Further evidence for the negative relationship between stock returns and volatility

Abstract

Using a simple autoregression with exogenous variables (and its transformed error-correction model), we investigate relationships between realized return and risk measured by realized volatility. The empirical results obtained from analysing the German Stock Index (DAX) and the Dow Jones Index (DJ) show a negative relationship between the realized return and risk and also between changes of the realized return and risk for both monthly and quarterly frequencies. There is also some weak evidence of a negative impact of large volatility on changes of the return that can be detected.

The views expressed in this article are those of the author and not necessarily those of the Deutsche Bundesbank.

Notes

¹For various definitions of volatilities, see Bollerslev and Zhou (2006).

²See Li et al. (2005) for a literature survey.

³So far, our model is more similar to the Threshold GARCH (TGARCH) model considered by Glosten et al. (1993).

⁴This reparametrization is commonly used in the context of co-integration and error correction (Banerjee et al., 1990). We use this reparametrization for stationary variables, which has a long tradition in econometric modelling in the field of control theory (Davidson et al., 1978).

⁵The TGARCH models capture volatility processes having two-regime threshold dynamics depending on the sign of the shocks. Our empirical results show that such a sign-asymmetry is not significant for any data or any frequency and is therefore excluded in the following theoretical and empirical discussions.

⁶This phenomenon was already observed in the early 1960s by Mandelbrot (1963) and was denoted as the Joseph Effect that symbolizes a switched persistence between large and small volatilities.

⁷This data series is the longest DAX series, which covers whole years from its beginning. For a better comparison, we take the DJ for the same period.

⁸Andrews (1993) proposes a sample range of 70% as optimal to control the trade-off between zero power when and and zero detection of a possible threshold effect when .