An Analysis of the Beta Anomaly from an Idiosyncratic Volatility Perspective

Master's Thesis, 2017

58 Pages, Grade: 1,7


I. Table of Contents

I. Table of Contents

II. List of Tables

III. List of Figures

IV. List of Abbreviations
1. Introduction
2. Theoretical Background
3. Empirical Results
3.1 Data and Methodology
3.2 The Beta Anomaly
3.3 Betting-Against-Correlation
3.3.1 BAC Leverage Effects
3.3.2 BAC and Size Effects
3.3.3 BAC and Sentiment Effects
3.4 BAB versus BAI
4. Conclusions

V. Bibliography

VI. Appendices

II. List of Tables

T1 – Portfolio sorts on mispricing and beta

T2 – Alphas on portfolio sorts (mispricing / beta)

T2 A – Average IVOL for each portfolio sort in %

T3 – Alphas on portfolio sorts (deleting overpriced high-IVOL stocks)

T4 – Alphas on portfolio sorts (deleting high-beta stocks)

T4 A – Alphas on portfolio sorts (deleting high- & low-beta stocks)

T5 – Alphas on portfolio sorts (IVOL / beta)

T6 – Alphas on portfolios (mispricing / IVOL-adjusted beta)

T7 – Beta anomaly in sentiment and correlation regimes

T8 – Sources of BAB profits

T9 – Alphas on portfolio sorts (mispricing / correlation) by Asness et al. (2017)

T10 – Alphas on portfolio sorts (mispricing / correlation)

T11 – Average Beta per portfolio sort (mispricing / correlation)

T12 – BAC performance analysis

T13 – BAC Leverage Effects

T14 – Alphas on portfolio sorts (volatility / correlation)

T15 – BAC performance analysis (small-price stocks added)

T16 – Alphas on portfolio sorts (marketcap / correlation)

T17 – Alphas on portfolio sorts (mispricing / correlation) equal-weight

T18 – BAC under different sentiment and correlation regimes

T19 – BAC Alphas in top and bottom sentiment regimes

T20 – BAI Alphas

T21 – BAB Alphas

T22 – BAB/BAI controlling for each other

T23 – BAI according to Ang et al. (2006)

A1 – Alphas on portfolio sorts (mispricing / correlation) (1. Variation – CAPM Alphas, 5 sorts)

A2 – Alphas on portfolio sorts (mispricing / correlation) (2. Variation – CAPM Alphas, 5 sorts - conditional)

A3 – Alphas on portfolio sorts (mispricing / correlation) (3. Variation – CAPM Alphas, 5 sorts, NYSE breakpoints)

III. List of Figures

F1 – Alphas on beta deciles within each mispricing level

F2 – Investor Sentiment and Beta-IVOL Correlation

IV. List of Abbreviations

Abbildung in dieser Leseprobe nicht enthalten

1. Introduction

The beta anomaly is most likely one of the most widely known empirical phenomena and observations, not only in the academic world but also in the financial services industry and among financial practitioners and investors. Liu et al. (2017, p.1) even describe it as the longest-standing empirical challenge to the CAPM and other asset-pricing models which followed. Business students around the world learn the concepts and academic explanations of the low-beta anomaly in finance courses, which are based on findings developed over decades. As a prominent example of a practical implementation based on these findings and with a solid academic and empirical background, stands the paper by Frazzini and Pedersen (2014). By going long low-beta assets and short corresponding high-beta assets, one can yield significant positive risk-adjusted returns. With the rise of smart beta strategies in the form of various investment vehicles such as mutual funds, hedge funds and ETFs, investors have now received the opportunity to exploit this anomaly in some form or another as well. This seemingly easy strategy appears to work because high-beta stocks are observed to return less than predicted by common asset-pricing models, as well known observed by Black et al. (1972) and many other academic contributions which followed and pointed to the same direction. Frazzini and Pedersen (2014) provide an overview on the fundamental explanations for this anomaly which belongs to the consensus line of reasoning. Among them is the main idea that a sufficient number of investors and market participants face funding constraints due to institutional properties, restricted or expensive access to leverage. Constrained investors, who require a higher risk profile, drive up equilibrium prices of high-beta assets through excessive demand or an overweighting of high-beta securities to compensate for the inability to apply leverage as a means of increasing risk. These investors require lower risk-adjusted returns and drive future returns of high-beta assets lower relative to low-beta assets (Frazzini and Pedersen (2014)). Unconstrained investors, able to short assets and apply leverage to low-beta securities to generate a market neutral position as implemented in the betting-against-beta framework by Frazzini and Pedersen (2014), can exploit the beta anomaly and earn positive risk-adjusted returns. Given this academic backdrop and decade-long standing, the very recent empirical contributions by Liu et al. (2017) appear to be a revolutionary break to the consensus paradigm and underlying understanding of this anomaly. In plain words, they appear to have come across observations which would suggest that beta in fact is not the driving characteristic behind the beta anomaly and that their evidence is not in line with leverage-based explanations. From today’s perspective, it is already clear that these findings will contribute to an academic debate in the foreseeable future and that more research will be required to align different empirical results. Because of the far-reaching applications of the beta anomaly throughout the past decades, the academic discussion and its empirical analysis may change the way the beta anomaly is viewed and increase market efficiencies and improve investment decisions that are based on long-standing market anomalies and an understanding of their characteristics and behaviour. These aspects have been an important motivation to pursue a closer analysis of the beta anomaly which will be of value for all who study and act upon it. This thesis aims to provide its contribution to this process from various perspectives and shed more light on the existing literature and its empirical results. More specifically, this paper will tackle a number of tasks to answer various existing and upcoming questions that have emerged during the analysis of this problem, which will serve as clarifying contributions in this field. We believe that this analysis can add essential value, as it was surprising to see how many opposing and sometimes contradicting empirical views actually exist on the same topics and research questions. And for many of them, it was possible to find aligning explanations and reasons. The foundation of this contribution will be the verification of the empirical results presented by Liu et al. (2017). We will start by summarizing their findings and explanations in Chapter 2 (Theoretical Background) and build the theoretical foundation upon which the empirical interpretation will rest. We actually view the compilation of the theoretical explanations in an intuitive line of reasoning as our first valuable contribution for our readers, as an understanding of the topic is not as easily to grasp after a first consultation of the papers. In Chapter 3 (Empirical Results) we address the actual calculations and most important summary statistics that will either speak in support or against our interpretations. In 3.1 (Data and Methodology), we present our dataset important technical notes and in Chapter 3.2 (The Beta Anomaly) we summarize and try to replicate the empirical findings of Liu et al. (2017), which will serve as the bedrock of interpretations we will arrive at in following pages. In Chapter 3.3 (Betting-against-Correlation) and its sub-sections on leverage, size and sentiment, we turn our attention to a paper by Asness et al. (2017) which contributed and responded to the findings of Liu et al. (2017), but with often diametrically contradicting views. The reference date of 2017 suggests that both papers belong to a highly new set of research and we believe that by covering those two and adding our own findings we can add significant clarity to actually understand what is going in the space of the beta anomaly. In Chapter 3.4 (BAB vs. BAI), we finalise the empirical analysis by contrasting opposing views on strategies that are designed to exploit the low-beta anomaly and realign the findings that we came across. We will summarize the newly gained insights for our readers in our last Chapter 4.

2. Theoretical Background

We refer to the beta anomaly as the empirical observation that low-beta securities in general, not necessarily only restricted to stocks as found by Frazzini and Pedersen (2014), earn more than predicted by asset-pricing models and that high-beta securities earn less than predicted by asset-pricing models, or that, in other words, low- (high)-beta securities yield positive (negative) alphas (Liu et al. (2017)). There is a vast number of academic contributions to the beta anomaly as extensions to the findings of Black et al. (1972) and we will not go into the details of the beta anomaly in this paper, but only point out that real-world deviations from the assumption that investors can borrow and lend at a riskless rate without frictions (Black 1972), serves as the consensus explanation for this anomaly, also known as leverage-based explanations (Liu et al. (2017)). Constrained investors, with restricted access to leverage, drive up equilibrium prices of high-beta assets through excessive demand as a way to increase risk. This lowers risk-adjusted returns of high-beta assets relative to low-beta assets (Frazzini and Pedersen (2014)). An important practical extension is the construction of a betting-against-beta (BAB) strategy, as for example described by Frazzini and Pedersen (2014): By going long a portfolio of the lower half of stocks ranked and weighted according to their beta and levered to a portfolio beta of one, and simultaneously going short a portfolio of the top half of stocks ranked and weighted according to their beta and delevered to a portfolio beta of one, creating an ex-ante market-neutral position rebalanced each month, one can yield significant risk-adjusted returns as an exploitation of the anomaly.

As a strong contradiction to these observations and explanations, Liu, Stambaugh and Yuan (2017) find evidence that the beta anomaly in fact is not primarily driven and explained by beta itself and that the findings are not in line with leverage-based explanations. To understand this, we first need to refer to a finding that was made by most of the above-mentioned authors earlier. Stambaugh, Yu and Yuan (2015) conceptually linked arbitrage asymmetry, arbitrage risk and mispricing to find a strong and negative relation between idiosyncratic volatility (IVOL) of stocks and their corresponding returns, as explained in the following. As mispricing of stocks occurs, either under- or overpricing, arbitrage activity of going long underpriced stocks and going short overpriced stocks drives prices back to a more efficient equilibrium, earning excess returns (alpha) as a result. The economic intuition is, in other words, that underpriced stocks appreciate in value relative to the market because arbitrage activities bid up prices, so holding these stocks earns positive alpha. While overpriced stocks depreciate relative to the market because arbitrage activities push down prices of these stocks, so holding these stocks earns negative alpha, while shorting overpriced stocks would yield positive alpha as the other side of the trade. One could assume that these two effects are balanced for the entire stock universe of underpriced, fairly-priced and overpriced stocks. However, arbitrage activity comes with a risk. While systematic risk can be hedged away relatively easy, the greatest arbitrage risk is idiosyncratic stock behaviour, proxied and measured as idiosyncratic volatility. The higher IVOL, the riskier it becomes to conduct arbitrage trades and the riskier it becomes to trade against mispricing. As a result, mispricing can extend and survive for longer and overpriced stocks tend to be more overpriced and underpriced stocks tend to be more underpriced for higher IVOL and a reversal of this larger mispricing tends to come with larger return opportunities. The result is that mispriced stocks with higher IVOL earn greater alpha relative to those with lower IVOL as a form of compensation. The final missing aspect after mispricing and arbitrage risk, is arbitrage asymmetry. From an economic and practical perspective, arbitrage of underpriced stocks going long is not equivalent to arbitrage overpriced stocks going short. There is more economic capital available to act on underpricing by buying than there is to act on overpricing by selling, as short selling is often not allowed in institutional mandates, regulatory frameworks and if even if legally possible, often times less easy and more costly to implement among private and institutional market participants. A consequence of this is that overpricing tends to be more extended and more pronounced in absolute terms than underpricing, because there is more capital able to correct any underpricing by buying these mispriced securities. The economic intuition is that there is a greater negative return potential for overpriced stocks or that there is less capital that competes for alpha in the overpriced relative to the underpriced space as there are fewer agents acting against it. This arbitrage asymmetry leads to the observation that the negative alpha relation for overpriced stocks is stronger in absolute terms than the positive alpha relation for underpriced stocks, which results in a negative alpha-IVOL relation for the entire stock universe overall. All of the above is a summary of the findings by Stambaugh et al. (2015) and Liu et al. (2017) and to wrap up, underpriced stocks earn positive alpha while overpriced stocks earn negative alpha (the mispricing effect), and this relation tends to be stronger for higher IVOL among both mispricing directions (arbitrage risk effect), but asymmetrically so as the negative alpha for overpriced stocks is more pronounced than the positive alpha for underpriced stocks (arbitrage asymmetry effect). This leads to the result that IVOL is positively related to alpha among underpriced stocks (increasing IVOL increases alpha among underpriced stocks) and IVOL is, vice versa, negatively related to alpha among overpriced stocks, but the effect overall is stronger for overpriced securities, which is in line with the observation that the alpha-IVOL relation is negative for the entire stock universe, as the negative effects dominates.

What is the final link to the beta anomaly? Liu et al. (2017) demonstrated that the negative alpha-beta relation (higher beta leads to lower and up to negative alpha - the “beta anomaly”) is a misleading association driven by a positive beta-IVOL correlation of about 0,33 (p. 1) (our sample shows a correlation of 0,27). IVOL is the actual factor that is negatively related to alpha and in crude words one could say that it is an “IVOL anomaly”, but what has been seemingly observed was a pure beta anomaly because of the positive correlation between IVOL and beta. As explanations for why IVOL is positively correlated to beta, Liu et al. (2017), for example, reason that an increase in company leverage is typically associated with an increase in both IVOL and beta, or both idiosyncratic risks and market sensitivity. They also point to a sentiment relation, in that stocks that have higher IVOL are more susceptible to mispricing (as already mentioned with reference to arbitrage risk above), and that these stocks are expected to be more exposed to market related shifts. One further important conclusion that Liu et al. (2017) came to and observed is that the betting-against-beta strategy conclusively is not primarily driven by the spread between betas. To analyse this theoretical basis, we perform a number of empirical tests that we will present in the following.

3. Empirical Results

3.1 Data and Methodology

The analysis is to a large degree based on mispricing data. We refer to the data as defined in Stambaugh and Yuan (2015) and applied in Stambaugh et al. (2015), which we downloaded from Yu Yuan’s academic website of the Shanghai Advanced Institute of Finance. As we depend on these measures for the calculations, it is also the most restrictive data set in that it defines the range of dates and stocks of the remaining data. The mispricing scores are available from July 1965 to December 2016 for each month and for each stock for which a mispricing score could be calculated according to the criteria of Stambaugh and Yuan. (2015). The mispricing score is a simple average of 11 of mispricing measures and the higher the mispricing score, the more overpriced the security is relative to other stocks in that month (Liu et al. (2017)). From their sample, stocks below a stock price of $5 and with less than 7 of 11 available mispricing measures are excluded, which according to Liu et al. (2017, p. 4) eliminates about 10% of the analysed stock universe that consists of all NYSE/AMEX/NASDAQ common stocks. We use the data from Yu Yuan to extract the relevant stocks of the future sample (in the form of permnos), for all stocks that satisfied these conditions. We retrieved stock return, market capitalization (as the number of outstanding stocks times the stock price), the daily-IVOL for the past month (regression residual on daily Fama and French 3-factor returns (1993) for the past 21 trading days) and Fama and French 3-factor returns (1993) with monthly frequency from the WRDS database. The WRDS data starts earlier, in January 1960, as we require a pre-sample range of observations to calculate betas for our stock universe. We also merged our set with investor sentiment scores from Baker and Wurgler (2006) (August 1965 - Sep 2015) and Fama and French (2015) 5-factor monthly returns (July 1963 - December 2016) from French’s research data website, which we will require in a later step of the analysis. For each stock, we estimate the market beta through a regression of monthly excess stock returns on monthly market and lagged market excess returns. We apply the summed-slope procedure and Vasicek shrinkage exactly as lined out in Liu et al. (2017, p.5), the only difference being that instead of an estimate of the cross-sectional variance of betas, adjusted for the cross-sectional mean of squared standard errors of beta, we calculate the actual sample cross-sectional variance of betas, which is similar to the approach by Frazzini and Pedersen (2014, p. 17, footnote 14). Although the first approach takes care of outliers in beta, it created outliers of its own at some points of the sample where the shrinkage factor sometimes turned negative, which is most likely due to the fact that we depend on a slightly different estimate of covariance between beta and lagged beta to come up with an estimate of standard errors of summed-sloped betas. As detailed in Liu et al. (2017), stocks receive a beta measure based on the last 60 months of observations, requiring at least 36 months of data. We hence drop all observations where the regression was based on less than 36 observations. Our final sample ranges from July 1965 to December 2015, IVOL data restricting the set on the upper limit as available until the end of 2015, with 1.549.721 observations for 13.584 stocks. Unless stated differently, all calculations refers to the above mentioned time horizon and t-statistics will be reported in parentheses based on heteroscedasticity-consistent standard errors. Regressions are performed on monthly Fama and French 3-factor realizations (1993).

3.2 The Beta Anomaly

As Liu et al. (2017), we start by sorting our sample independently on mispricing quintiles and on our stock’s estimated pre-ranking betas into deciles, generating market capitalization-weighted portfolios and their respective monthly returns. We determine the average number of stocks for each of our 5x10 portfolios (Table 1, Panel A) and regress portfolios returns on FF-3 factors and report the market beta coefficient in Panel B of Table 1.

T1 – Portfolio sorts on mispricing and beta

Abbildung in dieser Leseprobe nicht enthalten

Our portfolios are relatively well-balanced in terms of number of stocks per portfolio and our sample exhibits similar skews as the sample by Liu et al. (2017). E.g. for the most underpriced quintile, the number of stocks falls in beta (60 to 35) and increases in beta for the most overpriced quintile (42 to 74). Not surprisingly, the estimated portfolio betas increase in beta deciles with a spread between highest and lowest beta decile of about 0,80 on average. In comparison to the Liu et al. (2017) sample, our estimate is less extreme in terms of difference between lowest and highest decile and in terms of minima and maxima for each mispricing quintile. Their spread is closer to 1,00 and ranges from 0,55 to 1,67. We can infer that our beta methodology and sample is more conservative for a beta anomaly analysis in that it is based on a tighter spread in betas.

Liu et al. (2017) then looked at how each of those 5x10 portfolios performed relative to the FF-3 factor models. As a reminder, each month the strategy sorts the entire stock universe independently into five mispricing quintiles and ten beta deciles and then allocates each stock into the intersections of these sorts of in total 5x10 portfolios (Liu et al. (2017)). In Table 2 we report alphas on our 50 mispricing and beta portfolio sorts.

T2 – Alphas on portfolio sorts (mispricing / beta)

Abbildung in dieser Leseprobe nicht enthalten

This is certainly among the most important pieces of analysis which empirically supports the aspects that were discussed in the theoretical discussion of Chapter 2. If we look at the bottom row of Table 2, which represents our entire stock sample sorted into 10 beta deciles without an independent sort on mispricing, we observe that alphas fall for increasing beta, or in other words, the typical empirical beta anomaly observation. A strategy that goes long the highest beta decile (high-beta stocks) and goes short the lowest beta decile (low-beta stocks), which is equivalent to the “H-L column” of the table on the right, earns significant negative alphas of 32 bps. The opposite implementation of this trade (L-H), would earn significantly positive alphas, which is a way of implementing a strategy that seeks to exploit the beta anomaly. In Figure 1 we summarize these effects visually and plot the alpha of each portfolio, grouped into the five mispricing deciles.

F1 – Alphas on beta deciles within each mispricing level

Abbildung in dieser Leseprobe nicht enthalten

We can see and also confirm that underpriced stocks (lowest mispricing quintile), yield strongly significant positive alphas, while the most overpriced stocks earn strongly significant negative alphas, which is also in line with our theoretical discussion. As predicted, it is also observable that the effect is much stronger in absolute terms for overpriced than for underpriced stocks (arbitrage asymmetry). With a few exceptions, there seems to be no discernible effect among the “middle” mispricing quintiles. When we look at the beta anomaly and the alpha spreads between highest and lowest beta deciles for each mispricing quintile instead of the overall stock universe, we see that the effect is only prevalent, and significantly so among most overpriced stocks and in Figure 1 visible as a steeply falling alpha. The difference between high and low deciles, although significant for single portfolios, is not significant enough even for most underpriced stocks to explain the beta anomaly for the entire sample. The vast majority of this effect is driven by the overpriced sample. This is also in line with the findings by Liu et al. (2017), who argue against beta as the factor explaining the beta anomaly and against leverage-based explanations. If beta is the driving force, there is no reason why it should be confined to overpriced stocks only and why leverage-constrained investors would prefer overpriced stocks to increase stock-market exposure, if one could do so more optimally and profitably with underpriced stocks (Liu et al. (2017)). They note that this apparent contradiction can be resolved if we incorporate the findings by Stambaugh et al. (2015) as summarized in the theoretical discussion of Chapter 2, who confirm that the negative alpha-IVOL relation is most dominant among overpriced stocks and that a positive correlation between IVOL and beta creates the appearance of a beta anomaly, which in fact is rather resembling a IVOL anomaly confined to overpriced stocks. Such an explanation would be in line with the findings by Liu et al. (2017) and us and speak against leverage-based explanations. To further develop this thought, we present and add a table that cannot be found in Liu et al. (2017) which plots the average IVOL for the portfolio sorts we developed above in Table 2.


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An Analysis of the Beta Anomaly from an Idiosyncratic Volatility Perspective
University of Mannheim
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analysis, beta, anomaly, idiosyncratic, volatility
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Mark Matern (Author), 2017, An Analysis of the Beta Anomaly from an Idiosyncratic Volatility Perspective, Munich, GRIN Verlag,


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