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Optimal Portfolios for the Long Run

Our study shows that equity risk declines as time horizon expands.

David Blanchett, CFA, 12/10/2013

This article originally appeared in the December/January 2014 issue of MorningstarAdvisor magazine. To subscribe, please call 1-800-384-4000. 

There is surprisingly little agreement among academics about the existence of time diversification, which we define as the anomaly where equities become less risky over longer investment periods. The primary critique of time diversification is theoretical and the primary defense empirical.

Samuelson (1963) and Bodie (1995) pointed out that if stocks are less risky in the long run there is a free lunch for long-run equity investors. Bodie (1995) emphasized the fact that time diversification violates the Black-Scholes option pricing model. If time diversification exists, then options hedging long-run equity risk should reflect a decreasing likelihood of loss over longer time periods (they don’t).1 Campbell and Viceira (2003) argued that empirical evidence shows that stock returns are not independent and identically distributed over time (they tend to mean revert), which implies that long-run stock returns may be predictable and that long-run investors should overweight equities.2

Our Analysis
The majority of past empirical research on time diversification has been based on U.S. stock returns using historical periods either from 1926 to present (Ibbotson data) or 1802 to present (Siegel data). For our analysis, we use historical real stock returns created by Dimson, Marsh, and Staunton (the DMS dataset), obtained from Morningstar Direct. The DMS dataset consists of historical annual returns from 20 different countries3 from 1900 to 2012 (113 years of data). This results in a total of 2,260 years of return data, which are roughly 10 times the annual returns reviewed by Siegel (2008) and approximately 25 times the annual returns available in the Ibbotson data series.

We focus on the annual real returns earned by local investors in bills (cash), bonds, and stocks for the 20 respective countries in the DMS dataset. We use real returns under the assumption that investors in each country seek to maintain some level of inflation-adjusted wealth within that country.

We also use overlapping rolling returns for our analysis, both overlapping and distinct periods. For overlapping analysis, we use the maximum number of return years available for each test period that role forward through time. For example, our one-year return model would include each year from 1900 to 2012; however, for the 20-year period, the last rolling set of returns would be assumed to begin in 1993. The use of overlapping periods results in underweighting the earliest and latest returns in the dataset, since, for example, the years 1900 and 2012 will only be used in a single 20-year simulation while the middle years (for example, 1950) would be used in 20 different rolling periods. This is important given the poor relative performance of 2008 since it will show up less frequently than other periods.

We use the cumulative real growth of the portfolio value (that is, the final inflation-adjusted wealth over the period) to represent the “return” of the portfolio. Instead of using a definition of risk such as standard deviation, which treats outcomes above and below the target goal as equally risky, we use a utility function, which we believe better approximates how investors feel about good and bad outcomes. A utility function also allows us to consider cumulative wealth as the outcome versus annualized return dispersion. More specifically, we use a Constant Relative Risk Aversion utility function, as depicted in Equation 1:

David Blanchett, CFA is Director of Retirement Research with Morningstar Investment Management.  

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