This paper examines the intertemporal relation between risk and return for the aggregate stock market using high-frequency data. We use daily realized, GARCH, implied, and range-based volatility estimators to determine the existence and significance of a risk-return trade-off for several stock market indices. We find a positive and statistically significant relation between the conditional mean and conditional volatility of market returns at the daily level. This result is robust to alternative specifications of the volatility process, across different measures of market return and sample periods, and after controlling for macro-economic variables associated with business cycle fluctuations. We also analyze the risk-return relationship over time using rolling regressions, and find that the strong positive relation persists throughout our sample period. The market risk measures adopted in the paper add power to the analysis by incorporating valuable information, either by taking advantage of high-frequency intraday data (in the case of realized, GARCH, and range volatility) or by utilizing the market's expectation of future volatility (in the case of implied volatility index).
