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Exhibit 10.7 The Interval Effect
Year HM Return Arithmetic average Geometric average
i S 10 one-year intervals 4.70 4.17
2 -10 5 two-year intervals 4.51 4.17
3 8 2 five-year intervals 4.19 4.17
4 16 1 ten-year interval 4.17 4.17
Exhibit 10.8 shows the market risk premium for U.S. large capitalization stocks using the arithmetic mean for different return periods. For example, for the three-year periodicity, we calculated the three-year returns for 24 periods and then took the arithmetic average of the three-year returns (annualized to one year). The results show that the estimated arithmetic average declines as you average over longer intervals. There is no guidance or intuition that would lead us to conclude that the CAPM, a one-period model, is necessarily a one-year model. Note that the arithmetic risk premium, based on two-year intervals, is a full one percent less than the premium based on one-year intervals. Given the large gap between one- and two-year intervals compared with the gap between two years and all other intervals, we chose to base our market risk premium estimate on the two-year interval.
Our choice of a two-year or greater interval is supported by evidence that historical returns are not independent draws from a stationary distribution. Empirical research by Fama and French (1988), Lo and MacKinlay (1988), and Poterba and Summers (1988)10 indicates that a significant long-term negative
Exhibit 10.8 Arithmetic Average for Various Intervals
1926-1998 Large company Long-term Market risk
stocks government bonds premium
Arithmetic mean of 1-year returns Arithmetic mean of 2-year returns Arithmetic mean of 3-year returns Arithmetic mean of 4-year returns Geometric mean
13 2 11. 'J 11.6 11.4 11-2
7.5 6 5 6.3 6.1 5.9
autocorrelation exists in stock returns. The implication is that the true market risk premium lies between the arithmetic and geometric averages.
Brown, Goetzmann, and Ross first raised survivorship bias as an issue (1995), claiming that survival imparts a bias to ex post returns.11 If the market risk premium were zero, a substantial upward bias would be imparted on markets that survive over a century without going under. Jorion and Goetzmann (1999) have attempted to estimate the survivorship bias by collecting monthly rate of return data from 1921 to 1996 for 39 stock market indices.12 If one looks at geometric returns, the United States outperformed all others during the twentieth century, averaging 6.9 percent in nominal terms annually, or 4.3 percent in real terms (deflating by the wholesale price index) between January 1926 and December 1996. Of the group of 24 markets that existed in 1931, only seven experienced no interruption in trading (the United States, Canada, the United Kingdom, Australia, New Zealand, Sweden, and Switzerland), seven suspended trading for less than a year, and the remaining 10 suffered long-term closure. The breaks were not favorable events. Over World War II the Japanese market fell 95 percent in real terms, and the German market fell 84 percent.
It is unlikely that the U.S. market index will do as well over the next century as it has in the past, so we adjust downward the historical arithmetic average market risk premium. Using the tables in Jorion and Goetzmann, we find that between 1926 and 1996, the U.S. arithmetic annual return exceeded the median return on a set of 11 countries with continuous histories dating to the 1920s by 1.9 percent in real terms, or 1.4 percent in nominal terms. If we subtract a 1 1/2 percent to 2 percent survivorship bias from the long-term arithmetic average of 6.5 percent, we conclude that the market risk premium should be in the 4 1/2 percent to 5 percent range.
Ex Ante Estimates of the Market Risk Premium
An alternative to the historically estimated market risk premium is an ex ante estimate, one based on the current value of the share market relative to projections of earnings or cash flows. One approach estimates the expected rate of return on the market portfolio, E(rm), by adding the analysts' consensus estimate of
10 E. Fama and K. French, ''Dividend Yields and Expected Stock Returns," Journal of Financial Economics (October 1988), pp. 3-26; A. Lo and C. MacKinlay, "Stock Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test," Review of Financial Studies (1988), pp. 41-66; J. Poterba and L. Summers, "Mean Reversion in Stock Prices," Journal of Financial Economics (October 1988), pp. 27-60.
11 S. Brown, W. Goetzmann, and S. Ross, "Survivorship Bias," Journal of Finance (July 1995), pp. 853- 873.
12 P. Jorion and W. Goetzmann, "Global Stock Markets in the Twentieth Century," Working Paper (New Haven,
CT: Yale School of Management, 1999).
growth in the dividend of the S&P 500 Index, g, to the dividend yield for the index, Div/S:
The risk-free rate is then subtracted from the expected return on the market to obtain the forecast of the market risk premium. Analysts have shown limited skill in forecasting price changes in the S&P 500. In addition, the formula that provides the basis for this approach implicitly assumes perpetual growth at a constant rate, g. This is a particularly stringent assumption.