"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

SHANE F. WHELAN*

Derrig and Orr provide a comprehensive overview of current estimates of the equity risk premium (ERP), carefully distinguishing between the many different definitions in common use that lead to much confusion. Their survey concentrated almost entirely on the U.S. markets, with the statistical analysis based primarily on the 77 annual returns over the period 1926-2002 (inclusive) given in Ibbotson Associates (2003), with occasional reference to a longer series of annual returns from 1871 to 2002 in Wilson and Jones (2002) or data over the years 1802-2001 in Siegel (2002). In this discussion I address two issues:

1. I draw attention to another strand of research in this area that demonstrates that returns from capital markets are not a stationary series. If returns are nonstationary, then this undermines the direct use of simple historical averages or estimating the future ERP based on projections from stationary models fitted to the data. As the approach outlined by the authors as well as many of those surveyed assumed returns are stationary, this is a particularly devastating critique.

2. I explore a little further the alternative way of viewing the historical market returns suggested by the authors (section 6), when the U.S. experience is treated as just one realized path of the grand stochastic process that is the capital markets. The past performance of other national capital markets traces other paths, which, though perhaps neither independent nor equally likely, can be used to shed light on the process of asset price formation and the evolving market price of risk. To provide added contrast to the Derrig and Orr study, I treat the experience of the smallest national market with a history as long as the U.S. market: the Irish capital markets. The Irish experience reinforces the earlier remarks on the nonstationarity of the ERP.

IS THE PATH TRACED BY THE U.S. EQUITY RISK PREMIUM WEAKLY STATIONARY?

Derrig and Orr do provide some tests for the ERP being weakly stationary (Sections 6-9) but fail, in my opinion, to interpret them correctly. First, they report that equality of the sample variances over two subperiods can be rejected at the 1% significance level under a standard F-test (footnote 16). This is evidence, insofar as the normality assumption under the F-test is tenable, that the annual ERP does not form a stationary series and, in particular, cannot adequately be modeled as independent and identically distributed as suggested (sections 6 and 9). Second, the t-test they employ to test equality of means in Table 6 (or more strictly, that the mean of the subperiod 1960-2002 equals the mean of the total period 1926-2002) is questionable in light of the reported difference of variances. However, even if the variances were equal and best estimated with just the 1960-2002 data, the test they employ has such low power that it could not reject the null at the 5% critical level if the true ERP in the 1926-1959 fell anywhere in the range -5.7% to 16.3%.1 As this range encompasses all reasonable values for the ERP, the failure to reject the null of constancy of the ERP is really saying more about the paucity of data than about the structure of the data.

If we move from annual return to monthly return data, then the data set increases 12-fold, and statistical testing can be more discriminating between alternative hypotheses. I am unaware of studies testing properties of the monthly ERP, but a considerable literature exists on the properties of monthly returns from both bonds and equities. Loretan and Phillips (1994) is a particularly pertinent study as it demonstrates that U.S. monthly stock returns (from January 1834 to December 1987) are not weakly stationary (even when allowance is made for the well-documented seasonality in means, second moment dependencies, and failure of the fourth moment of the unconditional return distribution). This finding is especially general as it rules out many classes of models popularly used to characterize return data, such as the ARMA suite, ARCH and GARGH processes where the unconditional second moment is constant, and many types of regime-switching models (where the unconditional model found from integrating over all possible regimes is stationary). As the ERP is the difference between volatile stock returns and less volatile cash (or bond) returns, one would expect the ERP series to inherit noncovariance stationarity from the stock return series.

Loretan and Phillips's testing procedure reports that the failure of the ERP's being weakly stationary is due to the nonconstancy of the unconditional variance of the return series, so their finding does not preclude the constancy of the unconditional mean of the returns (and thereby the ERP) over the period. However, if the ERP is a premium for assuming equity risk, and equity risk is measured by the volatility of excess returns,2 then, on economic grounds, one would predict a higher ERP in those times when the equity return series exhibit higher volatility. Hence, we can infer nonconstancy of the ERP from the nonconstancy of the unconditional variance of the stock returns. Since Loretan and Phillips (1994) a number of papers, using different approaches, have appeared that confirm their finding of the nonstationarity of returns from capital markets, although most such studies are based on daily or higher-frequency returns. see, for instance, Ibrahim (2003) for another direct testing procedure that reports failure of weak stationarity in daily returns of the S&P500, or, more indirectly, the very considerable empirical evidence presented in Plerou et al. (1999a, 1999b), and the supporting evidence in Pagan (1996), based on the monthly returns of the S&P Composite Price Index in the period 1928-87, that the fourth moment of the unconditional return distribution of U.S. stocks and stock indices fails-a finding inconsistent with a weakly stationary series where the fourth moment of the innovations exist.