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.

Do stock market liberalizations cause investment booms?

Henry, Peter Blair, 2000, “Do stock market liberalizations cause investment booms?” Journal of Financial Economics 58 (2000), 301-334.

Purpose:  To show that liberalizing a country’s stock market leads to increased private investment.

Motivation:  International asset pricing theory predicts that a stock market liberalization will be accompanied by a rise in the liberalizing country’s equity prices and by increased investment in physical capital.  Prior research has empirically confirmed the first prediction.  This paper investigates the second.

Findings:  In countries that liberalize their equity markets, where the marginal product of capital is high and domestic cost of capital exceeds the world average, private investment significantly and meaningfully rises.

Data/Methods:  This is an event study of liberalization in a sample of 11 emerging-market countries.

  • Determine dates of liberalization by using date of government mandate, date of first country mutual fund, or date of a jump in the IFC’s Investability Index.
  • Obtain private investment data from the World Bank’s STARS database (Socioeconomic Time Series Access and Retrieval).
  • Find stock returns in local currencies (including dividends) in the IFC Global Index, from the IFC’S Emerging Markets Database (EMDB).
  • Regress changes in log investment on dummies for the year of liberalization and the two following.
    • Include calendar year dummies to control for global macroeconomic trends.
  • Regress changes in log investment on stock returns and lagged stock returns.
    • Again include calendar year dummies.
  • Also use real U.S. interest rates and OECD output growth rates to control for world business cycles.
  • Use dummies to control for other simultaneous reforms: macroeconomic stabilization programs, trade liberalizations, privatization programs, and reductions of exchange controls.
  • Also control for domestic fundamentals, such as GDP growth.


  • Market liberalization leads to increased stock prices.
  • Growth in private investment is strongly correlated with changes in stock prices.
  • The correlation is stronger for valuation changes related to liberalization.
  • Private investment increases after liberalization, even after controlling for global cycles.

Investment and Financing Constraints: Evidence from the Funding of Corporate Pension Plans

Rauh, Joshua 2006, “Investment and Financing Constraints: Evidence from the Funding of Corporate Pension Plans,” The Journal of Finance 61 (1), 33-71.

Purpose:   This paper evaluates the dependence of corporate investing on internal financial resources, using mandatory defined-benefit contributions (MCs) as an instrument for internal resources.

Findings:  Capital expenditures fall by $0.60-$0.70 for $1 in mandatory pension contributions (MCs).  The size of this coefficient is inversely related to credit rating, and is clearest among firms with observable financial constraints.

Motivation:  The Miller-Modigliani model predicts that internal financing and external financing are perfect substitutes.  However, with taxes, agency costs, and asymmetric information, internal financing may be less expensive.  If so, then financial constraints should cause a drop in investment in capital or R&D (internal “financing”).  Previous research observes firms’ response to exogenous funding shocks in small samples.  This paper uses a much larger sample and an instrument for exogenous funding shocks.  There is also a prior debate about the correlation between investment and cash flow, which this paper also addresses


  • Data:  1,522 Compustat firms making DB contributions between 1990 and 1998 (8,030 firm years)
  • Define cash flow as net income plus depreciation/amortization plus pension expense minus pension plan contribution from IRS forms 5500, since pension expense does not equal cash contribution
  • Regress investment on cash flow and pension contribution variables
  • DB pension funding may be endogenous to the firm’s investing opportunities
    • Overcome endogeneity by looking at contributions to funds around the underfunded threshold
    • Identifying assumption is that the relationship between unobserved investment opportunities and funding status is not the same as that between funding status and required contributions.
  • Separate MCs into unexpected and “predictable”
    • Use predictable, unexpected, and both contribution types as independent variables
  • Divide the sample along several observable dimensions (firm age, credit rating, etc)
    • Run the baseline regression (investment on MCs) on each division
  • Look at whether MCs in one period affect investment in surrounding periods
  • Look at whether MCs in constrained firms leads to higher investment in non-constrained firms


  • Capital expenditures decrease $0.60 for $1 in MCs, which is 4-7x the magnitude of the effect of cash flows
  • MCs also have marginal effects on acquisitions and dividends, but not on borrowing
  • Unexpected and predictable MCs each account for equal portions of the variance in total MCs
  • Firms that do not sponsor defined benefit (DB) plans take up about 12% of the capital expenditure slack that constrained DB firms leave on the table when MCs are high

Enjoying the Quiet Life? –commentary

This paper uses a clever check for reverse causality.  The authors find that for firms whose incorporating state adopts anti-takeover laws in the preceding twelve months, wages rise and productivity falls.  The reverse causality story is that rising wages (perhaps due to union pressure) and falling productivity threaten firms in a certain state, and therefore make that state more likely to enact anti-takeover laws.  The authors’ story depends on causality in the right direction.

The paper’s main tests use a binary variable BC to indicate whether a firm’s incorporating state enacted anti-takeover laws in the prior 12 months.  The coefficients on BC in the several regressions are the paper’s main findings.

In the reverse-causality check, the authors replace BC with four dummy variables:

  • Before-1 for firms incorporated in states that had not yet passed legislation but would do so in the next 12 months
  • Before0 for firms incorporated in states that had passed legislation in the prior 12 months
  • After 1 for firms incorporated in states that had passed legislation between 12 and 24 months prior
  • After2+ for firms incorporated in states that had passed legislation two or more years previously

Significant coefficients on Before-1 would have suggested that legislation was passed in response to an already-changing business environment.

Before0 was the variable of primary interest.

Small and Insignificant coefficients on After 1 and After2+ would have suggested that the effects of legislation were short-lived and therefore economically much less interesting.

In this case, the coefficients on Before-1 were small and statistically insignificant, ruling out the reverse causality.  The coefficients on After 1 and After2+ were significant and even larger than the coefficients on Before0, strengthening support for the authors’ story since their reported effects of anti-takeover laws continued to grow in the longer term.

Enjoying the Quiet Life? Corporate Governance and Managerial Preferences

Bertrand, Marianne, and Sendhil Mullainathan, 2003, “Enjoying the Quiet Life?  Corporate Governance and Managerial Preferences,” The Journal of Political Economy 111 (5), 1043-1075.

Purpose:  This paper examines the effect of anti-takeover laws on a variety of firm behaviors.

Findings:  Following the passage of anti-takeover laws, blue-collar wages rise 1%, white-collar wages rise 4%, and plant creation and destruction both fall so that firm size does not significantly change.  Capital expenditures are unaffected.  Total factor productivity falls.   Return on capital falls by 1%.  Findings contradict stakeholder theory that proposes increased efficiency when workers are paid more.  Findings contradict “empire-building” theories of corporate governance that suggest unfettered managers opt to increase firm size.

Motivation:  The reduced-form agency problem is our assumption that managers desire to pursue their own goals, which may not align with shareholders’ best interests.  There are many theories—but no consensus—regarding what managers’ personal goals actually are.


  • Longitudinal Research Database details plant-level employment and wages and plant creation and destruction.
    • The LRD does not include data on workers’ age, education, or tenure.
  • Compustat includes firm-level financial data, and each firm’s state of incorporation.
  • The Census of Auxiliary Establishments has better data on white-collar workers than the LRD.
    • These data are limited to the firm level (matching to specific plants is not possible)
    • These data are not available in every year, so we cannot analyze the trend
  • Corporate governance and firm behavior are endogenous, but this is overcome by studying anti-takeover laws passed by several states at different times.  The laws weakened governance by limiting the threat of hostile takeover, but were not driven by specific characteristics of any firm.  Laws were also passed at different times, so many firms belong both to the treatment group and to the control group in different years.
  • Analysis of firm-level outcomes
    • Difference-in-differences using firms incorporated in states that recently passed anti-takeover laws as the treatment group
  • Analysis of plant-level outcomes
    • The laws passed affected all firms incorporated in the passing state, regardless of the actual plant location
    • Control for regional economic and political variation by considering plants located in the same regions, one incorporated in a passing state, and one incorporated in some other state
  • Check for reverse causality, whereby states with rising wage pressures are more likely to pass anti-takeover legislation.  We do not find significant evidence of this.

Conclusions:  Anti-takeover legislation does change firm behavior.  Managers pay blue-collar workers more and pay white-collar workers much more, thus transferring more benefits to stakeholders.  Contrary to stakeholder theory, this benefit to stakeholders does not create overall improvement, as firm efficiency declines.  Managers avoid either opening or closing plants, undermining the “empire-building” view of manager preferences.  Managers appear to prefer “the quiet life,” with less employee conflict and fewer hard decisions.


The Cross-Section of Expected Stock Returns

Fama, Eugene F. and Kenneth R. French, 1992, “The Cross-Section of Expected Stock Returns,” The Journal of Finance 47 (2), 427-465.

Purpose:  This paper evaluates the joint effect of market beta, firm size, E/P ratio, leverage, and book-to-market equity in explaining the cross-section of average stock returns on NYSE, AMEX, and NASDAQ.

Findings:  Beta does not explain the cross-section of average returns.  Size and book-to-market equity each have explanative power both when used alone and in the presence of other variables.

Motivation:  The Sharpe, Lintner, and Black asset pricing model (beta) has been very influential, but there are notable exceptions to its premises.  Banz (1981) finds a significant size effect.  Bhandari (1988) finds a leverage effect.  Others have argued for effects of the book-to-market equity ratio and the earnings-to-price ratio.  Furthermore, Reinganum (1981) and Lakonishok and Shapiro (1986) find that the beta-return relationship disappears after 1963.


  • Data:  Nonfinancial NYSE, AMEX, and NASDAQ firms from 1962-1989
    • Monthly return data from CRSP
    • Annual accounting data from COMPUSTAT
  • Create portfolios based on size and pre-ranked beta (using trailing data)
  • Calculate the beta for each portfolio-year and assign it to each stock in that portfolio-year
  • Fama-MacBeth Regressions
    • Beta-size portfolios
      • For each month, for the entire cross-section, regress average return on beta, ln(ME), ln(BE/ME), ln(A/ME), ln(A/BE), and E/P
      • Sort stocks into 10 size deciles and then into 100 sub-deciles on “pre-ranking” beta
        • pre-ranking beta is each security’s beta for the 60 months prior to portfolio creation (requiring at least 24 months of data for inclusion in any portfolio)
        • Pre-ranking beta cutoffs are established using only NYSE stocks
    • Book-to-market portfolios and E/P portfolios
      • formed in a similar manner, with stocks sorted on either BE/ME or E/P
    • Size & book-to-market portfolios
      • Match accounting data for fiscal year-ends in calendar year t-1 to returns for the period starting in July of year t and ending in June of year t+1.
      • Use market equity in December of year t-1 to calculate leverage, book-to-market, and E/P ratios.
      • Use market equity in June of year t to measure size.
      • sort stocks into 10 market equity deciles, then into 100 book-to-market sub-deciles.


  • Controlling for size, there is no relationship between beta and average return
  • Size is significant in predicting average returns
  • Book-to-market equity is also significant in predicting average returns, and has an even bigger effect than size
  • The effects of leverage and E/P are captured by size and book-to-market equity

A Further Empirical Investigation of the Bankruptcy Cost Question

Altman, Edward I., 1984, “A Further Empirical Investigation of the Bankruptcy Cost Question,” The Journal of Finance 39 (4), 1067-1089.

Purpose:  This paper measures the direct and indirect costs of bankruptcy, and compares the present value of expected bankruptcy costs to the present value of the tax benefits of leverage to evaluate whether an optimal capital structure exists.

Findings:  Bankruptcy costs, both direct and indirect, are economically significant in the years prior to bankruptcy, and they have a significant impact on determining a firm’s optimal capital structure.

Motivation:  There is very little empirical evidence regarding the costs of bankruptcy, and much debate on whether expected bankruptcy costs are relevant.  Indirect bankruptcy costs have been identified as relevant by some theorists, but never measured.  This paper assumes all bankruptcy costs are relevant, and seeks empirical evidence regarding the size of those costs—whether they are significant or trivial.

Data/Methods:  Measure direct and indirect costs as a percentage of firm value in the years leading up to bankruptcy.

  1. Direct Costs
    1. gathered from files in the U.S. District Bankruptcy Courts
  2. Indirect Costs – compare predicted profits to actual profits
    1. Regression technique
      1. For each forecast year, regress firm sales on industry sales over the prior 10 years
      2. Predict firm sales for each of the three years preceding bankruptcy
      3. Multiply predicted sales by average historical profit margins to get expected profits
      4. Compare expected profits to actual profits to get total indirect costs
      5. Results:  direct costs averaged 6% of firm value in each of the five years prior to bankruptcy; firm value was relatively stable before falling in the final year before bankruptcy; average total bankruptcy costs were 12.1% of firm value in year t-3 and 16.7% in year t
    2. Analyst estimates technique
      1. Data from I/B/E/S on a sample of seven large bankruptcies
        1. Most of the sample still in reorganization, so direct costs unknown
      2. Indirect costs are calculated as forecasted earnings minus actual earnings
      3. Results:  Indirect costs averaged 20% of firm value for years t-3 to t-1 (no forecasts for year t)
    3. Check for bias and reverse causality:  did low earnings cause bankruptcy?
      1. Compare unexpected earnings for high- and low-risk firms (Altman Zeta risk measure)
      2. High-risk firms who do not go bankrupt also have higher unexpected losses
    4. Compare PV(expected bankruptcy costs) to PV(tax benefits of debt)
      1. Use Altman Zeta-model to calculate probability of bankruptcy
      2. PV(bankruptcy costs) > PV(tax benefits) means too much debt

Conclusions:  There is strong evidence that bankruptcy costs are not trivial.  Also, most of the sample firms had a present value of expected bankruptcy costs that exceeded the tax benefits of their debt.  This indicates that these firms were over-leveraged, and that bankruptcy costs are an important factor to consider when analyzing capital structure.  Also, high-risk firms who do not enter bankruptcy also tend to have more unexpected losses, supporting the theory of significant indirect costs tied to bankruptcy risk.