The Cross-Section of Volatility and Expected Returns

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.


  • 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.


  • 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.

Are Seasonal Anomalies Real?

Lakonishok, Josef, and Seymour Smidt, “Are Seasonal Anomalies Real?” The Review of Financial Studies, Vol 1, No 4 (1988), 403-425.

If researchers analyze data using 100 different hypotheses, then formulate a theory based on results and then test the theory using the same data, they are very likely to get significant results for the theory.  This problem frequently arises due to the limited scope of stock return data (only a few standard sources).  Phenomena that are actually just noise get reported as asset-pricing anomalies.

In addition, rational efficient-market economists don’t want to publish or read papers that claim the market is efficient and investors are rational.  Therefore, a type of selection bias can occur when the majority of publications show anomalies, even if the majority of evidence argues against them.

This paper studies anomalies using “new” data to avoid the first problem.  The data are the daily Dow Jones Industrial Average returns, from January 1, 1897 to June 11, 1986.  Recent anomalies studies were done with post-1962 or post-1927 data; thus, using the DJIA since its inception adds 30-65 years of new data.

The 30 firms in the DJIA compose almost 25% of the entire NYSE.  The stocks of these very large firms are highly liquid, and so are unlikely to suffer from issues of nonsynchronous trading, which makes the DJIA a good measure of short-term market activity.  However, using the DJIA means that this study cannot test the January effect, which is observed in small stocks.


  • Monday returns are significantly negative (-0.14%).
  • Turn-of-month price increases are greater than the price increase for the entire month.
  • Prices increase 1.5% between Christmas and New Year’s.
  • Rates of return before holidays is 20x the normal rate of return.
  • Most anomalies are quite small in magnitude.
  • There is no consistent monthly pattern in stock returns.
  • There is no significant evidence that returns in the first part of month are different from returns in the last part.

Macroeconomic Seasonality and the January Effect

Kramer, Charles, “Macroeconomic Seasonality and the January Effect,” The Journal of Finance, Vol 49, No 5 (1994), 1883-1891.

This paper seeks to explain the “January effect,” which is the phenomenon of small and low-priced stocks outperforming in January by a much wider and more significant margin than the rest of the year.

Kramer explains this effect by defining separate betas for January.