Individual investors often have long investment horizons. For certain saving purposes, like retirement savings, the relevant horizon can easily span several decades. Understanding the properties of stock returns over such horizons is therefore important.
A long-run investor collects the total compound returns over the investment horizon. Multiplicative compounding implies that the moments of long-run returns are non-trivial functions of the moments of short-run period-by-period returns. This is particularly true for higher-order moments such as skewness. We show in this study that skewness becomes a characterizing feature of long-run return distributions. Specifically, our main message is that compounding inevitably leads to (strongly) positively skewed long-run returns. The strength of the skew-inducing effect of compounding depends primarily on the level of volatility in the single-period return — the higher the volatility, the stronger the effect — and is not qualitatively affected by potential asymmetries in the single-period return distribution.
Large positive skewness implies that mean (expected) long-run returns are often considerably larger than median returns. Focusing on long-run expected returns can therefore be misleading. As a simple illustration, consider the rule-of-thumb that if an asset delivers a 7% annual expected return, it takes 10 years to double the initial investment. This statement is valid in expected terms, but it ignores the uncertainty in the 10-year outcome. Under standard assumptions, if the asset has a 17% annual volatility (like the U.S. market), there is a 50% chance that the initial investment is doubled only after 13 years, and there is a 30% chance that it takes at least 20 years. For more volatile portfolios, these effects become even more pronounced. Our results highlight that an understanding of the likely long-run outcomes of an investment requires considerably more mental effort than the corresponding short-run exercise. Transforming statistics on annual returns into a meaningful characterization of the distribution of 10- or 30-year returns is clearly beyond the capabilities of most investors.
More formal procedures can also be misleading when applied to long-run returns. Investor-preferences for skewness and higher-order moments are often captured by Taylor expansions of standard utility functions. The large effects of compounding on higher-order moments are shown to affect the validity of such Taylor expansions, when applied to returns of annual or longer horizons.
We also show that for horizons longer than a year, skewness in long-run returns is often impossible to empirically estimate with any reasonable accuracy. Indeed, skewness in compound returns turns out to be so difficult to estimate that care is needed also when running Monte Carlo simulations. Larger-than-normal sample sizes are often needed and, for sufficiently long horizons, obtaining reliable simulation results become virtually impossible.