310  Estimating the Cost of Capital to estimate growth,5 but many argue that analyst forecasts focus on the short term and are upward biased. In 2003, Eugene Fama and Kenneth French used long-term dividend growth rates as a proxy for future growth, but they focus on dividend yields, not on available cash flow.6 Therefore, we believe this implementation is best. To convert the real expected return into a nominal return appropriate for discounting, add an estimate of future inflation that is consistent with your cash flow projections. In the United States, the Federal Reserve Bank of Phila- delphia provides a long-run forecast of expected inflation.7 In December 2018, this equaled 2.3 percent. Alternatively, you can estimate expected long-term inflation using the spread between the yield on inflation-protected bonds and regular government bonds. In 2018, this spread was approximately 1.7 per- cent. When you add inflation in the range of 1.7 to 2.3 percent to a real return of 7 percent, you get an expected market return of 8.7 to 9.3 percent. Later in this chapter, we use the CAPM to adjust the market return for com- pany risk. The CAPM requires an estimate of the market risk premium, mea- sured as the difference between stock returns and the return on risk-free bonds. Using data from 1962 to 2018, we estimate the average inflation-adjusted stock market return at 7 percent and the average inflation-adjusted U.S. Treasury re- turn at 2 percent. The difference represents a market risk premium of 5 percent. 6 E. F. Fama and K. R. French, “The Equity Premium,” Journal of Finance 57, no. 2 (April 2002): 637–659. 5 J. Claus and J. Thomas, “Equity Premia as Low as Three Percent? Evidence from Analysts’ Earnings Forecasts for Domestic and International Stocks,” Journal of Finance 56, no. 5 (October 2001): 1629–1666; and W. R. Gebhardt, C. M. C. Lee, and B. Swaminathan, “Toward an Implied Cost of Capital,” Journal of Accounting Research 39, no. 1 (2001): 135–176. 7 See Federal Reserve Bank of Philadelphia, Survey of Professional Forecasters, www.philadelphiafed .org. EXHIBIT 15.2  S&P 500 Real and Nominal Expected Returns, 1962–2018 % 0 4 8 12 16 20 1962 1972 1982 1992 2002 2012 Nominal expected return Real expected return  Estimating the Cost of Equity  311 Alternatively, if we expect the market to earn 7 percent in real terms going forward and subtract the December 2018 inflation-adjusted interest rate of 1 percent, this implies a market risk premium going forward of 6 percent. While we are not averse to this larger-than-normal risk premium, our statistical tests do not provide confirming evidence that risk premiums have risen. If this were the case, low-risk stocks should increase in value relative to high-risk stocks, because as the price of risk rises, high-risk stocks require greater re- turns and consequently have lower valuations. When we examined the trend of P/Es for low-risk stocks versus high-risk stocks, we did not observe any widening of the spread as real interest rates fell, even to historical lows. Historical Estimates of the Market Risk Premium  A second method to esti- mate the expected market return starts with a historical estimate of the market risk premium and then adds this estimate to today’s long-term government bond rate. We add today’s rates so the estimate of the expected market return incorporates current interest rates, rather than those in the past. Estimating the historical risk premium properly requires some statistical sophistication. A full description of the most relevant issues is available in Ap- pendix F; we offer only a summary here. First, use as long a time period as possible. Our work relies on research by Elroy Dimson, Paul Marsh, and Mike Staunton, who provide market returns dating back to 1900.8 Although some argue that market risk premiums have dropped over time, a simple regression analysis does not support this. Therefore, we believe more data improve the quality of estimation. Second, neither the arithmetic average nor a geometric av- erage of past returns will estimate multiyear discount rates well. The best value falls somewhere between the two averages. While the arithmetic average is best for estimating a one-period return, compounding the average return also com- pounds any estimation error, causing the compounded number to be too high. To counter this bias, Marshall Blume created an estimator using a combination of the two averages.9 Exhibit 15.3 presents the average cumulative returns of the U.S. stock mar- ket, the U.S. bond market, and excess returns (stocks minus bonds) between 1900 and 2018. Using five- to ten-year holding periods, the average annual excess return is 5.5 to 5.7 percent. Blume’s estimator for longer-date cash flows is slightly higher, at just above 6 percent. Even with the best statistical tech- niques, however, this number is probably too high, because the observable sample includes only countries with strong historical returns.10 Statisticians 8 E. Dimson, P. Marsh, and M. Staunton, “The Worldwide Equity Premium: A Smaller Puzzle,” in Hand- book of Investments: Equity Risk Premium, ed. R. Mehra (Amsterdam: Elsevier Science, 2007). 9 D. C. Indro and W. Y. Lee, “Biases in Arithmetic and Geometric Averages as Estimates of Long-Run Expected Returns and Risk Premia,” Financial Management 26, no. 4 (Winter 1997): 81–90; and M. E. Blume, “Unbiased Estimators of Long-Run Expected Rates of Return,” Journal of the American Statistical Association 69, no. 347 (September 1974): 634–638. 10 S. Brown, W. Goetzmann, and S. Ross, “Survivorship Bias,” Journal of Finance (July 1995): 853–873.