## Betting Against Beta:

## An Empirical Analysis of European Equities

By

Junyu Wu, 12099902

BSc in Economics and Business Economics Faculty of Economics and Business

**University of Amsterdam **

Thesis supervisor: Dr. Mario Bersem

**June, 2021 **

**Abstract: With the assumption of unlimited borrowing and lending, the CAPM model is proven to **
overestimate the return of risky assets and underestimate the return of low-beta asset. Frazzini and
Pederson therefore try to exploit this potential beta arbitrage in their article “Betting Against Beta” in
2014. This thesis empirically examines the Betting-against-Beta model developed by Frazzini and
Pederson (2014) in the scope of ten European countries and test the effect with all common stocks
and index constituents separately. The author finds out the BAB strategy performs better with all
common stocks whereas higher returns for lower beta is more significant among the selected index
constituents.

**Keywords: Asset prices, Leverage constraints, Beta, CAPM, Eurozone Equities **
**JEL: G11 G12 G14 G15 **

**Words count: 7,605 **

**1. Introduction **

As a milestone in the world of financial economics, the Capital Asset Pricing Model (CAPM) portrays the connection between systematic market risk and expected return on assets, particularly stocks. It is widely used as a powerful tool to generate quantitative predictions about the potential return of a financial asset given the risk of it. However, behind the CAPM model, there are some theoretical assumptions which prevented researchers from consistent outcomes over different empirical tests.

One of the most criticized assumptions of CAPM is the unlimited access to leverage of investors.

In the modern financial environments, institutional investors may come across increasing

borrowing constraints because of the occurrence of credit crunches such as the financial crisis in 2007, whereas there are many types of borrowing limits for personal investors who can only borrow so much money from banks or other individuals.

With all the kinds of restrictions of borrowing, investors may be more inclined to buy high-beta assets in order to achieve higher returns, thereby pushing up the prices of high beta assets (Frazzini and Pedersen 2014). However, while the prices of high-beta assets grow too high, investors would not be rewarded enough alphas for holding the excess risk (Black, Jensen, and Scholes, 1972). This results in a possibility for an unconstrained arbitrageur to exploit the price disequilibrium by using leverage to go long on low-beta assets and sell high-beta assets.

Frazzini and Pedersen (2014) conduct a comprehensive research with expensive asset classes including equity, treasury bond, credit indices, currencies and commodities in both US and

international markets. They developed Betting-against-Beta (BAB) portfolios which buy low-beta assets and go short on high-beta assets and yield statistically significant returns. Their results are in line with the findings of Black (1972) on the capital market equilibrium with more restrictive borrowing or lending.

The objective of this thesis is to further examine Frazzini and Pedersen’s model in the stock markets within nine Eurozone countries and Switzerland separately over the period between January 2001 and March 2021 and test whether the results corroborate the existence of beta arbitrage. I break down the research question into two parts. Firstly, I examine whether BAB factors in the sample markets produce significant risk-adjusted returns and the results of their Sharpe ratios. I then analyze the pattern of the alphas across beta-sorted portfolios. Do portfolios with lower-beta produce higher alphas and Sharpe ratios?

In contrast with the international worldwide scope in Frazzini and Pedersen (2014), this paper aims to present more details on the performance of BAB portfolios within each individual market.

In this thesis, nine eurozone countries and Switzerland are investigated. Each country is also divided into two sections. One contains all listed common stocks and the other part includes only the constituents of the corresponding country index as the largest and most liquid fraction of a stock market. On top of that, an extensive European market index, STOXX 600, is also within the scope in order to provide a more general landscape of the whole European equity market. I intend to compare the significance of BAB factor with and without the presence of small- or micro-cap stocks. Noticeably, the constituents of each country index are rigorously filtered by the dates of their entrance and exit. Over the complete sample period, some stocks might be admitted to and expelled from an index for numerous times.

It appears that different counties behave differently in terms of both BAB factors and beta-sorted portfolios. For the majority of countries, all-common-stocks portfolios deliver higher BAB returns than the corresponding index-constituents portfolios. Four categorizations of the patterns of beta- sorted portfolios alphas are outlined in the analysis section.

From the finale of the Dot-com bubble in 2001 till the climax of the coronavirus pandemic in 2021, it covers a period of twenty years which extends across a few economic cycles. It is believed that this is an adequate period of time for the sake of the validity of conclusions.

After the introduction to the basic concepts which the thesis is based upon as the first part, the concepts and limitations of CAPM and its important alternative theories as well as the essential features of Frazzini and Pedersen’s (2014) BAB factor will be discussed in literature review. In the third part, the data sample and methodology will be thoroughly described. In the last part, the results of BAB factors and beta-sorted portfolios for each country will be summarized.

**2. Literature Review **

Following a brief introduction to this thesis, the next part the theoretical background of CAPM model and the limitations of it are explained. At last, I describe some basic ideas of beta arbitrage and Frazzini and Pedersen’s BAB factor.

**2.1. CAPM and its Limitations **

In 1952, Harry Markowitz’s “The utility of wealth” presents the famous Modern portfolio Theory (MPT) which later won him a Nobel Memorial Prize in Economic Sciences in 1990. This is a mathematical framework used to combine a portfolio of assets in order to optimize the expected returns at a given level of risk. The main idea behind Markowitz (1952) is that the diversified choice of financial assets helps lower the variance of portfolio prices as risk. The work

demonstrate that there is a compromise between the returns and the risk of a portfolio as a whole.

To reduce the risk, the returns will be discounted accordingly. In other words, additional risk should be compensated by additional returns.

Based upon Markowitz’s work, the Capital Asset Pricing Model (CAPM) was established by several economists independently. Sharpe (1964) and Lintner’s (1965) version of CAPM model estimates the cost of capital with the relation between the expected returns and systematic risk.

Sharpe (1964) assumed that all investors have access to funds at a common interest rate which is the risk-free rate, regardless the amount borrowed or lent as a basis of the CAPM model. On top of that, Sharpe further assumed investors have consent on a homogeneous anticipation of the expected values, standard deviations and correlation coefficients, which is, according to himself,

“highly restrictive and undoubtedly unrealistic”. (Sharpe, 1964, pp. 434)

Fama and French in 2004 discover that the Sharpe-Lintner’s CAPM model is not empirically valid. They first emphasize there are three implications of the relation between returns and market risk in CAPM. First, for all assets, the association between the expected returns and their betas is linear and there is no other explanatory variables. Second, a positive beta premium means that the expected return of the market portfolio exceeds the expected return of assets which has nothing to do with market returns. Third, when it comes to the assets which are independent to the market, their returns are the risk-free interest rate and the corresponding risk premium is the difference

between the expected market return and the risk-free rate. Accordingly, some problems arise. The imprecision of beta estimation for individual assets results in a measurement error. Moreover, common sources of variation affect the regression residuals. Upward or downward bias will be produced in the ordinary least squares (OLS) regression depending on the correlation in the residuals.

Fama and French (2004) conclude most application of CAPM are discredited due to its serious problems. For example, the typical choice of the market portfolio consist of all common stocks.

However, in practice the relation between beta and average return is weaker than the prediction of Sharpe-Linter version of CAPM. Subsequently, Friend and Blume (1970) prove that high beta assets are estimated to have too high cost of capital relative to historical average returns and for low beta assets, the cost is too low.

**2.2. Development of Arbitrage Pricing Theory **

Since the late 1970s, more expansionary variables such as size, various price ratios and momentum are added to analyze the average returns besides beta. The first multifactor model developed by Ross (1976) suggests that more accuracy of asset pricing can be brought by multiple macroeconomic and asset-specific factors rather than the beta alone.

Starting with observations, Fama and French (1993) discover two classes of stocks perform better than the market as a whole and add the characters as factors into the CAPM model to create a more precise three-factor model. These two characters are the size factor, Small Minus Big (SMB), which is small companies outperforming large companies over long terms and the value factor, High Minus Low (HML), which is value stocks outperforming growth stocks. SMB factor and HML factor are constructed by buying small-cap and selling large-cap stocks and buying high book-to-market and selling low book-to-market stocks respectively.

Furthermore, Carhart (1997) unveil that stocks have a tendency to remain the prices trends in the future, rising asset prices to keep rising, and falling prices to fall further. This is called the price momentum. Carhart exploits this market abnormality and creates a Up Minus Down (UMD) factor by buying positive momentum (recent winners) and selling negative momentum (recent losers) of the past 12 months.

Nonetheless, despite that further researches show there exist over three hundred potential variable factors which can statistically significantly explain the average returns by change (Harvey and Liu, 2014; Harvey, Liu and Zhu, 2016), doubts have been arisen that whether these multi-factor models are actually informative or they are merely statistical misinterpretations (Black, 1992).

**2.3. Funds Restrictions and Risky Assets Performance **

As mentioned above, CAPM has been criticized for assuming all investors are able to borrow or lend unlimited amount of money at a risk-free interest rate. Black (1972) tightens this assumption by testing the CAPM model in two cases. First, he examines the model with the absence of risk- free assets and no riskless borrowing or lending is allowed. The other case is that investors can only go long for the risk-free asset and are forbidden to go short for the riskless asset. In both cases, there is no constraints on borrowing and lending risky assets. Black then indicates that with the presence of restrictions, the expected return of a risk asset is still linearly related to its beta.

However, the slope of the line relating to the expected return of any risky asset is shown to be smaller and the intercept increases above the risk-free interest rate. This outcome is in line with Black, Jensen, and Scholes (1972) which point out that Sharpe-Linter’s CAPM model

overestimates the return of high-beta assets whilst underestimate the return of low-beta assets.

The same result is also achieved by Friend and Blume (1970).

Frazzini and Pedersen (2014) goes further to exploit the beta arbitrage by optimizing risky assets and less-risky assets in a portfolio. The authors construct a Betting-against-Beta factor by buying low-beta assets and selling high-beta assets and empirically examine the factor in different markets and asset classes. In their work, the BAB factor actually delivers statistically significant positive excess returns and risk adjusted returns across the whole sample.

**3. Data Samples **

*As exhibited in Table 3.1-1, the data of this paper can be divided into two parts. One part contains *
all common stocks of nine Eurozone countries and Switzerland from S&P Compustat Global
Security Daily (tcpi equal to zero). The other part contains only the constituents of the indices of
these countries and the Europe as a whole. These stocks are the largest or most liquid fraction
among their local markets, since I want to examine whether the betting-against-beta factor is also
significant among these stocks, due to fact that anomalies or price patterns are hardly

economically important if they are only obvious among small-cap or micro-cap stocks (Blitz and van Vliet, 2018). It is noticeable that sometimes when there are more than one index I can choose for a country. I try to use a representative index with approximately 10% of stocks relative to the total amount for each country. For example, among the family of DAX Index, I consider HDAX, the composition of DAX, MDAX, and TecDAX, would extensively represent the largest and most liquid stocks in the German market than DAX alone.

The data are collected from various sources. The prices of all common stocks are on a daily frequency and acquired from S&P Compustat. Prices of MSCI market indices are gathered from FactSet. The time period covers from January, 2001 till March, 2021. Stock returns are all in Euro and the excess returns are deducted by Euro Interbank Offered Rate (Euribor) with one month maturity. Historical exchange rates acquired from European Central Bank (ECB) are used to convert USD and CHF into EUR.

The historical constituents of country indices are also obtained from S&P Compustat. However, there are only a limited collection of indices in the database most of which are provided by local stocks exchanges such as Euronext and Deutsche Börse. I have no access to the historical

constituents of indices which are provided by private companies, hence for some countries which

*Table 3.1-1. Overview of Data Samples *
Assets

Class Country Country Index Market Index Sample Period Currency Source

Equity Austria (AUT) ATX MSCI Austria Jan, 2001-Mar, 2021 EUR(€) S&P Compustat
Equity Belgium (BEL) BEL 20 MSCI Belgium Jan, 2001-Mar, 2021 EUR(€) S&P Compustat
Equity Finland (FIN) OMX 25 MSCI Finland Jan, 2001-Mar, 2021 EUR(€) S&P Compustat
Equity France (FRA) SBF 120 MSCI France Jan, 2001-Mar, 2021 EUR(€) S&P Compustat
Equity German (DEU) HDAX MSCI German Jan, 2001-Mar, 2021 EUR(€) S&P Compustat
Equity Greece (GRC) ATHEX MSCI Greece Jan, 2001-Mar, 2021 EUR(€) S&P Compustat
Equity Netherlands (NLD) AEX MSCI Netherlands Jan, 2001-Mar, 2021 EUR(€) S&P Compustat
Equity Portugal (PRT) PSI-20 MSCI Portugal Jan, 2001-Mar, 2021 EUR(€) S&P Compustat
Equity Spain (ESP) IBEX 35 MSCI Spain Jan, 2001-Mar, 2021 EUR(€) S&P Compustat
Equity *(Europe) * STOXX 600 MSCI Europe (USD) Jan, 2001-Mar, 2021 EUR(€) S&P Compustat
Equity Switzerland (CHE) SMI MSCI Switzerland Jan, 2001-Mar, 2021 CHF S&P Compustat

have only FTSE Index have to be given up. For example, instead of the local stocks exchanges in the UK and Italy, FTSE Group provides FTSE 100 and FTSE MIB as the stock indices for these two markets. In the data sample, the index constituents are strictly filtered by the date of when they are selected into the index. Some stocks exit and enter an index multiple times during the sample period.

Betas of each stock are computed against its corresponding local MSCI market index which is adopted as the market prices. The betas of the constituents of STOXX 600 Index are computed with respect to MSCI Europe. The choice of MSCI indices as local market indices is based on the fact that some market indices provided by the local stock exchanges are price indices e.g. AEX and STOXX 600 while dividends are also included in MSCI indices. Further performance evaluations of BAB are constructed with other three risk factors, namely size (SMB), book-to- market (HML), and momentum (UMD). All of them are obtained from Ken French's data library.

A conversion to Euro is needed because these factors are calculated in US dollar. I use the
following formula to achieve it: 𝑟_{𝐸𝑈𝑅} = (𝑟_{𝑈𝑆𝐷}+ 1) ∗ (𝑟^{𝑈𝑆𝐷}

𝐸𝑈𝑅

+ 1) − 1, where 𝑟^{𝑈𝑆𝐷}

𝐸𝑈𝑅

is the rate of
change in the EUR-USD spot rate, 𝑟^{𝑈𝑆𝐷}

𝐸𝑈𝑅

= ^{𝑈𝑆𝐷/𝐸𝑈𝑅}^{𝑡}

𝑈𝑆𝐷/𝐸𝑈𝑅𝑡−1.

*Table 3.1-2. Statistics Summary *

Country Number of

stocks, total

Number of

stocks, mean Country Index Number of stocks, total

Number of stocks, mean

Austria 223 130 ATX 43 20

Belgium 380 222 BEL 20 40 20

Finland 348 247 OMX 25 40 25

France 1,691 969 SBF 120 210 120

Germany 1,761 969 HDAX 225 110

Greece 394 320 ATHEX 51 50

Netherlands 452 235 AEX 61 25

Portugal 110 74 PSI-20 41 20

Spain 469 309 IBEX 35 67 35

*(Europe) * STOXX 600 1,172 600

Switzerland 199 310 SMI 36 20

*The amounts of stocks for each country and the corresponding index are displayed in Table 3.1-2. *

For most of the countries, the total numbers of stocks over the whole sample period are between one hundred and five hundred, other than France and Germany which have about 1.7 thousand.

For each month, the proportions of index constituents in the all common stocks are around 10% to 20% depending on different countries.

Cumulatively, the data samples consist of seven thousand stocks and more than 16 million total observations covering ten European countries. All data manipulations are done with python.

**4. Methodology **

**4.1. Beta Estimation **

The estimated betas are calculated with daily data in order to improve the accuracy of covariance estimation (Merton, 1980) given by the following formula:

𝛽̂^{𝑡𝑠} = 𝜌̂ 𝜎̂_{𝑖}
𝜎̂_{𝑚}

where 𝜎_{𝑖} is the estimated volatility of stock 𝑖, 𝜎_{𝑚} is the estimated volatility of the market prices
and 𝜌 is the estimated correlation given by the ratio of the covariance of the return of the 𝑖^{𝑡ℎ} asset
with the return of the market, divided by the product of their corresponding volatilities.

I follow the notice of Frazzini and Pedersen (2014) that correlations usually tend to move slower
than volatilities, hence one-year (250 days) rolling standard deviations and correlations with five-
year (1,250 days) horizon are used in the paper. Moreover, volatilities and correlations are
computed respectively using one-day log return and three-day overlapping log returns, 𝑟_{𝑖,𝑡}^{3𝑑}=

∑^{2}_{𝑘=0}ln(1 + 𝑟_{𝑡+𝑘}^{𝑖} ), to control the potential effect of nonsynchronous trading on correlations.

At last, the time series of estimated betas (𝛽^{𝑡𝑠}) are shrunk towards the cross-sectional mean (𝛽^{𝑋𝑆})
to reduce the influence of outliers, following Vasicek (1973) and Elton, Gruber, Brown, and
Goetzmann (2009):

𝛽_{𝑖} = 𝑤_{𝑖}𝛽_{𝑖}^{𝑇𝑆}+ (1 − 𝑤_{𝑖})𝛽^{𝑋𝑆}

where 𝛽^{𝑋𝑆} is the cross sectional average market beta and 𝑤_{𝑖} is the shrinkage factor coefficient of
the i-th asset estimated beta 𝛽_{𝑖}^{𝑇𝑆}. For simplicity, 𝛽^{𝑋𝑆} and 𝑤 are set to 1 and 0.6 for all the whole
period and all countries in this paper.

**4.2. Betting-against-Beta Factor Construction **

The methodology to construct BAB factors is in line with Frazzini and Pedersen (2014). All stocks of each market are positioned into two portions, low-beta and high-beta, according to their

estimated beta. The stocks which have lower estimated beta than the country median are assigned to the low-beta portfolio and vice versa. In each portfolio, the weight of a stock are decided by how much its beta is away from the country median, i.e. the lower the estimated beta of a stock in the low-beta portfolio is, the higher the stock weights. And on the contrary, the higher the

estimated beta of a stock in the high-beta portfolio is, the higher the stock weights.

To put it in a mathematical formality, let 𝑧 be the 𝑛 × 1 vector of beta ranks 𝑧_{𝑖} = 𝑟𝑎𝑛𝑘(𝛽_{𝑖𝑡}) at
portfolio formation, and let 𝑧̅ be the average rank. The weight of a stock in the low-beta or high-
beta portfolios is given by:

𝑤_{𝐻,𝑖} = 𝑘|𝑧_{𝑖} − 𝑧̅|

𝑤_{𝐿,𝑖} = 𝑘|𝑧_{𝑖} − 𝑧̅|

*where k is a normalizing constant that sets the aggregate weights of each portfolio equal to 1 *
(𝑤_{𝐻}= 𝑤_{𝐿} = 1).

The BAB portfolio is designed to be self-financing and zero-beta portfolio. Both low- and high- beta portfolios are rescaled to have an ex ante beta of one at the combination. To achieve a market exposure neutrality, I go long the low-beta portfolio and short sell the high-beta portfolio. The BAB portfolio is rebalanced every month.

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 𝑟_{𝑡+1}^{𝐿} = 𝑟_{𝑡+1}^{′} 𝑤_{𝐿}, 𝑟_{𝑡+1}^{𝐻} = 𝑟_{𝑡+1}^{′} 𝑤_{𝐻}, 𝛽_{𝑡}^{𝐿} = 𝛽_{𝑡}^{′}𝑤_{𝐿} and 𝛽_{𝑡}^{𝐻}= 𝛽_{𝑡}^{′}𝑤_{𝐻}, 𝑟^{𝑓} is Euribor rate with one
month maturity.

**4.3. Beta-Sorted Portfolio Construction **

Within a market, stocks are assigned into five portfolios in the ascending order of estimated betas.

P1 is the portfolio containing stocks with the lowest betas and P5 contains the ones with the highest betas. Specially, due to the relatively large amount of stocks in France, Germany and STOXX 600 constituents, stocks are divided into ten portfolios. In these three markets, P10

includes the highest beta stocks. Each stock weights equally and the portfolios are refreshed monthly, based on the beta rankings at the end of each month.

It is notable that the returns I have now are daily returns. I convert them to monthly returns, 𝑟_{𝑡},
which is given by

𝑟_{𝑡}= ∏ (1 + 𝑟_{𝑡}_{𝑛}) − 1

𝑡_{𝑛}
𝑡𝑘

where 𝑡_{𝑛}* is the days in month t and 𝑟*_{𝑡}_{𝑛} is the daily return of day 𝑡_{𝑛}.

**4.4. Performance Evaluation **

In order to answer the research question, I assess the risk adjusted returns by regressing the excess returns of the BAB portfolio and the beta-sorted portfolios against three prevalent asset pricing models, namely the Capital Asset Pricing Model (CAPM), the Fama-French three-factor model, and the four-factor model with a momentum factor on top of the three-factor. The specific OLS regressions for each portfolio are presented below.

𝑟_{𝑡}^{𝑝}− 𝑟_{𝑡}^{𝑓}= 𝛼_{𝑡}^{𝑝}+ 𝛽^{𝑀}𝑀𝐾𝑇_{𝑡}

𝑟_{𝑡}^{𝑝}− 𝑟_{𝑡}^{𝑓}= 𝛼_{𝑡}^{𝑝}+ 𝛽^{𝑀}𝑀𝐾𝑇_{𝑡}+ 𝛽^{𝑆𝑀𝐵}𝑆𝑀𝐵_{𝑡}+ 𝛽^{𝐻𝑀𝐿}𝐻𝑀𝐿_{𝑡}

𝑟_{𝑡}^{𝑝}− 𝑟_{𝑡}^{𝑓} = 𝛼_{𝑡}^{𝑝}+ 𝛽^{𝑀}𝑀𝐾𝑇_{𝑡}+ 𝛽^{𝑆𝑀𝐵}𝑆𝑀𝐵_{𝑡}+ 𝛽^{𝐻𝑀𝐿}𝐻𝑀𝐿_{𝑡}+ 𝛽^{𝑈𝑀𝐿}𝑈𝑀𝐷_{𝑡}

where 𝑟_{𝑡}^{𝑝} is the BAB and beta-sorted portfolio return, 𝑟_{𝑡}^{𝑓} is Euribor rate with one month maturity
*at time t, MKT is the local MSCI market index, SMB, HML, and UMD are the risk factors from *
Ken French's data library for the Europe which are not country-specific.

Positive and statistically significant excess returns from the BAB factor and the portfolios with lower betas are expected. Significance level is at 5%.

**5. Results and Analysis **

In this section, the analysis and the results of BAB factor are presented collectively. The
*annualized Sharpe ratios of the BAB portfolios across all sample countries shown in Figure 1. *

*Table 4-1 and Table 4-2 exhibit the excess returns, alphas of CAPM, three-factor model and four-*
factor model and the corresponding t-statistics of the BAB portfolios across all sample countries.

The results of beta-sorted portfolios can be found in Appendix: Betting-against-Beta in Each Country.

*Table 4-1. BAB Portfolio, all common stocks *

Country Austria Belgium Finland France Germany Greece Netherlands Portugal Spain *(Europe) * Switzerland

Excess return 1.49% **0.73% ** **0.92% ** **1.42% ** **1.22% ** **1.36% ** **0.32% ** **1.27% ** **0.34% ** **0.27% **

(1.91) (1.30) (2.26) (4.34) (3.18) (2.40) (1.33) (2.13) (1.18) (1.09)

CAPM alpha **1.59% ** **0.73% ** **0.89% ** **1.44% ** **1.29% ** **0.96% ** 0.36% **1.22% ** 0.36% **0.32% **

(2.10) (1.31) (2.43) (4.61) (3.80) (1.91) (1.55) (2.04) (1.54) (1.32) Three-factor

alpha

**1.57% ** **0.74% ** **0.78% ** **1.43% ** **1.25% ** **0.90% ** **0.32% ** **1.25% ** 0.32% **0.32% **

(2.02) (1.29) (2.24) (4.71) (3.62) (1.74) (1.41) (2.03) (1.39) (1.33) Four-factor

alpha

**1.21% ** **0.69% ** **0.64% ** **1.32% ** **1.00% ** **0.82% ** **0.25% ** 1.11% 0.18% **0.28% **

(1.66) (1.16) (1.89) (4.06) (2.72) (1.63) (1.05) (1.74) (0.76) (1.08)

Beta (ex-ante) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Volatility 0.42 0.30 0.22 0.18 0.21 0.30 0.13 0.32 0.16 0.14

Sharpe ratio 0.46 0.30 0.53 1.05 0.76 0.59 0.30 0.51 0.27 0.25

*Table 4-2. BAB Portfolio, country index constituents *

Country ATX BEL 20 OMX

25

SBF

120 HDAX ATHEX AEX PSI-20 IBEX

35

STOXX

600 SMI

Excess return 0.64% **0.07% ** **1.04% ** 0.40% **0.68% ** **0.94% ** **0.13% ** **0.79% ** **0.64% ** 0.38% **0.24% **

(1.78) (0.29) (2.44) (1.90) (2.64) (2.37) (0.48) (2.83) (2.46) (1.89) (0.99)

CAPM alpha **0.70% ** 0.07% **1.04% ** **0.41% ** **0.72% ** **0.78% ** 0.19% **0.78% ** **0.65% ** 0.37% **0.28% **

**(2.02) ** **(0.31) ** (2.71) (2.15) **(3.09) ** (2.04) (0.73) (2.76) (2.66) (1.92) (1.2)
Three-factor

alpha

**0.64% ** **0.13% ** **0.98% ** **0.42% ** **0.74% ** **0.69% ** 0.26% **0.83% ** **0.64% ** **0.41% ** **0.35% **

(1.87) (0.56) (2.54) (2.23) (3.19) (1.81) (1.00) (2.95) (2.58) (2.21) (1.51) Four-factor

alpha

**0.63% ** **0.04% ** **0.74% ** **0.32% ** **0.66% ** **0.53% ** 0.16% **0.77% ** **0.51% ** 0.29% **0.26% **

(1.46) (0.14) (1.94) (1.60) (2.80) (1.32) (0.60) (2.70) (1.97) (1.47) (1.05)

Beta (ex-ante) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Volatility 0.19 0.14 0.23 0.11 0.14 0.21 0.15 0.15 0.14 0.11 0.13

Sharpe ratio 0.41 0.06 0.58 0.43 0.61 0.57 0.11 0.66 0.57 0.43 0.22

*Figure 1. Annualized Betting-against-beta (BAB) Sharpe ratios by country *

*Figure 1 illustrates the annualized Sharpe ratios of Betting-against-Beta factors across sample *
countries and their corresponding indices. All the BAB factors produce positive Sharpe ratios and
most of them fall between 0.2 and 0.6. In four countries, the BAB portfolios of all common stocks
deliver approximately same as the portfolios of index constituents, namely Austria, Finland,
Greece, and Switzerland. However, in Belgium, France, Germany, and Netherlands, the BAB
portfolios of all common stocks have significantly higher ratios than the portfolios of index
constituents. Specifically, except Germany, the BAB Sharpe ratios are more than doubled than the
index constituents. The highest Sharpe ratio is presented by the portfolio of all French common
stocks, which is 1.05 and the portfolio of BEL 20 constituents has the lowest which is 0.06.

*It is shown in Table 4-1 and Table 4-2 that among all ten countries and the European index, the *
BAB portfolios produce not only positive excess returns but also positive risk-adjusted returns.

The BAB portfolio is the self-financing market neutral portfolio and has an estimated beta of zero.

Except Spain, the excess returns of the BAB portfolios which consists of all common stocks have higher values with statistical significance. For example, in France, the monthly excess return of BAB of all common stocks is roughly 1.4% and significant in all three models whilst its

counterpart, the BAB of SBF 120 constituents, which contain only an monthly average of 118 stocks, present about 1% lower for both the excess returns and all the risk-adjusted returns. The situation for Spain is quite on the contrary of the others. Over the twenty years period, the

components of Spanish Exchange Index (IBEX 35) appear to deliver greater BAB returns than all Spanish stocks. None of the BAB returns of common stocks are significant and, in contrast, the

IBEX 35 components have averagely 0.3% higher monthly returns. As an aggregate, the six hundred European stocks in STOXX 600 give a significant 0.38% BAB returns monthly with a Sharpe ratio of 0.43.

In the section with country specific results above, I present the excess returns and risk-adjusted returns of three asset pricing models of each beta-sorted portfolio and the BAB portfolios.

Followed by figures with visualized excess returns and CAPM alphas composed with the estimated betas and the realized CAPM betas respectively for both all stocks and indices.

Frazzini and Pedersen (2014) present in their work that for the international market, “the alpha and Sharpe ratios of the beta-sorted portfolios decline (although not perfectly monotonically) with the betas” (Frazzini and Pedersen, 2014, pp. 10). Nevertheless, the results of this thesis

demonstrated very different patterns of alphas for each country in the European market. In general, I classify the behaviours of alphas with the increase of portfolio betas into four

categorizations, namely alphas declining from low-beta to high-beta portfolios, alphas peaking in the median-beta portfolios, alphas increasing from low- to high-beta portfolios, and no apparent relation between alphas and portfolio betas.

Most of country indices and one country with all common stocks can be categorized as alphas declining from low-beta to high-beta portfolios, including OMX 25, SBF 120, HDAX, AEX, PSI- 20, IBEX 35, SMI, and STOXX 600 on top of Portugal. The 25 components of OMX Helsinki 25 and the European overall index, STOXX 600 have the most monotonical trend of alphas related to portfolio betas. SBF 120 appears a stage decrease – from about 0.6% for P1 and P2 to roughly 0.4% for P3 to P6 and at last almost zero for P9 and P10. The first six beta-sorted portfolios of HDAX remains at the same level of 0.65% while the returns drop more steeply when it comes to the last four portfolios from 0.4% to -0.4%.

* Figure 2. CAPM alpha and realized beta of STOXX 600 constituents*

*Figure 3. Excess returns and estimated beta of Portugal and PSI-20 constituents *

In the second categorization, alphas peaking in the median-beta portfolios, I have indices ATX, BEL 20 and countries Finland, Germany, Spain, Netherlands, and Switzerland. Except BEL 20 and Germany, in other markets alphas increase from the low-beta portfolios and peak in the meddle portfolios – P3. Alphas of which the portfolios with higher estimated betas lower consequently.

*Figure 4. CAPM alpha and realized beta of Finland and OMX 25 constituents *

Thirdly, in contrast to the anticipation, the beta-sorted portfolios of all common stocks in Austria and Greece have a roughly increasing alphas with beta. Starting with a high 0.89% alphas from P1 in Austria, the number drops to 0.48% for P2 and then the alphas monotonously grow to 0.83%

for P5. In Greece, alphas increase from 0.32% for P1 to 0.53% for P4 before it resets to zero for the highest-beta portfolio P5.

*Figure 5. CAPM alpha and realized beta of Greece and ATHEX constituents *

At last, the remaining markets such as Belgium, France and ATHEX index occur no apparent alphas patterns in relation to beta. With an increasing beta, the return of Belgium stocks fluctuate between 0.36% and 0.63%. And in France, there is a spike appearing in the lowest-beta portfolio and the rest remains at approximately 0.6% level.

*Figure 6. CAPM alpha and realized beta of Belgium and BEL 20 constituents *

Overall, other than Portugal, none of the country with all common stocks has a declining alpha trend as obvious as what was presented in Frazzini and Pedersen (2014). Whereas the constituents country indies display a rather consistent pattern with the authors work across countries.

**5. Conclusion **

In this thesis, I empirically test Frazzini and Pedersen’s Betting-against-Beta strategy in several European countries. I find that when all common stocks in a country are included, the portfolios of high beta stocks does not necessarily have lower alphas. Moreover, different countries behave differently which are categorized into four classes. However, when it comes to the selected stocks of the index constituents, the effect described by Frazzini and Pedersen seems to be strengthened.

I consider seven out of ten country indices and STOXX 600, which represents the complete European market, perform in line with the pattern of “high beta, low alpha”.

Furthermore, in all sample countries, the BAB factor is able to deliver positive returns and positive Sharpe ratio. And I find that the country indices usually produce smaller BAB returns than the all-stocks portfolio.

**6. References **

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Fama, E. F., & French, K. R. (2004). The capital asset pricing model: Theory and evidence. Journal of economic perspectives, 18(3), 25-46.

Fama, E. F., & French, K. R. (2021). Common risk factors in the returns on stocks and bonds. The Fama Portfolio, 392-449.

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Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. The journal of finance, 19(3), 425-442.

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The Journal of Finance, 28(5), 1233-1239.

**Appendix: Betting-against-Beta in Each Country **

In this section, results are presented by the sample countries. The tables shows estimated betas, excess returns, alphas, volatilities, and Sharpe ratios of beta-sorted portfolio and BAB portfolio.

Portfolios are ranked by the estimated betas. P1 includes the lowest estimated bate stocks and P5/P10 contains the highest betas. Germany, France and STOXX 600 constituents have ten beta- sorted portfolios due to the large amount of stocks and other markets have five beta-sorted portfolios. The returns of the Betting-against-Beta (BAB) factor are demonstrate in the rightmost column. Returns and alphas are in monthly percent. Below the excess returns and alphas show t- statistics and 5% statistical significance is indicated in bold. Beta (ex-ante) is the average estimated betas at portfolio formation. Volatilities and Sharpe ratios are annualized.

**1. Austria **

*Table 1-1. Austria (AUT), all common stocks *

Portfolio P1

(low beta)

P2 P3 P4 P5

(high beta)

BAB

Excess return **0.89% ** **0.48% ** **0.48% ** **0.60% ** **0.83% ** 1.49%

**(2.97) ** **(3.16) ** (2.85) (2.75) (3.12) (1.91)

CAPM alpha **0.89% ** **0.40% ** **0.34% ** 0.42% **0.58% ** **1.59% **

(2.74) (2.28) (1.55) (1.51) (1.84) (2.10)

Three-factor alpha **0.88% ** **0.37% ** 0.30% 0.40% 0.56% **1.57% **

(2.59) (2.07) (1.36) (1.41) (1.78) (2.02)

Four-factor alpha **0.74% ** 0.34% 0.26% 0.44% 0.65% 1.21%

(2.39) (1.74) (1.15) (1.36) (1.87) (1.66)

Beta (ex-ante) 0.36 0.56 0.74 0.98 1.44 0.00

Volatility 0.75 0.37 0.41 0.53 0.65 0.42

Sharpe ratio 0.15 0.16 0.14 0.14 0.16 0.46

*Table 1-2. Austrian Traded Index (ATX) Constituents *

*Portfolio * P1

(low beta)

P2 P3 P4 P5

(high beta)

BAB

Excess return **0.65% ** **0.88% ** **0.82% ** **0.75% ** 0.39% 0.64%

(2.17) **(3.02) ** **(2.64) ** (2.00) (0.99) (1.78)

CAPM alpha 0.56% **0.66% ** **0.68% ** 0.39% -0.04% **0.70% **

(1.79) **(2.39) ** **(2.65) ** (1.26) (-0.13) (2.02)

Three-factor alpha 0.52% **0.57% ** **0.71% ** 0.37% -0.05% 0.64%

(1.65) (2.12) (2.78) (1.21) (-0.16) (1.87)

Four-factor alpha 0.52% **0.56% ** 0.50% 0.45% 0.01% 0.63%

(1.43) (2.00) (1.88) (1.38) (0.04) (1.46)

Beta (ex-ante) 0.80 1.02 1.18 1.36 1.70 0.00

Volatility 0.32 0.29 0.31 0.38 0.41 0.19

Sharpe ratio 0.25 0.38 0.33 0.25 0.12 0.41

*Figure 1-1. Excess returns and estimated beta of Austria and ATX constituents *

*Figure 1-2. CAPM alpha and realized beta of Austria and ATX constituents*

**2. Belgium **

*Table 2-1. Belgium (BEL), all common stocks *

Portfolio P1

(low beta)

P2 P3 P4 P5

(high beta)

BAB

Excess return 0.38% **0.57% ** **0.48% ** **0.63% ** 0.36% 0.73%

(1.93) **(4.65) ** **(3.03) ** **(3.64) ** (1.65) (1.30)

CAPM alpha **0.36% ** **0.54% ** **0.42% ** **0.63% ** **0.39% ** 0.73%

(1.39) (3.13) (1.97) (2.56) (1.12) (1.31)

Three-factor alpha 0.34% **0.50% ** 0.38% **0.56% ** 0.26% 0.74%

(1.27) (2.91) (1.82) (2.31) (0.74) (1.29)

Four-factor alpha 0.33% **0.43% ** 0.36% **0.60% ** 0.25% 0.69%

(1.26) (2.34) (1.59) (2.36) (0.64) (1.16)

Beta (ex-ante) 0.34 0.57 0.76 0.99 1.45 0.00

Volatility 0.63 0.39 0.50 0.55 0.70 0.30

Sharpe ratio 0.07 0.18 0.12 0.14 0.06 0.30

*Table 2-2. BEL 20 Constituents *

*Portfolio * P1

(low beta)

P2 P3 P4 P5

(high beta)

BAB

Excess return 0.12% 0.40% 0.32% 0.32% 0.03% 0.07%

(0.73) (1.54) (1.25) (0.93) (0.07) (0.29)

CAPM alpha 0.12% 0.43% 0.35% 0.33% 0.08% 0.07%

(0.72) **(1.79) ** **(1.45) ** (1.02) (0.23) (0.31)

Three-factor alpha 0.12% 0.42% 0.31% 0.22% -0.06% 0.13%

(0.71) (1.73) (1.30) (0.66) (-0.15) (0.56)

Four-factor alpha 0.12% 0.37% 0.28% 0.21% 0.13% 0.04%

(0.66) (1.50) (1.14) (0.61) (0.30) (0.14)

Beta (ex-ante) 0.87 1.07 1.20 1.37 1.81 0.00

Volatility 0.18 0.25 0.26 0.33 0.44 0.14

Sharpe ratio 0.08 0.20 0.15 0.12 0.01 0.06

*Figure 2-1. Excess returns and estimated beta of Belgium and BEL 20 constituents *

*Figure 2-2. CAPM alpha and realized beta of Belgium and BEL 20 constituents *

**3. Finland **

*Table 3-1. Finland (FIN), all common stocks *

Portfolio P1

(low beta)

P2 P3 P4 P5

(high beta)

BAB

Excess return **0.51% ** **0.73% ** **0.90% ** **0.73% ** 0.22% **0.92% **

(3.99) (6.32) (7.50) (5.73) (1.40) (2.26)

CAPM alpha **0.52% ** **0.71% ** **0.87% ** **0.67% ** **0.12% ** **0.89% **

(2.25) (2.81) (3.28) (2.52) (0.34) (2.43)

Three-factor alpha **0.42% ** **0.59% ** **0.73% ** **0.57% ** -0.01% **0.78% **

(1.88) (2.43) (2.83) (2.19) (-0.04) (2.24)

Four-factor alpha 0.31% **0.49% ** **0.57% ** 0.47% -0.09% 0.64%

(1.42) (1.96) (2.07) (1.68) (-0.25) (1.89)

Beta (ex-ante) 0.51 0.68 0.81 1.00 1.43 0.00

Volatility 0.40 0.36 0.37 0.39 0.49 0.22

Sharpe ratio 0.16 0.25 0.30 0.23 0.05 0.53

*Table 3-2. OMX Helsinki 25 Constituents *

*Portfolio * P1

(low beta)

P2 P3 P4 P5

(high beta)

BAB

Excess return **0.89% ** **0.69% ** **0.63% ** 0.35% -0.05% **1.04% **

(3.93) (2.53) (2.10) (0.95) (-0.13) (2.44)

CAPM alpha **0.85% ** 0.63% 0.57% 0.41% -0.03% **1.04% **

(3.21) **(1.77) ** **(1.65) ** (1.03) (-0.07) (2.71)

Three-factor alpha **0.77% ** 0.54% 0.50% 0.30% -0.10% **0.98% **

(2.89) (1.52) (1.51) (0.75) (-0.27) (2.54)

Four-factor alpha **0.62% ** 0.47% 0.53% 0.31% 0.01% 0.74%

(2.37) (1.35) (1.43) (0.66) (0.02) (1.94)

Beta (ex-ante) 0.83 0.98 1.15 1.31 1.66 0.00

Volatility 0.27 0.29 0.33 0.39 0.47 0.23

Sharpe ratio 0.42 0.30 0.24 0.11 -0.01 0.58

*Figure 3-1. Excess returns and estimated beta of Finland and OMX 25 constituents *

*Figure 3-2. CAPM alpha and realized beta of Finland and OMX 25 constituents *

**4. France **

*Table 4-1. France (FRA), all common stocks *

Portfolio P1

(low beta)

P2 P3 P4 P5 P6 P7 P8 P9 P10

(high beta)

BAB

Excess return **1.16% ** **0.68% ** **0.70% ** **0.68% ** **0.67% ** **0.60% ** **0.57% ** **0.54% ** **0.37% ** **0.73% ** **1.42% **

(6.71) (5.92) (6.98) (6.93) (6.37) (5.94) (5.24) (4.62) (3.16) (4.49) (4.34)

CAPM alpha **1.10% ** **0.63% ** **0.63% ** **0.60% ** **0.59% ** **0.54% ** 0.46% 0.41% 0.21% 0.59% **1.44% **

(4.85) (3.43) (3.88) (3.04) (2.79) (2.50) (1.93) (1.69) (0.86) (1.55) (4.61)
Three-factor alpha **1.08% ** **0.59% ** **0.61% ** **0.55% ** **0.55% ** **0.48% ** 0.36% 0.35% 0.15% 0.50% **1.43% **

(4.80) (3.42) (3.72) (2.81) (2.57) (2.21) (1.57) (1.41) (0.60) (1.37) (4.71)
Four-factor alpha **1.04% ** **0.53% ** **0.55% ** **0.47% ** **0.46% ** 0.40% 0.24% 0.29% 0.14% 0.53% **1.32% **

(4.50) (2.79) (3.29) (2.34) (2.01) (1.81) (1.05) (1.14) (0.54) (1.42) (4.06)

Beta (ex-ante) 0.29 0.47 0.57 0.66 0.77 0.88 1.01 1.18 1.39 1.86 0.00

Volatility 0.83 0.55 0.48 0.47 0.51 0.48 0.52 0.57 0.56 0.78 0.18

Sharpe ratio 0.18 0.15 0.18 0.18 0.16 0.15 0.14 0.12 0.08 0.12 1.05

*Table 4-2. SBF 120 Constituents *

Portfolio P1

(low beta)

P2 P3 P4 P5 P6 P7 P8 P9 P10

(high beta)

BAB

Excess return **0.66% ** **0.72% ** **0.35% ** **0.43% ** **0.43% ** 0.28% **0.49% ** **0.49% ** -0.02% 0.23% **0.40% **

(5.24) (5.33) (2.24) (2.55) (2.54) (1.45) (2.35) (2.16) (-0.07) (0.71) (1.90)

CAPM alpha **0.60% ** **0.66% ** 0.28% 0.33% **0.33% ** 0.18% 0.36% 0.37% -0.16% 0.06% **0.41% **

**(4.06) ** **(3.98) ** (1.79) (1.88) **(2.00) ** (0.95) (1.48) (1.51) (-0.48) (0.14) (2.15)
Three-factor alpha **0.57% ** **0.67% ** 0.28% 0.29% 0.31% 0.17% 0.31% 0.38% -0.21% -0.01% **0.42% **

(3.89) (4.01) (1.81) (1.63) (1.88) (0.89) (1.31) (1.58) (-0.62) (-0.03) (2.23)
Four-factor alpha **0.48% ** **0.61% ** 0.25% 0.23% 0.21% 0.07% 0.21% 0.35% -0.21% 0.13% 0.32%

(3.22) (3.47) (1.57) (1.24) (1.27) (0.37) (0.86) (1.38) -(0.55) (0.31) (1.60)

Beta (ex-ante) 0.84 0.99 1.10 1.21 1.29 1.38 1.49 1.63 1.81 2.18 0.00

Volatility 0.22 0.24 0.27 0.30 0.30 0.33 0.37 0.40 0.47 0.57 0.11

Sharpe ratio 0.36 0.38 0.16 0.18 0.18 0.10 0.17 0.15 0.00 0.05 0.43

*Figure 4-1. Excess returns and estimated beta of France and SBF 120 constituents *

*Figure 4-2. CAPM alpha and realized beta of France and SBF 120 constituents *