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Traditional theory has often found that an investor will experience reduced volatility over the long-term.  The implication has led investors to buy into various long-term investment strategies that imply reduced risk.  This study (thanks to Abnormal Returns), however, from Lubos Astor and Robert Stambaugh of the Chicago School and Wharton, finds that stocks are more volatile over the long-term.  While volatility does not necessarily imply risk, the findings are interesting nonetheless.   I’ll expand on the findings in the coming days:

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 it is more than offset by various uncertainties faced by the investor, especially uncertainty about the expected return. The same uncertainties reduce desired stock allocations of long-horizon investors contemplating target-date funds.


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 post-war sample spanning 55 years, we find that our main results get even stronger.

Changing the sample is only one of many robustness checks performed in the paper. We have considered 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 post-war 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.

Read the full paper here.

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