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.

Dangers of data mining: The case of calendar effects in stock returns

Sullivan, Ryan, Allan Timmerman, and Halbert White, “Dangers of data mining:  The case of calendar effects in stock returns,” Journal of Econometrics 105 (2001), 249-286.

Using the same set of data to both formulate and test a hypothesis introduces data-mining biases.  Calendar effects in stock returns are an outstanding instance of data-driven findings.  Evaluated correctly, however, these calendar effects are not statistically significant.

Researchers have documented day of the week effects, week of the month effects, month of the year effects, and effects for turn of the month, turn of the year, and holidays, none of which was predicted ex ante by theory.  By pure statistical chance, when enough theories are tested on the same set of U.S. publicly-traded common stock returns, some of them are bound to outperform a benchmark, no matter which criteria are used to compare performance.

This paper uses 100 years of data to examine a “full universe” of 9453 calender-based investment rules, and a “reduced universe” of 244 rules.  Investment strategies are tested jointly with many other similar strategies.  Report nominal p-values, and White’s reality check p-value for each null hypothesis of no effect.  White’s p-value adjusts for the data-mining bias.

Conclusions:  Nominal p-values are highly significant for many strategies, but White’s reality check p-values are not significant for any calendar-based strategy.

Stock Market Prices do not Follow Random Walks: Evidence from a Simple Specification Test

Lo, Andrew W., and A. Craig MacKinlay, “Stock Market Prices do not Follow Random Walks:  Evidence from a Simple Specification Test,” The Review of Financial Studies, Vol 1 No 1 (1988), 41-66.

Most tests of the efficient market hypothesis have assumed that common stock returns follow a random walk.  However, some papers, including this one, have presented evidence against the random walk hypothesis.

Methods & Data:  Lo & Mackinlay use weekly return data from September 6, 1962 to December 26, 1985.  The test relies on the characteristic of a random walk whereby the variance between increments is linear in the interval between increments.  In other words, the variance of monthly observations should be about four times the variance of weekly observations.


  • The random walk hypothesis is rejected for weekly stock market returns.
  • The rejection is especially strong for small stocks, but is not entirely explained by infrequent trading or time-variation in volatility.
  • Weekly stock returns do not appear to be mean-reverting.
  • This does not mean the market is not efficient, but it does indicate that the random-walk model is not correct.

Risk, Return, and Equilibrium Empirical Tests

Fama, Eugene F., and James D. MacBeth, “Risk, Return, and Equilibrium Empirical Tests,” The Journal of Political Economy, Vol 81, No 3 (1973), 607-636.

Purpose:  To test market efficiency, the two-period portfolio model, and the tradeoff between risk and expected return.


  • In the two-parameter portfolio model, the risk of an individual asset is proportional to its contribution to the total portfolio’s ratio of expected value to dispersion (typically, standard deviation).
    • Each asset’s risk depends upon its weight in the portfolio, its covariance with other portfolio assets, and its standard deviation.  The same asset, therefore, can have different risk levels in different portfolios.
    • Risk-averse investors choose assets and weights to form an “efficient portfolio,” or one that maximizes expected return for any level of dispersion.

Theoretical Background:

  • The expected return-dispersion model, E(R_i) = E(R_0) + \beta[E(R_m) - E(R_0)], makes three testable predictions:
    •  In an efficient portfolio, an asset’s relationship between expected return and risk should be linear.
    • \beta_i = \frac{cov(R_i, R_m)}{\sigma^2(R_m)} should measure the total risk of security i in the portfolio m.
    • Higher risk should be associated with higher expected return.
  • In an efficient capital market, investors should form efficient portfolios that fit the model given above.


  • Data are NYSE monthly stock returns for 1926-1968
  • Market return is estimated by the equal-weighted NYSE return
  • Form portfolios of stocks
    • estimated portfolio betas exhibit less error than the sum of individual security betas if the individual betas are not perfectly positively correlated
    • to avoid bunching positive and negative errors in the portfolios, stocks are sorted into portfolios by their betas in one period, then data from a different period are used to calculate portfolio betas.
      • Use 1926-1929 data to sort NYSE stocks into 20 portfolios
      • Use 1930-1934 data to calculate portfolio betas in 1935; use 1930-1935 data to calculate portfolio betas in 1936, etc.
      • portfolios are reformed every four years
      • e.g, regressions in 1950 are on portfolios formed using 1935-1941 data but with portfolio returns calculated using 1942-1949 data.


  • The beta-return relationship is linear for all periods except the five-year post-war period 1951-1955
  • Beta appears to be a very complete measure of risk in all periods.
  • The expected return-beta relationship is positive for all periods but the five years 1956-1960, where it is slightly negative.
  • Findings are consistent with an efficient capital market, where risk-averse investors assemble efficient portfolios.