Ang, Andrew, Robert J. Hodrick, Yuhang Xing, and Xiaoyan Zhang, “The Cross-Section of Volatility and Expected Returns”, *The Journal of Finance, Vol 61, No 1 (2006)*, 259-299.

**Purpose:** To show that stocks with high volatility have low average returns.

**Findings:**

- Stocks that are sensitive to aggregate volatility earn low average returns.
- Stocks with high idiosyncratic volatility also earn low average returns.
- This effect cannot be explained by exposure to aggregate volatility risk, size, book-to-market, momentum, or liquidity.

**Methods/Data:** The first part of the paper looks at stocks’ sensitivity to aggregate volatility risk. The second and more interesting part concerns idiosyncratic volatility. Data are NYSE stocks for the period 1963-2000.

- Aggregate Volatility
- Create 5 portfolios, and measure their “beta_vix” as the sensitivity of their returns to changes in the VXO (the paper calls it “VIX,” after the newer volatility index that replaced the VXO in 2003).
- The VIX is very highly autocorrelated–0.94 at the daily frequency–so the authors’ assumption that daily changes in the VIX proxy for shocks to volatility is probably justified.
- Use beta_vix from month t-1 to predict returns in month t.

- Idiosyncratic Volatility
- Measure i.vol. as the standard deviation of the residuals on a Fama-French 3-Factor model.
- Compare returns of volatility- and size-ranked portfolios.

**Results:**

- High sensitivity to aggregate volatility is related to lower earnings, since a stock’s high volatility is a hedge against market volatility. The stock becomes volatile at the same time the broader market does, making the stock less likely to fall or rise simultaneously with the market.
- The aggregate volatility results are robust to controlling for liquidity, volume, and momentum, but not to time period. The effect disappears if volatility from month t-2 is used to predict month t returns, or if month t-1 volatility is used to predict t+1 returns.
- High idiosyncratic volatility means lower returns. This result is robust to controls for size, book-to-market, leverage, liquidity risk, volume, share turnover, bid-ask spread, coskewness, dispersion of analyst forecasts, momentum, aggregate volatility risk, and–unlike the aggregate volatility effect–to different time periods.
- volatility in month t-1 explains returns in month t+1.
- volatility for month t-1 explains returns for months 2-12.
- volatility for months t-12 to t-1 explain returns in month t+1.
- volatility for months t-12 to t-1 explain returns for months 2-12.
- The effect is present in every decade of the sample period, and are stronger in the more recent half of the full period.
- The effect is also significant both in periods of high aggregate volatility and in stable periods, in periods of recession and expansion, and in bull and bear markets.

- Authors cannot rule out the Peso problem.
- The Peso Problem comes from a study testing the efficient markets hypothesis in the Mexican stock market. The data rejected market efficiency, the authors believed, due to investors expectation of a coming devaluation of the Peso. The data ended in June without any devaluation observed, and the Peso was devalued two months later in August. The Peso problem can be stated as the latent (leading or lagged) of something just outside the data window that affects statistical inference.