The Cross-Section of Volatility and Expected Returns

 Andrew Ang, Robert J. Hodrick, Yuhang Xing, Xiaoyan Zhang. "The Cross-Section of Volatility and Expected Returns." Journal of Finance (2006): 259-299.

In this paper, the authors seek to examine the effects of aggregate volatility on the cross section of stock returns. They hypothesize that while the time-series relation of volatility and expected market return has been thoroughly researched, further consideration with respect to the cross section of stock returns should show that the volatility of the market return is a systematic risk factor and should be priced in the cross section of stock returns. With this, they then suggest that stocks with different sensitivities to aggregate volatility innovations should have different expected returns. To test this hypothesis, the authors look to show whether the aggregate market volatility is a priced risk factor and, if so, determine the price of this factor. They also examine the cross-sectional relationship between idiosyncratic volatility and expected returns to test whether such a risk factor is orthogonal to existing factors.

Theoretical Motivation

The model that they look to test is of the following form,

where (in order from left to right) the loading on the excess market return, asset sensitivity to volatility risk, and loadings of K other factors are regressed on excess stock returns.

They then simplify the above to arrive at the following with respect to expected returns,


Empirical Test

Because we do not yet know the true set of risk factors and, as a result, cannot observe such loadings of factors, the authors design a practical empirical framework to test their hypothesis.

As a measure of innovations in aggregate volatility, the authors use daily changes in the VIX index. This is represented in their pre-formation regression,

which is used to test the difference in average returns of stocks with different sensitivities to innovations in aggregate volatility. Quintile portfolios are made by sorting from lowest to highest the regressed results of the loadings on the volatility factor. The stocks of each portfolio are then value-weighted. They measure the difference of average returns between portfolios with the highest and lowest volatility coefficients and find a difference of -1.04% per month that is statistically significant at the 1% level. This can be seen in the mean column of Table 1 below,


Factor Risk Explanation

The authors then look to create a factor from aggregate volatility innovations to suggest a factor risk-based explanation for the above results. This factor, called FVIX, is then used to measure ex post exposure to aggregate volatility risk. The following regression is used for the creation of FVIX by estimating the coefficient of the returns on the base assets X,

where the previous quintile portfolios are used for the base assets.

After constructing FVIX, the authors then substitute FVIX for the daily change in VIX in their pre-formation regression to get the following cross-sectional regression,

however, to test this model a contemporaneous relationship between factor loadings and average returns must be shown. The authors do this by observing that the pre-formation FVIX factor loadings and pre-formation VIX factor loadings are very similar. This can be seen in the second panel of Table 1 as follows,

Post-formation factor loadings are also shown in Table 1. It is the full sample post-formation FVIX betas that are then used to examine ex post factor exposure to aggregate volatility risk. To control for market, size, and value factors from the Fama French 3 Factor model, the authors use the following regression,

From Table 1, it can be seen that the full sample post-formation loadings on FVIX are significantly different across the quintile portfolios.

Pricing Aggregate Volatility Risk

The full regression specification used to estimate the unconditional price of the aggregate volatility risk fact is as follows,

which includes momentum and liquidity factors UMD and LIQ, respectively.

The Fama-MacBeth regression results of the above are,


Regression I from above shows the estimated price of volatility risk is -0.08% per month. The effects of volatility can be measured by multiplying the -0.08 from above with the 13.13 ex post spread in FVIX betas from Table 1. This results in a difference in average returns of -1.05%, which is almost the same as the -1.04% difference in the raw average returns, suggesting the difference in raw average returns can be attributed to exposure to aggregate volatility risk.

Idiosyncratic Volatility

The authors then sort portfolios by idiosyncratic volatility to observe its effects on cross-sectional average returns. They define idiosyncratic volatility relative to the Fama French 3 Factor model as the square root of the variance of the residual. This is to control for systematic risk, however, they also sort portfolios by total volatility. As seen by Table VI below, average returns for portfolios sorted by total volatility and idiosyncratic volatility are very similar and the difference in highest and lowest volatility portfolios are statistically significant.



Robustness tests suggest these results cannot be explained by exposures to size, book-to-market, leverage, liquidity, volume, turnover, bid-ask spreads, coskewness, or dispersion in analysts' forecasts characteristics.

Conclusion

The authors conclude that aggregate market volatility is a risk factor that can be observed in the cross-section of stock returns, and that the price of this risk factor is negative and statistically significant.

Conditional Skewness in Asset Pricing Tests

Conditional Skewness in Asset Pricing Tests

By Campbell Harvey and Akhtar Siddique

The single factor CAPM has been proven not to be a reliable tool in predicting stock returns.  Because of this, some have created factors to help explain more of the variation in stock returns such as the SMB and HML factors created by Fama and French. Harvey and Siddique examine the linkage between the empirical evidence on these additional factors and systematic coskewness. Since there is considerable evidence that returns cannot b e adequately characterized by mean and variance alone, this leads to the inclusion of the next moment – skewness.  The hypothesis is that investors will attempt to avoid stocks that are left-skewed due to risk aversion, and these stocks should command higher expected returns as a result.

            To illustrate the relationship between skewness and expected returns, Figure 1 shows the trade-offs between mean, variance, and skewness.  Panel B includes the risk-free rate. Any points that are tangent to the risk-free plane are considered efficient portfolios.  The figure shows that expected return should increase as variance and skewness increase.  Also, the portfolios that are tangent to the risk-free plane are when skewness and variance are the highest.

The formula for finding the coskewness beta is

 

Using these betas, Harvey and Siddique create three portfolios.  The 30 percent of stocks with the most negative coskewness fall in the S- portfolio.  The 30 percent of stocks with the most positive coskewness fall in the S+ portfolio.  Based on the hypothesis and figure 1, the S- portfolio should have higher expected returns. They find the average annualized spread between the returns on S- and S+ portfolios is 3.60 percent from July 1963 to December 1993.  This result is statistically significant.  They compute the coskewness for a risky asset from its beta with the spread between the returns on the S- and S+ portfolios and call this measure B_SKS.  Using the summary stats in table 1, it is apparent that coskewness plays a role in explaining the cross section of asset returns.  Table 2 shows that conditional coskewness can explain a significant part of the variation in returns even when factors based on size and book/market like SMB and HML are added to the asset pricing model.  There is a significant correlation between the pricing errors of these factor models and the S- portfolio.

            Table 3 shows the R-squares of several regressions. It first shows the R-squared of the traditional CAPM and the 3-factor model.  Then it incorporates the S- portfolio and the SKS portfolio.  Both a cross-sectional regression (CSR) approach using rolling betas and a full-information maximum likelihood (FIML) method using constant betas are used.  Overall, when the S- or SKS portfolio are incorporated with the CAPM or 3-factor model, the R-squared increases.  This shows that including coskewness in the model can help explain stock returns over and above what the 3-factor model can.

            They move on to show that the relevance of the SMB and HML factors appears to be dependent on how old the stock is.  SMB is only significant for stocks with less than 60 months of returns.  This could be an IPO effect, where factors other than market (like SMB and HML) may be more useful in predicting the returns on firms with a short return history.  Skewness on the other hand, remains significant across almost all return length groups.

            In table 5, Harvey and Siddique show the momentum factors is related to systematic skewness.  They consider several different definitions of momentum where they vary the horizon (36-2, 24-2, 12-2, 6-2, and 3-2 months) and holding period (1, 3, 6, 12, 24, and 36 months).  All the portfolios in table 5 show a clear relation between average return and skewness.  The portfolios with the higher average returns also have the more negative skewness.

Figure 3 shows the necessity of negative skewness to have higher mean returns when using momentum strategies. The y-axis shows the mean annualized return from the strategy while the length of the line on the x-axis shows the difference in skewness between the winners and losers.  The negative slope of each line indicates that in a momentum-based trading strategy, buying the winner and selling the loser requires acceptance of substantial negative skewness.

            These findings support the hypothesis that skewness helps explain variation in asset pricing and supports the theory that having negative skewness should command higher abnormal returns because investors will naturally avoid those assets because of risk aversion.  Overall, coskewness provides us with some insight into why variables like size and book-to-market value are important and that the momentum effect is related to systematic skewness.  Harvey and Siddique concede that measurement of ex ante skewness may be difficult due to data limitations.

Does Corporate Headquarters Location Matter for Stock Returns?

Pirinsky, C., Wang, Q. “Does corporate headquarters locating matter for stock returns?” Journal of Finance 61.4 (2006): 1991-2015. (link)

This paper explores the connection between corporate headquarter location and stock comovement.  The authors find that stocks in the same region demonstrate comovement with each other that is not explained by the general market or industry comovement.  Furthermore, companies that move their headquarters show less stock comovement with companies in the previous location and greater comovement with those in the new location.  This comovement could be the result of either local fundamental factors or geographic segmentation of investors.  The paper tests and finds evidence against the idea that stock comovement is determined by local economic conditions and instead argues in favor of geographic investor segmentation.

The paper considers the location of a firm to be the location of its headquarters since firms often choose to locate their headquarters in a region that is close to their main operations.  Of 272 Metropolitan Statistical Areas (MSA) used to define location, about 90 have at least five publicly traded companies over the sample period of 1988 to 2002.  This set of MSAs is used in the empirical tests.

To measure a stocks comovement with other stocks in the same MSA the authors construct equal weighted portfolios for each MSA.  They then regress the excess returns of each stock on the return of this local portfolio, $R_i^{LOC}$, (excluding the stock used as the dependent variable) and on the market factor.

\[ R_{t} = \alpha_i + \beta^{LOC}R_i^{LOC} + \beta^{MKT}R_i^{MKT} + \epsilon_{i,t} \]

This regression is also expanded to include an equal weighted industry portfolio.  To address the concern that industries, as defined by CRSP, do not accurately reflect the interaction of local companies, the authors also include whichever two additional industries they find exhibit the greatest comovement with the returns of the particular stock in the regression.

\[ R_{t} = \alpha_i + \beta^{LOC}R_i^{LOC} + \beta^{MKT}R_i^{MKT} + \beta^{IND}R_i^{IND} +  \sum_{K=1}^{2} \beta^{IND_K}R_i^{IND_K} + \epsilon_{i,t} \]

Table 2 presents the results from three regressions over the full sample and two sub-periods.  The location beta is positive and statistically significant across all periods and regressions.  Although including industry returns decreases the magnitude of the location beta, stocks continue to exhibit significant comovement with local portfolios.  When additional industries are included, which are independently correlated with the stock movement, their betas are found to be negative, adding to the strength of the local comovement result.

Table 2

Next, they identify 118 firms that relocate their headquarters during the period 1992-1997.  Many of these firms are small and cite reasons for relocation such as moving closer to customers or the means of production.  The following regression is run for the five years before and after the relocation, not including the year of the relocation.  The returns of the old MSA’s index are represented by $R_t^{LOCO}$ and the returns of the new MSA’s index by $R_t^{LOCN}$.

\[ R_{t} = \alpha_i + \beta^{LOCO}R_i^{LOCO} + \beta^{LOCN}R_i^{LOCN} + \beta^{MKT}R_i^{MKT} + \beta^{IND}R_i^{IND} + \epsilon_{i,t} \]

As shown in table 3, the change in comovement with the old index is negative and the change in comovement with the new index is positive with both statistically significant.  This holds both without the industry factor (2) or with (1).  In fact, the comovement with the old index is reduced by nearly half in the five years after the move.  The authors note that it is unlikely that the change in comovement is due to changes in the fundamental aspects of the firm since the firms in the sample generally did not make any changes in their production process.

Table 3

The authors further test if local comovement of stock prices can be explained by local economic variables.  They find no comovement in local earnings and no explanatory power in a measure of local economic fundamentals.

Finally, the paper address the cross-section of stocks to determine the relationship between common firm characteristics and local comovement.  As seen in table 7, size, return on asset (ROA), and institutional ownership are all negatively and generally significantly correlated with comovement throughout the full sample and three sub-periods.  The interpretation is that smaller stocks, as well as less profitable stocks and stocks with more individual ownership experience greater local comovement.

Table 7

Given the lack of evidence for comovement of local fundamentals, as well as the finding that comovement is stronger for small firms with greater individual investor ownership, the authors suggest local comovement is due to geographic segmentation.  This indicates that geographic diversification is an important consideration.  However the finding that investors’ biases contribute to the local comovement effect indicates a general lack of geographic diversification.  The authors conclude by noting that geographic diversity could play an important role in pension plans and 401(k) accounts.

Price-based Return Co-movement.

Green, T.C. and Hwang, B.H., 2009. Price-based return co-movement. Journal of Financial Economics, 93(1), pp.37-50.

Research Question and General Results:

The paper documents a source of return co-movement related to stock price. Specifically, stocks correlate with their counterparts in the same price category. Using the stock-split setting and extending the result to all stocks, authors find that stocks experiencing splits positively co-move with low-priced stocks and negatively co-move with high-priced stocks. Since the price-based co-movement is not germane to various firm characteristics, authors conclude that investors categorize securities based on their prices.

Empirical Results:

Stock-split serves a relatively clean circumstance for evaluating the price-based co-movement. It decreases a stock’s nominal prices without altering its firm fundamentals. In this paper, authors concentrate on 2-for-1 stock splits and construct corresponding low and high price indices. In the stock-split setting (a decrease in a stock’s nominal price), beta coefficients on low-priced index and high-price index are significantly positive and negative, respectively. This result suggests that a decrease in a stock’s nominal price by its split makes it become more correlated with its low-priced stocks and less with high-priced stocks. Besides, authors find a similar result for all stocks that stocks have an increased co-movement with their counterparts in the same price category. The result is robust to common variations in industry, firm size, transaction cost, and return momentum.

Determinants of Price-based Return Co-movement:

The empirical results demonstrate investors categorize stocks given their prices. Thus, authors conjecture that investors adopt price a naïve proxy for firm size and expect more upside potentials in low-priced stocks. The empirical test also confirms behavioral explanations. The weak relation between institutional ownership and price-based co-movement indicate that the co-movement is hardly driven by market frictions.

Volatility-Managed Portfolios

Moreira, Alan and Tyler Muir. “Volatility-Managed Portfolios” The Journal of Finance Vol. 72 No. 4 (2017) 1611-16430.

“There is little relation between lagged volatility and average returns but there is a strong relationship between lagged volatility and current volatility.” The authors exploit this relationship to show that significant gains can be made through volatility timing. Using last month’s realized variance as a proxy for a portfolio’s conditional variance, they show that, whatever the desired exposure to a variety of different portfolios, Sharpe ratios can be improved by levering down the exposure when last month’s volatility is high and levering up the exposure when last month’s volatility is low. They determine volatility-managed portfolio exposure (LHS of the equation) through weighting by the inverse of the previous month’s realized variance (RHS of the equation).

 

There is a constant, c, which is chosen by the authors to set the unconditional standard deviations of the managed and buy-and-hold portfolios equal, but it does not influence the Sharpe ratio of the managed portfolio.

Managed vs. Original Portfolio:

Here they regress the returns of the managed portfolio on those of the original portfolio. Positive alpha indicates an improvement in the Sharpe ratio. Of the chosen portfolios, only SMB and CMA from Fama and French show no significant improvement at least at the 10% level. When regressing managed portfolio returns on the Fama-French three factor model, the currency carry trade joins SMB and CMA as insignificant but the rest still show positive and significant alpha.

Expanding the Mean-Variance Frontier:

The authors use different factor models which have been shown to be effective in explaining the cross-section of stock return to construct in-sample mean-variance efficient portfolios. That is, for each model considered they find the combination of factors which produces the largest Sharpe ratio. They then run similar analysis as with the individual factors. Here, however, a positive and significant alpha also indicates an overall expansion of the mean-variance frontier. For each model considered, volatility-managed mean-variance efficient portfolios show significant improvement. They also show similar results for sub-sample analysis.

Utility Gains in the Presence of Leverage Constraints:

Relative to a risk-averse investor’s desired exposure to the market return, the authors also show here that for each desired exposure considered (x-axis) there is a positive improvement to the utility of the investor using volatility-managed portfolios even when there are leverage constraints. They consider leverage constraints at 1.5 times (consistent with a 50% margin requirement) and 1 times the portfolio. That is, the volatility managed exposure cannot exceed 1.5 or 1 times the market, respectively. This indicates that there are implementable utility gains for mean-variance investors.

Macro-Finance

John H. Cochrane. "Macro-Finance." Review of Finance (2017): 945-985.

The author discusses different macro-finance models. He contrasts their strengths and weaknesses, suggests paths for further research, and even describes their potential in furthering macroeconomic research related to recessions.

To sample from the different preferences and market structures, he considers ten cases:

1. Habits
2. Recursive Utility
3. Long-run risks
4. Idiosyncratic risk
5. Heterogeneous preferences
6. Rare disasters
7. Utility non separable across goods
8. Leverage; balance sheet; institutional finance
9. Ambiguity aversion, min-max preferences
10. Behavioral finance; probability mistakes

While varying in assumptions regarding markets and consumption preferences, all models are a generalization of marginal utility or discount factor of the form,

where C is consumption, gamma is the risk aversion coefficient, and Y represents additional risk.

In each case the author describes the central idea being that the market fears recessions as well as assets with decreasing values during recessions, and that the market's risk-bearing capacity falls during recessions as a result. Each case differentiates from the rest with respect to the exact state variable for expected returns including consumption related to recent values, news regarding long-run future consumption, cross-sectional risk, or leverage. The author reasons that these state variables are highly correlated and that each resulting model has helped to describe the same underlying idea of risk perceptions and how these perceptions relate to falling investments as well as our understanding of the mismatch between the riskiness of investment projects and the higher risk aversion of those who save. It is also noted, of course, that no model has succeeded in fully describing the equity premium/risk free rate puzzle as well as the excess volatility and business cycle risk premium.

Of the cases, particularly motivating is the idiosyncratic risk model. This model is similarly represented with a discount factor and a state variable,

with y representing the cross sectional variance of individual consumption growth. It's simplicity comes from the need to only assume a cross sectional variance process (y of t+1), however, this process must satisfy certain constraints for return predictability due to the model's simplicity. The volatility of cross sectional consumption must be large and vary greatly over time (as well as at the right times). While this isn't necessarily intuitive (empirical results aren't encouraging), the solution is to have a variance of future consumption that varies over time, or as the author states, "time variation in the conditional variance of the conditional variance of cross sectional risks". This has been explored, as in Constantinides and Ghosh (2017), with further opportunities for research relating to appropriate time-varying moments in micro data. Still, the model represents a simple explanation that people fear times of large idiosyncratic consumption risk (i.e. recessions) and this exacerbates a fear of poorly performing assets during these times. The author suggests this is in line with the reasoning of the other models discussed, with differences (such as the state variable being exogenous and requiring special assumptions) boiling down to esthetic preference of the researcher.

The author concludes by advocating for the consideration of macro-finance models and their potential usefulness in macroeconomic research related to recessions. He believes that research relating to macro-finance models may help to lessen the divide between macroeconomics and macro-finance due to the strong relationship between recessions and risk premiums, risk aversion, risk-bearing capacity among investors, as well as the shift to assets perceived safe during recessions. It will be our understanding of how risk perceptions affect investors, and not risk-free interest rates or inter-temporal substitution, that allow us to better understand recessions.

A Survey of Behavioral Finance

A Survey of Behavioral Finance

By: Nicholas Barberis and Richard Thaler

            Barberis and Thaler discuss the results of the behavioral finance, a fairly new field of finance compared to the traditional finance paradigm.  The key difference between behavioral finance and the traditional finance paradigm is rationality.  The traditional finance paradigm assumes rational individuals.  Rationality means two things.  First, when presented with new information, agents update their beliefs accordingly.  Second, given their beliefs, agents make choices that are normatively acceptable.  Behavioral finance considers when some agents are not rational.  Specifically, when one of the two tenets of rationality are relaxed.

            One of the two building blocks of behavior finance is limits to arbitrage.  A classic objection to behavioral finance is that when some agents don’t act rationally, the agents who do act rationally will prevent them from influencing prices for long due to arbitrage.  A series of theoretical papers suggests otherwise, showing that in an economy where rational and irrational investors interact, irrationality can have a long and substantial impact on prices.  This stands counter to the efficient markets hypothesis (EMH), where put simply, the “prices are right.” Limits to arbitrage suggest that mispricing does occur and can persist.  Some reasons these limits exist are the fundamental risks the arbitrageur faces, noise trader risks, and implementation costs.  Some evidence of the limits to arbitrage are twin shares, index inclusions, and internet carve-outs.

            The other building block of behavior finance is psychology.  The theory of limited arbitrage shows irrational traders can cause deviations from fundamental value, but psychology seeks to tell us why those traders act irrationally.  For guidance, economists consider extensive evidence compiled by cognitive psychologists on the systematic biases that arise when people form beliefs, and on people’s preferences.  Some of these beliefs that are particularly useful to behavioral finance are overconfidence, optimism and wishful thinking, representativeness, conservatism, belief perseverance, and anchoring.  A couple things to focus on when considering preferences are prospect theory and ambiguity aversion.

            Barberis and Thaler follow this overview by explaining some applications of behavioral finance.  The first is the aggregate stock market, where behavioral finance may help solve the equity premium puzzle and the volatility puzzle.  The second is the cross-section of average returns.  Behavioral finance may help explain some anomalies, such as the size premium, long-term reversals, the predictive power of scaled-price ratios, momentum, and event studies of earnings announcements, dividend initiations and omissions, stock repurchases, and primary and secondary offerings.  The third application is closed-end funds and comovement.  The fourth is investor behavior where irrationality and psychology can explain insufficient diversification, naïve diversification, excessive trading, the selling decision, and the buying decision.  The last application mentioned is corporate finance, where behavioral finance can address security issuance, capital structure, investment, and dividends.

            When De Bondt and Thaler (1985) published their paper, many scholars felt the best explanation was programming error.  Since then, most of the empirical facts are agreed upon but the interpretation of those facts in still in dispute.  Behavioral finance has been very helpful in understanding possible limits of arbitrage and bounded rationality.  Barberis and Thaler concede there are numerous degrees of freedom but note that rational modelers have just as many options to choose from.  There is only one scientific way to compare alternative theories, behavioral or rational, and that is with empirical tests.  They say we should be skeptical of theories based on behavior that is undocumented empirically.  Barberis and Thaler conclude by giving two predictions.  First, we will find out that most of our current theories, both rational and behavioral, are wrong.  Second, substantially better theories will emerge.