Some Analytical and Empirical Results on the Relation between Idiosyncratic Volatility and Expected Stock Return

Abstract

We show analytically that the cross-sectional relation between idiosyncratic volatility estimated as the variance of the residuals in a single factor model and expected stock return may be represented by a truncated parabola that opens to the left and has horizontal axis. This relation is uncovered for stocks of similar volatility and no abnormal return estimated in the factor model. The sensitivity of the relation between idiosyncratic volatility and expected stock return to these restrictions are discussed. Our findings may be extended to the multi-factor case as well. A non-linear inverse relation between idiosyncratic volatility and expected return is more likely to be observed when portfolios are formed controlling for abnormal return and total risk. Our interpretation of the idiosyncratic volatility and expected return relation help explain the inconsistency in the cross-sectional relation between idiosyncratic volatility and expected stock return observed in empirical studies. We provide empirical evidence to suggest that the relation between (i) idiosyncratic volatility and expected return is generally positive and is robust to the portfolio formation scheme and sample period and (ii) lagged idiosyncratic volatility and expected return is sensitive to the sample period.