The Risk–Return–Sentiment Nexus: Dealing with Low Power and Big Bias

When regressing return on variance, does a low coefficient necessarily indicate low risk-aversion? Considering CAPM tests conditional on investor sentiment, like in Yu and Yuan [2011], we find that the familiar power issue in single-equation CAPM tests is exacerbated when sentiment is high: the expected return is obscured by a higher variance, the predictors of risk exhibit less variation over time, and even more of that variation is noise (attenuation bias). When, following French, Schwert, and Stambaugh [1987], we add the change of risk as a regressor (to control for flight-for-quality effects and obtain 'indirect evidence' of risk aversion) the conclusions of the regression even self-contradict. For a cleaner answer we propose to start, instead, from a Taylor expansion of the stock's price, which induces as regressors the changes in variance, expected earnings, the risk-free rate, and longer-term earnings growth. The coefficient of the change of risk is closer to zero than it is in the extended-CAPM regression, and implies a plausible level RRA. It is also closer to zero when sentiment is high, but this can be fully explained by a lower and shorter-lived predictive power of the proxy conditional on high sentiment; we do not need lower risk aversion to explain this, in short. In fact, the implied point estimate of RRA for high sentiment is higher, not lower.