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.

Mean Reversion in Stock Prices?

Kim, Myung Jig, Charles R. Nelson, and Richard Startz, “Mean Reversion in Stock Prices?  A Reappraisal of the Empirical Evidence,” The Review of Economic Studies, Vol 58, No 3 (1991), 515-528.

Background:  In the 1970s and -80s, stock returns were thought to follow a random walk.  Researchers in the late 1980s began to question this view, and used a variance ratio method to show that autocorrelation did exist in stock returns.  Define the “variance ratio” as the return over K periods divided by the product of the return over one period and K.  If returns follow a random walk, this ratio must equal 1.

However, this assumption is not borne out by the data.  The variance ratio is higher than 1 for periods shorter than a year (positive autocorrelation) and is less than one for periods longer than a year (negative autocorrelation).  A common interpretation of this negative autocorrelation over longer periods is to say that returns are mean-reverting.

Fama & French’s approach is to regress the returns from period t to t+k on the return from period t-1 to t:

r_{k,t+K} = \alpha_K + \beta_Kr_{K,t} + \varepsilon_{K,t+K}

In this model, a negative beta indicates mean-reversion, and a zero beta, a random walk.  This model is also better suited to predicting future returns

Purpose:  This paper re-examines the data and finds no evidence of mean reversion after WWII.  Stock returns in the post-war period are actually mean-averting, meaning that disturbances are too persistent to support a mean-reversion theory. Furthermore, indicators of post-WWII mean-aversion are as statistically significant as indicators of mean-reversion for the whole 1926-1986 period.  The comparison of pre- and post-war returns do not support the random-walk hypothesis, but point to a fundamental change occurring at the end of the war.

Method:  Use statistical methods that do not assume returns are normally distributed.


  • Returns are only mean-reverting pre-WWII.
  • Post-war returns are, if anything, mean-averting.
  • The change may have accompanied the resolution of uncertainties surrounding the duration of the Great Depression, the outcome of WWII, and fears of another post-war depression.

Corporate Finance Over the Past 25 Years

Brennan, Michael J., 1995, “Corporate Finance Over the Past 25 Years,” Financial Management, Vol. 24, No. 2, Summer 1995, 9-22.

Purpose:  To review the development of corporate finance research over the period 1970-1995.

Four Chief Developments:

  • One development has relaxed the constraints of the classical MM (Modigliani-Miller) paradigm.
    • The MM theories about dividends and capital structure take firm cash flows as given, and study how the distribution of that income affects its total value.
    • Analysis later shifted to asking how the structure of claims against the cash flow affects the flow itself, which led to studies of the several types of securities issue.
    • Research in 1970 examined the overall corporation; in 1995, it looks more at specific events or transaction types, such as IPOs, takeovers, repurchases of equity, etc.
  • Another development recognizes that managers are not perfect agents (the agency problem).
    • Newly recognized facets of agents’ opportunities and preferences include their information endowments, their discretionary powers, the nature of the incentives embodied in their contracts, their non-financial compensation (perquisites, reputation, power, etc.), and aversion to effort.
  • A third development has replaced the traditional assumption of actors as price-takers with analyses of games under incomplete information. Analysis has begun to move away from discounted cash-flow modeling, and toward techniques similar to those used in pricing options.
  • A fourth development has realigned the aims of argument:
    • Scholars in 1970 were concerned with the implications of the structure of institutions. Their counterparts in 1995 are more interested in institutional structures as responses to specific problems, and in either defending the structures or proposing new ones.


Other Trends:

  • In 1970, bankruptcy was largely ignored, held as synonymous with liquidation, or treated as a cost to offset the tax savings of debt in determining optimal capital structure. Research in 1995 separates direct from indirect bankruptcy costs (and recognizes the latter as substantial); distinguishes between reorganization, bankruptcy, and liquidation; and examines the connection between bankruptcy laws and efficient liquidation.
  • A general trend in corporate finance has been a move away from efforts to prescribe how financial decisions ought to be made, and toward descriptions or explanations of how and why actual decisions are made.

 Unexplored Territory:

As of 1995, little attention had been given to the analysis of knowledge-based firms whose most important capital is autonomous employees (corporate finance research of the past generally assumed that firms relied on physical capital), or to the implications of increased globalization on corporate financial decisions and structures.