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

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

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 ^{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

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

^{4}Importantly,

The first term in this decomposition is the expectation of the conditional variance of

To assess the likely magnitude of this effect, consider the following back-of-the-envelope calculation. If uncertainty about

When returns are predictable, so that

^{5}

### A. *Conditional Variance*

The conditional variance in (6) consists of three terms. The first term,

Figure 1 plots the variance in (6) on a per-period basis (i.e., divided by ^{6} The contribution of the mean reversion term, and thus the extent of the U-shape, is stronger when

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

### B. *Components of Long-Horizon Variance*

The variance of interest,

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

## II. Empirical Framework: Predictive Systems

It is commonly assumed that the conditional expected return