be non-monotonic. In particular, when markets can support self-enforcing risk-sharing arrangements, a high degree of risk aversion improves the extent of risk sharing in markets and makes governments less necessary. The same pattern may also arise because of “income effects” on labor supply. Consequently, the welfare gains of governments relative to markets may have an inverse U-shape as a function of the degree of risk aversion of individuals.     

Optimal Life-Cycle Asset Allocation: Understanding the Empirical Evidence

We show that a life‐cycle model with realistically calibrated uninsurable labor income risk and moderate risk aversion can simultaneously match stock market participation rates and asset allocation decisions conditional on participation. The key ingredients of the model are Epstein–Zin preferences, a fixed stock market entry cost, and moderate heterogeneity in risk aversion. Households with low risk aversion smooth earnings shocks with a small buffer stock of assets, and consequently most of them (optimally) never invest in equities. Therefore, the marginal stockholders are (endogenously) more risk averse, and as a result they do not invest their portfolios fully in stocks.     

Generalized Disappointment Aversion and Asset Prices

We characterize generalized disappointment aversion (GDA) risk preferences that can overweight lower‐tail outcomes relative to expected utility. We show in an endowment economy that recursive utility with GDA risk preferences generates effective risk aversion that is countercyclical. This feature comes from endogenous variation in the probability of disappointment in the representative agent's intertemporal consumption‐saving problem that underlies the asset pricing model. The variation in effective risk aversion produces a large equity premium and a risk‐free rate that is procyclical and has low volatility in an economy with a simple autoregressive endowment‐growth process.     

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.     

Identity Theft and Consumer Payment Choice: Does Security Really Matter?

In their paper “Spectral Risk Measures: Properties and Limitations”, Dowd et al. (J Financ Serv Res 341:61–75, 2008) introduce exponential and power spectral risk measures as subclasses of spectral risk measures (SRMs) to the literature, and claim that they are subject to three serious limitations: First, for these subclasses, the spectral risk may be counterintuitively decreasing when the user’s risk aversion is increasing. Second, these subclasses, and power SRMs in particular, become completely insensitive to market volatility when the respective parameters of risk aversion tend to their lower and upper boundaries. Third, exponential SRMs exhibit constant absolute risk aversion, while constant relative risk aversion better meets the empirical evidence. Consequently, “users of spectral risk measures must be careful to select utility functions that fit the features of the particular problems they are dealing with, and should be especially careful when using power SRMs.” (p. 61). In this comment, we show that the findings of Dowd et al. (J Financ Serv Res 341:61–75, 2008) suffer from misinterpretations and wrong conclusions.     

A Shortcut Way of Pricing Default Risk Through Zero-Utility Principle

Pricing the default risk is a hot challenge for every risk manager. The problem is tackled in the framework of the zero‐utility principle. According to Pratt (1964) , an approximation of the risk premium should be proportional to the Arrow–Pratt absolute risk aversion coefficient and the variance. Is that still true as a default risk is concerned? The answer appears to be negative, because the variance does not look to be an appropriate tool for asymmetrical risk. On the other hand, fear of ruin coefficient and probability of default are proved to be well‐tailored tools for a preliminary pricing. Bid and ask price approximations are both elicited and a necessary condition for risk exchange set out.     

CONVEXITY,MAGNIFICATION, AND TRANSLATION: THE EFFECT OF MANAGERIAL OPTION‐BASED COMPENSATION ON CORPORATE CASH HOLDINGS

Using the distinctions among the convexity, magnification, and translation effects, we identify the pertinent parameters and examine empirically the relation between cash holdings and option‐based managerial compensation. We show that changes in delta reduce the effects of magnification and convexity on managerial risk aversion. We also provide evidence that there is a negative relation between the option‐based incentives delta and vega and cash holdings. These results are robust when incentives are extended to include all executive board members and when the sample is broken down according to different risk characteristics.     

How suboptimal are linear sharing rules?

The objective of this paper is to analyze criteria for portfolio choice when two investors are forced to invest in a common portfolio and share the proceeds by a linear sharing rule. A similar situation with many investors is typical for defined contribution pension schemes. The restriction implies two sources of suboptimal investment decisions as seen from each of the two investors individually. One is the suboptimal choice of portfolio, the other is the forced linear sharing rule. We measure the combined consequence for each investor by their respective loss in wealth equivalent. We show that significant losses can arise when investors are diverse in their risk attitude. We also show that an investor with a low degree of risk aversion, like the logarithmic or the square root investor, often applied in portfolio choice models, can either inflict or be subject to severe losses when being forced to participate in such a common investment pool.     

Auctions with selective entry and risk averse bidders: theory and evidence

We study auctions with selective entry and risk averse bidders. Our model accounts for risk averse bidders' endogenous participation decision and thus encompasses the existing entry models. We establish entry and bidding equilibrium in first‐price auction and ascending auction mechanisms and show that bidders' entry behavior differs between these two mechanisms with different forms of risk aversion. Our approach provides testable implications of risk aversion in terms of entry behavior. We analyze a timber auction data set and propose a simple test for the form of bidders' risk aversion based on our model implications.