**Betting Against Beta: Evidence from the U.S. & Eurozone Region **

### by

**Lu Huixi Madeline **

Student Number: 12074357

Program: Economics and Business Economics (BSc ECB) Track: Finance

Supervisor: Mario Bersem

Abstract

Frazzini and Pedersen (2014) proposed that leveraged constrained investors overweight risky stocks in portfolio construction, resulting in lower alphas for high-beta stocks. This leads to their development of the betting-against-beta (BAB) factor, which supports the low-beta phenomenon and adds an extension to the widely known Capital Asset Pricing Model. They observed that the BAB portfolio, that is long in low- beta stocks and short in high-beta stocks, generates a significant positive risk-adjusted return. This study explores the low-beta phenomenon using beta-sorted portfolios, as well as the persistence of the BAB factor in the U.S. and in selected Eurozone markets in a more recent period of January 2007 to December 2020. My findings suggest that the BAB factor is associated with a positive average monthly excess return in all markets. However, the results are only significant in France, Germany and the U.S. Based on the beta-sorted portfolios, the low-beta phenomenon is only significantly evident in the U.S. Thus, the conclusion of the low-beta anomaly as a global phenomenon cannot be drawn.

This document is written by Lu Huixi Madeline who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

**Introduction **

The low-beta anomaly first documented by Black, Jensen and Scholes (1972) starts from the observation that the relationship between the expected return of a security and its beta is flatter than the capital asset pricing model (CAPM) predicts. Building on this observation, multiple studies have found that low-beta securities generate higher abnormal returns than high-beta securities on a risk-adjusted basis, including a study by Blitz and Vliet (2007) that proved the results to be robust across different regions as they found significant evidence in the U.S., European and Japanese markets. This well- established phenomenon has been persistent over the past decades, adding as an extension to the widely used CAPM, which has an underlying concept that higher expected and realised returns compensate investors for taking on the non-diversifiable market risk.

In particular, Frazzini and Pedersen (2014) conducted a comprehensive research to document the low-beta anomaly over various asset classes. They developed a market-neutral betting against beta (BAB) strategy that has a long levered position on low-beta assets and a short delevered position on high-beta assets. This strategy, in fact, was found by the authors to produce statistically significant positive risk- adjusted returns in equities, bonds, credits, commodities and foreign exchange. Following this discovery, further studies illustrated that positive BAB returns on global equities are robust not just to size and momentum, but also to industry classification and the presence of transaction costs (Asness, Frazzini &

Pedersen, 2014).

As this beta arbitrage theory has garnered much attention, this thesis aims to contribute to existing literature by examining the persistence of the BAB factor. While multiple studies on this strategy have been done, the lack of study that investigates whether this phenomenon still holds in more recent years, specifically data including 2020, forms the purpose of this research. With the focus on equities, I aim to verify the empirical findings of Frazzini and Pedersen (2014) in the U.S. and the Eurozone regions for a shorter sample period, as well as dividing it further into pre- and post-publication periods for comparison.

The explanation behind a shorter period is because the U.S. equity and non-U.S. equity data from the main paper covers the period from 1926 to 2012 and 1989 to 2012 respectively, in which the positive BAB factor could be heavily attributed to the much older data points. By adjusting to a shorter period, I would be able to analyse the effects of the BAB factor in the more recent years. This leads to the central research question:

**Does the BAB factor hold to generate significant positive risk-adjusted returns in the U.S. and the ****Eurozone region from January 2007 to December 2020? **

To answer this question, the methodology of Frazzini and Pedersen (2014) will be replicated to examine the BAB investment strategy in the U.S. and the Eurozone regions. This finding will be supplemented by the construction of five beta-sorted portfolios to further investigate the validity of the low-beta phenomenon. I try to answer the research question by analysing the entire sample period as a whole, as well as the pre-publication sample period covering data from January 2007 to December 2013, the post-publication sample period covering data from January 2014 to December 2020. This allows me to compare and have a deeper analysis to test the presence of the BAB factor upon its publication in January 2014.

With the retrieved sample data, I construct the BAB factor following Frazzini and Pedersen (2014). All stocks within a market are sorted in ascending order at the beginning of each calendar month based on their beta estimates at the end of the previous month and allocated to either the low-beta portfolio or high-beta portfolio. The low-beta portfolio is then leveraged while the high-beta portfolio is deleveraged, to achieve a market-neutral portfolio at formation. The results exhibit that the BAB factor remains valid in recent years, with positive risk-adjusted returns found across all markets based on the period of 2007 to 2020. However, it is likely that the small number of stocks that serve as the sample in most markets make it tough for the findings to be significant.

As I further investigate the low-beta phenomenon, stocks are similarly ranked in ascending order at the beginning of each calendar month, based on the beta estimate at the end of the previous month. The ranked stocks are then allocated to one of the five quintiles portfolios, which ranges from low-beta to high-beta quintiles. Each portfolio consists of approximately the same number of stocks and within a portfolio, stocks are equally weighted. The overall results do not suggest evidence supporting the low- beta phenomenon. Apart from the U.S. market that depicts a declining alpha from low-beta to high-beta portfolio, the risk-adjusted returns are very mixed across the low-beta to high-beta portfolios in the other markets and no trend of such is observed. As such, I cannot reject the null hypothesis that low-beta assets are associated with higher alphas.

The remainder of the paper is organized as follows. Section 2 reviews literature pertaining to the low-beta anomaly, asset pricing models that estimate stock returns, as well as various alternative explanations to the low-beta anomaly. This leads up to the formulation of the research hypothesis. Section 3 describes the research setting in detail, the data to be used for the analysis and the methodology. The methodology covers the construction of the BAB portfolio, the construction of the beta-sorted portfolio and the performance metrics used to evaluate the portfolios. Section 4 presents the results while Section 5 concludes the paper.

**Literature Review **

In this section, I summarise the low-beta anomaly that violates the CAPM. Following that, I introduce some multi-factor models, in which other risk factors to estimate expected stock returns are discussed. Next, I delve deeper into the BAB factor developed by Frazzini and Pedersen (2014) that was built upon the low-beta anomaly, while providing further literature that supports their findings. Several possible reasons for this anomaly will also be examined to provide more insights. Based on these relevant reviews, I will develop a hypothesis that I will test in this study.

**Relationship between Risk and Return **

The CAPM, developed by Sharpe (1964) and Lintner (1965), forms the basic risk-return

relationship. It proposed that the expected return of a stock is based on the expected risk-free rate, the risk premium of a diversified market portfolio, and the stock’s exposure to market volatility. Since

idiosyncratic risk is diversifiable, the model rationalises that investors require only compensation for systematic risk that is non-diversifiable and measured by beta. While CAPM is still commonly used as a basis to introduce portfolio theory concepts and asset pricing due to its simplicity (Fama & French, 2014), early studies have found limitations and violations to this risk-return trade-off and deem insufficient empirical evidence for CAPM.

In the early 1970’s, an empirical study by Black, Jensen and Scholes (1972) observed a flatter security market line (SML) for the U.S. market than what CAPM suggests, indicating a low-beta anomaly. This implies that high beta assets actually require less risk-adjusted return than indicated by CAPM. As the authors relaxed the unconstrained leverage assumption of the CAPM, they proposed that borrowing restrictions, such as margin requirements, drive low-beta stocks to do relatively well (Black, Jensen & Scholes, 1972). Subsequently, Haugen and Heins (1975) further showed that more volatile assets do not necessarily deliver higher returns. They provided empirical evidence that in the long run, low volatility stock portfolios in fact outperformed high volatility stock portfolios in the U.S. market.

Apart from the U.S. market, Blitz and Vliet (2007) further found significant evidence in the European and Japanese markets.

Similarly, Ang et al. (2006) discovered that equities with low idiosyncratic risk tend to have much higher returns than equities with high idiosyncratic risk. The effect was found to remain robust in bull and bear markets, expansions and recessions, as well as stable and volatile periods. In 2009, the same authors supplemented their findings by conducting their study across 23 developed countries (Ang et al.,

2009), in which they concluded the phenomenon to be global. Their results also persisted when several factor loadings and firm characteristics were controlled for. While most studies delved into developed countries, Blitz, et al. (2013) found the low-beta anomaly also present in 30 different emerging markets.

With the sample period from 1988 to 2010, they further observed that it was getting more prominent over the years. This supports the statement by Ang et al. (2009) regarding the worldwide existence of this phenomenon.

**Multi-Factor Models **

Fama and French (1992) observed not only a flat relation between average return and beta relative to the notion led by CAPM, they also found that the market factor produced little information about stock returns. This led to their formulation of a more precise three-factor model to improve stock return estimations. Alongside the existing market factor, size factor (Small Minus Big) and value factor (High Minus Low) were added to the model. Their research deduced that small-cap firms outperform large-cap firms, while value stocks outperform growth stocks. As such, these added risk factors were proven to have significant explanatory power in the cross-section of average stock returns (Fama and French, 1993).

Thereafter, Carhart (1997) proposed an additional momentum factor, referred to as Up Minus Down (UMD), as an extension to the three-factor model. This accounts for the tendency of stock prices to persist recent price trends in the future. Stocks that perform well in the past will continue to do so in the future. Conversely, stocks with declining prices continue to decline. This factor is formulated by going long positive momentum and short negative momentum of the past 1 year.

Considering that these multifactor models provide better explanation to the variance in stock returns than CAPM, their application in the methodology is pertinent in adjusting returns and affirming whether the BAB factor does indeed result in positive returns.

**BAB Investment Strategy **

A more recent study by Frazzini and Pedersen (2014) contributed largely to the research on the low volatility anomaly. Their first proposition hypothesises that high beta is associated with low alpha, which can be explained by investor-specific margin and leverage constraints.

This proposition was tested by constructing portfolios sorted by betas. The authors presented significant empirical evidence of higher alphas and higher Sharpe ratios in low-beta portfolios relative to high-beta portfolios. This was found consistent across their sample of developed markets as well as various asset classes, which includes equities, bonds, foreign exchange, and commodities. They proved that as constrained investors seek higher returns, they tend to bid up and overweight high-beta assets (assets with beta above 1 on average), resulting in lower risk-adjusted return for high-beta assets. In other words, these investors do not hold the efficient market portfolio, which gives rise to arbitrage

opportunities due to market inefficiency.

As such, this notion provided the groundwork for Frazzini and Pedersen (2014) to develop the BAB factor as a strategy to illustrate the effect of the funding constraints on asset pricing, as well as to exploit this inefficiency. To simplify, the strategy involves a self-financing portfolio that utilises leverage and deleverage. It goes long on the low-beta assets and short on the high-beta assets, whereby both are rescaled to an estimated ex–ante beta of one. This brings about their second proposition that predicts a positive risk-adjusted return of the BAB factor.

To examine this proposition, the authors formulated the BAB factor, in which the step-by-step approach is provided in the paper, and empirically tested them. The approach will be covered in the methodology section. They documented significant positive BAB returns across different countries and asset classes, demonstrating that the phenomenon is likely applicable worldwide. Specifically, Frazzini and Pedersen found 0.70% excess monthly return in U.S. equities and 0.64% excess monthly return in the international equities respectively. Additionally, the overall findings remained robust when controlling for the size, value, momentum, and liquidity factor.

Following that, several studies based their research on this strategy. Asness, Frazzini and Pedersen (2014) showed that the BAB factor provided positive returns within each industry (industry- neutral bet) and across industries (pure industry bet). The authors also took into account transaction costs.

While this expense naturally reduces the returns, the net result still remained significant for the standard BAB, the industry-neutral BAB, as well as the pure industry BAB.

Adrian, Etula and Muir (2014) further investigated the theory regarding the BAB factor and funding friction. By constructing a leverage factor to measure funding constraints, the authors found that the leverage factor indeed has a positive correlation with the BAB factor returns. This indicates that a decreasing leverage, possibly due to tighter funding constraints, leads to the lower returns of low-beta securities since leverage is required to invest in them. Under this condition, the BAB returns become

negative and high-beta securities perform better instead. This is in line with the reasoning of Frazzini and Pedersen (2014) about the role of leverage constraints on the low-beta anomaly and BAB factor returns.

**Possible Explanations to the Low-Beta Anomaly **

To better understand the low-beta anomaly in asset pricing, I discuss other possible reasons apart from the explanation offered by Frazzini and Pedersen (2014) Baker, Bradley and Wurgler (2011) and Bali et al. (2017) propose behavioural factors that drive the “overbuying” of high-beta securities.

Baker, Bradley and Wurgler (2011) pointed out that many investors have too narrow confidence intervals, creating overconfidence in their judgement. These investors are prone to overvalue risky stocks, which causes lower future expected returns. The authors also attributed representativeness heuristic as a factor, explaining that investors are likely to overpay for volatile stocks in hopes of higher returns. That, however, ignores that a large proportion of speculative investments fail. This is supported by Hong and Sraer (2016), who hypothesise that high-beta assets are prone to speculative overpricing.

Similarly, Bali, et al. (2017) introduced a concept called “lottery demand”, that could partially explain the low-beta anomaly. It is the demand generated by investors' preference for assets with high probabilities of short-term gains, in hopes of making swift and large profits. Since these stocks generally have high sensitivity to market movements, they possess high betas. Thus, due to the disproportionate price pressure for assets with lottery-like features, prices of high-beta assets increase, which lowers their potential returns. With this reasoning, the authors showed that the inverse relationship between alpha and systematic risk cannot be found when characteristics of lottery stocks were controlled for.

It is worth noting that Bali et al. (2017) and Liu et al. (2018) argued that the beta anomaly is attributed to idiosyncratic risk instead of systematic risk. Demonstrating a positive relation between the CAPM beta factor and idiosyncratic risk, Liu et al. (2018) claimed that the results of the BAB factor is likely not due to the low-beta anomaly. As such, Asness, et al. (2018) responded by conducting further tests to separate the idiosyncratic risk explanations and the leverage constraint explanation of Frazzini and Pedersen (2014). They found stronger evidence for their theory.

**Hypothesis **

Putting together all the literature discussed, it seems that the low-beta phenomenon remains recognised and popular in the past decades, with many studies documenting their findings around it and

developing possible explanations for it. While there is no clear consensus on a likely explanation behind the positive BAB returns and the low-beta anomaly in general, many studies seem to have found

significant and robust evidence supporting the presence of both.

Therefore, it is rational to assume that both the BAB factor and the low-beta anomaly still continue to persist. Specifically, I hypothesise that the BAB factor does hold to generate significant positive risk-adjusted returns and that low-beta assets have a higher alpha relative to high-beta assets. The empirical study will be based on the U.S. market and selected Eurozone markets from the sample period of January 2007 to December 2020.

**Data and Methodology **

In this section, I will report the data sample collected to construct the BAB factor and beta-sorted portfolios. This also includes how the data is filtered. I will then formally explain the methodology behind the construction of the BAB factor and the beta-sorted portfolios, in accordance with the methodology of Frazzini and Pedersen (2014). All calculations were done in Python.

**Data **

The data construction closely follows Frazzini and Pedersen (2014) to ensure a fair comparison.

Table 1 reports the sample statistics of the U.S. and Eurozone stocks.

Country Local Market Index

No. of Stocks, Initial

No. of Stocks, Final

Start Year

End Year

Austria MSCI Austria 226 107 2007 2020

Belgium MSCI Belgium 356 197 2007 2020

Finland MSCI Finland 372 174 2007 2020

France MSCI France 1,728 988 2007 2020

Germany MSCI Germany 1,886 975 2007 2020

Italy MSCI Italy 814 306 2007 2020

Netherlands MSCI Netherlands 484 183 2007 2020

Spain MSCI Spain 533 296 2007 2020

United States CRSP value-weighted index 14,515 6,831 2007 2020
*Table 1: Summary statistics of dataset. All the listed firms that are available in the selected databases within the *
*sample period were included. The final number of stocks refer to the total number of stocks left in the sample after *
*filtering the original data and performing the rolling regression. *

The data in this study are collected from several sources. The U.S. stock market data is obtained from The Center for Research in Security Prices (CRSP) database. This includes all common stocks (SHRCD variable equals to 10 or 11) traded on the NYSE, AMEX and NASDAQ exchanges, and the CRSP value-weighted index that serves as the market index. In this database, the daily returns have already been computed.

On the other hand, the individual stock price data for all eight Eurozone markets is retrieved from the Compustat Global database via the Wharton Research Data Services (WRDS). This replaces the constituents of the MSCI developed universe used in the Frazzini and Pedersen (2014) paper due to data access limitations. The resulting stock data is achieved by filtering the type of issue (TPCI variable equals to 0) to include only common stocks and filtering the exchange code (EXCHG variable) to reflect only equities listed in the selected Eurozone market. That is to exclude securities of a company that has a headquarter in that selected country, but not necessarily listed there. For each Eurozone market, the MSCI local index obtained from Factset is used as the market index.

The data for all markets comprise of daily observations between 1 January 2007 to 31 December 2020, with the pre-publication period spanning from 1 January 2007 to 31 December 2013, and the post- publication period spanning from 1 January 2014 to 31 December 2020. The daily frequency helps improve estimation accuracy (Merton, 1980). Additionally, to avoid survivorship bias, both currently listed and delisted stocks are included in the sample. However, for the price data, I remove stocks with prices below US$5 to diminish the impact of outliers, since changes in daily price may cause huge swings in daily returns. Daily returns of over 500% are also removed.

For consistency purposes, all returns are in US dollars. This implies that the price data for the stocks and markets in the Eurozone region will be converted from Euros to US dollars using the daily Euro-Dollar exchange rate retrieved from Factset, before computing the returns. The computation of the daily returns of each Eurozone stock, as well as the market index returns is as follows:

𝑟_{𝑡} =^{𝑃}^{𝑡+1}^{∗} ^{−𝑃}^{𝑡}^{∗}

𝑃_{𝑡}^{∗} , where 𝑃^{∗}= ^{𝑃}

𝐴𝐹⋅ 𝐹𝑋

Where P* is the adjusted close price, P is the close price, AF is the adjustment factor and FX is the EUR/USD exchange rate that quotes the US dollar per 1 euro.

Excess returns in the U.S. market is calculated on top of the 1-month US Treasury bill rate, which is sourced from the Federal Resource Economic Data (FRED). On the other hand, excess returns in the Eurozone markets are calculated on top of the 1-month EURIBOR rate, which is obtained from the

European Money Market Institute (EMMI). As the risk-free rates are given in annual form, they are converted to daily rates.

The size factor (SMB) and value factor (HML) from Fama and French (1993), and the Carhart (1997) momentum factor (MOM), are sourced from the Ken French Data Library. To be precise, I use the North American factor data for the U.S. market and the European factor data for the Eurozone markets.

All factors are in US dollars as well and provided in monthly frequency. This means that for the regression analyses, the daily excess returns will be converted to monthly data eventually. This will be detailed in the methodology.

**Methodology **

**Estimating Ex-Ante Betas **

To construct the portfolios, the ex-ante beta for each stock needs to be obtained first. The ex-ante beta is estimated using daily rolling regressions of excess returns on excess market returns, in which is derived from the following formula:

𝛽̂_{𝑖} *= 𝜌̂* *𝜎̂*_{𝑖}
*𝜎̂*_{𝑚}

Where 𝜎̂_{𝑖} and 𝜎̂_{𝑚} refers to the estimated volatilies of stock i and the market respectively, and 𝜌̂ is their
correlation.

I use one-year rolling data, assuming 250 trading days in a year, for standard deviation and five-
year rolling data for correlation, as correlation seems to move slower than volatilities. Due to the shorter
time span of this sample period, I include a minimum window of one year of non-missing data for the
rolling correlation. Volatilities are estimated with simple one-day log return, while correlations are
estimated using overlapping three-day log returns, 𝑟_{𝑖,𝑡}* ^{3𝑑}*= 𝛴

_{𝑘=0}

^{2}*𝑙𝑛(1 + 𝑟*

_{𝑡+𝑘}

^{𝑖}). This controls for

nonsynchronous trading as it takes into account information delays. According to Hou and Moskowitz (2005), share prices have some delayed reaction to new information due to factors that include

information capacity constraints and limited participation in the stock market.

Finally, to minimise the influence of outliers, I will shrink the calculated time-series betas towards the cross-sectional mean using the following formula:

𝛽̂_{𝑖} = 𝑤_{𝑖}𝛽̂_{𝑖}*+ (1 − 𝑤*_{𝑖})𝛽̂_{𝑖}^{𝑋𝑆}

Where 𝛽̂_{𝑖} is the calculated time-series beta for stock i and 𝛽̂_{𝑖}^{𝑋𝑆} is the cross-sectional mean of beta. In line
with the literature of the followed paper, I set 𝑤 as 0.6 and 𝛽_{𝑖}^{𝑋𝑆}as 1 in all periods.

**Constructing BAB Portfolio **

Replicating the approach of Frazzini and Pedersen (2014), all stocks within a market are allocated to the low-beta portfolio or high-beta portfolio. The stocks are sorted in ascending order at the beginning of every month, based on the beta estimate at the end of the previous month. Stocks with beta ranked below its country median were assigned to the low-beta portfolio, while stocks with beta ranked above its country median were assigned to the high-beta portfolio. The weight of each stock within a portfolio is dependent on the ranked betas. Stocks with a lower beta will make up a higher weight in the low-beta portfolio. Conversely, stocks with a higher beta will make up a higher weight in the high-beta portfolio.

Both portfolios are rebalanced at the beginning of each calendar month, using the ranked beta at the end of the previous month. This also updates the composition of each portfolio since the ex-ante beta estimate of a security may vary from month to month.

The weight of security i within the high or low portfolio is determined by:

𝑤_{𝐿}= 𝑘(𝑧 − 𝑧̅)
𝑤_{𝐻}= 𝑘(𝑧 − 𝑧̅)

Where z is the rank of its ex-ante beta at the time of portfolio assignment, 𝑧̅ =^{∑}^{𝑛}^{𝑖=1}_{𝑛} ^{𝑧} is the average rank
and n is the total number of stocks. Additionally, 𝑘 = ^{2}

∑^{𝑛}_{𝑖=1}|𝑧−𝑧̅| is the normalizing factor in which the
denominator equates to the sum of the absolute deviation between the individual stock rank and the
average rank. The factor k ensures the sum of weights always equal to one.

To get the BAB factor, the portfolios are rescaled to achieve an ex-ante beta of one at portfolio formation through leverage for the low-beta portfolio and through deleverage for the high-beta portfolio.

This creates a neutral market exposure through a self-financing zero-beta portfolio.

The daily excess returns from the BAB portfolio is formally calculated as follows, in line with Frazzini and Pedersen (2014):

𝑟_{𝑡+1}^{𝐵𝐴𝐵}= *1*

𝛽_{𝑡}^{𝐿}(𝑟_{𝑡+1}^{𝐿} − 𝑟_{𝑓}) − *1*

𝛽_{𝑡}^{𝐻}(𝑟_{𝑡+1}^{𝐻} − 𝑟_{𝑓})

Where 𝛽_{𝑡}^{𝐿}= 𝛽_{𝑡}* ^{′}*𝑤

_{𝐿 }is the daily weighted beta and denominator of the leverage factor for the low-beta portfolio, 𝑟

_{𝑡+1}^{𝐿}= 𝑟

_{𝑡+1}*𝑤*

^{′}_{𝐿 }is the daily weighted return of the low-beta portfolio, 𝛽

_{𝑡}

^{𝐻}= 𝛽

_{𝑡}

*𝑤*

^{′}_{𝐻 }is the daily weighted beta and denominator of the deleverage factor for the high-beta portfolio, 𝑟

_{𝑡+1}^{𝐻}= 𝑟

_{𝑡+1}*𝑤*

^{′}_{𝐻 }is the daily weighted return of the high-beta portfolio, and 𝑟

_{𝑓}is the 1-month US Treasury bill rate for the U.S.

market and 1-month EURIBOR rate for the Eurozone markets.

To exemplify the methodology behind the BAB factor, consider stock W, X, Y and Z as the only
stocks in the sample at period t, with an ex-ante beta and return of 0.4, 0.8, 1.2, 1.6 and 6%, 5%, 6% and
5% respectively. This would mean that the stocks are ranked 1, 2, 3, and 4 respectively on the basis of
their beta, and have an average rank of 2.5. Given this information, stock W and X constitute the low-beta
portfolio while stock Y and Z constitute the high-beta portfolio. The individual beta rank deviates from
the average rank by -1.5, -0.5, 0.5 and 1.5 respectively. This brings the sum of absolute deviation between
beta rank and average rank to |-1.5|+|- 0.5|+|0.5|+|1.5| = 4. With the normalising factor k equated as 2
divided by the sum of absolute deviation, k therefore equals to ^{2}* _{4}*= 0.5.

Subsequently, the weight of each stock in a portfolio is calculated by the product of the

normalising factor and its deviation. Based on this example, stock W and X in the low-beta portfolio has a
weight of 0.75 (0.5 × 1.5) and 0.25(0.5 × 0.5)respectively, while Stock Y and Z in the high-beta portfolio
has a weight of 0.25 (0.5 × 0.5) and 0.75(0.5 × 1.5) respectively. This illustrates that lower beta securities
hold higher weight in the low-beta portfolio while higher beta securities hold higher weight in the high-
beta portfolio. With the above information, the portfolio beta and return can be calculated. For the low-
beta portfolio, the beta will be 0.45(0.75 × 0.4 + 0.25 × 0.6)and the return will be 5.75%(0.75 × 6% +
*0.25 × 5%). For the high-beta portfolio, the beta will be 1.30(0.75 × 1.2 + 0.25 × 1.6) and the return will *
be 5.25%(0.75 × 5% + 0.25 × 6%).

Assuming a risk-free rate of 1%, the excess return is 4.75% and 4.25% for the low-beta and high-
beta portfolio respectively. The rescale factor will then be obtained to derive the BAB return. The
leverage factor equates to _{0.45}^{1}*= 2.22 for the low-beta portfolio, and the deleverage factor equates to *_{1.3}* ^{1}* =

*0.77 for the high-beta portfolio. This shows that the overall BAB portfolio comprises of a long position in*the low-beta portfolio and a short position in the high-beta portfolio. Specifically, it is long $2.22 of low- beta stocks W and X that is financed by short selling $2.22 of risk-free assets, and it is short $0.77 of high-beta stocks Y and Z, whereby $0.77 earns the risk-free rate. With that, the BAB return is found to be 7.28%(2.22 × 4.75% − 0.77 × 4.25%).

**Formulating Beta-Sorted Portfolios **

To test the low-beta phenomenon, I construct five portfolios. This is tested on the entire sample
period of 2007 to 2020. In order to assign them, the stocks are ranked in ascending order at the beginning
of every month, based on the beta estimate at the end of the previous month. The ranked stocks are then
allocated to one of the five quintile portfolios, in which 𝑃*1* contains the lowest beta stocks and 𝑃*5* contains
the highest beta stocks. Each portfolio consists of approximately the same number of stocks and is
rebalanced monthly to maintain equal weights and update its composition. This means that their
allocation at the beginning of every month is always according to the ranked beta at the end of the
previous month. Within a portfolio, stocks are equally weighted for simplicity. Based on these
specifications, the excess daily returns of each portfolio can be computed.

**Evaluating Portfolio Performance **

To evaluate the constructed portfolios, I test the beta-sorted portfolio and BAB factor returns against the three aforementioned asset pricing models discussed earlier in the literature review to determine the risk adjusted returns, known as alphas. Namely, I use the standard CAPM, Fama and French (1993) three-factor model and Carhart (1997) four-factor model. As the factors have a monthly frequency, I converted the daily returns to monthly returns. This is computed using:

*𝑚𝑜𝑛𝑡ℎ𝑙𝑦 𝑟*_{𝑡}= ∏(

𝑡_{𝑛}

*𝑖=1*

*1 + 𝑟*_{𝑡,𝑖}*) − 1 *

Where 𝑡_{𝑛}is the number of the days within each month of the time series.

For the beta-sorted portfolio, the following OLS regressions are performed for each portfolio P:

𝑟^{𝑃}− 𝑟_{𝑓} = 𝛼_{𝐶𝐴𝑃𝑀}^{𝑃} + 𝛽_{𝑚}⋅ 𝑀𝑘𝑡 + 𝜀^{𝑃}

𝑟^{𝑃}− 𝑟_{𝑓}= 𝛼_{3−𝑓𝑎𝑐𝑡𝑜𝑟}^{𝑃} + 𝛽_{𝑚}⋅ 𝑀𝑘𝑡 + 𝛽_{𝑆𝑀𝐵}⋅ 𝑆𝑀𝐵 + 𝛽_{𝐻𝑀𝐿}⋅ 𝐻𝑀𝐿 + 𝜀^{𝑃}

𝑟^{𝑃}− 𝑟_{𝑓} = 𝛼_{4−𝑓𝑎𝑐𝑡𝑜𝑟}^{𝑃} + 𝛽_{𝑚}⋅ 𝑀𝑘𝑡 + 𝛽_{𝑆𝑀𝐵}⋅ 𝑆𝑀𝐵 + 𝛽_{𝐻𝑀𝐿}⋅ 𝐻𝑀𝐿 + 𝛽_{𝑈𝑀𝐷}⋅ 𝑈𝑀𝐷 + 𝜀^{𝑃}
Where 𝑟^{𝑃}is the beta-sorted portfolio return, 𝑟_{𝑓} is the risk-free rate, Mkt is the excess market return over
the used risk-free rate, 𝛽 is the factor loading for SMB, HML and UMD.

The ex-ante beta of each portfolio is measured as the average beta estimate at portfolio formation.

In other words, the average beta on the last day of every month. Annualised volatilities and Sharpe ratios are additionally calculated to supplement the performance evaluation across the five portfolios. The Sharpe ratio measures risk-adjusted return and is calculated by excess portfolio return over its standard deviation (Sharpe, 1994). In simpler words, it describes how much excess return an investor receives for the extra volatility taken due to the holding of riskier assets.

For the BAB portfolio, the following OLS regressions are performed:

𝑟^{𝐵𝐴𝐵} = 𝛼_{𝐶𝐴𝑃𝑀}^{𝐵𝐴𝐵} + 𝛽_{𝑚}⋅ 𝑀𝑘𝑡 + 𝜀^{𝐵𝐴𝐵}

𝑟^{𝐵𝐴𝐵} = 𝛼_{3−𝑓𝑎𝑐𝑡𝑜𝑟}^{𝐵𝐴𝐵} + 𝛽_{𝑚}⋅ 𝑀𝑘𝑡 + 𝛽_{𝑆𝑀𝐵}⋅ 𝑆𝑀𝐵 + 𝛽_{𝐻𝑀𝐿}⋅ 𝐻𝑀𝐿 + 𝜀^{𝐵𝐴𝐵}

𝑟^{𝐵𝐴𝐵}= 𝛼_{4−𝑓𝑎𝑐𝑡𝑜𝑟}^{𝐵𝐴𝐵} + 𝛽_{𝑚}⋅ 𝑀𝑘𝑡 + 𝛽_{𝑆𝑀𝐵}⋅ 𝑆𝑀𝐵 + 𝛽_{𝐻𝑀𝐿}⋅ 𝐻𝑀𝐿 + 𝛽_{𝑈𝑀𝐷}⋅ 𝑈𝑀𝐷 + 𝜀^{𝐵𝐴𝐵}

As BAB returns are formulated from the differences between the low-beta portfolio excess return
and the high-beta portfolio excess return, 𝑟^{𝐵𝐴𝐵}is already the excess return over the 1-month US treasury
bill rate. Similar to the beta-sorted portfolio, the annualised volatility and Sharpe ratio of the BAB
portfolio will also be reported.

**Results **

**BAB Return **

Table 2 presents the results of the market-neutral BAB factor for all tested markets in the period 2007 to 2020. I report the average excess returns, alphas, factor loadings, realised betas, volatilities, and Sharpe Ratio. To further supplement the overview, the annualised Sharpe ratios of all the BAB portfolios are plotted in Figure 1.

Consistent with the findings of Frazzini and Pedersen (2014), the BAB factor generates a high average excess return and high alphas in the three regression models. Specifically, the average Sharpe Ratio across all countries is 0.47. The highest positive returns are found in Germany, with a risk-adjusted return ranging between 0.70% to 0.926%, depending on the risk adjustment. Conversely, the BAB factor is least prominent in Belgium. It has a four-factor alpha of around 0%. That being said, it should be highlighted that only results in France, Germany and the U.S. are found to be statistically significant, with p < 0.0.1 for all the abnormal returns and alphas. This denotes that I am unable to assume that the BAB factor yields significant positive returns globally.

*Table 2: Summary of BAB portfolio performance in all markets from 2007 to 2020. Returns are in US dollars. *

*Returns and alphas are in monthly percent, with the t-statistics in parenthesis. $Short (Long) shows the average *
*dollar value of the short (long) position. The realised beta is the realised loading from the Standard CAPM. *

*Annualised volatilities and Sharpe ratios are also shown. *** indicates 1% statistical significance, ** indicates 5% *

*statistical significance, and * indicates 10% statistical significance. *

*Figure 1: Annualised Sharpe ratios of BAB portfolio in all markets from 2007 to 2020. *

I also consider the results separately for the 2007 to 2013 pre-publication period and 2014 to 2020 post-publication period across all of the markets in Table 3 and Table 4 respectively. Overall, the positive returns of the BAB factor still seem to be present in the shortened periods. This indicates that when focusing only on the period after the main paper was published, the BAB factor still persists. That being said, apart from Italy, the countries report relatively different results between both periods. Seven out of

the nine countries observe a higher excess return from the BAB factor in the post-publication period. This is visualised in Figure 2. Interestingly, this could imply that the positive outcome of the BAB factor is more evident after the paper has been published, though there is no clear answer to explain this. Finland even reported a negative excess return of -0.45% in the pre-publication period, while reporting a positive excess return of 1.06% in the post-publication period. In addition, the BAB returns are only found to be statistically significant in the U.S in the pre-publication period, while the BAB returns are only

statistically significant in France, Germany and the U.S. in the post-publication period.

This low frequency of significant results could be explained by the small sample size used, which makes it hard to reject the null hypothesis that suggest zero return on the BAB portfolio in each country. I can only support the hypothesis of positive risk-adjusted BAB return in the U.S. market. As such, I am unable to accept the hypothesis as a whole to conclude that the BAB factor does deliver significant positive risk-adjusted returns.

*Table 3: Summary of BAB portfolio performance in all markets from 2007 to 2013. Returns are in US dollars. *

*Returns and alphas are in monthly percent, with the t-statistics in parenthesis. $Short ($Long) shows the average *
*dollar value of the short (long) position. The realised beta is the realised loading from the Standard CAPM. *

*Annualised volatilities and Sharpe ratios are also shown. *** indicates 1% statistical significance, ** indicates 5% *

*statistical significance, and * indicates 10% statistical significance.*

*Table 4: Summary of BAB portfolio performance in all markets from 2014 to 2020. Returns are in US dollars. *

*Returns and alphas are in monthly percent, with the t-statistics in parenthesis. $Short (Long) shows the average *
*dollar value of the short (long) position. The realised beta is the realised loading from the Standard CAPM. *

*Annualised volatilities and Sharpe ratios are also shown. *** indicates 1% statistical significance, ** indicates 5% *

*statistical significance, and * indicates 10% statistical significance. *

*Figure 2: Annualised Sharpe ratios of BAB portfolio in all markets for the pre- and post-publication of 2007 to 2013 *
*and 2014 to 2020 respectively. *

**Returns across Beta-Sorted Portfolios **

Table 5 exhibits the results of the beta-sorted portfolios in all markets in the period of 2007 to 2020. Similarly, I report the average excess returns, alphas, factor loadings, realised betas, volatilities, and Sharpe Ratio. My overall findings, however, contrast the findings of Frazzini and Pedersen (2014), whereby they observed that the alphas decrease nearly monotonically from the low-beta to high-beta

portfolios. This observation is only seen in the U.S. market, in which the CAPM, three-factor and four- factor models depict a declining alpha from low-beta to high-beta portfolio. The Sharpe ratio also decreases across the low-beta to high-beta portfolio.

Despite statistically significant CAPM alphas found in all beta-sorted portfolios in Finland, France, Germany and Italy, the risk-adjusted returns are very mixed across the low-beta to high-beta portfolios in the Eurozone markets and no trend of such is observed. In fact, there is no strikingly consistent pattern across all markets that allows for an inference about the relationship between beta and alpha. The graph illustrating the relationship between the realised beta and the CAPM alpha across the five ex-ante beta-sorted portfolios for each market can be found in Appendix 1.

Due to the mixed results that do not illustrate any relationship between beta and alpha, my results are thus inconsistent with the low-beta phenomenon. As such, I cannot reject the hypothesis that low-beta assets are associated with higher alphas. It is also worth mentioning that I found the results to be rather sensitive to changes in the settings surrounding the methodology. For instance, the choice of rolling and minimum window for the rolling regression to estimate the ex-ante betas, and the choice of reference point at which extreme returns are filtered out. Therefore, it is important to note that my results are subjected to my settings mentioned previously in the data section.

** **

Austria

Portfolio P1 P2 P3 P4 P5

Excess Return 0.32 0.30 0.22 0.15 0.43

(0.63) (0.58) (0.42) (0.26) (0.76)

CAPM Alpha 0.45 0.43 0.36 0.30 0.59**

(1.51) (1.47) (1.28) (1.08) (2.27)

3-Factor Alpha 0.39 0.39 0.22 0.18 0.47

(1.39) (1.36) (0.80) (0.67) (1.86)

4-Factor Alpha 0.52* 0.35 0.20 0.25 0.49

(1.85) (1.22) (0.72) (0.89) (1.92)

Beta (ex-ante) 0.49 0.58 0.68 0.79 0.99

Beta (realised) 0.55 0.56 0.59 0.66 0.67

Volatility 22.15 22.22 22.89 24.80 24.70

Sharpe Ratio 0.18 0.16 0.11 0.07 0.21

Belgium

Portfolio P1 P2 P3 P4 P5

Excess Return 0.32 0.13 0.24 0.35 0.38

(0.80) (0.32) (0.53) (0.78) (0.82)

CAPM Alpha 0.34 0.16 0.27 0.38 0.41

(1.40) (0.65) (1.07) (1.49) (1.58) 3-Factor Alpha 0.28 0.13 0.35 0.46** 0.48**

(1.24) (0.59) (1.63) (2.01) (2.06) 4-Factor Alpha 0.30 0.17 0.35 0.49** 0.54**

(1.29) (0.76) (1.57) (2.14) (2.29)

Beta (ex-ante) 0.51 0.61 0.70 0.79 1.00

Beta (realised) 0.56 0.59 0.67 0.65 0.68

Volatility 17.41 17.87 19.79 19.53 20.18

Sharpe Ratio 0.22 0.09 0.15 0.22 0.23

Finland

Portfolio P1 P2 P3 P4 P5

Excess Return 0.61 0.57 0.84 0.59 0.55

(1.13) (1.09) (1.57) (1.04) (1.09) CAPM Alpha 0.66** 0.62** 0.89*** 0.65** 0.60**

(2.11) (2.16) (2.78) (2.12) (2.36) 3-Factor Alpha 0.58** 0.48* 0.82*** 0.57** 0.60**

(2.06) (1.86) (2.75) (2.01) (2.50) 4-Factor Alpha 0.62** 0.51* 0.87*** 0.63** 0.68***

(2.21) (1.89) (2.85) (2.22) (2.83)

Beta (ex-ante) 0.60 0.70 0.78 0.88 1.07

Beta (realised) 0.75 0.76 0.74 0.83 0.75

Volatility 23.21 22.82 23.09 24.68 21.82

Sharpe Ratio 0.31 0.30 0.43 0.29 0.30

France

Portfolio P1 P2 P3 P4 P5

Excess Return 0.69 0.69 0.68 0.54 0.62

(1.52) (1.45) (1.41) (1.10) (1.25) CAPM Alpha 0.59*** 0.58*** 0.57*** 0.42** 0.51***

(2.72) (2.58) (2.72) (2.27) (2.82) 3-Factor Alpha 0.39** 0.41** 0.36** 0.26* 0.35***

(2.41) (2.41) (2.30) (1.91) (2.59) 4-Factor Alpha 0.38** 0.45*** 0.40** 0.28** 0.38***

(2.34) (2.60) (2.48) (2.07) (2.82)

Beta (ex-ante) 0.51 0.63 0.72 0.83 1.07

Beta (realised) 0.75 0.78 0.82 0.85 0.88

Volatility 19.60 20.44 20.87 21.09 21.64

Sharpe Ratio 0.42 0.40 0.39 0.30 0.35

** **

*Table 5: Summary of beta-sorted portfolio performance of each market from 2007 to 2020. Returns are in US *
*dollars. Returns and alphas are in monthly percent, with the t-statistics in parenthesis. The ex-ante beta is the *
*average beta at portfolio formation and the realised beta is the realised loading from the Standard CAPM. *

*statistical significance, and * indicates 10% statistical significance. *

**Conclusion **

As the low-beta anomaly is still very much a pertinent area of research, this paper extends the literature by investigating the persistence of the significantly positive BAB factor in the U.S. and selected

Germany

Portfolio P1 P2 P3 P4 P5

Excess Return 0.83 0.82 0.75 0.83 0.77

(1.91) (1.92) (1.67) (1.82) (1.60) CAPM Alpha 0.73*** 0.71*** 0.63*** 0.71*** 0.64***

(3.15) (3.62) (3.23) (3.50) (3.56) 3-Factor Alpha 0.58*** 0.59*** 0.50*** 0.58*** 0.56***

(2.77) (3.41) (3.03) (3.28) (3.77)

4-Factor Alpha 0.57 0.59 0.49 0.56 0.58

(2.68) (3.37) (2.90) (3.14) (3.90)

Beta (ex-ante) 0.52 0.64 0.75 0.87 1.10

Beta (realised) 0.67 0.68 0.72 0.73 0.80

Volatility 18.91 18.43 19.37 19.70 20.77

Sharpe Ratio 0.53 0.53 0.46 0.51 0.44

Italy

Portfolio P1 P2 P3 P4 P5

Excess Return 0.41 0.65 0.78 0.54 0.58

(0.76) (1.22) (1.42) (0.96) (1.00) CAPM Alpha 0.59** 0.84*** 0.98*** 0.73*** 0.79***

(2.25) (3.09) (3.73) (3.17) (3.59) 3-Factor Alpha 0.38 0.60** 0.75*** 0.55** 0.57***

(1.56) (2.46) (3.11) (2.51) (3.02) 4-Factor Alpha 0.38 0.66*** 0.71*** 0.58*** 0.57***

(1.56) (2.66) (2.95) (2.62) (2.97)

Beta (ex-ante) 0.57 0.69 0.78 0.87 1.03

Beta (realised) 0.74 0.73 0.77 0.80 0.84

Volatility 23.39 23.22 23.99 24.25 24.99

Sharpe Ratio 0.21 -0.01 0.13 -0.14 -0.04

Netherlands

Portfolio P1 P2 P3 P4 P5

Excess Return 0.42 0.73 0.35 0.56 0.38

(0.83) (1.24) (0.71) (1.01) (0.74)

CAPM Alpha 0.04 0.32 -0.02 0.14 -0.01

(0.18) (1.01) (-0.10) (0.61) (-0.04) 3-Factor Alpha 0.06 0.44 -0.03 0.13 -0.05 (0.31) (1.63) (-0.18) (0.62) (-0.27)

4-Factor Alpha 0.14 0.51 -0.03 0.19 0.00

(0.70) (1.87) (-0.19) (0.89) (-0.02)

Beta (ex-ante) 0.63 0.78 0.87 0.99 1.22

Beta (realised) 0.92 1.00 0.89 1.00 0.93

Volatility 22.12 25.41 21.10 23.79 22.04

Sharpe Ratio 0.23 0.34 0.20 0.28 0.21

Spain

Portfolio P1 P2 P3 P4 P5

Excess Return 0.19 0.04 0.34 0.33 0.08

(0.47) (0.08) (0.73) (0.65) (0.15)

CAPM Alpha 0.30 0.15 0.46 0.46 0.22

(1.30) (0.62) (1.99) (1.83) (1.10)

3-Factor Alpha 0.18 -0.11 0.27 0.27 0.10

(0.82) (-0.52) (1.35) (1.19) (0.53)

4-Factor Alpha 0.09 -0.08 0.25 0.28 0.03

(0.40) (-0.37) (1.23) (1.21) (0.14)

Beta (ex-ante) 0.50 0.61 0.73 0.84 1.03

Beta (realised) 0.54 0.60 0.62 0.68 0.74

Volatility 17.90 19.80 19.93 21.65 22.20

Sharpe Ratio 0.13 0.02 0.20 0.18 0.04

United States

Portfolio P1 P2 P3 P4 P5

Excess Return 0.87** 0.94** 1.05** 1.01* 0.98 (2.44) (2.22) (2.09) (1.72) (1.24)

CAPM Alpha 0.36 0.29 0.26 0.09 -0.20

(1.92) (1.76) (1.50) (0.44) (-0.55) 3-Factor Alpha 0.41** 0.33*** 0.34*** 0.27** 0.16 (2.41) (2.64) (2.95) (2.10) (0.62) 4-Factor Alpha 0.40** 0.32*** 0.33*** 0.24** 0.11 (2.37) (2.60) (2.97) (2.17) (0.48)

Beta (ex-ante) 0.64 0.89 1.04 1.20 1.55

Beta (realised) 0.77 0.99 1.19 1.39 1.77

Volatility 15.40 18.40 21.70 25.44 34.02

Sharpe Ratio 0.68 0.62 0.58 0.48 0.34

Eurozone markets. Using the framework developed by Frazzini & Pedersen (2014), I construct the BAB portfolios in nine markets based on a sample period from January 2007 to December 2020.

The results are consistent with the findings of the main authors, as positive risk-adjusted returns and positive Sharpe ratios are seen in all of the markets. This suggests that experienced and unconstrained investors with regards to leverage can exploit the anomaly to gain excess returns by formulating BAB portfolios that go long on low-beta assets and short on high-beta assets. The results are statistically significant in three countries. The non-significance in the remaining countries could be attributed to the low number of stocks in the sample used. As such, despite the findings of positive BAB factors in each market, the hypothesis of zero return on the BAB portfolio can only be rejected in France, Germany, and the U.S when considering the whole period of 2007 to 2020. This also means that I cannot conclude the BAB factor to be a global phenomenon.

In addition, I divide the entire period into the pre- and post- publication period for comparison and to solely analyse the results post-publication. Again, positive risk-adjusted returns are observed in the post-publication. This could further support the notion that the BAB factor still remains. Furthermore, it seems that the BAB factor has become more evident in the post-publication period than the pre-

publication period, reporting higher Sharpe ratios in seven countries.

For a more holistic approach, I also construct beta-sorted portfolios to test the low-beta phenomenon. The low-beta anomaly can be observed in the U.S., depicting declining alphas and

decreasing Sharpe ratios from low-beta to high-beta portfolio. The resulting risk-adjusted returns for other markets, however, are very mixed across the low-beta to high-beta portfolios and not statistically

significant. There is no strikingly consistent trend that allows for an inference about the relationship between beta and alpha. Therefore, the hypothesis that is contrary to the low-beta anomaly can only be rejected in the U.S. and not in the other eight markets. Similarly, I cannot conclude the low-beta phenomenon to be a global phenomenon.

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**Appendix **

** **

** **

** **

** **

*Appendix 1: Alphas of beta-sorted portfolios based on the sample period of 2007 to 2020. Alphas are in monthly *
*percent and the realised beta is the realised loading from the Standard CAPM. *