What's the Long-Term Expected Return to Your Portfolio?

Abstract

Many investors and investment consultants use mean-variance optimizers to select efficient portfolios. Mean-variance optimization is inherently a single-period technique. Most investors use it in a multiyear context by assuming annual rebalancing and independence and stationarity of returns. These assumptions are often reasonable, and they enable investors to estimate a portfolio's future growth rate, or long-term expected return. Investors such as definedbenefit plan sponsors and foundations have a keen interest in the long-term expected returns to their portfolios. Sponsor contributions and foundation budgets are often tied directly to this estimate of the future growth rate of assets. This focus on a future portfolio growth rate appears to lead many practitioners mistakenly to use past growth rates to asset classes as inputs to their mean-variance optimizers. The consequence of this error is an estimated longterm expected return to the portfolio that is likely to be at least 50 basis points lower than the correct estimate. Correct estimation of a portfolio's future multiperiod growth rate is a two-step procedure. First, determine the single-period characteristics of portfolio returns. Then, estimate the multiperiod characteristics of those singleperiod returns. A simple example illustrates this point. From Ibbotson Associates' Stocks, Bonds, Bills, and Inflation 1994 Yearbook, the compound annual growth rates to largecompany stocks and long-term corporate bonds from December 31, 1925, through December 31, 1993, were 10.3 percent and 5.6 percent, respectively. These numbers could be used as estimates of future long-term expected returns to stocks and bonds. To estimate the future growth rate to a portfolio of stocks and bonds, rebalanced annually to 50/50, one might be tempted simply to take the arithmetic average of the stock and bond growth rates. This calculation is quick and easy, and it yields an estimated future portfolio growth rate of 7.9 percent. One might think 7.9 percent is a reasonable estimate, but it is not. Using the two-step procedure outlined in the preceding paragraph, first calculate all 68 annual portfolio returns to stocks and bonds. In the second step, calculate the annual growth rate from these 68 annual portfolio returns. That growth rate is 8.4 percent, 50 basis points higher than the quick and easy estimate of 7.9 percent! The stock/bond version of the erroneous calculation is given in Equation 1. Equation 1 is appropriate for calculating the singleperiod, say annual, expected return to a portfolio, because it holds for actual annual rates of return. Skipping Step 1 by using growth rates in Equation 1 is a mistake, however, because a portfolio growth rate is not a linear combination of the growth rates of the assets in the portfolio. An annualized growth rate calculated from a series of annual returns will be lower than the arithmetic average of the returns because of volatility. A simple approximation of this relationship is given in Equation 4. Use of Equation 1 with long-term expected returns ignores the reduction in volatility coming from diversification between the two assets and yields a long-term expected return that is too low. The more diversified the portfolio, the larger the error. To convince yourself about this, just follow through the details of the stock and bond example below. This is the same example as above, except that Equation 4 is used for Step 2 rather than a calculation of all 68 annual portfolio returns.