Over 1960 to 2017, we show that a positive risk premium from holding high-beta stocks (versus low-beta stocks) and small-cap stocks (versus large-cap stocks) is reliably earned only after the expected stock-market volatility breaches an approximate top-quintile threshold. The high conditional...

A measurement error in beta that arises from changes in leverage during the beta estimation window contributes in explaining the size effect. Simulations of asset returns show that the magnitude of the bias in equity returns is proportional to the stock market-induced changes in leverage. We...

Using high-frequency data, we decompose the time-varying beta for stocks into beta for continuous systematic risk and beta for discontinuous systematic risk. Estimated discontinuous betas for S&P500 constituents between 2003 and 2011 generally exceed the corresponding continuous betas. We...