Morningstar Advisor - December 2013/January 2014 - (Page 67)

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 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: 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. For our first test, we examine how the optimal allocation to equities changes across varying test periods for each of the different countries. We do this by determining the optimal allocation to equities for each country over each investment period (which is defined as the portfolio with the highest utility) and then running an ordinary least squared regression where we regress the optimal equity allocations (Eq%) against the respective time periods (t ), as noted by Equation 2 below. For example, we find: Eq%t   t  t 1 Ut  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 inflationadjusted wealth within that country. The Results Wt 1 Equation 1 allows us to estimate the utility (U ) received for a given value of wealth (W) for a given investment period (t). We estimate the ending inflation-adjusted value of the portfolio for a given asset allocation at the end of the period for a given risk aversion coefficient (). The risk aversion coefficient measures the degree to which the investor (or decision-maker) is averse to taking risks. The larger the coefficient, the more risk averse the investor. We test levels of risk aversion from 14 to 20 in increments of 1 for this analysis. We determine the optimal allocation to cash, bonds, and stocks for each scenario using a nonlinear optimization routine. The goal of the optimizer is to solve for the allocation that maximizes the resulting utility in the equation, for a given investment period, risk aversion, and historical country returns. The two constraints for our optimization are that the total allocation across the three asset classes must be equal to 100% and that none of the asset class weights can be negative. The intercept ( ) can be interpreted as the optimal equity allocation if there is neither an advantage nor a disadvantage in long holding periods. It can also be interpreted as the optimal equity allocation for a single period since the first observation of the independent variable t1. A positive slope ( ) indicates a positive relationship between investment horizon and utility and a negative slope, the opposite. A negative slope would be consistent with the work of Pastor and Stambaugh (2012). The intercept and slope values for the regressions for four different risk aversion coefficients () for each of the 20 different countries illustrate our results. For the risk aversion levels y2 (low risk aversion), y4 (moderate), and y16 (extremely high), 90% of the slopes are greater than 0, while for a risk aversion level of y8 (high), 95% of the slopes are greater than 0. Those countries with negative slopes or slopes that are zero tend to be relatively small in absolute terms and already have relatively high intercepts (that is, base equity allocation where 3 Austria, Australia, Belgium, Canada, Denmark, Finland, France, Germany, Ireland, Italy, Japan, Netherlands, New Zealand, Norway, South Africa, Spain, Sweden, Switzerland, United Kingdom, and the United States. 4 Technically, the lowest value is 1.001, because a value of 1 would result in an infinite negative utility in our utility function. MorningstarAdvisor.com 67 http://www.MorningstarAdvisor.com

Table of Contents for the Digital Edition of Morningstar Advisor - December 2013/January 2014

Morningstar Advisor - December 2013/January 2014
Contents
Contributors
Letter From the Editor
What’s Your Purpose?
Working for Gen Y
How to Allocate College Savings
Mobius Looks to a New Frontier
Investments á la Carte
Investment Briefs
How to Manage Bonds for Today and Tomorrow
Cloud Is the New Engine of Growth
Knowing Where to Look
Economic Vulnerability Varies by Country
Factor Investing in Emerging Markets
Following the Rules
Exploring Indexing’s Next Frontiers
Frequent Fliers
Family Blind Spots
Optimal Portfolios for the Long Run
Finding Value in a Pricey Sector
Our Favorite Mutual Funds
50 Most-Popular Equity ETFs
Undervalued Stocks With Wide Moats
The Emerging-Markets Roller Coaster

Morningstar Advisor - December 2013/January 2014