Consumption Risk over the Frequency Domain Sarantis Kalyvitis

In this paper we evaluate the Consumption Capital Asset Pricing Model by employing as an explanatory variable consumption risk over the frequency domain. We modify the standard two-step methodology (Fama and French, 1992) to account for the spectral properties of consumption risk and we nd that lower frequencies of consumption risk explain up to 98% of the cross-sectional variation of expected returns and that the equity premium puzzle is resolved. These results are robust to the de nitions of the variables, the sample span and the set of portfolios utilized, and the maturity of interest rates. JEL classi cation: G11, G12, C13. Keywords: C-CAPM, consumption risk, frequency domain, equity premium. Acknowledgements: We have bene ted from comments and suggestions by B. Candelon, A. Michaelides, S. Panageas and seminar participants at the 11th International Congress on Insurance, Mathematics and Economics in Piraeus, the 1st International Workshop on Economics and Finance in Tripolis, the 11th International Conference on Macroeconomic Analysis and International Finance in Crete, the VIII Workshop on Quantitative Finance in Venice, the University of Macedonia in Thessaloniki, the National University of Ireland, Maynooth, and the Athens University of Economics and Business. Part of the project was conducted when Kalyvitis was visiting the Institute of International Integration Studies, Trinity College Dublin, whose hospitality is gratefully acknowledged. The usual disclaimer applies. (corresponding author) Department of International and European Economic Studies, Athens University of Economics and Business, Patision Str 76, Athens 10434, Greece. e-mail: skalyvitis@aueb.gr. Tel-Fax: 0030 210 8203137. y Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli & Dimitriou str. , 18534, Greece. email: apano@unipi.gr. Tel: 0030 210 4142728. Fax: 0030 210 4142340. 1. Introduction A persistent puzzle in the macroeconomics and nance literature has been the failure of the Consumption Capital Asset Pricing model (C-CAPM), which measures risk by consumption beta, rst to explain empirically the di¤erences in expected stock returns by the variation in the covariance of consumption and returns, and second to provide plausible levels of risk aversion. However, several studies have emphasized that this puzzle is likely to emerge primarily at short horizons due for instance to habits, information delays, and transaction costs, and that short-run consumption should be replaced with a portfolio that exhibits higher correlations with long-run movements in consumption.1 Singleton (1990) has rst pointed out that the inclusion of consumption growth over longer time intervals improves the empirical t of the single-good asset pricing model. Brainard et al. (1991) have shown that the performance of the C-CAPM improves as the horizon increases, a nding con rmed by Daniel and Marshall (1997) who have found that at lower frequencies aggregate returns and consumption growth are more correlated and the behavior of the equity premium becomes less puzzling. In a series of papers, Parker (2001, 2003) and, in particular, Parker and Julliard (2005) have allowed for longterm consumption dynamics by focusing on the ultimate risk to consumption, de ned as the covariance between an assets return during a quarter and consumption growth over the quarter of the return and several following quarters, and have found that it explains the cross-sectional variation in returns surprisingly well, but also show that the equity premium puzzle persists and high levels of risk aversion are required to line up the model with the data. Bansal and Yaron (2004), Bansal et al. (2005), and Hansen et al. (2008) show that when consumption risk is measured by the covariance between long-run cashows from holding a security and longrun consumption growth in the economy, the di¤erences in consumption risk provide useful information about the expected return di¤erentials across assets. In this paper we assess the explanatory power of consumption risk over the frequency domain by performing within the C-CAPM a dynamic analysis of consumption risk over the spectrum rather than over the time domain. As pointed out by Granger and Hatanaka back in 1964, according to the spectral representation theorem a time series can be seen as the sum of waves of di¤erent periodicity and, hence, there is no reason to believe that economic variables should See, among others, Mankiw and Shapiro (1986), Grossman et al. (1987), Wheatley (1988), and Breeden et al. (1989). Mehra (2003) and Cochrane (2005) provide extensive surveys of the relevant literature.