In fact, if annual returns are uncorrelated over time, and our objective was to estimate the risk premium for the next year, the arithmetic average is the best and most unbiased estimate of the premium. There are, however, strong arguments that can be made for the use of geometric averages. First, empirical studies seem to indicate that returns on stocks are negatively correlated 64 over time. Consequently, the arithmetic average return is likely to overstate the premium. Second, while asset pricing models may be single period models, the use of these models to get expected returns over long periods (such as five or ten years) suggests that the estimation period may be much longer than a year. In this context, the argument for geometric average premiums becomes stronger. Indro and Lee (1997) compare arithmetic and geometric premiums, find them both wanting, and argue for a weighted average, with the weight on the geometric premium increasing with the time horizon. 65 63 The compounded return is computed by taking the value of the investment at the start of the period (Value 0 ) and the value at the end (Value N ), and then computing the following: 64 In other words, good years are more likely to be followed by poor years, and vice versa. The evidence on negative serial correlation in stock returns over time is extensive and can be found in Fama and French (1988). While they find that the one-year correlations are low, the five-year serial correlations are strongly negative for all size classes. Fama, E.F. and K.R. French, 1992, The Cross-Section of Expected Returns , Journal of Finance, Vol 47, 427-466. 65 Indro, D.C. and W. Y. Lee, 1997, Biases in Arithmetic and Geometric Averages as Estimates of Long-run Expected Returns and Risk Premium , Financial Management, v26, 81-90. Geometric Average = Value N Value 0 ! " # $ % & 1/ N − 1
Electronic copy available at: 33 In closing, the averaging approach used clearly matters. Arithmetic averages will be yield higher risk premiums than geometric averages, but using these arithmetic average premiums to obtain discount rates, which are then compounded over time, seems internally inconsistent. In corporate finance and valuation, at least, the argument for using geometric average premiums as estimates is strong. Estimates for the United States The questions of how far back in time to go, what risk free rate to use and how to average returns (arithmetic or geometric) may seem trivial until you see the effect that the choices you make have on your equity risk premium. Rather than rely on the summary values that are provided by data services, we will use raw return data on stocks, treasury bills and treasury bonds from 1928 to 2017 to make this assessment. 66 In figure 3, we begin with a chart of the annual returns on stock, treasury bills and bonds for each year: 66 The raw data for treasury rates is obtained from the Federal Reserve data archive ( ) at the Fed site in St. Louis, with the 3-month treasury bill rate used for
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