### ABSTRACT

- Top of page
- ABSTRACT
- I. Long-Horizon Variance and Parameter Uncertainty
- II. Empirical Framework: Predictive Systems
- III. Components of Predictive Variance (System 1)
- IV. Perfect Predictors versus Imperfect Predictors (System 2)
- V. Robustness
- VI. Predictive Variance versus True Variance
- VII. Long-Horizon Variance: Survey Evidence
- VIII. Target-Date Funds
- IX. Conclusions
- Appendix
- REFERENCES

According to conventional wisdom, annualized volatility of stock returns is lower over long horizons than over short horizons, due to mean reversion induced by return predictability. In contrast, we find that stocks are substantially more volatile over long horizons from an investor’s perspective. This perspective recognizes that parameters are uncertain, even with two centuries of data, and that observable predictors imperfectly deliver the conditional expected return. Mean reversion contributes strongly to reducing long-horizon variance but is more than offset by various uncertainties faced by the investor. The same uncertainties reduce desired stock allocations of long-horizon investors contemplating target-date funds.

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.

Our main object of interest is the *predictive* variance of , the -period return starting at time . Predictive variance, denoted by , conditions on , the data available to investors at time . From an investor’s perspective, predictive variance is the relevant variance—the one suitable for portfolio decisions. Readers might be more familiar with *true* variance, which conditions on , the parameters of the return-generating process. Investors realize they do not know , and predictive variance incorporates that parameter uncertainty by conditioning only on . In contrast, true variance conditions on , regardless of whether it also conditions on . The true unconditional variance, , is estimated by the usual sample variance, as in Siegel (2008). The true conditional variance, , is estimated by Campbell and Viceira (2002, 2005). True variance is the more common focus of statistical inference. For example, an extensive literature uses variance ratios and other statistics to test whether is the same for every investment horizon .^{1} We focus on instead. That is, we compare long- and short-horizon predictive variances, which matter to investors. Investors might well infer from the data that the true variance is lower at long horizons while at the same time assessing the predictive variance to be higher at long horizons.

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:

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 . Components (i), (ii), and (iii) all condition on the current value of . 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 . 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 , 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 is persistent, uncertainty about the current contributes to uncertainty about in multiple future periods, on top of the uncertainty about future ’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 . The above studies employ VAR approaches in which observed predictors perfectly capture , 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 is not observed, learning about its persistence and predictive ability is more difficult than when 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 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

- Top of page
- ABSTRACT
- I. Long-Horizon Variance and Parameter Uncertainty
- II. Empirical Framework: Predictive Systems
- III. Components of Predictive Variance (System 1)
- IV. Perfect Predictors versus Imperfect Predictors (System 2)
- V. Robustness
- VI. Predictive Variance versus True Variance
- VII. Long-Horizon Variance: Survey Evidence
- VIII. Target-Date Funds
- IX. Conclusions
- Appendix
- REFERENCES

The first term in this decomposition is the expectation of the conditional variance of -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 and . Investors who do not know and are interested in the expected value of this conditional variance, and they also account for the variance of the conditional expected -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 , which accounts for uncertainty about and , increases with the investment horizon.

#### A. *Conditional Variance*

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

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

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 , 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.

### II. Empirical Framework: Predictive Systems

- Top of page
- ABSTRACT
- I. Long-Horizon Variance and Parameter Uncertainty
- II. Empirical Framework: Predictive Systems
- III. Components of Predictive Variance (System 1)
- IV. Perfect Predictors versus Imperfect Predictors (System 2)
- V. Robustness
- VI. Predictive Variance versus True Variance
- VII. Long-Horizon Variance: Survey Evidence
- VIII. Target-Date Funds
- IX. Conclusions
- Appendix
- REFERENCES

It is commonly assumed that the conditional expected return is given by a linear combination of a set of observable predictors, , so that . This assumption is useful in many applications, but we relax it here because it understates the uncertainty faced by an investor assessing the variance of future returns. Any given set of predictors is likely to be *imperfect*, in that is unlikely to be captured by any linear combination of (). The true expected return generally reflects more information than what we assume to be observed by the investor—the histories of and . To incorporate the likely presence of predictor imperfection, we employ a predictive system, defined in Pástor and Stambaugh (2009) as a state-space model in which , , and follow a VAR with coefficients restricted so that is the mean of .^{7} As noted by Pástor and Stambaugh, a predictive system can also be represented as a VAR for , , and an unobserved additional predictor. We employ both versions here, as each is best suited to different dimensions of our investigation. Our two predictive systems are specified as follows:

System 1 is well suited for analyzing the components of predictive variance discussed in the previous section, because the AR(1) specification for in equation (16) is the same as that in equation (5). Pástor and Stambaugh (2009) provide a detailed analysis of System 1, and we apply their econometric methodology in this study. In the next section, we investigate empirically the components of predictive variance using System 1.

We conduct analyses using both annual and quarterly data. Our annual data consist of observations for the 206-year period from 1802 through 2007, as compiled by Siegel (1992, 2008). The return is the annual real log return on the U.S. equity market, and contains three predictors: the dividend yield on U.S equity, the first difference in the long-term high-grade bond yield, and the difference between the long-term bond yield and the short-term interest rate.^{8} We refer to these quantities as the “dividend yield,” the “bond yield,” and the “term spread,” respectively. These three predictors seem reasonable choices given the various predictors used in previous studies and the information available in Siegel’s data set. Dividend yield and the term spread have long been entertained as return predictors (e.g., Fama and French (1989)). Using post-war quarterly data, Pástor and Stambaugh (2009) find that the long-term bond yield, relative to its recent levels, exhibits significant predictive ability in predictive regressions. That evidence motivates our choice of the bond-yield variable used here. All three predictors exhibit significant predictive abilities in a predictive regression as in (20), with an in that regression of 5.6%.^{9} Our quarterly data consist of observations for the 220-quarter period from 1952Q1 through 2006Q4. We use the same three predictors in as Pástor and Stambaugh (2009): dividend yield, CAY, and bond yield.^{10}

### IV. Perfect Predictors versus Imperfect Predictors (System 2)

- Top of page
- ABSTRACT
- I. Long-Horizon Variance and Parameter Uncertainty
- II. Empirical Framework: Predictive Systems
- III. Components of Predictive Variance (System 1)
- IV. Perfect Predictors versus Imperfect Predictors (System 2)
- V. Robustness
- VI. Predictive Variance versus True Variance
- VII. Long-Horizon Variance: Survey Evidence
- VIII. Target-Date Funds
- IX. Conclusions
- Appendix
- REFERENCES

Even when investors assess potential predictor imperfection to be relatively modest, the imperfection has important consequences for the predictive variance of long-horizon returns. Predictive variances for horizons up to 50 years are shown in Panel A of Figure 8 for the annual data, while Panel B shows the corresponding results for the quarterly data. The importance of recognizing predictor imperfection emerges clearly from these results. In Panel A, the predictive variances at the longest horizons are about 1.3 times higher when predictor imperfection is recognized than when predictors are assumed to be perfect. For the quarterly results in Panel B, that ratio is well over 2.0.

We also see in Figure 8 that predictive variances are substantially greater at long horizons than at short horizons, once predictor imperfection is recognized. Thus, the results for System 2 deliver the same overall message as the earlier results for System 1. In Panel A, using annual data, the predictive variance at the 50-year horizon is 1.4 to 1.5 times the 1-year variance, depending on the degree of predictor imperfection. In Panel B, using quarterly data, the 50-year variance is 1.3 to 1.4 times the 1-year variance.

Stambaugh (1999) and Barberis (2000) investigate the effects of parameter uncertainty using data beginning in 1952, the same year that our quarterly data begin. With these data, predictor imperfection plays an especially large role—more than doubling the variance at long horizons. With perfect predictors, consistent with Stambaugh and Barberis, predictive variance is substantially lower at long horizons: the 50-year variance ratio is then 0.6. In contrast, when predictor imperfection is incorporated, the 50-year variance ratio is 1.3 to 1.4, as observed above. Thus, when using post-1951 data, accounting for predictor imperfection rather dramatically reverses the answer to the question of whether stocks are less volatile in the long run.

We also see that the findings of Stambaugh and Barberis, which indicate that stocks are less volatile at longer horizons even after incorporating parameter uncertainty, do not obtain over the longer 206-year period. The predictive variances in Panel E are actually higher at long horizons, given perfect predictors, with a 50-year variance ratio just below 1.2. In all of our results, however, admitting predictor imperfection produces long-run variance that substantially exceeds not only short-run variance but also long-run variance computed assuming perfect predictors.

### VI. Predictive Variance versus True Variance

- Top of page
- ABSTRACT
- I. Long-Horizon Variance and Parameter Uncertainty
- II. Empirical Framework: Predictive Systems
- III. Components of Predictive Variance (System 1)
- IV. Perfect Predictors versus Imperfect Predictors (System 2)
- V. Robustness
- VI. Predictive Variance versus True Variance
- VII. Long-Horizon Variance: Survey Evidence
- VIII. Target-Date Funds
- IX. Conclusions
- Appendix
- REFERENCES

This section provides further perspective on our results by distinguishing between two different measures of variance: predictive variance and true variance. The predictive variance, our main object of interest thus far, is the variance from the perspective of an investor who conditions on the historical data but remains uncertain about the true values of the parameters. The true variance is defined as the variance conditional on the true parameter values. The predictive variance and the true variance coincide if the data history is infinitely long, in which case the parameters are estimated with infinite precision. Estimates of the true variance can be relevant in some applications, such as option pricing, but the predictive variance is relevant for portfolio decisions.

When conducting inference about the true variance, a commonly employed statistic is the sample long-horizon variance ratio. Values of such ratios are often less than one for stocks, suggesting lower unconditional variances per period at long horizons. Figure 9 plots sample variance ratios for horizons of 2 to 50 years computed with the 206-year sample of annual real log stock returns analyzed above. The calculations use overlapping returns and unbiased variance estimates.^{21} Also plotted are percentiles of the variance ratio’s Monte Carlo sampling distribution under the null hypothesis that returns are i.i.d. normal. That distribution exhibits positive skewness and has nearly 60% of its mass below one. The realized value of 0.28 at the 30-year horizon attains a Monte Carlo -value of 0.01, supporting the inference that the true 30-year variance ratio lies below one (setting aside the multiple-comparison issues of selecting one horizon from many). Panel A of Figure 10 plots the posterior distribution of the 30-year ratio for true unconditional variance, based on the benchmark priors and System 1. Even though the posterior mean of this ratio is 1.34, the distribution is positively skewed and 63% of the posterior probability mass lies below one. We thus see that the variance ratio statistic in a frequentist setting and the posterior distribution in a Bayesian setting both favor the inference that the true unconditional variance ratio is below one.

The second and larger point is that inference about true variance, conditional or unconditional, is distinct from assessing the predictive variance perceived by an investor who does not know the parameters. This distinction can be drawn clearly in the context of the variance decomposition,

- (25)

The variance on the left-hand side of (25) is the predictive variance. The quantity inside the expectation in the first term, , is the true conditional variance, relevant only to an investor who knows the true parameter vector (but not , thus maintaining predictor imperfection). The data can imply that this *true* variance is probably *lower* at long horizons than at short horizons while also implying that the *predictive* variance is *higher* at long horizons. In other words, investors who observe can infer that, if they were told the true parameter values, they would probably assess 30-year variance to be less than 1-year variance. These investors realize, however, that they do not know the true parameters. As a consequence, they evaluate the posterior mean of the true conditional variance, the first term in (25). That posterior mean can exceed the most likely values of the true conditional variance, because the posterior distribution of the true variance can be skewed (we return to this point below). Moreover, investors must add to that posterior mean the posterior variance of the true conditional mean, the second term in (25), which is the same as the estimation-risk term in equation (12). In a sense, investors do conduct inference about true variance—they compute its posterior mean—but they realize that this estimate is only part of predictive variance.

The results based on our 206-year sample illustrate how predictive variance can be higher at long horizons while true variance is inferred to be most likely higher at short horizons. Panel B of Figure 10 plots the posterior distribution of the variance ratio

- (26)

for years. The posterior probability that this ratio of true variances lies below one is 76%, and the posterior mode is below 0.5. In contrast, recall that 30-year predictive variance is substantially greater than 1-year variance, as shown earlier in Figure 6 and Table I.

The true conditional variance is the sum of four quantities, namely, the first four components in equation (12) with the expectations operators removed. The posterior distributions of those quantities (not shown to save space) exhibit significant asymmetries. As a result, less likely values of these quantities exert a disproportionate effect on the posterior means and therefore on the first term of the predictive variance in (25). The components reflecting uncertainty about current and future are positively skewed, so their contributions to predictive variance exceed what they would be if evaluated at the most likely parameter values. This feature of parameter uncertainty also helps drive predictive variance above the most likely value of true variance.

### VII. Long-Horizon Variance: Survey Evidence

- Top of page
- ABSTRACT
- I. Long-Horizon Variance and Parameter Uncertainty
- II. Empirical Framework: Predictive Systems
- III. Components of Predictive Variance (System 1)
- IV. Perfect Predictors versus Imperfect Predictors (System 2)
- V. Robustness
- VI. Predictive Variance versus True Variance
- VII. Long-Horizon Variance: Survey Evidence
- VIII. Target-Date Funds
- IX. Conclusions
- Appendix
- REFERENCES

Our empirical results show that investors should view stocks as more volatile over long horizons than over short horizons. Corporate CFOs indeed appear to exhibit such a view, as can be inferred from survey results reported by Ben-David et al. (2010). Their survey asks each CFO to give the 10th and 90th percentiles of a confidence interval for the annualized (average) excess equity return to be realized over the upcoming 10-year period. The same question is asked for a 1-year horizon. For each horizon (), the authors use the 10th and 90th percentiles to approximate , the variance of the CFO’s perceived distribution of the annualized return. The resulting standard deviations are then averaged across CFOs. If we treat the averaged standard deviations as those perceived by a “typical” CFO, we can infer the typical CFO’s views about long-horizon variance.

The relation between and the annualized variance of the -year return, , which is our object of interest, must obey

- (27)

If CFOs perceive stocks as equally volatile at all horizons, as in the standard i.i.d. setting with no parameter uncertainty, then and . In that case, the perceived standard deviation of the 1-year return should be 3.2 (=) times the perceived standard deviation of the annualized 10-year return. In the survey results reported by Ben-David et al., we observe that the ratios of the 1-year standard deviation to the 10-year standard deviation are substantially below 3.2. Across 33 quarterly surveys from the first quarter of 2002 through the first quarter of 2010, the ratio ranges from 1.25 to 2.14, and its average value is 1.54. Even the maximum ratio of 2.14 implies

- (28)

or, applying (27), a 10-year variance ratio given by

- (29)

as compared to the value of 1.0 when stocks are equally volatile over long and short horizons. In other words, the typical CFO appears to view stock returns as having at least twice the variance over a 10-year horizon than over a 1-year horizon.

### VIII. Target-Date Funds

- Top of page
- ABSTRACT
- I. Long-Horizon Variance and Parameter Uncertainty
- II. Empirical Framework: Predictive Systems
- III. Components of Predictive Variance (System 1)
- IV. Perfect Predictors versus Imperfect Predictors (System 2)
- V. Robustness
- VI. Predictive Variance versus True Variance
- VII. Long-Horizon Variance: Survey Evidence
- VIII. Target-Date Funds
- IX. Conclusions
- Appendix
- REFERENCES

This section explores the long-run riskiness of stocks from the perspective of a very popular investment strategy. Target-date funds, also known as life-cycle funds, represent one of the fastest-growing segments of the investment industry. Since the inception of these funds in the mid-1990s, their assets have grown to about $280 billion in 2010, including a net cash inflow of $42 billion during the tumultuous year 2008. About 87% of target-date fund assets are held in retirement accounts as of third-quarter 2010 (Investment Company Institute (2011)).

Target-date funds follow a predetermined asset allocation policy that gradually reduces the stock allocation as the target date approaches, with the aim of providing a more conservative asset mix to investors approaching retirement.^{23} A predetermined allocation policy is not optimal because it sacrifices the ability to rebalance in response to future events, an ability analyzed in numerous studies of dynamic asset allocation.^{24} We venture off the well-trod path of that literature to consider a long-horizon strategy that, while suboptimal in theory, has become important in practice. We do not attempt to explain why so many real-world investors desire a predetermined path for their asset allocations. We simply take that fact as given and analyze the asset allocation problem within that setting. This focus also seems natural in the context of our study, since long-horizon equity volatility is relevant for investors making long-horizon equity decisions.

Target-date funds are often motivated by arguments related to human capital and labor income. A typical argument goes as follows.^{25} Human capital is bond-like as it offers a steady stream of labor income. Younger people have more human capital because they stand to collect labor income over a longer time period. Younger people thus have a larger implicit position in bonds. To balance that position, younger people should invest a bigger fraction of their financial wealth in stocks, and they should gradually reduce their stock allocation as they grow older.

We consider two frameworks that differ in their treatment of labor income. In the first framework, presented in Section VIII.A, the investor invests an initial nest egg and does not invest additional savings from any labor income. In the second framework, presented in Section VIII.B, the investor also saves a fraction of his labor income. Both frameworks lead to the same conclusions regarding the effects of parameter uncertainty on the stock allocations of long-horizon investors.

#### A. *No Savings from Labor Income*

Panels A and B of Figure 11 plot the investor’s optimal initial and final stock allocations, (solid line) and (dashed line), for investment horizons ranging from 1 to 30 years. In Panel A, parameter uncertainty is ignored, in that the parameters characterizing the return process are treated as known and equal to their posterior means. In Panel B, parameter uncertainty is incorporated by using the posterior distributions. These come from our baseline setting: System 1 implemented on the 1802 to 2007 sample with three predictors and the benchmark prior.

The optimal allocations in Panel A of Figure 11 are strikingly similar to those selected by real-world target-date funds. The initial allocation decreases steadily as the investment horizon shortens, declining from about 85% at long horizons such as 25 or 30 years to about 30% at the 1-year horizon, whereas the final allocation is roughly constant at about 30% to 40% across all horizons. Investors in real-world target-date funds similarly commit to a stock allocation schedule, or glide path, that decreases steadily to a given level at the target date. The final stock allocation in a target-date fund does not depend on when investors enter the fund, but the initial allocation does—it is higher for investors entering longer before the target date. Not only the patterns but also the magnitudes of the optimal allocations in Panel A resemble those of target-date funds. For example, Viceira (2008) reports that the target-date funds offered by Fidelity and Vanguard reduce their stock allocations from 90% at long horizons to about 30% at short horizons. In addition, Vanguard’s stock allocations equal 90% for all horizons of 25 years or longer (see Viceira’s Figure 5.2), which corresponds nicely to the relatively flat portion of the solid line in Panel A.^{27} In short, target-date funds seem appealing to investors who maximize expected power utility of wealth at the target date and who ignore parameter uncertainty.

#### B. *Labor Income*

We assume the following simple process for labor income growth:

- (32)

where is a constant, denotes the investor’s age in year , and is drawn randomly from . We set , which is equal to the estimate of the annualized standard deviation of wage income growth reported by Heaton and Lucas (2000). The motivation for the age-related term in equation (32) is the evidence that expected labor income exhibits a hump-shaped pattern over a typical investor’s life cycle. For example, Figure 1 in Cocco et al. (2005) shows that labor income is an inverse U-shaped function of age, for each of three different groups of households sorted by their education level. To capture the concave pattern in the level of labor income, we assume that the growth rate of labor income is a linearly decreasing function of age. We calibrate this function to the middle line in Cocco, Gomes, and Maenhout’s Figure 1, according to which expected labor income grows until age 43 and declines thereafter. We set , so that initial labor income growth at age 20 is 10%, as in Cocco, Gomes, and Maenhout’s Figure 1. We assume that the investor retires at age 65, which is also the end of his investment horizon, so that .

Note that labor income growth in equation (32) is uncorrelated with stock market returns. This assumption is motivated by the evidence that the correlation between wage growth and the stock market is generally close to zero. For example, Heaton and Lucas (2000) report a correlation of 0.07, and Cocco et al. (2005) report correlations ranging from 0.02 to 0.01 across three different education levels. However, the assumption of zero correlation is not necessary for our conclusions. In an earlier version of the paper, we modeled labor income growth as a convex combination of returns on the stock market and the T-bill, and we found that our conclusions were unaffected by relatively large changes in the weight on the stock market.

Panels C and D of Figure 11 plot the investor’s optimal initial and final stock allocations, and , as a function of the investment horizon. These panels are constructed in the same way as Panels A and B, except that the investor’s financial wealth follows equation (31) rather than equation (30). Parameter uncertainty is incorporated in Panel D but not in Panel C.

The results in Figure 11 demonstrate how parameter uncertainty makes target-date funds undesirable when they would otherwise be virtually optimal for investors who desire a predetermined asset allocation policy. It would be premature, however, to conclude that parameter uncertainty makes target-date funds undesirable to such investors in all settings. The above analysis abstracts from many important considerations faced by investors, such as intermediate consumption, housing, etc. Our objective in this section is simply to illustrate how parameter uncertainty can reduce the stock allocations of long-horizon investors, consistent with our results about long-horizon volatility.

### IX. Conclusions

- Top of page
- ABSTRACT
- I. Long-Horizon Variance and Parameter Uncertainty
- II. Empirical Framework: Predictive Systems
- III. Components of Predictive Variance (System 1)
- IV. Perfect Predictors versus Imperfect Predictors (System 2)
- V. Robustness
- VI. Predictive Variance versus True Variance
- VII. Long-Horizon Variance: Survey Evidence
- VIII. Target-Date Funds
- IX. Conclusions
- Appendix
- REFERENCES

We use predictive systems and up to 206 years of data to compute long-horizon variance of real stock returns from the perspective of an investor who recognizes that parameters are uncertain and predictors are imperfect. Mean reversion reduces long-horizon variance considerably, but it is more than offset by other effects. As a result, long-horizon variance substantially exceeds short-horizon variance on a per-year basis. A major contributor to higher long-horizon variance is uncertainty about future expected returns, a component of variance that is inherent to return predictability, especially when expected return is persistent. Estimation risk is another important component of predictive variance that is higher at longer horizons. Uncertainty about current expected return, arising from predictor imperfection, also adds considerably to long-horizon variance. Accounting for predictor imperfection is key in reaching the conclusion that stocks are substantially more volatile in the long run. Overall, our results show that long-horizon stock investors face more volatility than short-horizon investors, in contrast to previous research.

In computing predictive variance, we assume that the parameters of the predictive system remain constant over 206 years. Such an assumption, while certainly strong, is motivated by our objective to be conservative in treating parameter uncertainty. This uncertainty, which already contributes substantially to long-horizon variance, would generally be even greater under alternative scenarios in which investors would effectively have less information about the current values of the parameters. There is, of course, no guarantee that using a longer sample is conservative. In principle, for example, the predictability exhibited in a given shorter sample could be so much higher that both parameter uncertainty as well as long-run predictive variance would be lower. However, when we examine a particularly relevant shorter sample, a quarterly postwar sample spanning 55 years, we find that our main results are even stronger.

Changing the sample is only one of many robustness checks performed in the paper. We also consider a number of different prior distributions and modeling choices, reaching the same conclusion. Nonetheless, we cannot rule out the possibility that our conclusion would be reversed under other priors or modeling choices. In fact, we already know that, if expected returns are modeled in a particularly simple way, assuming perfect predictors, then investors who rely on the postwar sample view stocks as less volatile in the long run. By continuity, stocks will also appear less volatile if only a very small degree of predictor imperfection is admitted a priori. Our point is that this traditional conclusion about long-run volatility is reversed in a number of settings that we view as more realistic, even when the degree of predictor imperfection is relatively modest.

Our finding that predictive variance of stock returns is higher at long horizons makes stocks less appealing to long-horizon investors than conventional wisdom would suggest. A clear illustration of such long-horizon effects emerges from our analysis of target-date funds. We demonstrate that a simple specification of the investment objective makes such funds appealing in the absence of parameter uncertainty but less appealing in the presence of that uncertainty. However, one must be cautious in drawing conclusions about the desirability of stocks for long-horizon investors in settings with additional risky assets such as nominal bonds, additional life cycle considerations such as intermediate consumption, and optimal dynamic saving and investment decisions. Investigating asset allocation decisions in such settings while allowing the higher long-run stock volatility to enter the problem is beyond the scope of this study but offers interesting directions for future research.