Nov 9, 2009
The realized equity premium for U.S. stocks relative to long-term government bonds has been negative for the 5, 10, 15, 20, and 25-year periods ending in 2008 despite substantially greater standard deviation for stocks. How do I use this information to develop a sensible portfolio based on mean-variance optimization?

EFF: We have emphasized in previous posts that there is substantial uncertainty about the size of the expected equity premium, that is, the true expected return on stocks less the expected return on riskless bonds. Whatever estimate you use, 5, 10, or even 15 years of recent evidence should not change your estimate much. 20 or 25 years of data are more serious, but then there is another issue. 

Specifically, if the expected equity premium changes through time, a 25-year period with a negative realized premium may signal that the expected premium is higher going forward. In other words, stock prices may have fallen relative to bond prices in large part to generate higher expected stock returns going forward. If this is correct, then a bad 25-year period for the equity premium is probably a signal to raise your estimate of the expected equity premium going forward. 

Optimizers are quite sensitive to estimates of expected returns. Because the estimates are unavoidably quite noisy, optimizers are probably a waste of time, except perhaps as a rough guide. 

One of the benefits of asset pricing models is to make optimizers irrelevant if markets are efficient. For example, in the CAPM, optimal portfolios are just combinations of riskless bonds and the market portfolio of risky assets, and there is no need to run an optimizer to identify them. In a multifactor model, like the Fama-French three-factor model, optimal portfolios are combinations of riskless bonds, the market portfolio, and diversified portfolios that provide tilts toward other relevant risk factors in returns. Again, there is no need to run an optimizer. There is still a ton of unavoidable uncertainty about the size of the expected premiums for different sources of risk, but an optimizer is no help with that. 

KRF: The question seems driven by a common misconception about how one should use mean-variance optimizers to construct portfolios. As Gene emphasizes, small changes in the inputs can produce huge changes in the outputs. Two investors with slightly different estimates of expected returns, for example, may find that the "best" portfolios identified by their optimizers are quite different. The problem is particularly important because expected returns are notoriously hard to estimate. But let's ignore this problem. Doesn't the negative equity premium for the last 25 years make it impossible to use an optimizer anyway? Of course not. 

If you insist on using a mean-variance optimizer, your inputs should describe the future, not the past. You want the best forecasts of each asset's expected return and variance, and of the covariances between each pair of assets. Past returns might help you forecast, but it would be crazy to use them mechanically as your best guess of future returns - particularly when that would cause you to forecast a negative equity premium. You might instead combine the average premium from a much longer prior period with estimates from a different approach, such as the one Gene and I use in "The Equity Premium" (Journal of Finance, 2002).

Eugene F. Fama
The Robert R. McCormick Distinguished Service Professor of Finance at the University of Chicago Booth School of Business
Kenneth R. French
The Roth Family Distinguished Professor of Finance at the Tuck School of Business at Dartmouth College
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Eugene Fama and Ken French are members of the Board of Directors of the general partner of, and provide consulting services to Dimensional Fund Advisors LP.