An analytical derivation of the relation between idiosyncratic volatility and expected stock return

Modelling stock return generating process as a single factor model, we show analytically that the relation between idiosyncratic volatility measured as variance of the residuals and expected stock return in the cross-section may be represented by a parabola that opens to the left and has horizontal axis. This relation is uncovered for stocks of similar volatility and no abnormal return. The sensitivity of the derived relation when these restrictions are relaxed is discussed. Our findings, to a great extent, help uncover the shape of the non-linear inverse relation between idiosyncratic volatility and expected stock return observed in the cross-section in previous empirical studies.