Expected stock returns and volatility

Kenneth R. French, G. William Schwert, Robert F. Stambaugh. "Expected stock returns and volatility." Journal of Financial Economics 19 (1987): 3-29.

In this paper, the authors find that the expected market risk premium is positively related to the predicted volatility of stock returns. The authors also find that there is evidence of a negative relation between unexpected stock market returns and unexpected change in volatility of stock returns. This, in turn, is evidence that expected market risk premiums and volatility are positively related.

Daily values of the S&P composite portfolio are used to compute the monthly standard deviation of returns from 1928 through 1984. The standard deviation is computed as follows,

 

A log function is then used to adjust for a positive skewness in the standard deviations.

Conditional forecasts of the S&P return standard deviation and variance are done as follows,

where the log of the variance is the result of a third order moving average process,

The error term, u, in these equations appears normally distributed. As a result, the predicted standard deviations closely follow the real standard deviations.

S&P Monthly Standard Deviations

Predicted S&P Monthly Standard Deviations

To represent returns as a series with changing volatility, the authors use an autoregressive conditional heteroskedasticity (ARCH) model of the form,

This is done as the return and variance processes are jointly estimated. Using a first order moving average process for the error terms, the model is generalized for daily premiums so as to account for positive first order serial correlations in returns induced by non-synchronous trading,

where the moving average coefficient, theta, is negative.

The ARCH model is then generalized by predicting the variance of epsilon using the average of the previous 22 squared errors (comparable to monthly variance estimates as there are approximately 22 trading days in a month),

It is further generalized using a generalized autoregressive conditional heteroskedasticity (GARCH) model of the form,

From Table 2, it can be seen from both ARCH and GARCH models that the variance of daily risk premiums is highly autocorrelated.

The authors then look to estimate the relations between premiums and volatility. They do this by running a regression of excess holding period returns on the forecasted standard deviation or variance,

That is, if alpha=0 and beta>0 then the expected risk premium is proportional to the standard deviation (p=1) or variance (p=2) of returns.

Regressions are then used to measure the relation between excess holding period returns and contemporaneous unexpected changes in volatility (gamma) as follows,

Table 4 summarizes these results. As can be seen, as beta estimates are within 1 standard error of zero.

Tbe volatility regressions show little direct evidence of a relation between expected premiums and volatility.

The authors then use GARCH-in-mean models that allow the conditional mean return to be a function of volatility. These are of the form,

Table 5 results suggest a positive relation between expected premiums and predicted volatility. The volatility beta is 0.073 with a standard error of 0.023. This supports the author's notion of a negative relation between realized premiums and unexpected change in volatility.

It is noted that a negative relation between excess holding period returns and unexpected change in volatility is consistent with a positive ex ante relation between premiums and volatility. It is also shown (similarly in Black (1976)) that the effects of leverage are not enough to account for the magnitude of the negative relation between returns and changes in volatility. As such, it can then be seen as supportive evidence of a positive relation between expected premiums and ex ante volatility.