Volatility factor in asset pricing model

Volatility Factor in Asset Pricing Model

Student number: 10630287

Programme: BSc Economics and Business

Track: Finance and Organization

Supervisor: Dr. Esther Eiling

Word count: 5772

Date: 29 June 2016

Abstract

Studies show that higher expected stock returns are associated with lower risk, which

contradicts the simple capital asset pricing model (CAPM). The conducted research confirms the presence of the low volatility anomaly for the U.S. market for the period between January 1960 and December 2015. It is found that the volatility effect is apparent only for small and medium companies in terms of market equity. The volatility (NMV) factor is found to be statistically significant for more than half of the U.S. industries and for nine Fama and French 25 size-BE/ME portfolios. The cross-sectional regression results show that the market β has a statistically insignificant coefficient, while exposure to SMB, HML and NMV factors is priced.

Statement of Originality

This document is written by Student Simonas Deksnys who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document is 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.

Table of Contents

1. Introduction ___________________________________________________________ 1 2. Literature______________________________________________________________ 3

a. Low Risk Anomaly _________________________________________________________ 3 b. Asset Pricing Model ________________________________________________________ 5 3. Methodology ___________________________________________________________ 6 4. Results and Analysis ____________________________________________________ 10

a. Low Volatility Anomaly ____________________________________________________ 10 b. NMV factor ______________________________________________________________ 12

c. Main Findings ____________________________________________________________ 14

5. Conclusion ___________________________________________________________ 15 6. References ____________________________________________________________ 17 7. Appendix A ___________________________________________________________ 18

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1. Introduction

Several studies show that stocks with lower risk historically outperform the ones with higher risk, where risk is measured in different ways. Blitz and Van Vliet (2007) find that stocks with lower volatility historically outperform market and stocks with higher volatility. Frazzini and Pedersen (2014) show that lower market β is associated with higher returns. Ang et al. (2009) discover the negative relation between stock returns and idiosyncratic volatility, calculated as a standard deviation of residuals from Fama and French 3-factor model. All of these authors indicate that the difference between risky and non-risky stocks are neither captured by simple CAPM model nor by Fama and French 3-factor model.

As mention in Fama and French (1996) stock returns that are not predicted by CAPM are called anomalies. Therefore, the differences in average returns of stocks with low risk and high risk is called low volatility anomaly. In their paper, Fama and French (1992) claim that stocks with low market equity historically outperform stock with high market equity based on average returns. Similarly stocks with high book-to-market equity ratio historically

outperform stock with low book-to-market equity ratio. The former stocks are called growth stocks and the latter are called value stocks. Fama and French (1992) find that these

differences in average returns are not captured by simple CAPM model, which measure sensitivity of stock returns to market returns. Thus, these finding can be regarded as

anomalies. Fama and French (1992) exploit these difference in returns to build two additional factors that are added to CAPM regression model extending it to 3-factor model. They claim that small minus big (SMB) and high minus low (HML) factors “capture the cross-sectional variation in average stock returns associated with market β, size, leverage, book-to-market equity and earnings-price ratios” (Fama & French, 1992). They build SMB and HML portfolios by calculating the gap between the returns of small and big firms based on market equity and the difference between value and growth stocks respectively. They find that both factors have statistically significant coefficients in multiple regression models, when

predicting the returns of various portfolios built by applying different investment strategies. Following the reasoning behind Fama and French SMB and HML factors, new non-volatile minus non-volatile (NMV) factor will be created to test whether low volatility anomaly can be exploited to improve the 3-factor model in terms of predicting returns of Fama and French (1997) industry portfolios and Fama and French (1992) 25 size and BE/ME portfolios.

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Hence, the research question of the paper is formulated as follows: Does adding a Volatility

Factor (NMV) improve the 3-factor multiple regression model for the U.S. industry, Fama and French 25 and Volatility portfolios?

The research question contributes to former literature as it builds upon the idea of Fama and French SMB and HML factors and exploits the volatility anomaly described by the other authors in order to create a new (volatility) factor, therefore, combining two research topics on investment into one. The paper investigates both the possibility to improve the 3-factor model and gives some insight on the volatility anomaly, which is still a topic with ample opportunities for research as there is no particular explanation for it. The topic is relevant as research suggests that even 3-factor model has volatile factor slopes (Fama & French, 1997), thus the predicted cost of equity might be inaccurate. It is important to investigate possible improvements of asset pricing models to make cost of equity estimates more accurate, which are used in discounting future cash flows by financial managers. Having a strong model would help managers to make more efficient investment decisions. Finally, the model, which can better predict industries’ returns, would benefit mutual and hedge funds as their portfolios tend to track specific industries.

The goal of the paper is to investigate U.S. stock data to verify whether the low volatility anomaly exists and if it does, use the acquired results to calculate portfolio returns based on volatility in order to create another factor for Fama and French model and thus improve the predictive power of the multiple regression model. Firstly, the U.S. stock returns will be divided into quintile portfolios based on historic volatility. Then the descriptive statistics will be calculated in order to examine the volatility anomaly. The volatility factor will be built by going long on lowest quintile volatility portfolio and short on higher quintile volatility portfolio, hence the factor name non-volatile minus volatile (NMV). Then the 3-factor model will be compared with the new 4-factor model in terms of predicting stock returns of 43 industry portfolios built by Fama and French and their 25 size-BE/ME portfolios. Finally cross-sectional regression on portfolio average portfolio returns will be used to check, whether the exposure to the volatility factors is priced.

The outline of the paper is as follows. First, existing literature will be reviewed to explain the reasoning behind the research question and hypothesis will be posed based on findings in the literature. In the third part, research method, data used in the paper and the reasoning behind these choices will be presented. The fourth section will provide the results of the

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research and implications based on those results. Finally the last section will include short recap of the main ideas, results and limitations of the paper and proposal for the possible future research.

2. Literature

a. Low Risk Anomaly

In their paper Blitz and Van Vliet (2007) present the findings that stocks with lower historical volatility, as measured by standard deviation, outperform the market and high volatility stocks. Less volatile stocks have higher Sharpe ratios and higher average historical returns. They test the difference between the Sharpe ratios using Jobson and Korkie [1981] test (Blitz & Van Vliet, 2007). Authors find the presence of the volatility anomaly globally on large-cap stocks, which indicates that it is not only common to the U.S. market. Their approach is to rank stocks based on 3 year historical volatility, using weekly data, into decile portfolios. They find that the lowest decile portfolio has higher average return than the top decile portfolio. Blitz and Van Vliet (2007) indicate that in order to exploit the volatility effect it is necessary to apply high leverage to the portfolio. However, there are leverage constraints as private investors are unable or unwilling to apply high leverage to their portfolios and mutual funds are restricted from applying leverage by regulatory authorities. To add, the volatility effect can be driven by the investors’ preference for more volatile stocks as investors tend to hold few stocks in their portfolios and in order to achieve high returns, they require risky stocks (Blitz & Van Vliet, 2007). These investor characteristics lead to higher volume of investment into volatile stocks, making the market more efficient on the volatile side and less efficient on the less-volatile side due to lower demand for safer stocks. Hence positive alphas are observed for low volatility portfolios throughout the

studies, which indicate that these portfolios are underpriced. Finally Blitz et al. (2014) discuss that the volatility effect arises due to failure in simple CAPM assumptions that there are no constraints, for instance on leverage or borrowing stocks to short sell, investors are risk averse, existence of perfect information and market efficiency.

Baker et al. (2011) support the findings of the volatility anomaly. They approach the issue by splitting their stock data into quintile portfolios based on standard deviation of 60 months historic returns and adjusting the portfolios every month. They also contribute to the research by sorting data into quintiles based on market beta and finding that lower beta stocks have higher returns than higher beta stocks. This implies that the volatility anomaly is persistent to

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other risk factors, thus riskier stocks underperform less risky stocks. Baker et al. (2011) provide two explanations for the low volatility anomaly based on private investors’ behavior. First they claim that investors irrationally invest in high volatility stocks, which leads to underinvestment in low volatility ones. This is caused due to the preference for lotteries as named by the authors. Investors prefer to invest in relatively cheap stocks, which can provide high returns in short-periods, but smaller returns on average. This behavior leads to

overinvestment and overpricing of volatile stocks, thus reducing their average returns. Another reason for preference of volatile stocks is representativeness. As described in paper, for instance, firms that work with technological innovations are represented by big

corporations such as Microsoft, which makes uniformed investors believe that, technology firms can provide high returns, while they do not know that the majority technological startups fail. Finally, the preference for high volatility stocks is driven by the overconfidence of investors. Optimistic investors tend to overestimate the future returns of stocks, thus expecting to capture high profits from volatile stocks. This lead to overvaluation of stocks and thus they are sold for higher prices, leading to lower returns. Another reason indicated by Baker et al. (2011) is underinvestment in low volatility stocks by institutional investors. One reason for this is that high volatility stocks tend to be small stocks, which are hard to short sell and thus exploiting their low returns. Another reason is that investment funds face leverage constraints as managers are expected to maximize risk-adjusted returns, without applying leverage to their portfolios, which is a necessary condition to exploit the volatility anomaly as stated by Blitz and Van Vliet (2007). These behavioral patterns of individual and institutional investors drive up the demand for more volatile stocks, which in turn reduces their returns.

Frazzini and Pedersen (2014) also find negative relation between market beta and average stock returns. They also contribute towards the academic field by proposing an investment strategy of buying low beta stocks and short selling high beta stocks. They find that portfolios built based on this strategy have higher Sharpe ratios than all of the quintile portfolios ranked by market beta.

Ang et al. (2009) investigate the risk anomaly by sorting stocks based on idiosyncratic volatility as measured by the standard deviation of residuals from Fama and French 3-factor regression model. They perform the research globally and find the negative relation between the idiosyncratic volatility and average returns across different world markets. Unlike Fama and French approach to subtract the returns of portfolio with smaller average from the

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portfolio with higher average returns, Ang et al. (2009) derive their volatility factor by long buying the U.S. stocks of the highest idiosyncratic volatility quintile portfolios and short selling stocks of the lowest quintile, where bottom quintile portfolio has higher average return than the top quintile portfolio. They find that this factor has statistically significant

coefficients globally in predicting stock returns. Even though the volatility factor was created from the U.S. data only, it also has predictive power in global markets.

The former researches on the low risk anomaly provide evidence that the CAPM and Fama and French 3-factor models fail to accurately predict stock returns globally. In addition, there is a strong evidence of negative relation between various measures of risk and stock returns, which is driven by investors’ behavior. These studies imply that standard deviation, idiosyncratic volatility and market betas are closely related. What is important to mention, the bottom quintile or decile volatility portfolios were not necessarily found to be the ones with highest returns, but they outperformed the top volatility portfolio in all of the studies.

b. Asset Pricing Model

The simple CAPM also fails to explain other variations in returns as even earlier researches suggest. Banz (1981) as mentioned in Fama and French (1992) finds that the CAPM does not accommodate the variation of returns due to the size calculated as shares outstanding times the market value of the share. More specifically, smaller companies’ shares tend to outperform their bigger counterparts. It is found that controlling for size in linear regression model, improves the fit of the model, when predicting stock returns. Furthermore, the CAPM fails to account for stock return differences based on book-to-market equity ratio (BE/ME). Stocks with higher BE/ME, called value stocks, tend to historically outperform the ones with low BE/ME, which are referred to as growth stocks. It is also found that BE/ME can be used to explain the variation in stock returns. Following these finding Fama and French (1992) add 2 additional factors to the simple CAPM model, namely small-minus-big (SML) and high-minus-low (HML). The approach is explained in Fama and French (1996). The authors test SMB and HML factors by running regression on 25 portfolios built based on size and BE/ME and other portfolios created based on various investment strategies, such as momentum. They find statistically significant coefficients and better fits of the data in the model than compared to the simple CAPM. Finally, the cross-sectional regression results performed by Fama and French (1992) show that market β has a flat relation to the stock returns, while exposure to size and BE/ME factors has statistically significant coefficients.

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Fama and French (1997) indicate that current asset pricing models are imprecise when estimating cost of equity across different U.S. industries. They find that the coefficients estimated by the multiple regression 3-factor model are inconsistent between different time periods, which makes it biased estimators in predicting stock returns. This is a major issue as financial managers heavily rely on pricing models, when estimating the cost of capital, used to discount future cash flows, when evaluating investment opportunities for their companies. This can cause inefficient projects undertaken by companies, thus deteriorating society’s welfare. Also as some of the mutual and hedge funds build their portfolios by tracking specific industries, failure to estimate the returns accurately can lead to unprofitable investments and thus decrease in the wealth of investors.

Several studies provide evidence that current asset pricing models fail to explain stock returns accurately in many ways, due to investor behavior, restrictions in the market and market inefficiency in particular. This makes it important to investigate possible opportunities to improve the asset pricing models by adding additional factors that could capture the

anomalous returns. The factor under investigation in this paper is the volatility factor, which controls for differences between low and high volatility stock returns. By combining findings on the low risk phenomenon and Fama and French approach for SMB and HML factors based on anomalous differences in stock returns, it is expected that the volatility factor can help better explain the variations in the returns of the U.S. industries’ portfolios.

3. Methodology

The data used in the research is the whole CRSP database universe, accessed through (Center for Research in Security Prices, 2016), which includes AMEX, NYSE and NASDAQ stock indexes. The U.S. data is used mainly due to its availability, however it is expected that findings in the U.S. would persist in other markets as the volatility effect is present there as well. The time period under consideration is from 31 January, 1960 to 31 December, 2015 and monthly returns data is used. The period was chosen arbitrarily, but two points were important. First, it is important to use the period long enough to have enough data points for significant research and second, to consider the recent years, as all the former researches were conducted prior to the 2008 credit crisis or used the time period prior to it. Thus, investigating this specific time span also contributes towards the academic field as it takes into account the data, which has not been considered yet in terms of volatility effect.

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Before working with the CRSP data, outliers were deleted. The data points, which were considered outliers were calculated by the following equations: 𝐿𝐵 = 𝑄1 − 𝐼𝑄𝑅 × 1.5 for the lower boundary, where Q1 is the first quartile of returns and IQR is interquartile range and 𝑈𝐵 = 𝑄3 + 𝐼𝑄𝑅 × 1.5 for the upper boundary, where Q3 is the third quartile. After omitting the outliers, the monthly stock returns ranged from -0.2260 to 0.2326. The outliers were deleted as there were monthly returns in the data such as 700%, which were considered unrealistic. Even if such returns were actually observed it is extremely unlikely for such returns to occur, therefore, it is believed that they should not be considered as they distort the data.

The first part of the research was to verify the existence of the volatility effect as the reasoning behind the NMV factor relies on it. To do so, the data was split into 672 groups of 36 months, readjusting monthly. Every group was divided into 3 size groups based on bottom 30%, middle 40% and top 30% market equity breakpoints. Then as in Blitz and Van Vliet (2007), for each data size group, the historic volatility, as measured by the standard deviation, was calculated for each individual stock for 36 months and then the next month, the data was split into quintile portfolios based on this volatility estimate. The portfolios are split into quintiles following the approach of Baker et al. (2011) and Ang et al. (2009) since these researches show that this is enough to observe the risk anomaly. Then, for every portfolio the value-weighted monthly average returns were calculated. This process was repeated for every 672 months from January 1960 to December 2015, so the portfolios were adjusted monthly. Small, medium and big ME quintile portfolio returns were then averaged for every month to come up with the portfolios, which are uninfluenced by the size factor. The data was not ranked by BE/ME ratio, which would reduce the influence of HML factor on the volatility factor due to the simplicity of the research and unavailability of the data. It is also assumed that the HML factor does not influence the volatility factor based on Ang et al. (2009) paper, where they do not adjust the data for neither of the Fama and French factors. For every portfolio, average returns, standard deviations, excess returns and Sharpe ratios are calculated to observe, whether the volatility effect exists. To test for differences between Sharpe ratios of portfolios, the test statistic is calculated according to the equation 3.1. Based to Blitz and Van Vliet (2007) this test statistics is asymptotically standard normally distributed.

𝑧 = 𝑆𝑅1− 𝑆𝑅2

√_{𝑁 × (2 × (1 − 𝜌}1 _1,2) +1_{2 × (𝑆𝑅1}2_{× 𝑆𝑅}

22× 1 + 𝜌1,22 ))

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Where 𝑆𝑅 is a Sharpe ratio of the portfolio, 𝑁 is the number of observations, 𝜌 is the correlation between the portfolios and 𝑧 is the test statistics, which is standard normally distributed.

To add, for all portfolios 3-factor model coefficients are estimated using Ordinary Least Squares (OLS) multiple regression model as in the equation 3.2.

𝑅𝑖 − 𝑅𝑓 = 𝛼𝑖 + 𝛽𝑖 × (𝑅𝑀− 𝑅𝑓) + 𝑠𝑖 × 𝑆𝑀𝐵 + ℎ𝑖 × 𝐻𝑀𝐿 + 𝜀𝑖 (3.2)

Where dependent variable (𝑅𝑖 − 𝑅𝑓) are time-series returns of portfolio excess of the

risk-free rate, (𝑅_𝑀− 𝑅_𝑓) is an excess market return, 𝑆𝑀𝐵 is Fama and French Short-minus-Big factor and 𝐻𝑀𝐿 is Fama and French High-minus-Low factor. 𝛼, 𝛽, 𝑠 𝑎𝑛𝑑 ℎ are regression coefficients, 𝜀 is the error term of the regression and 𝛼 is a deviation of the portfolio return from historic average, which occurs due to the returns that are not explained by the model. The regression is used to observe any nonzero alphas in portfolios, as the former research suggests that low volatility portfolios have positive alphas and high volatility portfolio have negative alphas (Blitz & Van Vliet, 2007). The data for excess market returns, SMB, HML factors and risk-free rate are retrieved from Kenneth R. French database (French, 2016).

Based on the findings of the low risk anomaly, non-volatile minus volatile (NMV) factor is created by subtracting for every month the returns of the top quintile volatility portfolio from the bottom quintile volatility portfolio. This approach of building the factors for asset pricing model based on anomalous returns is taken from Fama and French (1992). However, there’s a small deviation from their approach as the data is split into quintiles and only extreme portfolios are used, unlike in SMB and HML factors, where data is split into two parts based on median and 3 parts respectively. This approach is taken because researches find that portfolios with lower risk not necessarily have higher returns than portfolios with higher risk, when middle deciles or quintiles portfolios are observed as in Blitz and Van Vliet (2007) and Baker et al. (2011). Though, the bottom portfolio always outperformed the top risk portfolio. The choice for using historic standard deviation as a risk measure to exploit the low risk effect is mainly driven by the simplicity of calculating it and the fact that the low risk anomaly exists across different types of risks, whether it be market beta (Frazzini & Pedersen, 2014) , idiosyncratic volatility (Ang, Hodrick, Xing, & Zhang, 2009) or historic standard deviation (Blitz & Van Vliet, 2007).

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The significance of NMV factor is then tested by running 2 multiple regression models on the U.S. 43 industries’ and 25 ME-BE/ME portfolios’ monthly value weighted average returns acquired from Kenneth R. French database (French, 2016). Only 43 out of 49 industries are considered as the rest did not have returns data available. First 3-factor model coefficients are estimated by the equation 3.2. Then NMV factor is added to the model and coefficients are again estimated for all portfolios of interest by the equation 3.3.

𝑅𝑖 − 𝑅𝑓 = 𝛼𝑖 + 𝛽𝑖 × (𝑅𝑀− 𝑅𝑓) + 𝑠𝑖 × 𝑆𝑀𝐵 + ℎ𝑖 × 𝐻𝑀𝐿 + 𝑣𝑖 × 𝑁𝑀𝑉 + 𝜀𝑖 (3.3)

Where the variables are as in equation 3.2, 𝑣 is a regression coefficient and 𝑁𝑀𝑉 is a non-volatile-minus-volatile factor calculated as described above.

For each industry, two regression outputs are compared to observe, whether NMV factor improves the regression model. In order to check that, t-test is used with robust standard errors, in order to avoid biased estimation due to possible heteroscedasticity in regression model error term. Hence the following hypotheses are formulated:

𝐻₀: 𝑣_𝑖 = 0; 𝐻₁: ≠ 0. For volatile portfolios it is expected to observe negative NMV coefficient and for non-volatile portfolios the coefficient is expected to be positive. This reasoning follows from Fama and French (1992) SMB and HML factors.

The estimated regression coefficients are then tested by running cross-sectional regression on average returns of 73 portfolios (43 industries, 25 ME-BE/ME portfolios, 5 volatility portfolios). The regression model used is shown in equation 3.4 below.

𝑅𝑖− 𝑅𝑓

̅̅̅̅̅̅̅̅̅̅ = 𝛼_𝑖 + 𝛾𝑚× 𝛽𝑖 + 𝛾𝑠× 𝑠𝑖 + 𝛾ℎ× ℎ𝑖 + 𝛾𝑣× 𝑣𝑖 + 𝜀𝑖 (3.4)

Where 𝑅̅̅̅̅̅̅̅̅̅̅ is a cross-sectional average excess returns of 73 portfolios, 𝛾_𝑖− 𝑅_{𝑓 𝑚}, 𝛾_𝑠, 𝛾_ℎ, 𝛾_𝑣 are regression coefficients for market, size, BE/ME and volatility time-series regression

coefficients from equation 3.3 respectively, which are used as independent variables in this regression. This method is a simplified Fama and Macbeth (1973) regression model, where they test the factors for different time periods, while here the whole period averages are tested at once. It is expected to find 𝛾_𝑣 > 0, which would indicate that the NMV (volatility) factor is priced and improves the predictive power of the asset pricing model. In order to test this expectation t-test is used with robust standard errors.

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4. Results and Analysis

a.

Low Volatility Anomaly

The research begins by investigating the differences between descriptive statistics of quintile volatility portfolios. The statistics for the 5 portfolios are shown in Table 1 in

increasing order, where L represent the lowest volatility portfolio and H represent the highest volatility portfolio.

All statistics were calculated using monthly portfolio returns. It can be observed from Table 1 that higher standard deviation is associated with lower return, when considering top and bottom volatility portfolios. Even though, the difference in Sharpe ratio between bottom and top portfolios is obvious, statistical test is performed for robustness. Using the equation 3.1, the test statistic z is estimated to be 5.7038, which rejects the null hypothesis 𝑆𝑅𝐿 = 𝑆𝑅𝐻

in favor of the alternative hypothesis 𝑆𝑅_𝐿 > 𝑆𝑅_𝐻 at 1% significance level, with p-value close to 0, where 𝑆𝑅𝐿 and 𝑆𝑅𝐻 are the Sharpe ratios of bottom and top quintile portfolios

respectively. Hence, the low volatility anomaly is statistically verified for the U.S. stock data for the period January 1960 – December 2015.

The 5 volatility portfolios for three size groups are also presented in Tables 2, 3 and 4 as the results provide some important insights about the volatility effect.

Portfolios L 2 3 4 H

ER 0.0100 0.0108 0.0101 0.0084 0.0071

STD 0.0307 0.0346 0.0386 0.0414 0.0455

ER-rf 0.0061 0.0069 0.0062 0.0045 0.0032

Sharpe 0.1998 0.1998 0.1616 0.1098 0.0706

Table 1. Quintile Portfolios Based on Standard Deviation

Portfolios L 2 3 4 H

ER 0.0077 0.0081 0.0065 0.0027 0.0001

STD 0.0351 0.0374 0.0401 0.0436 0.0458

ER-rf 0.0039 0.0042 0.0026 -0.0012 -0.0038

Sharpe 0.1099 0.1132 0.0646 -0.0264 -0.0831

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As it can be seen from the tables above, the volatility effect is clearly present for the small firms, where expected returns and Sharpe ratios are higher for the bottom portfolio when compared with the top portfolio. Sharpe ratio test statistic is estimated to be 7.0122 for the difference between Sharpe ratios of bottom and top volatility portfolios in the small ME group, which indicates statistical significance at 1% significance level. However, for the companies with high ME, the expected returns are higher for the top volatility portfolio than for the bottom volatility portfolio. Although, the Sharpe ratio remains the highest for the bottom portfolio of the big companies the difference is not statistically significant at 10% significance level as the test statistic for difference between the two Sharpe ratios is estimated to be 1.0519. Another interesting observation is that the expected returns and Sharpe ratios for all portfolios of big companies are higher than compared with the small companies’ portfolios. This is due to the fact, that big companies outperformed the small companies on average during the period January 1960 – December 2015, which is a contradicting result to the Fama and French SMB portfolio, which has an expected return of 0.0020 during the same time period, indicating that, the small companies outperformed the big companies. This difference might have occurred due to the different approach in splitting data based on size. In Fama and French (1992) the companies are ranked based on ME every July and then the portfolios are created in June. In this paper the data was split into three ME groups every month for the previous three years of monthly returns. The test statistics for the differences in Sharpe ratios between small companies’ and big companies’ portfolios can be found in Appendix A, Table A1. It is found that all Sharpe ratios are significantly higher for big portfolios compared to small ones.

Portfolios L 2 3 4 H

ER 0.0119 0.0128 0.0123 0.0105 0.0089

STD 0.0341 0.0385 0.0425 0.0455 0.0488

ER-rf 0.0080 0.0090 0.0084 0.0066 0.0050

Sharpe 0.2354 0.2327 0.1981 0.1449 0.1020

Table 3. Quintile Portfolios Based on Standard deviation for Medium Companies

Portfolios L 2 3 4 H

ER 0.0104 0.0114 0.0116 0.0121 0.0123

STD 0.0340 0.0381 0.0421 0.0461 0.0516

ER-rf 0.0065 0.0076 0.0077 0.0082 0.0085

Sharpe 0.1924 0.1983 0.1829 0.1777 0.1642

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Next 3-factor multiple regression is performed on excess returns of all 5 portfolios using the equation 3.2. The results are provided in Table 5. This is done to observe the alphas of the portfolios. As it was found by previous researches bottom quintile portfolios had positive alphas and top quintile portfolios had negative alphas, meaning that the low volatility portfolios were underpriced, while high volatility portfolios were overpriced. This can be explained in terms of investor behavior as described by Baker et al. (2011), investors in general tend to have higher preference for more volatile stocks, which drives their price up. As it can be seen from the Table 5 the two lowest volatility portfolios’ alphas are

significantly higher than zero at 1% significance level, which indicates that the portfolios are underpriced according to the 3-factor model. However, the rest of the portfolios have

statistically insignificant alphas, which means that they are priced correctly due to higher volume of investment in higher volatility stocks.

‘*’ – indicates 10% statistical significance level; ‘**’ – indicates 5% statistical significance level; ‘***’ – indicates 1% statistical significance level

b.

NMV factor

After verifying the volatility anomaly, NMV factor was built by subtracting monthly returns of the top portfolio from the bottom one. The volatility factor is first tested with the volatility portfolios by running multiple regression models on all 5 portfolios using the equation 3.3. The results of the 4-factor model (3-factors + NMV) are presented in Table 6. The NMV factor is statistically significant for 4 out of 5 volatility portfolios and as expected has negative slope with volatile portfolios and positive with less volatile portfolios. The 4-factor model regression output also provides an interesting result: all the regression coefficients were found to be identical for bottom and top portfolios apart from the NMV

Portfolios L 2 3 4 H α 0.0021116*** 0.0024821*** 0.0014076 -0.00028934 -0.0015098 t 2.8643 2.6661 1.3281 -0.2426 -1.0149 β 0.80465*** 0.89576*** 0.95942*** 0.95993*** 0.97598*** t 45.5442 40.1456 37.7708 33.5800 27.3753 s 0.44149*** 0.50832*** 0.57085*** 0.67977*** 0.77445*** t 17.5400 15.9907 15.7745 16.6914 15.2474 h 0.025913 -0.026583 -0.04106 -0.1027** -0.21177*** t 0.9512 -0.7727 -1.0484 -2.3299 -3.8524 R^2 0.8317 0.7990 0.7828 0.7594 0.6990

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factor coefficient. This indicates that the difference in portfolio returns was influenced by the difference in the volatility of the returns. However, the regression model with NMV factors still leaves some significant alphas, which means that the factor does not fully account for all the variation in portfolio returns above the risk-free rate.

‘*’ – indicates 10% statistical significance level; ‘**’ – indicates 5% statistical significance level; ‘***’ – indicates 1% statistical significance level

The volatility factor is then tested by comparing both 3-factor and 4-factor multiple regression outputs on 43 Fama and French (1997) industry portfolios monthly returns and 25 Fama and French (1992) portfolios constructed by size and BE/ME. The results can be found in Appendix 1, Tables A2, A3, A4 and A5. It was found that NMV factor is statistically significant at 5% significance level for 24 out of 43 industries and for 19 industries at 1% significance level. For Fama and French 25 portfolios the NMV factor is statistically significant for 9 out of 25 portfolios at 5% significance and only for 7 at 1% significance level, this is mainly because the returns of portfolios created by sorting data based on market equity and BE/ME ratio are explained by SMB and HML factors.

The time-series regression results do not tell whether the NMV factor is actually priced, in other words, whether NMV factor improves the predictive power of the asset pricing model, as the factor‘s coefficient is not significant for all the portfolios. By performing cross-sectional regression of average portfolio returns on the coefficients of time-series regression as in equation 3.4, it is tested whether knowing the exposure of stock returns to specific factor, can help predict its return. It was expected to find positive relation between the volatility factor‘s coefficient 𝑣 and returns of 73 portfolios (43 industries, 25 Fama and

Portfolios L 2 3 4 H α 0.00080946* 0.0021918*** 0.0016341*** 0.00041457 0.00080946* t 1.7543 4.7625 3.1918 0.6996 1.7543 β 0.68607*** 0.72931*** 0.76852*** 0.74644*** 0.68607*** t 59.7136 63.6439 60.2871 50.5887 59.7136 s 0.2054*** 0.17921*** 0.19422*** 0.25924*** 0.2054*** t 12.0457 10.5373 10.2659 11.8386 12.0457 h 0.19074*** 0.20465*** 0.22409*** 0.1938*** 0.19074*** t 10.8701 11.6933 11.5101 8.5997 10.8701 v 0.28794*** 0.0085631 -0.13415*** -0.26599*** -0.71206*** t 13.6483 0.4070 -5.7310 -9.8176 -33.7516 R^2 0.8608 0.8913 0.8917 0.8742 0.9368

14

French portfolios and 5 volatility portfolios). The results of the cross-sectional regression are shown in Table 7.

‘*’ – indicates 10% statistical significance level; ‘**’ – indicates 5% statistical significance level; ‘***’ – indicates 1% statistical significance level

As in Fama and French (1992) it is found that market β has a flat, not significantly different from zero relation to the portfolio returns, while the sensitivities to other factors have positive regression coefficients. The sensitivity to the NMV factor has a cross-sectional regression coefficient of 0.2%, which means that increase in portfolios sensitivity 𝑣 to the NMV factor by one, would lead to the increase in the average portfolio returns by 0.2 percentage points. The coefficient 𝛾_𝑣 was found to be statistically different from zero at 1% significance level. This means, that the NMV factor does improve the asset pricing model, when estimating prices for the U.S. industry portfolios, 25 Fama and French portfolios and five volatility portfolios.

c.

Main Findings

The findings of the research verify the low volatility portfolio returns anomaly for the U.S. market for the period between January 1960 and December 2015. It was also found that the volatility anomaly occurs for the small ME companies’ group, while for the big ME companies’ group bottom volatility portfolio outperforms top volatility portfolio only in terms of Sharpe ratio, for which the difference is statistically insignificant. Furthermore, the volatility (NMV) factor portfolio created by subtracting bottom portfolio returns from top portfolio returns provides statistically significant explanation in variability for the returns of 24 out of 43 U.S. industries and nine out of 25 Fama and French size – book to market portfolios. The cross-sectional regression results confirms the findings by Fama and French (1992) of the flat market β, and also answers the research question as the sensitivity to the NMV factor was found be priced, having statistically significant cross-sectional regression coefficient at 1% significance level.

Coefficient γ_m γ_s γ_h γ_v

Estimate -0.0013289 0.00091374* 0.0016253*** 0.0023457***

t -0.926229772 1.856152567 2.841982798 2.671871494

15

5. Conclusion

The research was conducted in order to answer the research question: Does adding a

Volatility Factor (NMV) improve the 3-factor multiple regression model for the U.S. industry, Fama and French 25 and Volatility portfolios? It was hypothesized that the

NMV portfolio returns have statistically significant coefficient in multiple regression asset pricing model for the U.S. industries and 25 Fama and French portfolios.

The conducted research was based on the low risk anomaly findings by other authors and Fama and French 3-factor asset pricing model (1992) approach of building SMB and HML portfolios. The research contributes to the academic world by verifying the low volatility anomaly for the U.S. stock market for the period January 1960 to December 2015. It also shows that the anomaly is present for the standard deviation as a risk measure, which satisfies the assumption that risks investigated by other researchers are related to the standard

deviation as well. The paper also provides additional insights about the anomaly, as it is more apparent in small companies than big companies in terms of market equity. To add, the big companies’ volatility portfolios tend to outperform small companies’ volatility portfolios during the period of consideration. The findings also provide statistical evidence for the NMV portfolio explanatory capability of stock returns of the U.S. industries and Fama and French 25 portfolios. This is important as financial managers require accurate asset pricing models to make profitable investment decisions, thus contributing towards the welfare of their clients and society as a whole. The cross-sectional regression results also show that the portfolio returns exposure to the NMV factor is priced and thus can be used in predicting the future returns.

The research has its limitations as the NMV factor is not adjusted for book-to-market equity measure, which might impose bias on 4-factor multiple regression coefficients, when predicting the returns of the U.S. industries. Also the volatility effect, is found only among small and medium U.S. companies which partially contradicts the hypothesized volatility effect. Also it is observed that big companies outperform small companies during the period of the consideration, which contradicts the findings by Fama and French (1992) as their SMB factors has positive expected return during the same period. Finally, the outliers were deleted from the CRSP data returns, which might have influenced the significant results found on the NMV factor as high returns, which were considered unrealistic could have eliminated the volatility effect and also caused the higher average returns of the big companies.

16

For the future research it is proposed to investigate the low risk anomaly for different market equity company groups. Also SMB and HML factors are advised to be adjusted for volatility as well, the way Fama and French (1992) eliminated the influence of the factors on each other. Also, it advised to consider whether these findings apply to global markets as well. The variation in the NMV factor coefficient should also be tested by splitting the period under consideration into several periods, and checking whether the exposure to the factor is consistent over different times. Also periodical Fama-Macbeth regression should be used to further investigate the extent to which the NMV factor is actually priced.

To conclude, the research question is answered as the sensitivity of portfolios’ returns to the NMV factor have a significant cross-sectional regression coefficient, which in other words means that this exposure is priced. This is to the extent that the limitations do not influence the final results. Therefore, the findings should be addressed carefully and with further investigation.

17

6. References

Ang, A., Hodrick, R. J., Xing, Y., & Zhang, X. (2009). High idiosyncratic volatility and low returns: International and further U.S. evidence. Journal of Financial Economics, 91(1), 1-23.

Baker, M., Bradley, B., & Wurgler, J. (2011). Bechmarks as Limits to Arbitrage: Understading the Low-Volatility Anomaly. Financial Analysts Journal, 67(1), 40-54.

Blitz, D. C., & Van Vliet, P. (2007). The Volatility Effect. The Journal of Portfolio Management, 34(1), 102-113.

Blitz, D., Falkenstein, E., & Van Vliet, P. (2014). Explanations for the Volatility Effect: An Overview Based on the CAPM Assumptions. Journal of Portfolio Management, 40(3), 61-76. Center for Research in Security Prices. (2016). CRSP monthly stock. Retrieved May 14, 2016, from

Wharton Research Data Services:

https://wrds-web.wharton.upenn.edu/wrds/ds/crsp/stock_a/dsf.cfm?navId=128

Fama, E. F., & French, K. R. (1992). The Cross-Section of Expected Stock Returns. Journal of Finance, 47(2), 427-465.

Fama, E. F., & French, K. R. (1996). Multifactor Explanations of Asset Pricing Anomalies. The Journal of Finance, 51(1), 55-84.

Fama, E. F., & French, K. R. (1997). Industry costs of equity. Journal of Financial Economics, 43(2), 153-193.

Fama, E. F., & MacBeth, J. D. (1973). Risk, Return, and Equilibrium: Empirical Tests. Journal of Political Economy, 81(3), 607-636.

Frazzini, A., & Pedersen, L. H. (2014). Betting against beta. Journal of FInancial Economics, 111(1), 1-25.

French, K. R. (2016). Data Library: Kenneth R. French database. Retrieved May 30, 2016, from Kenneth R. French Web site:

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7. Appendix A

For all the tables in the Appendix A ‘*’ – indicates 10% statistical significance level; ‘**’ – indicates 5% statistical significance level; ‘***’ – indicates 1% statistical significance level.

Portfolios L 2 3 4 H

Difference 0.0834*** 0.0846*** 0.1172*** 0.2029*** 0.2450***

z 2.3042 2.4708 3.9191 6.3025 8.5446

19 Industry α t β t s t h t R^2 Agric -0.0002401 -0.12 0.836*** 17.21 0.4286*** 6.19 0.09981 1.33 0.4012 Food 0.002999** 2.49 0.7757*** 26.90 -0.1593*** -3.88 0.1651*** 3.71 0.5282 Beer 0.002939* 1.96 0.8111*** 22.52 -0.1086** -2.12 0.1125** 2.03 0.4483 Smoke 0.00569*** 2.78 0.7844*** 16.01 -0.2397*** -3.43 0.2037*** 2.70 0.2795 Toys -0.0032 -1.60 1.095*** 23.02 0.5253*** 7.75 0.1854** 2.53 0.5359 Fun 0.0010 0.59 1.311*** 30.75 0.4734*** 7.79 0.191*** 2.91 0.6576 Books -0.0011 -0.86 1.091*** 34.55 0.2212*** 4.91 0.2406*** 4.94 0.6838 Hshld 0.0016 1.37 0.8764*** 31.34 -0.1629*** -4.09 -0.0220 -0.51 0.6206 Clths -0.0006 -0.41 1.106*** 30.06 0.3783*** 7.22 0.3738*** 6.59 0.6345 MedEq 0.003841*** 2.81 0.8603*** 26.26 0.0628 1.35 -0.237*** -4.69 0.5867 Drugs 0.004561*** 3.57 0.8421*** 27.48 -0.3355*** -7.68 -0.3303*** -6.99 0.5908 Chems -0.0016 -1.44 1.127*** 41.78 -0.077** -2.00 0.3383*** 8.14 0.7342 Rubbr -0.0009 -0.68 1.019*** 33.99 0.6271*** 14.68 0.3415*** 7.39 0.7273 Txtls -0.0027 -1.60 1.147*** 28.07 0.6128*** 10.53 0.7687*** 12.20 0.6272 BldMt -0.002455** -2.15 1.198*** 43.67 0.2639*** 6.76 0.4646*** 10.99 0.7722 Cnstr -0.003083* -1.91 1.263*** 32.68 0.498*** 9.04 0.3337*** 5.60 0.6825 Steel -0.006219*** -3.74 1.274*** 31.93 0.407*** 7.16 0.3933*** 6.39 0.6600 Mach -0.0018 -1.57 1.187*** 43.63 0.2763*** 7.13 0.163*** 3.89 0.7819 ElcEq 0.0007 0.53 1.205*** 40.30 0.0591 1.39 0.0210 0.45 0.7433 Autos -0.004182** -2.56 1.214*** 30.98 0.1469*** 2.63 0.6482*** 10.73 0.6176 Aero 0.0002 0.12 1.148*** 28.16 0.203*** 3.49 0.3335*** 5.30 0.5827 Ships -0.0016 -0.79 1.158*** 24.14 0.144** 2.11 0.4952*** 6.70 0.4958 Mines -0.0011 -0.54 1.11*** 22.10 0.3276*** 4.58 0.4182*** 5.40 0.4756 Coal -0.0016 -0.47 1.142*** 13.97 0.3453*** 2.96 0.3876*** 3.07 0.2679 Oil 0.0011 0.73 0.9037*** 24.74 -0.2411*** -4.63 0.311*** 5.52 0.4797 Util 0.0003 0.23 0.658*** 23.77 -0.1878*** -4.76 0.4099*** 9.60 0.4647 Telcm 0.0009 0.77 0.8475*** 29.99 -0.2013*** -5.00 0.08222* 1.89 0.5869 PerSv -0.003302* -1.82 1.055*** 24.23 0.4627*** 7.46 0.1229* 1.83 0.5584 BusSv -0.0004 -0.43 1.064*** 50.45 0.3879*** 12.91 -0.0239 -0.74 0.8457 Hardw 0.0023 1.40 1.076*** 27.39 0.1547*** 2.76 -0.6232*** -10.29 0.6550 Chips 0.0000 0.01 1.229*** 36.18 0.4112*** 8.49 -0.4398*** -8.39 0.7634 LabEq 0.0013 0.95 1.132*** 34.28 0.482*** 10.24 -0.4303*** -8.45 0.7556 Paper -0.0011 -0.85 1.067*** 34.77 0.0103 0.23 0.43*** 9.09 0.6606 Boxes 0.0004 0.25 1.005*** 29.80 -0.0712 -1.48 0.1022** 1.97 0.5946 Trans -0.0015 -1.19 1.091*** 36.58 0.1902*** 4.48 0.3668*** 7.98 0.7003 Whlsl -0.0002 -0.16 1.004*** 39.82 0.463*** 12.89 0.1318*** 3.39 0.7753 Rtail 0.0015 1.28 0.9873*** 34.13 0.08075* 1.96 0.0189 0.42 0.6782 Meals 0.0008 0.54 1.034*** 28.26 0.2699*** 5.17 0.1394** 2.47 0.6047 Banks -0.0019 -1.50 1.201*** 39.48 -0.1441*** -3.33 0.5707*** 12.17 0.7064 Insur -0.0005 -0.32 1.086*** 32.38 -0.1432*** -3.00 0.401*** 7.75 0.6179 RlEst -0.007305*** -4.19 1.176*** 28.12 0.9256*** 15.54 0.7566*** 11.73 0.6675 Fin 0.0000 -0.03 1.243*** 49.31 0.1412*** 3.93 0.1841*** 4.74 0.8108 Other -0.003714** -2.17 1.167*** 28.42 0.255*** 4.36 0.165*** 2.61 0.6012

20 Industry α t β t s t h t v t R^2 Agric 0.0000 0.02 0.82308*** 16.08 0.40347*** 5.31 0.1177 1.51 -0.0753 -0.80 0.4018 Food 0.0017 1.41 0.83865*** 28.53 -0.0369 -0.85 0.077743* 1.73 0.3677*** 6.81 0.5589 Beer 0.0014 0.94 0.88435*** 23.98 0.0338 0.62 0.0108 0.19 0.42778*** 6.32 0.4794 Smoke 0.0042416** 2.07 0.8529*** 16.74 -0.1065 -1.41 0.1086 1.40 0.3999*** 4.27 0.2987 Toys -0.0027 -1.36 1.0746*** 21.45 0.4861*** 6.54 0.21341*** 2.79 -0.1179 -1.28 0.5371 Fun 0.0014 0.75 1.2965*** 28.87 0.44476*** 6.67 0.21147*** 3.08 -0.0861 -1.04 0.6581 Books -0.0014 -1.03 1.1031*** 33.15 0.24371*** 4.93 0.2245*** 4.42 0.0677 1.11 0.6844 Hshld 0.0009 0.78 0.90846*** 31.10 -0.10051** -2.32 -0.0665 -1.49 0.18727*** 3.49 0.6274 Clths -0.0010 -0.63 1.1228*** 28.99 0.41131*** 7.16 0.3503*** 5.92 0.0990 1.39 0.6355 MedEq 0.003842*** 2.77 0.86019*** 24.91 0.0627 1.22 -0.23692*** -4.49 -0.0004 -0.01 0.5867 Drugs 0.0039466*** 3.06 0.87113*** 27.13 -0.279*** -5.85 -0.37061*** -7.56 0.16954*** 2.88 0.5958 Chems -0.0012 -1.08 1.1085*** 39.11 -0.11244*** -2.67 0.36362*** 8.40 -0.10644** -2.05 0.7358 Rubbr -0.0014 -1.12 1.0457*** 33.26 0.67882*** 14.55 0.30457*** 6.34 0.15526*** 2.69 0.7302 Txtls -0.0028 -1.64 1.1513*** 26.73 0.62142*** 9.72 0.76255*** 11.59 0.0260 0.33 0.6273 BldMt -0.0023597** -2.03 1.1931*** 41.27 0.25516*** 5.95 0.47083*** 10.66 -0.0263 -0.50 0.7723 Cnstr -0.0022 -1.35 1.2211*** 30.21 0.41646*** 6.94 0.39192*** 6.35 -0.24487*** -3.30 0.6876 Steel -0.0045231*** -2.75 1.1934*** 29.19 0.25111*** 4.14 0.50459*** 8.08 -0.46832*** -6.24 0.6787 Mach -0.0006 -0.53 1.1309*** 40.62 0.16752*** 4.05 0.24062*** 5.66 -0.32667*** -6.39 0.7945 ElcEq 0.0011 0.90 1.1821*** 37.65 0.0152 0.33 0.0523 1.09 -0.13174** -2.29 0.7453 Autos -0.0033617** -2.04 1.1751*** 28.64 0.0715 1.17 0.70201*** 11.20 -0.22641*** -3.01 0.6227 Aero 0.0013 0.78 1.0954*** 25.77 0.1001 1.59 0.40694*** 6.27 -0.30916*** -3.96 0.5923 Ships -0.0008 -0.37 1.1182*** 22.22 0.0676 0.90 0.54972*** 7.15 -0.22959** -2.48 0.5004 Mines 0.0008 0.41 1.0156*** 19.66 0.14491* 1.89 0.54856*** 6.95 -0.54854*** -5.78 0.5007 Coal 0.0008 0.24 1.0271*** 12.08 0.1211 0.96 0.5476*** 4.22 -0.67327*** -4.31 0.2877 Oil 0.0017 1.08 0.87785*** 22.87 -0.29141*** -5.12 0.34695*** 5.92 -0.15106** -2.14 0.4833 Util -0.0011 -0.97 0.72258*** 25.73 -0.0624 -1.50 0.32035*** 7.47 0.37667*** 7.30 0.5043 Telcm 0.0006 0.49 0.86246*** 29.00 -0.17224*** -3.90 0.0615 1.35 0.0873 1.60 0.5885 PerSv -0.0037445** -2.03 1.0762*** 23.48 0.50332*** 7.40 0.0939 1.34 0.1221 1.45 0.5598 BusSv -0.0005 -0.54 1.0691*** 48.10 0.39785*** 12.06 -0.0310 -0.91 0.0299 0.73 0.8458 Hardw 0.0035574** 2.17 1.0162*** 24.93 0.0380 0.63 -0.53986*** -8.67 -0.35074*** -4.69 0.6660 Chips 0.0016 1.17 1.1531*** 33.35 0.26302*** 5.13 -0.33404*** -6.33 -0.44491*** -7.01 0.7796 LabEq 0.0019 1.34 1.1062*** 31.90 0.43149*** 8.38 -0.39417*** -7.44 -0.15179** -2.38 0.7576 Paper -0.0008 -0.65 1.0549*** 32.65 -0.0126 -0.26 0.44631*** 9.05 -0.0685 -1.16 0.6613 Boxes 0.0001 0.05 1.0182*** 28.66 -0.0460 -0.87 0.0842 1.55 0.0758 1.16 0.5954 Trans -0.0008 -0.62 1.0576*** 33.93 0.12603*** 2.72 0.41257*** 8.67 -0.19259*** -3.36 0.7053 Whlsl -0.0002 -0.14 1.0033*** 37.75 0.46189*** 11.71 0.13262*** 3.27 -0.0034 -0.07 0.7753 Rtail 0.0007 0.60 1.0261*** 34.06 0.15618*** 3.49 -0.0349 -0.76 0.22654*** 4.10 0.6861 Meals 0.0005 0.32 1.0494*** 27.22 0.29893*** 5.22 0.11867** 2.02 0.0873 1.23 0.6056 Banks -0.0028259** -2.23 1.2443*** 39.36 -0.0592 -1.26 0.51014*** 10.57 0.25494*** 4.39 0.7147 Insur -0.0008 -0.54 1.1009*** 31.18 -0.11412** -2.18 0.38025*** 7.05 0.0873 1.35 0.6189 RlEst -0.0070397*** -3.98 1.1633*** 26.40 0.90119*** 13.78 0.77406*** 11.50 -0.0733 -0.91 0.6680 Fin 0.0001 0.13 1.235*** 46.50 0.12559*** 3.19 0.19524*** 4.81 -0.0469 -0.96 0.8110 Other -0.0041904** -2.42 1.1891*** 27.53 0.29887*** 4.66 0.13373** 2.03 0.13166* 1.66 0.6029

21 Portfolio α t β t s t h t R^2 SMALL LoBM -0.0046127*** -4.61 1.0751*** 44.80 1.3718*** 40.12 -0.27739*** -7.50 0.8978 ME1 BM2 -0.0002 -0.33 0.96826*** 58.85 1.294*** 55.20 0.050869** 2.01 0.9351 ME1 BM3 -0.0001 -0.25 0.91661*** 71.21 1.0951*** 59.72 0.29322*** 14.77 0.9473 ME1 BM4 0.0015013*** 2.88 0.88472*** 70.85 1.0314*** 57.97 0.46205*** 24.00 0.9443 SMALL HiBM 0.0011474** 2.09 0.97576*** 74.32 1.0895*** 58.25 0.69664*** 34.41 0.9468 ME2 BM1 -0.0018976*** -2.94 1.1085*** 71.61 0.99503*** 45.12 -0.38312*** -16.05 0.9473 ME2 BM2 -0.0003 -0.55 1.0095*** 74.46 0.87199*** 45.14 0.12917*** 6.18 0.9414 ME2 BM3 0.00097191* 1.83 0.96644*** 75.91 0.77326*** 42.63 0.38561*** 19.64 0.9377 ME2 BM4 0.00086821* 1.72 0.97074*** 80.18 0.73257*** 42.47 0.56133*** 30.07 0.9411 ME2 BM5 -0.0005 -0.92 1.0841*** 83.22 0.87036*** 46.90 0.80534*** 40.10 0.9468 ME3 BM1 -0.0005 -0.93 1.0875*** 78.34 0.73402*** 37.12 -0.43857*** -20.49 0.9502 ME3 BM2 0.0006 0.95 1.0337*** 67.72 0.53651*** 24.67 0.18225*** 7.74 0.9111 ME3 BM3 0.0000 -0.04 0.99639*** 66.15 0.44175*** 20.59 0.43754*** 18.84 0.8985 ME3 BM4 0.0003 0.50 0.99881*** 68.78 0.40801*** 19.72 0.61639*** 27.53 0.9030 ME3 BM5 0.0005 0.76 1.0597*** 63.28 0.5596*** 23.46 0.78421*** 30.37 0.8953 ME4 BM1 0.0011619** 2.00 1.0613*** 76.39 0.38446*** 19.42 -0.41976*** -19.60 0.9371 ME4 BM2 -0.0007 -1.05 1.0704*** 67.31 0.21768*** 9.61 0.20586*** 8.40 0.8921 ME4 BM3 -0.0003 -0.45 1.0761*** 66.74 0.17729*** 7.72 0.44462*** 17.88 0.8847 ME4 BM4 0.0006 0.89 1.0146*** 66.29 0.22976*** 10.54 0.57287*** 24.28 0.8862 ME4 BM5 -0.0011 -1.40 1.1386*** 61.35 0.25646*** 9.70 0.81251*** 28.39 0.8710 BIG LoBM 0.0014477*** 3.26 0.97474*** 91.57 -0.23893*** -15.76 -0.36383*** -22.17 0.9422 ME5 BM2 0.0003 0.63 0.99794*** 76.96 -0.22017*** -11.92 0.095001*** 4.75 0.9038 ME5 BM3 -0.0003 -0.45 0.97349*** 63.43 -0.22742*** -10.40 0.29513*** 12.47 0.8594 ME5 BM4 -0.0013535** -2.42 0.98533*** 73.61 -0.19581*** -10.27 0.60723*** 29.42 0.8933 BIG HiBM -0.0016242* -1.88 1.0512*** 50.78 -0.078575*** -2.66 0.76218*** 23.88 0.8064

22

For all the tables in the Appendix A ‘*’ – indicates 10% statistical significance level; ‘**’ – indicates 5% statistical significance level; ‘***’ – indicates 1% statistical significance level. Portfolio α t β t s t h t v t R^2 SMALL LoBM -0.0040296*** -4.00 1.0475*** 41.78 1.3182*** 35.43 -0.23912*** -6.24 -0.16102*** -3.50 0.8996 ME1 BM2 -0.0001 -0.18 0.96376*** 55.58 1.2853*** 49.95 0.05712** 2.16 -0.0263 -0.83 0.9352 ME1 BM3 -0.0002 -0.35 0.91935*** 67.77 1.1004*** 54.66 0.28941*** 13.97 0.0160 0.64 0.9474 ME1 BM4 0.0013547** 2.57 0.89165*** 67.87 1.0449*** 53.59 0.45243*** 22.55 0.040471* 1.68 0.9445 SMALL HiBM 0.0011873** 2.14 0.97387*** 70.37 1.0859*** 52.87 0.69925*** 33.08 -0.0110 -0.43 0.9468 ME2 BM1 -0.0013518** -2.10 1.0827*** 67.64 0.94485*** 39.77 -0.3473*** -14.21 -0.15071*** -5.13 0.9493 ME2 BM2 -0.0002 -0.39 1.0055*** 70.39 0.86406*** 40.76 0.13483*** 6.18 -0.0238 -0.91 0.9415 ME2 BM3 0.00093605* 1.74 0.96813*** 72.14 0.77656*** 38.99 0.38325*** 18.70 0.0099 0.40 0.9377 ME2 BM4 0.0006 1.22 0.98238*** 77.45 0.75518*** 40.12 0.5452*** 28.14 0.067896*** 2.92 0.9418 ME2 BM5 -0.0006 -1.02 1.0871*** 79.18 0.87624*** 43.01 0.80115*** 38.21 0.0177 0.70 0.9469 ME3 BM1 -0.0002 -0.35 1.0717*** 73.89 0.70328*** 32.67 -0.41663*** -18.81 -0.092331*** -3.47 0.9511 ME3 BM2 0.0005 0.82 1.0374*** 64.50 0.54385*** 22.78 0.17701*** 7.21 0.0220 0.75 0.9112 ME3 BM3 0.0001 0.20 0.98921*** 62.39 0.4278*** 18.18 0.4475*** 18.48 -0.0419 -1.44 0.8988 ME3 BM4 0.0004 0.72 0.99233*** 64.90 0.39541*** 17.43 0.62538*** 26.78 -0.0378 -1.35 0.9032 ME3 BM5 0.0008 1.10 1.0479*** 59.56 0.53671*** 20.55 0.80055*** 29.79 -0.068738** -2.13 0.8961 ME4 BM1 0.0014973** 2.57 1.0455*** 72.02 0.35362*** 16.41 -0.39775*** -17.94 -0.092618*** -3.47 0.9383 ME4 BM2 -0.0006 -0.87 1.0651*** 63.58 0.20744*** 8.34 0.21317*** 8.33 -0.0308 -1.00 0.8923 ME4 BM3 -0.0002 -0.25 1.0698*** 63.00 0.16513*** 6.55 0.4533*** 17.48 -0.0365 -1.17 0.8849 ME4 BM4 0.0007 1.12 1.007*** 62.51 0.21496*** 8.99 0.58344*** 23.72 -0.0445 -1.50 0.8866 ME4 BM5 -0.0007 -0.91 1.1209*** 57.64 0.22204*** 7.69 0.83708*** 28.19 -0.10337*** -2.89 0.8726 BIG LoBM 0.0014185*** 3.15 0.97612*** 86.99 -0.23625*** -14.19 -0.36575*** -21.34 0.0081 0.39 0.9422 ME5 BM2 0.0002 0.31 1.0059*** 73.77 -0.20466*** -10.11 0.083931*** 4.03 0.046572* 1.86 0.9043 ME5 BM3 -0.0005 -0.79 0.9842*** 61.03 -0.20661*** -8.63 0.28028*** 11.38 0.062506** 2.11 0.8603 ME5 BM4 -0.0013008** -2.30 0.98284*** 69.66 -0.20065*** -9.58 0.61068*** 28.34 -0.0145 -0.56 0.8933 BIG HiBM -0.0007 -0.83 1.0078*** 47.63 -0.16287*** -5.19 0.82235*** 25.45 -0.25318*** -6.52 0.8180