"Equity Risk Premium: Expectations Great and Small," Richard A. Derrig and Elisha D. Orr, January 2004/AUTHORS' REPLY

North American Actuarial Journal,  Jan 2005  by Whelan, Shane F

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AUTHORS' REPLY

We welcome Whelan's discussion for highlighting the role that stationarity plays in discussing the theoretical formation of an equity risk premium (ERP) and for the introduction of data from the Irish capital markets. Our reply consists of three points. First, we do not view the assertion of the absence of weak stationarity as a "devastating critique," as Whelan concludes. Second, the principal thrust of our paper was the wide definitional disparity among the many studies of the ERP puzzle for the U.S. market and their subsequent expectations for the future, most of which avoided or ignored the question of stationarity in any form. Third, our recommendation to practicing actuaries was to use the Ibbotson-Chen building-block method to forecast the ERP, a tool that could be applied equally well to the Irish and other equity markets and that does not depend on stationarity but does depend on replication of the historical mean for each block, absent a rationale and an estimate for a change in the block value.

STATIONARITY

Strictly speaking, a time series is stationary if all of the statistical properties remain unchanged when the period of observation is shifted forward or backward, or equivalently, if the distribution functions of all consecutive subseries are independent of time (Kruskal and Tanur 1978, pp. 1168-69; Kendall and Stuart 1976, p. 424). Thus the mean, variance, and all other existing moments will remain the same when the period of observation is shortened or lengthened in a stationary series. Weakly stationary generally means that only the first two moments, mean and variance, need to be equal. The ERP puzzle literature we reviewed relates only to the expected mean and only incidentally to the other moments. Whelan discusses our test for stationarity of the Ibbotson 1926-2002 series (Derrig and Orr 2004, pp. 51-52), where we informally define stationarity as a mean value unchanging with time (Kendall and Stuart [1976, p. 424] define a separate "stationary in the mean" as the "customary" definition of stationarity of stochastic processes), in line with the ERP puzzle, and test for equal means for the entire series and the 1960-2002 subperiod of the Ibbotson annual data. We find that the t-test supports equal means whether or not the variances are assumed equal or not and that there is also some support for unequal variances: that is, the entire Ibbotson series is not weakly stationary. This result is due to the large volatility of the depression years of the 1930s (41.6% versus less than 20% for later decades; see Ibbotson Associates 2004, Yearbook, Table 6-1, p. 110), much as the latter years of the Irish market data appear to be more volatile. (Ibbotson Associates' [2004, Valuation Edition, pp. 85-86] graphic shows the large pre-World War II volatility similar to Whelan's post-1970s Irish market.) Table 1 indicates that beginning the Ibbotson series in 1943 (60 years) would give us an annual ERP series with about the same subperiod (30 years) means but equal variances.