The tradeoff between risk and return has long been an important topic in asset valuation research. Most of this research has examined the tradeoff between risk and return among different securities within a given time period. The intertemporal relation between risk and return has been examined by several authors—Fama and Schwert (1977), French, Schwert, and Stambaugh (1987), Harvey (1989), Campbell and Hentschel (1992), Nelson (1991), and Chan, Karolyi, and Stulz (1992), to name a few. This paper extends that research.

There is general agreement that investors, within a given time period, require a larger expected return from a security that is riskier. However, there is no such agreement about the relation between risk and return across time. Whether or not investors require a larger risk premium on average for investing in a security during times when the security is more risky remains an open question. At first blush, it may appear that rational risk-averse investors would require a relatively larger risk premium during times when the payoff from the security is more risky. A larger risk premium may not be required, however, because time periods which are relatively more risky could coincide with time periods when investors are better able to bear particular types of risk. Further, a larger risk premium may not be required because investors may want to save relatively more during periods when the future is more risky. If all the productive assets available for transferring income to the future carry risk and no risk-free investment opportunities are available, then the price of the risky asset may be bid up considerably, thereby reducing the risk premium.^{1} Hence a positive as well as a negative sign for the covariance between the conditional mean and the conditional variance of the excess return on stocks would be consistent with theory. Since there are conflicting predictions about this aspect of the tradeoff between risk and return, it is important to empirically characterize the nature of the relation between the conditional mean and the conditional variance of the excess return on stocks as a group.

The empirical literature on this topic has attempted to characterize the nature of the linear relation between the conditional mean and the conditional variance of the excess return on stocks. However, the reported findings are conflicting. For example, Campbell and Hentschel (1992) and French, Schwert, and Stambaugh (1987) conclude that the data are consistent with a positive relation between conditional expected excess return and conditional variance, whereas Fama and Schwert (1977), Campbell (1987), Pagan and Hong (1991), Breen, Glosten, and Jagannathan (1989), Turner, Startz, and Nelson (1989), and Nelson (1991) find a negative relation. Chan, Karolyi, and Stulz (1992) find no significant variance effect for the United States, but implicitly find one of the world market portfolio. Harvey (1989) provides empirical evidence suggesting that there may be some time variation in the relation between risk and return.

Most of the support for a zero or positive relation has come from studies that use the standard GARCH-M model of stochastic volatility.^{2} Other studies, using alternative techniques, have documented a negative relation between expected return and conditional variance. In order to resolve this conflict we examine the possibility that the standard GARCH-M model may not be rich enough to capture the time series properties of the monthly excess return on stocks. We consider a more general specification of the GARCH-M model. In particular, (1) we incorporate dummy variables in the GARCH-M model to capture seasonal effects using the procedure first suggested by Glosten, Jagannathan, and Runkle (1988), (2) we allow for asymmetries in the conditional variance equation, following the suggestions of Glosten, Jagannathan, and Runkle (1988), (3) we include the nominal interest rate in the conditional variance equation, and (4) we consider the EGARCH-M specification suggested by Nelson (1991) with the modifications mentioned in (1) through (3) above. These models suggest a weak but statistically significant negative relation between conditional variance and expected return.

Two of our findings are somewhat at odds with the existing literature. First, our data provide little evidence to support the belief that the conditional volatility of the monthly excess return on stocks is highly persistent, while Nelson (1991) finds high persistence in the volatility of daily returns. There are no theoretical reasons for the properties of the monthly and daily returns to be the same. In particular, Nelson (1991) argues that as the frequency at which data are sampled becomes very high, persistence should become larger. Second, both unexpected positive and negative excess returns on stocks change the next period's conditional volatility of the excess return on stocks. Unexpected positive returns result in a downward revision while unexpected negative returns result in an upward revision. In contrast, Nelson (1991) and Engle and Ng (1993), using daily data on stock index returns, find that large positive as well as negative unanticipated returns lead to an upward revision in the conditional volatility, although negative shocks of similar magnitude lead to larger revisions. Hence the time series properties of monthly excess returns are somewhat different from those of daily returns reported in Nelson (1991) and Engle and Ng (1993), and our results for monthly data along with the results for daily data reported by others provide a more complete characterization of the time series properties of stock index returns.

The remainder of the paper proceeds as follows. Section I describes the model that forms the basis for our empirical analysis. Section II discusses the econometric issues involved and our estimation methods. Section III contains the empirical results. Section IV concludes.