Conventional wisdom views stock returns as less volatile over longer investment horizons. This view seems consistent with various empirical estimates. For example, using two centuries of U.S. equity returns, Siegel (2008) reports that variances realized over investment horizons of several decades are substantially lower than short-horizon variances on a per-year basis. Such evidence pertains to unconditional variance, but a similar message is delivered by studies that condition variance on information useful in predicting returns. Campbell and Viceira (2002, 2005), for example, report estimates of conditional variances that decrease with the investment horizon.

We find that stocks are actually more volatile over long horizons from an investor's perspective. Investors condition on available information but realize their knowledge is limited in two key respects. First, even after observing 206 years of data (1802 to 2007), investors do not know the values of the parameters of the return-generating process, especially the parameters related to the conditional expected return. Second, investors recognize that observable “predictors” used to forecast returns deliver only an imperfect proxy for the conditional expected return, whether or not the parameter values are known. When viewed from this perspective, the return variance per year at a 50-year horizon is at least 1.3 times higher than the variance at a 1-year horizon.

The distinction between predictive variance and true variance is readily seen in the simple case in which an investor knows the true variance of returns but not the true expected return. Uncertainty about the expected return contributes to the investor's overall uncertainty about what the upcoming realized returns will be. Predictive variance includes that uncertainty, while true variance excludes it. Expected return is notoriously hard to estimate. Uncertainty about the current expected return and about how expected return will change in the future is the key element of our story. This uncertainty plays an increasingly important role as the investment horizon grows, as long as investors believe that expected return is “persistent,” that is, likely to take similar values across adjacent periods.

Under the traditional random walk assumption that returns are distributed independently and identically (i.i.d.) over time, true return variance per period is equal at all investment horizons. Explanations for lower true variance at long horizons commonly focus on “mean reversion,” whereby a negative shock to the current return is offset by positive shocks to future returns and vice versa. Our conclusion that stocks are more volatile in the long run obtains despite the presence of mean reversion. We show that mean reversion is only one of five components of long-run predictive variance:

  • (i)

    i.i.d. uncertainty

  • (ii)

    mean reversion

  • (iii)

    uncertainty about future expected returns

  • (iv)

    uncertainty about current expected return

  • (v)

    estimation risk.

Whereas the mean-reversion component is strongly negative, the other components are all positive, and their combined effect outweighs that of mean reversion.

Three additional components also make significant positive contributions to long-horizon predictive variance. One is simply the variance attributable to unexpected returns. Under an i.i.d. assumption for unexpected returns, this variance makes a constant contribution to variance per period at all investment horizons. At long horizons, this component (i), though quite important, is actually smaller in magnitude than components (ii) and (iii) discussed above.

Another component of long-horizon predictive variance reflects uncertainty about the current μt. Components (i), (ii), and (iii) all condition on the current value of μt. Conditioning on the current expected return is standard in long-horizon variance calculations using a vector autoregression (VAR), such as Campbell (1991) and Campbell, Chan, and Viceira (2003). In reality, though, an investor does not observe μt. We assume that the investor observes the histories of returns and a given set of return predictors. This information is capable of producing only an imperfect proxy for μt, which in general reflects additional information. Pástor and Stambaugh (2009) introduce a predictive system to deal with imperfect predictors, and we use that framework to assess long-horizon predictive variance and capture component (iv). When μt is persistent, uncertainty about the current μt contributes to uncertainty about μt in multiple future periods, on top of the uncertainty about future μt's discussed earlier.

The fifth and last component adding to long-horizon predictive variance, also positively, is one we label “estimation risk,” following common usage of the term. This component reflects the fact that, after observing the available data, an investor remains uncertain about the parameters of the joint process generating returns, expected returns, and the observed predictors. That parameter uncertainty adds to the overall variance of returns assessed by an investor. If the investor knew the parameter values, this estimation-risk component would be zero.

Parameter uncertainty also enters long-horizon predictive variance more pervasively. Unlike the fifth component, the first four components are nonzero even if the parameters are known to an investor. At the same time, those four components can be affected significantly by parameter uncertainty. Each component is an expectation of a function of the parameters, with the expectation evaluated over the distribution characterizing an investor's parameter uncertainty. We find that Bayesian posterior distributions of these functions are often skewed, so that less likely parameter values exert a significant influence on the posterior means, and thus on long-horizon predictive variance.

The effects of parameter uncertainty on the predictive variance of long-horizon returns are analyzed in previous studies such as Stambaugh (1999), Barberis (2000), and Hoevenaars et al. (2007). Barberis discusses how parameter uncertainty essentially compounds across periods and exerts stronger effects at long horizons. The above studies find that predictive variance is substantially higher than estimates of true variance that ignore parameter uncertainty. However, all three studies also find that long-horizon predictive variance is lower than short-horizon variance for the horizons considered—up to 10 years in Barberis (2000), up to 20 years in Stambaugh (1999), and up to 50 years in Hoevenaars et al. (2007).2 In contrast, we often find that predictive variance even at a 10-year horizon is higher than at a 1-year horizon.

A key difference between our analysis and the above studies is our inclusion of uncertainty about the current expected return μt. The above studies employ VAR approaches in which observed predictors perfectly capture μt, whereas we consider predictors to be imperfect, as explained earlier. We compare predictive variances under perfect versus imperfect predictors, and we find that long-run variance is substantially higher when predictors are imperfect. Predictor imperfection increases long-run variance both directly and indirectly. The direct effect, component (iv) of predictive variance, is large enough at a 10-year horizon that subtracting it from predictive variance leaves the remaining portion lower than the 1-year variance.

The indirect effect of predictor imperfection is even larger, stemming from the fact that predictor imperfection and parameter uncertainty interact—once predictor imperfection is admitted, parameter uncertainty is more important in general. This result occurs despite the use of informative prior beliefs about parameter values, as opposed to the noninformative priors used in the above studies. When μt is not observed, learning about its persistence and predictive ability is more difficult than when μt is assumed to be given by observed predictors. The effects of parameter uncertainty pervade all components of long-horizon returns, as noted earlier. The greater parameter uncertainty accompanying predictor imperfection further widens the gap between our analysis and the previous studies.3

Predictor imperfection can be viewed as omitting an unobserved predictor from the set of observable predictors used in a standard predictive regression. The degree of predictor imperfection can be characterized by the increase in the R2 of that predictive regression if the omitted predictor were included. Even if investors assign a low probability to this increase being larger than 2% for annual returns, such modest predictor imperfection nevertheless exerts a substantial effect on long-horizon variance. At a 30-year horizon, for example, the predictive variance is 1.2 times higher than when the predictors are assumed to be perfect.

Our empirical results indicate that stocks should be viewed by investors as more volatile at long horizons. Indeed, corporate Chief Financial Officers (CFOs) tend to exhibit such a view, as we discover by analyzing survey evidence reported by Ben-David, Graham, and Harvey (2010). In quarterly surveys conducted over 8 years, Ben-David, Graham, and Harvey ask CFOs to express confidence intervals for the stock market's annualized return over the next year and the next 10 years. From the reported results of these surveys, we infer that the typical CFO views the annualized variance of 10-year returns to be at least twice the 1-year variance.

The long-run volatility of stocks is of substantial interest to investors. Evidence of lower long-horizon variance is cited in support of higher equity allocations for long-run investors (e.g, Siegel (2008)) as well as the increasingly popular target-date mutual funds (e.g., Gordon and Stockton (2006), Greer (2004), and Viceira (2008)). These funds gradually reduce an investor's stock allocation by following a predetermined “glide path” that depends only on the time remaining until the investor's target date, typically retirement. When the parameters and conditional expected return are assumed to be known, we find that the typical glide path of a target-date fund closely resembles the pattern of allocations desired by risk-averse investors with utility for wealth at the target date. Once uncertainty about the parameters and conditional expected return is recognized, however, the same investors find the typical glide path significantly less appealing. Investors with sufficiently long horizons instead prefer glide paths whose initial as well as final stock allocations are substantially lower than those of investors with shorter horizons.

The remainder of the paper proceeds as follows. Section I derives expressions for the five components of long-horizon variance discussed above and analyzes their theoretical properties. Section II describes our empirical framework, which uses up to 206 years of data to implement two predictive systems that allow us to analyze various properties of long-horizon variance. Section III explores the five components of long-horizon variance using a predictive system in which the conditional expected return follows a first-order autoregression. Section IV then gauges the importance of predictor imperfection using an alternative predictive system that includes an unobservable predictor. Section V discusses the robustness of our results. Section VI returns to the above discussion of the distinction between an investor's problem and inference about true variance. Section VII considers the implications of the CFO surveys reported by Ben-David et al. (2010). Section VIII analyzes investment implications of our results in the context of target-date funds. Section IX summarizes our conclusions.

I. Long-Horizon Variance and Parameter Uncertainty

Let ◂◽.▸rt+1 denote the continuously compounded return from time t to time t+1. We can write
◂=▸◂◽.▸rt+1=μt+◂◽.▸ut+1, (1)
where μt denotes the expected return conditional on all information at time t and ◂◽.▸ut+1 has zero mean. Also define the k-period return from period T+1 through period T+k,
r◂,▸T,T+k=◂+▸◂◽.▸rT+1+◂◽.▸rT+2++◂◽.▸rT+k. (2)
An investor assessing the variance of r◂,▸T,T+k uses DT, a subset of all information at time T. In our empirical analysis in Section III, DT consists of the full histories of returns as well as predictors that investors use in forecasting returns.4 Importantly, DT typically reveals neither the value of μT in equation (1) nor the values of the parameters governing the joint dynamics of rt, μT, and the predictors. Let ϕ denote the vector containing those parameter values.
This paper focuses on Var◂()▸(r◂,▸T,T+k|DT), the predictive variance of r◂,▸T,T+k given the investor's information set. Since the investor is uncertain about μT and ϕ, it is useful to decompose this variance as
◂...▸Var◂()▸(r◂,▸T,T+k|DT)=◂f()▸E◂{}▸{Var◂()▸(r◂,▸T,T+k|μT,ϕ,DT)|DT}◂...▸+Var{◂f()▸E◂()▸(r◂,▸T,T+k|μT,ϕ,DT)|DT}. (3)

The first term in this decomposition is the expectation of the conditional variance of k-period returns. This conditional variance, which has been estimated by Campbell and Viceira (2002, 2005), is of interest only to investors who know the true values of μT and ϕ. Investors who do not know μT and ϕ are interested in the expected value of this conditional variance, and they also account for the variance of the conditional expected k-period return, the second term in equation (3). As a result, they perceive returns to be more volatile and, as we show below, they perceive disproportionately more volatility at long horizons. Whereas the conditional per-period variance of stock returns appears to decrease with the investment horizon, we show that ◂...▸(1/k)Var◂()▸(r◂,▸T,T+k|DT), which accounts for uncertainty about μT and ϕ, increases with the investment horizon.

The potential importance of parameter uncertainty for long-run variance is readily seen in the special case where returns are i.i.d. with known variance σ2 and unknown mean μ. In this case, the mean and variance of k-period returns conditional on μ are both linear in k: the mean is kμ and the variance is kσ2. An investor who knows μ faces the same per-period variance, σ2, regardless of k. However, an investor who does not know μ faces more variance, and this variance increases with k. To see this, apply the variance decomposition from equation (3),
◂...▸Var◂()▸(r◂,▸T,T+k|DT)=◂f()▸E◂{}▸{kσ2|DT}+Var{kμ|DT}◂...▸=◂+▸kσ2+k2Var{μ|DT}, (4)
so that ◂...▸(1/k)Var◂()▸(r◂,▸T,T+k|DT) increases with k. In fact, ◂...▸(1/k)Var◂()▸(r◂,▸T,T+k|DT) as k. That is, an investor who believes that stock prices follow a random walk but who is uncertain about the unconditional mean μ views stocks as more volatile in the long run.

To assess the likely magnitude of this effect, consider the following back-of-the-envelope calculation. If uncertainty about μ is given by the standard error of the sample average return computed over T periods, or σ/T, then ◂...▸(1/k)Var◂()▸(r◂,▸T,T+k|DT)=◂⋅▸σ2(1+k/T). With k=50 years and T=206 years, as in the sample that we use in Section III, ◂=▸(1+k/T)=1.243, so the per-period predictive variance exceeds σ2 by a quarter. Of course, if the sample mean estimate of μ is computed from a sample shorter than 206 years (e.g., due to concerns about nonstationarity), then uncertainty about μ is larger and the effect on predictive variance is even stronger.

When returns are predictable, so that μt is time-varying, Var◂()▸(r◂,▸T,T+k|DT) can be above or below its value in the i.i.d. case. Predictability can induce mean reversion, which reduces long-run variance, but predictability also introduces uncertainty about additional quantities, such as future values of μt and the parameters that govern its behavior. It is not clear a priori whether predictability makes returns more or less volatile at long horizons, compared to the i.i.d. case. At sufficiently long horizons, uncertainty about the unconditional expected return will still dominate and drive ◂...▸(1/k)Var◂()▸(r◂,▸T,T+k|DT) to infinity. At long horizons of relevance to investors, whether that per-period variance is higher than at short horizons is an empirical question that we explore.

In the rest of this section, we assume for simplicity that μt follows an AR(1) process,5
◂◽.▸μt+1=◂+▸◂⋅▸(1β)Er+βμt+◂◽.▸wt+1,0<β<1. (5)
The AR(1) assumption for μt allows us to further decompose both terms on the right-hand side of equation (3), providing additional insights into the components of Var◂()▸(r◂,▸T,T+k|DT). The AR(1) assumption also allows a simple characterization of mean reversion. Time variation in μt induces mean reversion in returns if the unexpected return ◂◽.▸ut+1 is negatively correlated with future values of μt. Under the AR(1) assumption, mean reversion requires a negative correlation between ◂◽.▸ut+1 and ◂◽.▸wt+1, or ◂◽.▸ρuw<0. If fluctuations in μt are persistent, then a negative shock in ◂◽.▸ut+1 is accompanied by offsetting positive shifts in the ◂◽.▸μt+i's for multiple future periods, resulting in a stronger negative contribution to the variance of long-horizon returns.

A. Conditional Variance

This section analyzes the conditional variance Var◂()▸(r◂,▸T,T+k|μT,ϕ,DT), which is an important building block in computing the variance in equation (3). The conditional variance reflects neither parameter uncertainty nor uncertainty about the current expected return, since it conditions on both ϕ and μT. The parameter vector ϕ includes all parameters in equations (1) and (5), ϕ=◂()▸(β,Er,◂◽.▸ρuw,σu,σw), where σu and σw are the conditional standard deviations of ◂◽.▸ut+1 and ◂◽.▸wt+1, respectively. Assuming that equations (1) and (5) hold and that the conditional covariance matrix of [◂◽.▸ut+1◂◽.▸wt+1] is constant, ◂...▸Var◂()▸(r◂,▸T,T+k|μT,ϕ,DT)=Var◂()▸(r◂,▸T,T+k|μT,ϕ). Furthermore, we show in the Appendix that
Var◂()▸(r◂,▸T,T+k|μT,ϕ)=◂⋅▸kσu2[1+◂⋅▸2dˉ◂◽.▸ρuwA(k)+◂⋅▸◂◽˙▸dˉ2B(k)], (6)
◂=▸A(k)=◂+▸1+1k◂()▸(1β◂/▸1◂◽˙▸βk11β) (7)
◂=▸B(k)=◂+▸1+1k◂()▸(◂−▸12β◂/▸1◂◽˙▸βk11β+β2◂/▸1β◂⋅▸2(k1)1β2) (8)
dˉ=◂◽˙▸[◂/▸1+β1+β◂/▸R21R2]1/2, (9)
and R2 is the ratio of the variance of μt to the variance of ◂◽.▸rt+1, based on equation (1).

The conditional variance in (6) consists of three terms. The first term, kσu2, captures the well-known feature of i.i.d. returns—the variance of k-period returns increases linearly with k. The second term, which contains A(k), reflects mean reversion in returns arising from the likely negative correlation between realized returns and expected future returns (◂◽.▸ρuw<0), and it contributes negatively to long-horizon variance. The third term, which contains B(k), reflects the uncertainty about future values of μt, and it contributes positively to long-horizon variance. When returns are unpredictable, only the first term is present (because R2=0 implies dˉ=0, so the terms involving A(k) and B(k) are zero). Now suppose that returns are predictable, so that R2>0 and dˉ>0. When k=1, the first term is still the only one present, because ◂=▸A(1)=B(1)=0. As k increases, though, the terms involving A(k) and B(k) become increasingly important, because both A(k) and B(k) increase monotonically from zero to one as k goes from one to infinity.

Figure 1 plots the variance in (6) on a per-period basis (i.e., divided by k) as a function of the investment horizon k. Also shown are the terms containing A(k) and B(k). It can be verified that A(k) converges to one faster than B(k). (See the Appendix.) As a result, the conditional variance in Figure 1 is U-shaped: as k increases, mean reversion exerts a stronger effect initially, but uncertainty about future expected returns dominates eventually.6 The contribution of the mean reversion term, and thus the extent of the U-shape, is stronger when ◂◽.▸ρuw takes larger negative values. The contributions of mean reversion and uncertainty about future ◂◽.▸μT+i's both become stronger as predictability increases. These effects are illustrated in Figure 2, which plots the same quantities as Figure 1, but for three different R2 values. Note that a higher R2 implies not only stronger mean reversion but also a more volatile μt, which in turn implies more uncertainty about future ◂◽.▸μT+i's.

The key insight arising from Figures 1 and 2 is that, although mean reversion significantly reduces long-horizon variance, that reduction can be more than offset by uncertainty about future expected returns. Both effects become stronger as R2 increases, but uncertainty about future expected returns prevails when R2 is high. A high R2 implies high volatility in μt and therefore high uncertainty about ◂◽.▸μT+j. In that case, long-horizon variance exceeds short-horizon variance on a per-period basis, even though ϕ and the current μT are assumed to be known. Uncertainty about ϕ and the current μT exerts a greater effect at longer horizons, further increasing the long-horizon variance relative to the short-horizon variance.

B. Components of Long-Horizon Variance

The variance of interest, Var◂()▸(r◂,▸T,T+k|DT), consists of two terms on the right-hand side of equation (3). The first term is the expectation of the conditional variance in equation (6), so each of the three terms in (6) is replaced by its expectation with respect to ϕ. (We need not take the expectation with respect to μT, since μT does not appear on the right in (6).) The interpretations of these terms are the same as before, except that now each term also reflects parameter uncertainty.

The second term on the right-hand side of equation (3) is the variance of the true conditional expected return. This variance is taken with respect to ϕ and μT. It can be decomposed into two components: one reflecting uncertainty about the current μT, or predictor imperfection, and the other reflecting uncertainty about ϕ, or “estimation risk.” (See the Appendix.) Let bT and qT denote the conditional mean and variance of the unobservable expected return μT:
bT=◂f()▸E◂()▸(μT|ϕ,DT) (10)
◂...▸qT=Var◂()▸(μT|ϕ,DT). (11)
The right-hand side of equation (3) can then be expressed as the sum of five components:
Var◂()▸(r◂,▸T,T+k|DT)=◂+▸◂f()▸E◂{}▸{kσu2|DT}◂,▸i.i.d.uncertainty+E{◂⋅▸2kσu2dˉ◂◽.▸ρuwA(k)|DT}◂...▸meanreversion+E{◂⋅▸kσu2◂◽˙▸dˉ2B(k)|DT}◂...▸future◂◽.▸μT+iuncertainty◂+▸+◂f()▸E◂{}▸{◂◽˙▸(◂/▸1βk1β)2qT|DT}◂...▸currentμTuncertainty+◂...▸Var{◂+▸kEr+◂/▸1βk1β◂()▸(bTEr)|DT}◂...▸estimationrisk. (12)

Parameter uncertainty plays a role in all five components in equation (12). The first four components are expected values of quantities that are viewed as random due to uncertainty about ϕ, the parameters governing the joint dynamics of returns and predictors. (If the values of these parameters were known to the investor, the expectation operators could be removed from those four components.) Parameter uncertainty can exert a nontrivial effect on the first four components, in that the expectations can be influenced by parameter values that are unlikely but cannot be ruled out. The fifth component in equation (12) is the variance of a quantity whose randomness is also due to parameter uncertainty. In the absence of such uncertainty, the fifth component is zero, which is why we assign it the interpretation of estimation risk.

The estimation risk term includes the variance of kEr, where Er denotes the unconditional mean return. This variance equals ◂...▸k2Var◂()▸(Er|DT), so the per-period variance ◂...▸(1/k)Var◂()▸(r◂,▸T,T+k|DT) increases at rate k. Similar to the i.i.d. case, if Er is unknown, then the per-period variance grows without bounds as the horizon k goes to infinity. For finite horizons that are typically of interest to investors, however, the fifth component in equation (12) can nevertheless be smaller in magnitude than the other four components. In general, the k-period variance ratio, defined as
◂=▸V(k)=◂...▸(1/k)Var◂()▸(r◂,▸T,T+k|DT)◂...▸Var(◂◽.▸rT+1|DT), (13)
can exhibit a variety of patterns as k increases. Whether ◂>▸V(k)>1 at various horizons k is an empirical question.

II. Empirical Framework: Predictive Systems

It is commonly assumed that the conditional expected return