Components of Market Risk and Return

This article proposes a flexible but parsimonious specification of the joint dynamics of market risk and return to produce forecasts of a time-varying market equity premium. Our parsimonious volatility model allows components to decay at different rates, generates mean-reverting forecasts, and allows variance targeting. These features contribute to realistic equity premium forecasts for the U.S. market over the 1840–2006 period. For example, the premium forecast was low in the mid-1990s but has recently increased. Although the market’s total conditional variance has a positive effect on returns, the smooth long-run component of volatility is more important for capturing the dynamics of the premium. This result is robust to univariate specifications that condition on either levels or logs of past realized volatility (RV), as well as to a new bivariate model of returns and RV. keywords: volatility components, long-run market risk premium, realized volatility The expected return on the market portfolio is an important input for many decisions in finance. For example, accurate measures or forecasts of the equity The authors thank the editor, René Garcia, two anonymous referees, Hakan Bal, Harjoat Bhamra, Chris Jones, Mark Kamstra, Raymond Kan, Benoit Perron, Daniel Smith, participants at the May 2005 CIREQ-CIRANO Financial Econometrics Conference, the 2005 NFA annual meetings, and seminar participants at Simon Fraser University and McMaster University for very helpful comments, as well as Bill Schwert for providing U.S. equity returns for the period 1802–1925. They are also grateful to SSHRC for financial support. An earlier draft of this paper was titled ‘‘The long-run relationship between market risk and return.’’ Address correspondence to Thomas H. McCurdy, Joseph L. Rotman School of Management, University of Toronto, 105 St. George Street, Toronto, ON M5S 3E6, 416-978-3425, and Associate Fellow, CIRANO, or e-mail: [email protected]. doi:10.1093/jjfinec/nbm012 Advance Access publication August 31, 2007 © The Author 2007. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. All rights reserved The online version of this article has been published under an open access model. Users are entitled to use, reproduce, disseminate, or display the open access version of this article for non-commercial purposes provided that: the original authorship is properly and fully attributed; the Journal and Oxford University Press are attributed as the original place of publication with the correct citation details given; if an article is subsequently reproduced or disseminated not in its entirety but only in part or as a derivative work this must be clearly indicated. For commercial re-use, please contact [email protected] Maheu & McCurdy Components of Market Risk and Return 561 premium are important for computing risk-adjusted discount rates, capital budgeting decisions involving the cost-of-equity capital, as well as optimal investment allocations. The simplest approach to measuring the market premium is to use the historical average market excess return. Unfortunately, this assumes that the premium is constant over time. If the premium is time varying, as asset pricing theory suggests, then a historical average will be sensitive to the time period used. For example, if the level of market risk were higher in some subperiods than others, then the average excess return will be sensitive to the subsample chosen. A better approach to estimating the premium is to directly incorporate the information governing changes in risk. For example, the Merton (1980) model implies that the market equity premium is a positive function of market risk, where risk is measured by the variance of the premium. Under certain conditions discussed in the next section, intertemporal asset pricing models (IAPM) reduce to a conditional version of Merton (1980). That is, if the conditional variance of the market portfolio return is larger, investors will demand a higher premium to compensate for the increase in risk.1 This positive risk–return relationship for the market portfolio has generated a large literature which investigates the empirical evidence. Historically, authors have found mixed evidence concerning the relationship between the expected return on the market and its conditional variance. In some cases a significant positive relationship is found, in others it is insignificant, and still others report it as being significantly negative.2 Recent empirical work investigating the relationship between market risk and return offers some resolution to the conflicting results in the early literature. Scruggs (1998) includes an additional risk factor implied by the model of Merton (1973), arguing that ignoring it in an empirical test of the risk–return relationship results in a misspecified model. Including a second factor, measured by long-term government bond returns, restores a positive relationship between expected return and risk. Campbell and Hentschel (1992), Guo and Whitelaw (2006) and Kim, et al. (2004) report a positive relationship between market risk and return when volatility feedbacks are incorporated. Using a latent VAR process to model the conditional mean and volatility of stock returns, Brandt and Kang (2004) find a negative conditional correlation between innovations to those conditional moments but a positive unconditional correlation due to significant lead–lag correlations. Lundblad (2007) reports a positive trade-off over a long time period, Engle and Lee (1999) find a positive relationship between return and the permanent volatility 1 There are many other models of asset premiums, many of which can be thought of as versions of a multi-factor approach, such as the three-factor model of Fama and French (1992), or the arbitrage pricing theory of Ross (1976). For example, Claus and Thomas (2001), Fama and French (2002), and Donaldson, et al. (2004) use earnings or dividend growth to estimate the market premium. Pastor, et al. (2007) use implied cost of capital as a measure of expected return. 2 Early examples include Campbell (1987), Engle, et al. (1987), French, et al. (1987), Chou (1988), Harvey (1989), Turner et al. (1989), Baillie and DeGennaro (1990), Glosten, et al. (1993) and Whitelaw (1994). Table 1 of Scruggs (1998) summarizes that empirical evidence. 562 Journal of Financial Econometrics component, and Ghysels, et al. (2005) find a positive trade-off using a mixed data sampling (MIDAS) approach to estimate variance. Pastor and Stambaugh (2001) find evidence of structural breaks in the risk–return relationship.3 Our article investigates a conditional version4 of the risk–return specification. We exploit improved measures of ex post variance and incorporate them into a new component forecasting model in order to implement a time-varying risk model of the equity premium. How do we achieve better forecasts of a time-varying market equity premium? First, we use a nonparametric measure of ex post variance, referred to as realized volatility (RV). Andersen and Bollerslev (1998) show that RV is considerably more accurate than traditional measures of ex post latent variance. Due to the data constraints of our long historical sample, in this article we construct annual RV using daily squared excess returns.5 Second, as in Andersen, et al. (2003), French, et al. (1987) and Maheu and McCurdy (2002), our volatility forecasts condition on past RV. Since RV is less noisy than traditional proxies for latent volatility, it is also a better information variable with which to forecast future volatility. Third, we propose a new volatility forecasting function which is based on exponential smoothing. Our model inherits the good performance of the popular exponential smoothing filter but allows for mean reversion of volatility forecasts and targeting of a well-defined long-run (unconditional) variance. This feature adds to the parsimony of our forecasting function, which is important in our case given the relatively low frequency data necessary to allow estimation over a long time period. It also allows for multiperiod forecasts. Fourth, motivated by the component-GARCH approach of Engle and Lee (1999) applied to squared returns, we extend our conditional variance specification, which conditions on past RV, to a component-forecasting model. This flexible conditioning function allows different decay rates for different volatility components. We also investigate whether or not total market risk or just some component of it is priced, that is, we allow our risk–return model to determine which components of the volatility best explain the dynamics of the equity risk premium. Finally, in one of our parameterizations, we generalize the univariate risk–return model for the market equity premium by estimating a bivariate stochastic specification of annual excess returns and the logarithm of RV. In this case, the conditional variance of excess returns is obtained as the conditional 3 There is also a literature which investigates a nonlinear relationship between market risk and return, for example, Pagan and Hong (1990), Backus and Gregory (1993), Whitelaw (2000) and Linton and Perron (2003). 4 As discussed below, our conditional parameterization is motivated by the IAPM of Campbell (1993) and Merton (1973), as well as the component model of Engle and Lee (1999). 5 Early empirical applications of this measure at low frequencies, for example, using daily squared returns to compute monthly volatility, included Poterba and Summers (1986), French, et al. (1987), Schwert (1989), Schwert and Seguin (1990) and Hsieh (1991). Maheu & McCurdy Components of Market Risk and Return 563 expectation of the RV process. Again, multiperiod forecasts are available from the assumed dynamics of the bivariate process. We focus on the dynamics of the premium over the 1840–2006 period. Our volatility specification, which only requires one parameter per volatility component, produces precise estimates of the risk–return relationship. The forecasts of a time-varying premium match important features of the data. For example, our Figure 9 shows how well our forecasts captured the declining equity premium in the mid-1990s. In summary, we use improved measures of volatility in a parsimonious forecasting model which allows components of volatility with different decay rates to be priced in a conditional risk–return model. This involves several new contributions. We introduce a new weighting function on past RV, and show how mean reversion can be imposed in the model to target the unconditional mean of RV. Building on Engle and Lee (1999), we focus on a multiple component formulation of our new-volatility forecasting function in order to allow components of volatility to decay at different rates and to investigate which component is priced. Exploiting our mean-reverting multiperiod variance forecasts, our models can generate multiperiod premium forecasts. We analyze a long, low-frequency dataset and show that our models produce realistic timevarying premium forecasts over the entire 1840–2006 time period. Our empirical results show that for 167 years of the U.S. equity market, there is a significant positive relationship between market risk and the market-wide equity premium. The equity premium varies considerably over time and confirms that the average excess return associated with subperiods can be misleading as a forecast. Nevertheless, long samples of historical information are useful as conditioning information and contribute to improved estimates of the time-varying market premium. In our two-component specifications of the conditional variance, one component tracks long-run moves in volatility while another captures the shortrun dynamics. The two-component conditional variance specification provides a superior variance forecast. Furthermore, it is the long-run component in the variance that provides a stronger risk–return relationship. The article is organized as follows. Section 1 introduces the models that motivate our empirical study, and discusses the importance of the measurement and modeling of the variance of market returns. Section 2 details our results on the significance of the risk–return relationship for several model specifications. We discuss the importance of volatility components, and the range of implied premiums that the models produce. Finally, Section 3 summarizes the results and future work. 1 THE RISK–RETURN MODEL