Journal of Financial Econometrics Advance Access originally published online on August 31, 2007
Journal of Financial Econometrics 2007 5(4):560-590; doi:10.1093/jjfinec/nbm012
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Components of Market Risk and Return
University of Toronto
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: tmccurdy{at}rotman.utoronto.ca
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Abstract |
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 Top Abstract Introduction1 The Risk-Return Model2 Results3 Summary and Future...ReferencesFootnotes |
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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
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Introduction |
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TopAbstractIntroduction1 The Risk-Return Model2 Results3 Summary and Future...ReferencesFootnotes |
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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 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 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 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 time-varying 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 short-run 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.
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1 The Risk–Return Model |
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TopAbstractIntroduction1 The Risk-Return Model2 Results3 Summary and Future...ReferencesFootnotes |
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1.1 Background
Both static and intertemporal models of asset pricing imply a risk–return relationship. Examples of intertemporal models which do not require consumption data are the IAPM proposed by Merton (1973
) and Campbell (1993), and also the conditional capital asset pricing model(CAPM).
The IAPM of Merton (1973) relates the expected market return and variance through a representative agent's coefficient of relative risk aversion and also allows sensitivity of the market premium to a vector of state variables (or hedge portfolios) which capture changing investment opportunities. Under some assumptions, the intertemporal model implies a market risk–return relationship with no additional factors, that is, market risk is captured by the variance of the market portfolio. Merton (1980) argues that this case will be a close approximation to the intertemporal asset pricing model in Merton (1973) if either the variance of the change in wealth is much larger than the variance of the change in the other factor(s), or if the change in consumption in response to a change in wealth is much larger than that associated with a change in other state variable(s). Sufficient conditions are if the investment opportunity set is essentially constant, or if the representative investor has logarithmic utility.
Campbell (1993) provides a discrete-time intertemporal model which substitutes out consumption. In this case, the expected market premium is a function of its variance as well as its covariance with news (revisions in expectations) about future returns on the market. As in Merton (1973), if the coefficient of relative risk aversion is equal to 1 or if the investment opportunity set is constant or uncorrelated with news about future market returns, the expected market premium will only be a function of the market return variance. However, the Campbell (1993) derivation provides an alternative, empirically plausible, condition under which that market risk–return relationship obtains. If the covariance of the market return with news about future investment opportunities is proportional to the variance of the market return, then the latter will be a sufficient statistic for market risk.6 Section III of Campbell (1993) provides conditions that produce this conditional market risk–return relationship.
This motivates a risk–return model
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| (1) |
is the conditional variance of the market excess return (that is, the forecast of the variance conditional on time t – 1 information). This model implies a proportional relationship between the market equity premium and its conditional variance. As discussed in introduction above, Merton (1980) argues that one should impose the prior that the expected equity premium is nonnegative. If Equation (1) were the risk–return model, this would be satisfied as long as 
1 > 0.
1.2 Measuring and Forecasting Volatility
In this section, we discuss how we measure and then forecast the volatility which drives the time-varying risk premiums. Note that, throughout the article, we use the term volatility to refer generically to either the variance or standard deviation. Where necessary for clarity, we refer specifically to whether it is an ex post (realized) measure or a conditional estimate (forecast); and whether we are referring to a variance or a standard deviation. For ease of notation, we also drop the subscript M on the market excess return and its conditional variance so that henceforth rt 
rM, t and 
.
1.2.1 Measuring volatility
In this article, we employ a nonparametric measure of volatility. A traditional proxy for ex post latent volatility has been squared returns or squared residuals from a regression model. As shown by Andersen and Bollerslev (1998), this measure of volatility is very noisy and of limited use in assessing features of volatility such as its time-series properties.
Better measures of ex post latent volatility are available. In this article, we use a measure of ex post variance, termed (RV), developed in a series of papers by Andersen, Bollerslev, Diebold and co-authors, and Barndorff-Nielsen and Shephard. The increment of quadratic variation is a natural measure of ex post variance over a time interval. RV is computed as the sum of squared returns over this time interval. As shown by Andersen, et al. (2001b), as the sampling frequency is increased, the sum of squared returns converges to the quadratic variation over a fixed time interval for a broad class of models. Thus RV is a consistent estimate of ex post variance for that period. The asymptotic distribution of RV has been studied by Barndorff-Nielsen and Shephard (2002b) who provide conditions under which RV is also an unbiased estimate. Recent reviews of this growing literature are by Andersen, et al. (2004) and Barndorff-Nielsen, et al. (2004).
Defining RV of the market return for year t as RVt, we construct annual RV as
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| (2) |
1.2.2 Forecasting volatility
Our time-varying risk model of the equity premium is forward looking. That is, the expected market equity premium is a function of market equity risk. According to our test equations, the latter is measured by the conditional variance of market excess returns. Therefore, we need a forecast of the time t volatility, conditional on information at time t – 1. Our volatility forecasts condition on past RV. Given that RV has a superior signal-to-noise ratio for measuring latent volatility, it should be a superior conditioning variable for forecasting future volatility.
What functional form should be used to summarize information in past RV? Given our objective to model risk and return at annual frequencies, parsimony is an obvious objective. One candidate is the infinite exponential smoothing function. In this case, estimates can be derived from the recursion
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| (3) |
< 1, is the smoothing parameter, and 
is the conditional variance. A small value of puts more weight on the most recent observable value of RV, that is RVt–1, and less weight on the past forecast 
. Conversely, an close to 1 puts less weight on recent observations and more weight on past forecasts which smooths the data. This model is analogous to the popular RiskMetrics filter, except for the fact that the standard practice is to smooth on lagged squared returns rather than on lagged RV. The recursion in Equation (3) implies the following weighting function on past RV,
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| (4) |
Although infinite exponential smoothing provides parsimonious estimates, it possesses several drawbacks. Given our long time period and low sampling frequency, we prefer to estimate the smoothing parameter rather than taking it as given as is common practice with filtering approaches such as RiskMetrics. The smoothing parameter can be estimated using an objective criterion, such as maximum likelihood. However, as Nelson (1990) has shown in the case of squared returns or squared innovations to returns, filters with no intercept and whose weights sum to one are degenerate in the asymptotic limit in the sense that the distribution of the variance process collapses to zero variance asymptotically. This is problematic for maximum-likelihood estimation of the smoothing parameter. To circumvent these problems, but still retain the parsimony and accuracy of exponential smoothing, we propose the following volatility specification,
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| (5) |
.7 In this specification the weights sum to less than one allowing mean reversion in volatility forecasts as shown below.
Based on Corollary 1 of Andersen, et al. (2003), we assume that the conditional expectation of annual quadratic variation (QVt) is equal to the conditional variance of annual returns,8 that is, 
.9 Assuming that RV is a unbiased estimator of quadratic variation it follows that 
, and we can derive the unconditional variance associated with specification (5) as10
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| (6) |
2 from the data and using
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| (7) |
In summary, this new specification is similar in spirit to exponential smoothing but allows for mean reversion in volatility forecasts. In addition, the finite unconditional variance allows for variance targeting which means that only one parameter needs to be estimated. Our specification is also more parsimonious than the covariance-stationary GARCH(1,1) model.11 As discussed later in the text, at least for our sample of annual data the more parsimonious specification is critical for precision of the estimates of the risk–return relationship and also for generating reasonable premium estimates.
As is evident from Equation (5), past data receive an exponentially declining weight. As we will see below, this is not flexible enough to capture the time-series dynamics of RV. A simple approach to providing a more flexible model is to allow different components of volatility to decay at different rates.12 This can be achieved with a component volatility function which estimates the conditional variance as the average of two or more components. Formally, define the k-component volatility model as
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| (8) |
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| (9) |
Related work on volatility modeling includes the component model of Engle and Lee (1999) and Ghysels, et al. (2005). Relative to component-GARCH models, our parameterization only requires 1 parameter per component rather than two. Another difference is that we smooth on past annual RV. Ghysels, et al. (2005) use a MIDAS approach to estimate volatility. In that paper, using data from 1928 to 2000, the monthly conditional variance of returns is modeled using a flexible functional form to estimate the weight given to each lagged daily squared return. They find that a two- parameter filter works well.13 Our decay rates are additive. That is, in our case with two components, the coefficient on RVt–j–1 is 
, allowing us to separate out and price short-lived versus slower-decaying components. In contrast, the smoothing coefficients in Ghysels, et al. (2005) interact (multiplicatively) in a way that makes it difficult to isolate their separate effects on pricing.14
Our conditional variance specification maintains the parsimony of smoothing models but allows mean reversion. This allows us to use variance targeting which may be important to gain precision in our application. In the next section, we extend the existing literature to investigate a bivariate risk–return specification. This joint stochastic specification of returns and RV allows for multiperiod forecasts of the premium.
Our objective is to have a parsimonious and flexible function that summarizes information in past RV that might be useful for forecasting changes in the market equity risk premium. We allow for alternative components of volatility with different decay rates. Not only is this a more flexible way to capture the time-series dynamics of volatility, but it also allows us to investigate whether a particular component, rather than the full conditional variance, is more important in driving the market premium.
1.3 The Empirical Risk–Return Models
As discussed in Section 1.1, our empirical models based on Equation (1) are motivated as special cases of an IAPM. Each of the empirical models implies a time-varying equity premium which is a function of its own conditional second moment, that is, a forecast of the equity premium's time t variance conditional on time t – 1 information.
The conventional univariate test equation for (1) is a linear model,
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| (10) |
) motivates the addition of an intercept to account for market imperfections, such as differential tax treatment of equity versus Treasury-bill returns, which might account for a constant equity premium unrelated to risk. In this subsection we introduce two alternative empirical specifications of the risk–return relationship. Each of our models jointly estimate the conditional mean and conditional variance parameters using maximum likelihood. We label the first specification univariate since it fits the stochastic excess return process by conditioning on variance forecasts which are estimated using a projection on past RV as in Equation (8).15 The second specification is bivariate since we estimate a bivariate stochastic specification of annual excess returns and log(RV). In that case, the conditional variance of excess returns is obtained as the conditional expectation of the RV process.
1.3.1 Univariate risk–return specifications
The conditional mean is,
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| (11) |
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| (12) |
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| (13) |
in the conditional mean, in which q denotes the number of volatility components that affect the market premium when the conditional variance follows a k-component model. Note that qk. For example, where we want the total variance to enter the mean, we set q = k. On the other hand, for models in which q = 1, k = 2, we let the maximum likelihood estimator determine which component of the variance is optimal in the mean specification.
The tth contribution to the loglikelihood is
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| (14) |
depends on the levels or log specification. The loglikelihood 
is maximized with respect to the parameters 0, 1, 1..., k.
1.3.2 Bivariate risk–return specification
In this case, we estimate a bivariate stochastic specification of annual excess returns and log(RV). The parameterization has conditional mean
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| (15) |
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| (16) |
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| (17) |
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| (18) |
Again, 
, represents the conditional variance component(s) that affect the conditional mean. lt consists of contributions from the return and volatility equation,
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| (19) |
In this case, by parameterizing the joint density of annual excess returns and log(RV), we explicitly model the annual log(RV) process as stochastic given the most recent information. However, making a conditional log-normal assumption for RV allows for a convenient calculation of the conditional variance as in Equation (16).
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2 Results |
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TopAbstractIntroduction1 The Risk-Return Model2 Results3 Summary and Future...ReferencesFootnotes |
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2.1 Data and Descriptive Statistics
We are evaluating the risk–return relationship associated with equity for the market as a whole. Therefore, we require data on a broad equity market index and on a riskfree security so that we can construct returns on the equity index in excess of the riskfree rate.
There are two considerations. First, as shown by Merton (1980) for i.i.d. returns, we can only increase the precision of expected return estimates by increasing the length of the time period (calendar span) covered by our historical sample. In other words, sampling more frequently (for example, collecting daily instead of monthly data) will not improve the precision of the market premium estimates. For this reason we want our historical sample to be as long as possible. Second, since we use a nonparametric measure of RV which is constructed as the sum of squared intraperiod returns, we will need data at a higher frequency than our estimation frequency in order to compute RV as in Equation (2). As noted in Section 1.2.1 above, we use monthly returns from 1802–1885 and daily returns from 1886–2006 to compute annual RV.
The U.S. market equity returns from 1802–1925 are from Schwert (1990). The Schwert equity index was constructed from various sources: railway and financial companies up to 1862 from Smith and Cole (1935); the Macaulay (1938) railway index (1863–1870); the value-weighted market index published by the Cowles Commission for the time period 1871–1885; and the Dow Jones index of industrial and railway stocks for 1885–1925.16 For the 1926–2006 period, we use returns (including distributions) for the value-weighted portfolio of NYSE, NASDAQ and AMEX stocks compiled by the Center for Research in Security Prices (CRSP). The equity return data are converted to continuously compounded returns by taking the natural logarithm of one plus the monthly or daily equity return.
Annual bill yield data (riskfree rate) for the 1802–1925 period are from Jeremy Siegel. Siegel (1992) describes how these annual yields were constructed to remove the very variable risk premiums on commercial paper rates in the 19th century.17 For the period 1926–2006, we use monthly bid yields associated with U.S. 3-month Treasury Bills from the Fama monthly riskfree file provided by CRSP. The annual riskfree yield was obtained by summing the 12 monthly yields.
Annual continuously compounded equity returns are computed as the sum of the monthly continuously compounded returns. Annual excess returns are then computed by subtracting the annual riskfree rate. Henceforth, unless otherwise indicated, returns and excess returns refer to continuously compounded rates.18
Figure 1 plots annual market excess returns for the entire sample from 1803–2006. Figures 2 and 3 plot two alternative measures of ex post volatility, the absolute value of the excess returns and the square root of RV, respectively. Notice how much smoother the square root of RV is than the absolute value of annual excess returns. It is also clear from these plots that the data for the period 1803–1834 have a very different structure than that for the remainder of the sample. As noted in Schwert (1990), data from only a small number of companies was available for that subperiod. Both Schwert (1989) and Pastor and Stambaugh (2001) drop this period. For the same reason, our analyses focus on the time period 1840–2006. Starting in 1840 provides presample values to condition our time-varying volatility model.
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.Table 1 reports summary statistics for the time period, 1840–2006. The average excess return is 4.27%.
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2.2 Univariate Risk–Return Results
As discussed in Section 1.3.1, the univariate specifications fit the stochastic excess return process by conditioning on variance forecasts which are estimated using a projection on past RV. We report results using both the levels of RV, as in Equations (8) and (11), and the natural logarithm of RV, as in Equations (12) and (11).19 Due to the importance of the length of the time period necessary for efficient conditional mean forecasts, we use all of our data and thus forecasts are in-sample. See Inoue and Kilian (2004
) for evidence in favor of using in-sample forecasts for model evaluation.
2.2.1 Using RV levels
Table 2 reports parameter estimates and likelihood ratio test (LRT) results associated with the risk–return relationship estimated using the alternative component volatility models applied to levels of RV. Note that 
is computed, as in Equation (7). To avoid influential observations in our small sample, we set the long-run target volatility, 2, to the sample median of annual RV over the period 1802–2006. For each model, U(1), U(2) and U(2s), we report both the linear parameterization of the risk–return relationship, Equation (11), and the proportional parameterization with 0 = 0, as motivated by Equation (1).20 Models U(2) and U(2s) have the same volatility specification but U(2s) allows the risk–return model to determine which volatility component has the most explanatory power for the dynamics of the equity premium. We find that, when given the choice, the maximum likelihood estimator always chooses to price the long-run or smooth component.
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It is clear from Figure 4 that the two-component volatility specification tracks RV better than the one-component version.21 Figure 5 plots the individual components of volatility from the U(2) specification, that is, Equation (9) for k = 1, 2. This shows that the smooth component is persistent, as expected from the smoothing coefficient estimate
reported in Table 2. Note in particular how long it takes the smooth component to decay from its high level in the 1930s. On the other hand, the second component has much lower persistence, 
, which implies that it is more influenced by the most recent RV.
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): U(2) model.Table 2 reveals that the risk–return relationship is positive for all specifications. For linear parameterizations of the premium, the LRT results reveal a statistically significant relationship between excess market returns and their conditional variance, although it is marginal (p-value = 0.06) for the U(2) case. For these linear parameterizations, the t-statistic on the risk–return slope coefficient,
1, is highest (2.624) for the U(2s) model. That version also has the highest log-likelihood, 41.64, and a LRT statistic of 7.54 which rejects the hypothesis that 1 = 0.
However, for all volatility specifications, the more parsimonious proportional premium model cannot be rejected by the data. That is, the intercept in the conditional mean is statistically insignificant, both from the perspective of t-tests on 0 and from the LRT reported in the second-last column of Table 2. As noted above, by restricting the intercept to be zero, the precision of the estimate 
improves. For instance, the t-statistics associated with the hypothesis 1 = 0 increase, as do the LRT statistics which are more than twice as large for the proportional as opposed to the linear model. The one exception is the U(2s) model which reveals a strongly positive market risk–return relationship for both the linear and the proportional parameterization of the premium.
The LRT statistics in the final column of the table all reject the null hypothesis that 1 = 0. This result is particularly strong for the proportional parameterizaton of the risk premium in which case the test statistics are all greater than 12 so that the p-value associated with the null hypothesis is very small. The t-statistic for 1 also supports this conclusion in that it is greater than 3.5 for the proportional model for all of the alternative volatility specifications.
Figure 6, plots the market equity premium forecast for the proportional risk premium parameterization. When total volatility from the two-component volatility specification is priced (the U(2) model), there is a high peak in the forecasted market equity premium which lasts for 2 years, 1933 and 1934. In contrast, pricing the smooth component of volatility (the U(2s) model) results in a lower peak which decays more slowly. In this case, the premium forecast ranges from 1.7% to 9.2% with the average forecasted premium over the 1840–2006 sample equal to 4.29 percent.
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Table 3 reports model diagnostics and statistics for volatility and premium fit. These include mean absolute error (MAE) and root mean squared error (RMSE), as well as the R2 from Mincer and Zarnowitz (1969
) forecast regressions for a particular model's premium and volatility forecasts.22 The results in Table 3 support the two-component models U(2) and U(2s) over the more restrictive one-component version U(1). For example, as suggested by Figure 4, the R2 increases from about 7% to 26% by adding a second volatility component. Note that the U(2s) model, which had the highest log-likelihood, also has the best premium fit whether measured from the perspective of MAE, RMSE, or R2.
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2.2.2 Using Log(RV)
We also fit the univariate risk–return model (11) using variance forecasts which are estimated from a projection on the natural logarithm (rather than levels) of past RV, that is, Equations (12) and (13). Tables 4 and 5 report these results. Again,
is computed to target long-run volatility, for which log 2 is set to the 1802–2006 average of log(RV).23
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The parameter estimates using log(RV), reported in Table 4, reveal that the risk–return relationship is again positive for all specifications and, with two exceptions, is very significant—even more so than when projecting on the levels of past RV. The two exceptions, with p-values of 0.08 and 0.05 respectively for the LR test of the restriction
1 = 0, are the linear parameterizations of the premium with one-component and two-component volatility specifications. However, the proportional parameterizations of the premium do result in very statistically significant positive risk–return relationships for all cases. The LRT statistics in the final column range from 16.00 to 18.42, even larger than in Table 2. The t-statistics associated with the risk–return slope parameter estimates, 
, range from 3.97 to 4.15 in this case. Table 5 shows that the two-component parameterizations of volatility dominate the one-component version from the perspective of all three criteria, MAE, RMSE, or R2, for volatility fit. As in the levels case, the Ulog(2s) model, which prices the smooth component of volatility, has the best overall fit.
Figure 8 shows that the smooth proportional premium forecasts using the univariate model that conditions on log(RV), that is, the Ulog(2s) model, range from 1.6% to 9.5% with an average of 4.33% over the period 1840 to 2006. This figure also shows that these premium forecasts are similar to the univariate case in which volatility forecasts were estimated from past levels of RV rather than log(RV), except that in the log(RV) case the peaks are slightly higher with faster decay.24
2.3 Bivariate Risk–Return Results
In this section, we generalize our risk–return model for the market equity premium by estimating a joint stochastic specification of the conditional mean and annual log(RV) of market excess returns. The logarithmic specification of RV ensures that volatility is nonnegative during estimation of the joint stochastic process. The system of test equations is (15) and (16) to (18); and the results are summarized in Tables 6 and 7, as well as Figures 7 to 8.
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): B(2) model.
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The results are very comparable to the univariate results reported above. Figure 7 plots the individual components of log volatility from the B(2) specification, that is, Equation (18) for k = 1, 2. Table 7 shows that the two-component volatility specification fits log(RV) better. For example, the R2 for volatility fit increases from about 23% to 28% by adding the second volatility component.
Table 6 reports that the risk–return relationship is positive for all models and statistically significant for all of the models except two. As in the log(RV) univariate case, the t-statistic associated with 1 is smaller for the linear risk premium parameterizations B(1) and B(2). However, since we are unable to reject that 0 = 0, either by t-tests or by the LR tests reported in the second-last column of the table, the data support a proportional parameterization of the risk premium. In that case, the risk–return relationship is significantly positive for all volatility specifications. Again, as in Tables 2 and 4, the LRT results in the final column indicate that the relationship between excess market returns and their conditional variance is very strong for proportional parameterizations of the risk premium.
Table 7 reports that the preferred model is, once again, the 2s specification, that is, two volatility components with the smooth component being priced in the conditional mean.25 As in the univariate cases, all of the premium fit statistics, MAE, RMSE, or R2, support this conclusion; as does the log likelihood which is –134.30 for the B(2s) model, as opposed to –143.72 for the B(1) model which has a one-component volatility specification.
As displayed in Figure 8, the proportional premium for the preferred B(2s) specification is similar to that for the univariate case which conditions on past log(RV). The premium forecasts from this bivariate stochastic model range from 1.6% to 9.7% with an average of 4.31%. Note that all of our specifications deliver reasonable estimates of the market premium.26
Figure 9 illustrates the premium for the 1990-2006 period generated by our bivariate specification B(2s). The time-varying premium estimates for the period 1990 to 2000 were below the long-run average of 4.27%, reaching a low of 3.15% in 1997. In addition, they are considerably lower than the 1980-2006 average continuously compounded excess return of 6.2%. This graphically illustrates the point, discussed in the introduction, that the average realized excess return is not a very reliable forecast of the market equity premium since it will be sensitive to the subsample chosen. Finally, the forecasted continuously compounded premium increased from the low of 3.15% in 1997 to 5.4% in 2004 then back down to 4.9% in 2006.
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2.4 Robustness Analyses
We now check the robustness of the above results to alternative parameterizations and various subsamples. These results are summarized in Table 8.
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The bivariate model with the most empirical support is labeled B(2s). This specification specializes the bivariate system in Section 1.3.2 to the case with two volatility components (k = 2); allowing one of those components to affect the conditional mean (q = 1). That is,
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| (20) |
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To check that our parsimonious B(2s) specification is able to adequately capture the potential differential effect of the smooth long-run volatility component on the dynamics of the premium, we introduce a new bivariate specification that replaces Equation (20) with
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| (24) |
2 = 1/2 corresponds to the total variance being priced, while restricting 2 = 1 is one component being priced. These cases were discussed in the previous subsection and Table 6 as models B(2) and B(2s) respectively.
The results for the less parsimonious specification in Equation (24) are reported in the first panel of Table 8, labeled Model B(2a). The estimate for parameter 2 is 1.035 with a standard error of 0.201. Therefore, we can reject that it is zero (the t-statistic is 5.149) which confirms the results from all of our earlier 2s specifications that the persistent volatility component has a positive and very significant effect on the dynamics of excess returns. On the other hand, the coefficient on the second, less persistent volatility component is not significantly different from zero. While it is useful to see how the two volatility components are separately priced, notice that our more parsimonious model B(2s) has an almost identical fit which confirms the robustness of the 2s parameterizations reported earlier.
The third and fourth panels of Table 8 report results for the B(2s) model with alternative values of , the number of lags used in the exponential smoother components. As compared to the default value, = 40, the = 5 case results in an insignificant risk–return relationship. As is increased, the t-statistic on the slope parameter 1 increases. We have found that the optimal is somewhat model and criterion specific. For example, a = 25 produces a significantly better loglikelihood (–131.57) and higher t-statistic (3.153) associated with 1 than our base B(2s) which has = 40. However, the univariate specifications favored higher values of . Therefore, we chose a common value that was high enough to capture the smooth volatility component with our limited number of data.
The next three panels of Table 8 report subsample results for model B(2s). This specification which prices the smooth volatility component of a two-component volatility model was able to deliver a positive and statistically significant risk–return relationship for subsamples such as 1886–2006, 1928–2006 and 1945–2006. Note that as the sample gets shorter, the persistence, 2, of the second volatility component goes to zero. Nevertheless, the B(2s) parameterization still always dominates a one-component volatility specification.
Finally, the last panel of Table 8 reports results for model B(2s) for which annual RV is computed using monthly squared returns for the entire 1840–2006 sample, rather than daily squared returns when they became available in 1886. Our default model which uses the latter has a higher loglikelihood but otherwise the results are similar. We also tried using an indicator variable in the volatility function for the 1840–1885 part of the sample for which daily data were not available. Again, although that indicator variable was significantly different from zero indicating that annual RV was slightly lower over that first part of the sample, the risk–return results were similar. Also, note that the results for the 1886–2006 subsample refer to the period over which annual RV is always computed from daily squared returns. Again, the positive and significant risk–return relationship still obtains.
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3 Summary and Future Directions |
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TopAbstractIntroduction1 The Risk-Return Model2 Results3 Summary and Future...ReferencesFootnotes |
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This article evaluates the market risk–return relationship for U.S. equity over the period 1840-2006 using a time-varying market premium for equity risk. We begin with a univariate specification of the risk–return relationship. This application models the stochastic market excess returns by conditioning on variance forecasts which are estimated by projecting onto past RV. We assess the robustness of those results by also estimating a univariate version which projects onto past log(RV).
We propose a parsimonious and flexible function that summarizes information in past RV that might be useful for forecasting changes in the market equity risk premium. We allow for alternative components of volatility with different decay rates. Not only is this a flexible way to capture the time-series dynamics of volatility, but it also allows us to investigate whether a particular component, rather than the full conditional variance, is more important in driving the market premium.
Our conditional variance specification maintains the parsimony of exponential smoothing functions but allows mean reversion in forecasts, and targets the implied long-run variance. In addition, we allow for increased flexibility by explicitly modeling more than one volatility component.
Finally, we generalize the risk–return model for the market equity premium by estimating a bivariate stochastic specification of annual excess returns and log(RV). In this case, the conditional variance of excess returns is obtained as the conditional expectation of the RV process.
In summary, in addition to our volatility specification, we consider two additional contributions to the literature. First, unlike the univariate specifications that dominate the literature, we extend the analysis to a bivariate risk–return model of returns and RV. Second, we investigate whether or not one volatility component is more important than total volatility in driving the dynamics of the equity premium.
All of the empirical specifications support a conclusion that the relationship between risk and return for the market equity premium is positive. The higher the expected market risk, the higher the market equity premium. This relationship is strongest for a two-component specification of volatility with the long-run smooth component being priced in the conditional mean.27 In fact, the preferred model from the perspective of overall as well as equity premium fit, for both the univariate and bivariate specifications, is the 2s model which uses a two-component volatility specification and prices the smooth component of volatility in the conditional mean. In future work, as in Engle and Rangel (2004) and Engle, et al. (2006), we plan to explore potential sources of this smooth component of volatility.
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Footnotes |
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TopAbstractIntroduction1 The Risk-Return Model2 Results3 Summary and Future...ReferencesFootnotes |
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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."
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. 