A Note on Heterogeneous Beliefs with CRRA Utilities

In this paper, expected utility, defined by a Taylor series expansion around expected wealth, is maximized. The coefficient of relative risk aversion (CRRA) that is commensurate with a 100% investment in the risky asset is simulated. The following parameters are varied: the riskless return, the market standard deviation, the market stock premium, and the skewness and the kurtosis of the risky return. Both the high extremes and the low extremes are considered. With these figures, the upper bound of the market CRRA is 3.021 and the lower bound is 0.466. Log utility, which corresponds to a CRRA of 1, is not excluded.

PurposeThis paper seeks to reconsider the Euler equation of the Consumption Capital Asset Pricing Model (CCAPM), to derive a regression‐based model to test it, and to present evidence that the model is consistent with reasonable values for the coefficient of relative risk aversion (CRRA). This runs contrary to the findings of the literature on the equity premium puzzle, but is in agreement with the literature that estimates the CRRA for the purpose of computing the social discount rate, and is in line with the research on labor supply. Tests based on General Method of Moments (GMM) for the same sample produce results that are extremely disparate and unstable. The paper aims to check and find support for the robustness of the regression‐based tests. Habit formation models are also to be evaluated with regression‐based and GMM tests. However, the validity of the regression‐based models depends critically on their functional forms.Design/methodology/approachThe paper presents empirical evidence that the conventional use of GMM fails because of four pathological features of GMM that are referred to under the general caption of “weak identification”. In addition to GMM, the paper employs linear regression analysis to test the CCAPM, and it is found that the regression residuals follow well‐behaved distributional properties, making valid all statistical inferences, while GMM estimates are highly unstable.FindingsFour unexpected findings are reported. The first is that the regression‐based models are consistent with reasonable values for the CRRA, i.e. estimates that are below 4. The second is that the regression‐based tests are robust, while the GMM‐based tests are not. The third is that regression‐based tests with habit formation depend crucially on the specification of the model. The fourth is that there is evidence that market stock returns are sensitive to both consumption and dividends. The author calls the latter “extra sensitivity of market stock returns”, and it is described as a new puzzle.Originality/valueThe regression‐based models of the CCAPM Euler equation are novel. The comparison between GMM and regression‐based models for the same sample is original. The regression‐based models with habit formation are new. The equity premium puzzle disappears because the estimates of the CRRA are reasonable. But another puzzle is documented, which is the “extra sensitivity of market stock returns” to consumption and dividends together.

Consumption and portfolio choice with time-inconsistent preferences

ABSTRACTFor homothetic time and state separable preferences, the coefficient of relative risk aversion (CRRA) is equal to the reciprocal of the elasticity of intertemporal substitution (EIS). This paper shows that, when the growth rate of consumption is i.i.d., asset pricing models based upon preferences in which the CRRA and the EIS are no longer linked do not have more explanatory power. Further, in these stochastic environments, estimates of the CRRA in the standard preferences are measures of the true CRRA and not the EIS. These results are fairly accurate descriptions of economies calibrated using United States annual data.

<p style="text-indent:20px;">This paper considers an optimal investment problem under CRRA utility with a borrowing constraint. We formulate it into a free boundary problem consisting of a fully nonlinear equation and a linear equation. We prove the existence and uniqueness of the classical solution and present the condition for the existence of the free boundary under a linear constraint on a borrowing rate. Furthermore, we prove that the free boundary is continuous and smooth when the relative risk aversion coefficient is sufficiently small.</p>

What do happiness and health satisfaction data tell us about relative risk aversion?

This paper shows that existing evidence on labor supply behavior places an upper bound on risk aversion in the expected utility model. I derive a formula for the coefficient of relative risk aversion (g) in terms of (1) the ratio of the income elasticity of labor supply to the wage elasticity and (2) the degree of complementarity between consumption and labor. I bound the degree of complementarity using data on consumption choices when labor supply varies randomly across states. Using labor supply elasticity estimates from thirty-three studies, I find a mean estimate of g = 1. I then show that generating g > 2 would require that wage increases cause sharper reductions in labor supply than estimated in any of the studies.

Individuals' risk preferences are estimated and employed in a variety of settings, notably including choices in financial, labor, and product markets. Recent work, especially in financial economics, provides estimates of individuals' coefficients of relative risk aversion (CRRA's) in excess of one, and often significantly higher. However, it can be shown that high CRRA's imply equally high values for the income elasticity of the value of a statistical life. Yet estimates of this elasticity, derived from labor and product markets, are in the range of 0.5 to 0.6. Furthermore, it turns out that even a CRRA below one is difficult to reconcile with these elasticity estimates. Thus, there appears to be an important (additional) anomaly involving individuals' risk-taking behavior in different market settings.

In the standard intertemporal specification of expected utility $$ U_t = (1 - \beta )E\left[ {\sum\limits_{t = 0}^\infty {\frac{{c_t^{1 - \sigma } }} {{1 - \sigma }}} \beta ^t } \right],$$ (1) the coefficient of relative risk aversion σ is the reciprocal of the rate of intertemporal substitution 1/σ. This has been suspected as a source of poor performance of the standard stochastic consumption model and the risk-premium puzzle in asset pricing 1. More generally, the problematic feature of expected utility applied to intertemporal settings is that its treatment of ‘gambling over time’ cannot distinguish risk aversion and intertemporal substitution 2.

I show existing evidence on labor supply behavior places an upper bound on risk aversion in the expected utility model. I derive a formula for the coefficient of relative risk aversion (γ) in terms of the ratio of the income elasticity of labor supply to wage elasticity and degree of complementarity between consumption and labor. I bound the degree of complementarity using data on consumption choices when labor supply varies across states. Using labor supply elasticity estimates, I find a mean estimate of [Formula: see text], then show generating γ > 2 requires that wage increases cause sharper labor supply reductions.

This paper investigates the testable restrictions on the time-series behavior of equity premia implied by a representative agent model whose state- and time-non separable preferences are subject to taste shocks. The model nests state- and time-separable preferences with and without taste shocks as special cases. Empirically, the linearized Euler equations are estimated through Kalman filtering, allowing for conditional heteroscedasticity via a common factor GARCH process. With or without conditional heteroscedasticity, (i) the hypothesis that preferences are separable cannot be rejected, (ii) taste shocks influences are statistically significant, and (iii) taste shocks yield reasonable estimates of the coefficient of relative risk aversion. This last result occurs because taste shocks reproduce the large observed equity premium by shifting weight away from consumption risk in favor to taste risk.

Optimal Diversification in the Presence of Parameter Uncertainty for a Risk Averse Investor

The effect of risk aversion on the optimal project resource rate

Rethinking the effect of risk aversion on the benefits of service innovations in public administration agencies

Risk aversion is an important research area in the field of agricultural economics in the last years. Creating effective and efficient risk management tools in an increasingly volatile economic and natural environment requires proper recognition of farmers’ behavior and attitudes towards risk. In this context, the main aim of the paper was to estimate farmers’ attitudes towards risk and identification of farm’s and farmer’s characteristics in dependency on risk aversion level. The assessment of farmers’ preferences towards risk was based on hypothetical games in a representative sample of 600 Polish farms—participants of Farm Accountancy Data Network (FADN). Based on the interviews with farmers, a relative risk aversion coefficient has been estimated. Results revealed that on average Polish farmers have quite a strong risk aversion. Their attitudes towards risk are strongly linked with their self-assessment regarding their way of making decisions under risk. Some relations between farmers’ risk aversion and perception of selected risk factors could also be observed. The results revealed that the application of specified risk management tools by farmers and their potential reaction to a significant income drop are related to risk aversion level.