The empirical evidence on the cross-sectional relation between idiosyncratic risk and expected stock returns is mixed. We demonstrate that the omission of the previous month's stock returns can lead to a negatively biased estimate of the relation. The magnitude of the omitted variable bias depends on the approach to estimating the conditional idiosyncratic volatility. Although a negative relation exists when the estimate is based on daily returns, it disappears after return reversals are controlled for. Return reversals can explain both the negative relation between value-weighted portfolio returns and idiosyncratic volatility and the insignificant relation between equal-weighted portfolio returns and idiosyncratic volatility. In contrast, there is a significantly positive relation between the conditional idiosyncratic volatility estimated from monthly data and expected returns. This relation remains robust after controlling for return reversals.

We propose a measure for extreme downside risk (EDR) to investigate whether bearing such a risk is rewarded by higher expected stock returns. By constructing an EDR proxy with the left tail index in the classical generalized extreme value distribution, we document a significantly positive EDR premium in cross-section of stock returns even after controlling for market, size, value, momentum, and liquidity effects. The EDR premium is more prominent among glamor stocks and when high market returns are expected. High-EDR stocks are generally characterized by high idiosyncratic risk, large downside beta, lower coskewness and cokurtosis, and high bankruptcy risk. The EDR premium persists after these characteristics are controlled for. Although Value at Risk (VaR) plays a significant role in explaining the EDR premium, it cannot completely subsume the EDR effect.