Asset Pricing

Endogenous Disasters and Asset Prices

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Petrosky-Nadeau, Nicolas, Lu Zhang, Lars-Alexander Kuehn, “Endogenous Disasters and Asset Prices,” Charles A. Dice Center Working Paper No. 2012-1 (October 1, 2013).

Purpose: This model produces a realistic equity premium and stock return variance, and endogenously leads to rare economic disasters, at the confluence of small corporate profits, large job flows, and frictions in the matching process that connects unemployed workers with job vacancies.

Model:

  1. A representative household, with both employed and unemployed members, chooses its optimal consumption and asset allocation (holdings of shares in a representative firm and of a risk-free bond).
  2. A representative firm posts job vacancies, and unemployed workers apply for them.
    1. vacancies are costly for the firm.
  3. The labor market is a matching function that produces jobs using vacancies and unemployed workers as inputs.
    1. Matching frictions are composed of fixed and variable hiring costs.
    2. The wage rate is determined by a Nash bargaining process.

Results:

  1. The model generates an equity premium of 5.70%, versus 5.07% in the data (adjusted for financial leverage).
  2. Annual stock market volatility in the model is 10.83%, versus 12.94% in the data.
  3. The model’s interest rate volatility is 1.34%, versus the observed 1.87%.
  4. The equity premium is countercyclical, both in the model and in the data.
  5. The ratio of vacancies to unemployed workers forecasts (with a negative slope) excess returns; this is confirmed in the data.
  6. Rare disasters are endogenous.
    1. The average peak-to-trough magnitude of a disaster is roughly 20%, both modeled and observed.
    2. The probability of a consumption disaster is 3.08% in the model and 3.63% in the data.
    3. The probability of a GDP disaster is 4.66% in the model and 3.69% in the data.
  7. Comparative statics
    1. The value of workers’ activities in unemployment are assumed to have a high value, which makes wages inelastic. When output falls in hard times, wages fall less, and so the cyclical nature of profits and dividends is magnified. This raises the equity premium and makes the stock market more volatile compared to other models.
    2. Job flows are assumed to be about 5%, consistent with previous literature (5% of the workforce quits each month), so frictions in the matching process contribute to macroeconomic risk.
    3. Matching frictions (especially fixed hiring costs) cause marginal hiring costs to fall slowly in a recession and to rise quickly in an expansion.
      1. In a recession, there are many unemployed workers and few vacancies. An additional vacancy has only a slight impact on the likelihood of an existing vacancy being filled, so hiring costs fall slowly.  As workers continue to attrite  at a 5% rate, hiring may not keep up and the economy may fall off a cliff.
      2. In an expansion, there are few unemployed workers and many vacancies.  An additional vacancy in an expansion has a large (negative) impact on the likelihood of a vacancy being filled, so marginal hiring costs rise quickly, hampering the expansion.

Cross-Sectional Dispersion in Economic Forecasts and Expected Stock Returns

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Bali, Turin G., Stephen J. Brown, and Yi Tang, “Cross-Sectional Dispersion in Economic Forecasts and Expected Stock Returns,” The American Finance Association 75th Annual Meeting, Boston (2015).

Purpose:  To show that economic uncertainty is an economically and statistically significant driver of the cross-section of stock returns.

Motivation:  In the ICAPM world, investors care not only about the expected payoff of their investments, but also about their portfolios’ covariances with state variables affecting both future consumption and opportunities for investment.

Data/Methods:

  • Measure economic uncertainty using
    • the dispersion of forecasts from the Survey of Professional Forecasters
      • real GDP growth and real GDP level
      • log (75th pctl forecast / 25th pctl forecast) * 100
    • cross-sectional dispersion in forecasts for output, inflation, and unemployment
  • Fama-MacBeth regressions
    • Sort into deciles based on market beta.
    • Find time-varying “uncertainty betas” of stocks using rolling regressions of stock excess returns on the uncertainty measure, and sort into subdeciles.
  • Economic Uncertainty Index
    • Use Principal Component Analysis to find the common component among seven different proxies for economic uncertainty.

Results:

  • Covariance with economic uncertainty is significantly negatively correlated with higher returns, after controlling for market beta, size, book-to-market, momentum, short-term reversal, illiquidity, co-skewness, idiosyncratic volatility, and the dispersion of analyst forecasts.
  • The beta of the proposed “uncertainty index” appears able to significantly predict future stock returns.

Stock Market Valuations across U.S. States

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Bekaert, Geert, Campbell R. Harvey, Christian T. Lundblad, and Stephan Siegel, “Stock Market Valuations across U.S. States” American Finance Association, 75th Annual Meeting, Boston (2015).

Purpose:  To show that state-specific regulatory environment affects valuation, and to estimate the marginal impact of regulation.

Findings:

  • After controlling for leverage and earnings growth volatility, PE ratios vary across states within the same industry (segmentation).
  • State-specific financial deregulation decrease segmentation.
  • Increased labor laws increase segmentation.
  • Higher state-specific unemployment is linked with higher segmentation.
  • Higher population density is linked with lower segmentation.
  • Segmentation has been decreasing since the mid-1970s.
  • Distance between a given state’s capital and New York’s capital is a statistically significant, but economically small, determinant of segmentation

Methods:

  • Calculate the absolute difference in P/E ratios between an industry in a given state and the same industry in New York (the financial center of the U.S.)
    • price data comes from CRSP, with earnings data from Compustat
    • noise biases the measure upwards, so the measure is smaller for years or states with smaller firms
  • A state’s level of segmentation is given by the value-weighted sum of the measure for all industries in the state.
  • Regress the segmentation measure on variables
    • difference in leverage between industries in the given state and the same industries in New York
    • difference in earnings growth
    • difference in return volatility
    • number of firms in the state
    • time
  • Classify a number of regulation changes that were made during the time sample, and conduct difference in differences tests.

The Cross-Section of Volatility and Expected Returns

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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.

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

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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.

Conclusions:

  • 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.

Models of Stock Returns–A Comparison

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Kon, Stanley J., “Models of Stock Returns–A Comparison,” The Journal of Finance, Vol 39, No 1 (1984), 147-165.

Purpose:  To explain the observed kurtosis (fat tails) and positive skewness in the distribution of stock returns.

Findings:  The discrete mixture of normal distributions proposed in this paper explains these moments better than existing models, including the Student-t.

Motivation:  Mean-variance portfolio theory, option pricing models, and empirical tests of capital asset pricing models and efficient markets make assumptions about the distribution of stock returns.  It has been shown that stock returns cannot be approximated using a single normal distribution, but the normal is just what the theoretical models assume.

Methods:  Returns may be driven by multiple normal distributions–one for random shocks, one for firm-specific information, one for macroeconomic information, etc.  In other words, it is not necessary that each observation of stock return be drawn from the same distribution as all others.  This paper uses a mix of up to five normal distributions and likelihood ratio tests to model the returns of the 30 Dow Jones stocks.

  • Let  r_t = \alpha_i + u_{it}, where r_t is the observed return for period t, and u_{it} is normally distributed with mean zero and variance \sigma_i^2.
  • Let \underline{\gamma}_i be a normal distribution with mean \alpha_i and mean \sigma_i^2.
  • Let \lambda_i = T_i/T be the observations associated with information set I_i over total observations, or the proportion of total observations that are associated with information set I_i.
  • Let \underline{\theta} = \{\alpha_i, ..., \alpha_N, \sigma_i^2, ..., \sigma_N^2, \lambda_i, ..., \lambda_{N-1}\} be a vector of parameters.
  • Let \underline{r} be the vector of returns.
  • The vector of parameters, given a vector of returns, can be found by solving

max \ell(\underline{\theta}/\underline{r}) = \Pi_{t=1}^T \left[\sum_{i=1}^N \lambda_i p(r_t|\underline{\gamma_i})\right].

Results:

  • All 30 Dow Jones stocks can be explained by a mixture of 2, 3, or 4 normals.
  • Returns of the S&P500, value-weighted, and equal-weighted market indices were also explained by mixtures of normals.
  • The mixture of normals describes 27 of the 30 stocks, and all three indices, better than does the student-t distribution, which also has fat tails and has been used to model stock returns.
  • Differences in the means of the normal distributions can explain the skewness of observed stock returns.
  • Differences in the variance of the normals can explain the kurtosis of stock returns.
  • Stock returns do appear to be normally distributed, but the distribution parameters are time-varying, and the timing of parameter shifts also varies across stocks.

Mean Reversion in Stock Prices?

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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.

Findings:

  • 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?

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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.

Conclusions:

  • 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.

The Cross-Section of Expected Stock Returns

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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/Methods:

  • 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.

Conclusions:

  • 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