Simulating the market coefficient of relative risk aversion

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.

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.

Individual Risk Attitudes: New Evidence from a Large, Representative, Experimentally-Validated Survey

A Note on Heterogeneous Beliefs with CRRA Utilities

Lucas (1987, 2003) estimates that the cost of economic fluctuations is low; a social planner would pay no more than 0.1% of (permanent) consumption to eliminate all future business cycle fluctuations. The current paper extends Lucas’ calculations by studying the costs of fluctuations arising from asset bubbles. We estimate two classes of costs: consumption volatility due to asset bubbles in a representative agent economy and consumption volatility that arises because households have heterogeneous exposure to the bubble assets. We show that the magnitude of welfare costs is primarily driven by the existence of heterogeneity. Our benchmark calibration implies that the asset bubbles of the last decade generated a social welfare cost equal to a permanent 3% reduction in the level of national consumption. If assets are held proportionately across the population, these welfare costs fall by an order of magnitude. Our calculations are sensitive to the details of the calibration, including the degree of balance sheet and trading heterogeneity, the coefficient of relative risk aversion, and the magnitude of the asset bubble. Our preferred specifications generate welfare costs ranging from 1% to 10% of (permanent) national consumption.

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.

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

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

13 - Heterogeneous Risk Tolerance and Time Preference

We consider the economy in which an agent faces, in addition to market risk, an additive independent background risk in consumption. In contrast to the Lucas (Econometrica 46:1429–1445, 1978) complete consumption insurance model, under plausible assumptions about the unconditional mean and variance of the agent’s subjective distribution of background risk the model with the additive independent background risk fits the historical average excess return on the US stock market with the coefficient of relative risk aversion (RRA) below five for the subsets of households designated as assetholders. The greater the size and/or the lower the expected value of background risk, the lower (compared to the Lucas (Econometrica 46:1429–1445, 1978) model) the value of the RRA coefficient needed for the model with background risk to match the historical average equity premium. Allowing for an extremely unlike large decrease in the agent’s consumption considerably decreases the required coefficient of RRA. It is concluded that the presence of the additive independent background risk in the consumption of assetholders can account for nearly 60 % of the historical average equity premium, hence rationalizing the equity premium puzzle of Mehra and Prescott (J Monet Econ 15:145–162, 1985). With RRA below five, the model with background risk is consistent with the historical average real interest rate if the agent has the subjective time discount factor lower than, but close to, 1. The findings are robust to the assumed type of background risk, the proxy for the market portfolio, and the threshold value in the definition of assetholders.

Optimal retirement consumption with a stochastic force of mortality

Can VAR Models Capture Regime Shifts in Asset Returns? A Long-Horizon Strategic Asset Allocation Perspective

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.