Consumption Risk Over the Frequency Domain

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.

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.

Output gap and consumption risk on the cross-section of stock returns in Korea

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.

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.

New Goods and Asset Prices

This article, Part 1 of 2, reviews the classical origins, development, and tests of consumption-based asset pricing theory, focusing mainly on the first two decades from 1976 to 1998. Starting with the original consumption capital asset pricing model (CCAPM) derivations, we review both theory and subsequent tests and provide some new applications. The consumption aggregation theorem and CCAPM are derived, and optimal consumption and portfolio strategies are discussed. The term structure of interest rates is derived from the term structures for expected growth, volatility, and inflation. Time aggregation biases in consumption betas as well as the usefulness of the “consumption-mimicking portfolio” are also derived. In addition to various empirical tests, models and tests of limited participation in asset markets as well as models of incomplete markets are presented. When certain measurement issues are taken into account, the CCAPM performs better than the original CAPM and nearly as well as the Fama-French three-factor model.

The purpose of this paper is to examine the theoretical relationship between the multidimensionality of risk and dividend policy, in an intertemporal context. After assuming that dividends are generated by a multifactor process, we use the fundamental framework of the consumption capital asset pricing model to explore the effect of long-run risk on dividend payout ratios (dividends divided by earnings). Our approach is similar to any multifactor model that, given the N factor process, derives useful equilibrium conditions. Our main result shows that the dividend payout ratio is negatively related to N sensitive coefficients, given by the long-run covariance between dividends and economic factors. This suggests that the multidimensionality of long-run consumption risk influences dividend policy. In brief, the model proposes that the target payout ratio can be determined with a simple and easy-to-apply formula that takes into account the long-run sensitivity of dividends to various economic factors.

We propose a novel consumption measure that has a daily frequency and is based on real time shopping data. Our measure explains the joint equity-premium‚ risk-free rate puzzle with a risk aversion coefficient much lower than any other consumption measures. It explains the cross-sectional variation of expected returns on various portfolios and is the only consumption measure that passes Kleibergen and Zhan (Journal of Finance, 2020) robust tests. Our model decomposes consumption shocks into different frequencies of volatility and shows that ignoring short-term dynamics and intra-annual fluctuations explains the much higher risk aversion from low-frequency consumption measures. At zip-code level daily consumption, (a) consumption in blue areas suggests higher risk aversion than that in red areas; (b) only Democratic consumption beta explains a variation of cross-sectional returns, and is more sensitive to overall industry performance.

Many of the empirical rejections of the consumption CAPM can be explained by the fact that the marginal rate of substitution between present and future consumption, which in the standard model functions as the pricing kernel for all assets for which a consumer is not at a corner, seems to vary too little to be consistent with sensible values of the parameters. Moreover, when one considers more than one asset at a time, one typically gets strong rejections of the overidentifying restrictions implied by the model. The failure seems to be both in terms of unconditional and conditional moments. Two recent papers Attanasio et al., 2002 [henceforth ABT]; Vissing-Jorgensen, 2002 [henceforth VJ] have shown that, if one focuses on the consumption of individuals participating in the stock market, one does not reject some implications of the model. In particular, both VJ and ABT find that, using the consumption of stockholders, conditional Euler equations lead to sensible preference parameters and, in the case of ABT, fail to reject the overidentifying restrictions even when considering two assets (stocks and bonds) at the same time. While these results constitute a first empirical success, they do not necessarily constitute a solution to the equity premium puzzle. As argued by Robert E. Hall (1988) and Attanasio and Guglielmo Weber (1989), the estimates of the coefficient on the interest rate in a loglinearized Euler equation should be interpreted as the elasticity of intertemporal substitution (EIS) and can only be informative about the degree of risk-aversion under more restrictive assumptions (CRRA utility). In this paper we argue that considering the consumption growth of stockholders and two asset returns not only yields values of the EIS that are plausible, but also helps explain the equity premium puzzle. The evidence we have on the consumption, income, and portfolios of stockholders is consistent with fairly plausible values of the coefficient of relative risk-aversion when using the preference specification of Larry G. Epstein and Stanley E. Zin (1991). We use three Euler equations, one for each of the two assets considered, and one for the household’s total wealth portfolio, whose return will be denoted as Rm. Rm is, in principle, an unobservable variable in that it depends on the returns on all assets an individual consumer holds, including human capital. In this paper, we specify the conditional expected return to human capital as a linear function of the conditional expected returns to stocks and bonds. We suggest a novel approach to estimating the coefficients in this function based on conditional Euler equations for stockholders with two asset returns on the right-hand side. Then, considering unconditional moments, we estimate the risk-aversion of stockholders using the log-linearized equation for the equity premium from Epstein-Zin utility also explored by John Y. Campbell (1996). Unlike Campbell, our preferred approach does not substitute out consumption. Furthermore, we emphasize the importance of allowing for bonds in households’ portfolios as well as not restricting the conditional expected human-capital return to equal that of stocks.

A New Perspective on the Equity Premium Puzzle

This paper analyzes asset prices in a representative agent exchange economy with habit-forming preferences. For a general class of utility indices and endowment processes, the authors characterize the optimal demand for consumption and derive explicit solutions for the interest rate and asset risk premia. They show that consumption smoothness may obtain even when the interest rate is stochastic. The consumption capital asset pricing model may not hold when the endowment process has stochastic coefficients; asset risk premia are larger under mild assumptions. The interest rate depends on the growth in the standard of living. Malliavin calculus is employed in the analysis. Copyright 1991 by The Econometric Society.

I. INTRODUCTION The consumption-based capital asset pricing model (CCAPM) has become the dominant paradigm in asset pricing theory because of its intuitive characterization of asset risk. That is, unlike expectations-based asset pricing theories (e.g. the present value model of stock prices) which inherently assume risk neutrality, the CCAPM is predicated upon risk-averse agents attempting to achieve a smooth consumption profile through the trading of assets. From this perspective, the risk of an asset is captured by the covariance of the asset's return with agents' marginal utility of consumption, or, more broadly, agents' intertemporal marginal rate of substitution of wealth. With this characterization, the CCAPM extends the results of the standard CAPM, i.e. the risk of an asset is reflected in the covariance of the asset's return with the market portfolio, to a general equilibrium setting. However, this intuitively appealing theory has not found much empirical support. Perhaps the most famous demonstration of inconsistency between the theory and data was Rajnish Mehra and Edward Prescott's [1985] analysis of the equity premium, i.e. the difference between the average return on equity and that on government bonds. Using calibration methods rather than a formal statistical approach, they showed that a standard CCAPM severely underpredicts the size of the risk premium on equity. The important implication of this finding is that the basic CCAPM incorrectly models the covariance structure between the return on equity and asset holders' intertemporal marginal rate of substitution. Economists' response to this empirical rejection has been to either identify new sources of risk (e.g., that implied by a particular monetary environment as in Labadie [1989]) or define alternative preferences toward risk (e.g., the use of non-expected utility theory as presented in Weil [1989] and Epstein and Zin [1989]). The potential for this research was illustrated recently by George Constantinides's [1990] resolution of the equity premium puzzle in a model that included habit formation in consumption. By relaxing the standard CCAPM's assumption of time-separable preferences, Constantinides demonstrated that a relatively smooth consumption profile and large risk premium on equity are consistent with equilibrium in a complete markets, rational-expectations framework.(1) I use this insight to study another risk premium puzzle; specifically the behavior of the nominal term premium. The existence of a nominal term premium puzzle was demonstrated in papers by Backus, Gregory, and Zin [1989] and Salyer [1990] in which the term structure implications within a Lucas [1982] type cash-in-advance (CIA) economy were studied. While some equilibrium characteristics of this model were consistent with the data, both papers demonstrated that the model predicts the wrong sign of the term premium: the theory implies the term premium should be negative while, based on quarterly data for yields of three- and six-month U.S. Treasury Bills over the period 1959.1-1989.4, the observed term premium is positive. Similar to the equity premium puzzle as posed by Mehra and Prescott [1985], the prediction of a negative average term premium signals a critical discrepancy between the risk associated with nominal bonds as perceived by market participants and as captured by the models. Moreover, this term premium puzzle is in some sense a stronger rejection of the underlying asset pricing model than that embodied by the equity premium puzzle. To see this, recall that within the consumption-based asset pricing model, the sign of the risk premium on any asset depends on the sign of the covariance between the marginal utility of consumption and that asset's return. Applying this reasoning to the case of the equity premium suggests that, while the discrepancy in size between the observed equity premium and that generated within the model is troubling, the model's prediction of a positive equity premium implies that the basic payout structure of equity within the model is consistent with that in the data (e. …

This article presents a pure exchange economy that extends Rubinstein [Bell J. Econ. Manage. Sci., 1976, 7, 407–425] to show how the jump-diffusion option pricing model of Black and Scholes [J. Political Econ., 1973, 81, 637–654] and Merton [J. Financ. Econ., 1976, 4, 125–144] evolves in gamma jumping economies. From empirical analysis and theoretical study, both the aggregate consumption and the stock price are unknown in determining jumping times. By using the pricing kernel, we determine both the aggregate consumption jump time and the stock price jump time from the equilibrium interest rate and CCAPM (Consumption Capital Asset Pricing Model). Our general jump-diffusion option pricing model gives an explicit formula for how the jump process and the jump times alter the pricing. This innovation with predictable jump times enhances our analysis of the expected stock return in equilibrium and of hedging jump risks for jump-diffusion economies.

We augment the standard Consumption Capital Asset Pricing Model (CCAPM) by the growth in money holdings and empirically investigate whether money is helpful for pricing a cross-section of US excess returns. We find that the growth in M2 significantly improves the fit of the CCAPM with R 2s well above 80% in a cross-section with the three Fama–French factors, the momentum portfolio, a contrarian portfolio and two bond portfolios as test assets.