BSc Economics & Business Economics

Major Business Economics, specialization Finance

**The volatility effect: common risk factor exposure or evidence on systematic market **
**mispricing? **

Author: Nick Blankvoort Student number: 11789638 Supervisor: Ben Tims

Topic: Low Volatility Anomaly Slot: C

Date: June 28, 2021

**Abstract **

In financial markets all across the world, a low volatility anomaly – low volatility stocks outperforming high volatility stocks – can be observed. This study researches, validates and tests the low volatility anomaly in the U.S. equity markets for the July 1963 – December 2020 time period, after controlling for the relatively new common risk factors included in the Fama-French 5-factor asset pricing model.

To test this, regression- and Sharpe Ratio analysis is performed on volatility sorted portfolios. In this study, two approaches are used to measure volatility: total- and idiosyncratic volatility. It is found that the low volatility anomaly is still present and statistically significant after controlling for the factors included in the Fama-French 5-factor model when total volatility sorting is used. The anomaly is also present and statistically significant when comparing portfolio Sharpe Ratios using idiosyncratic volatility, however becomes insignificant when a regression analysis of the portfolio returns on the Fama-French risk factors is carried out. This study therefore provides empirical evidence of systematic mispricing in financial markets, although when using idiosyncratic volatility to a limited extent.

**Statement of Originality **

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

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

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

**Table of Contents **

Abstract ... i

Statement of Originality ... ii

1 Introduction ... 1

1.1 Research Question ... 1

1.2 Motivation and Contribution to the Literature ... 2

1.3 Preview Methodology and Data... 3

1.4 Preview Most Important Findings ... 3

2 Literature Review ... 4

2.1 History of Asset Pricing Models ... 4

2.2 The Low Volatility Anomaly... 6

2.3 Possible Explanations ... 7

2.4 Alternative Explanation based on New Academic Insights ... 8

2.5 Hypotheses ... 8

3 Methodology and Data ... 10

3.1 Portfolio Construction ... 10

3.2 Methodology ... 11

3.3 Checking Robustness and Alternative Methodologies ... 14

3.4 Data ... 15

4 Results ... 16

4.1 Total Volatility ... 16

4.2 Idiosyncratic Volatility ... 20

4.3 Summary ... 24

5 Robustness Checks ... 26

5.1 Without Winsorizing ... 26

5.2 Quintile Portfolios ... 28

5.3 Adjusting Timeframes for Average Volatility Calculations ... 30

5.4 Double Sorting on Size and Volatility ... 35

6 Shortcomings and Future Research ... 38

7 Conclusion ... 39

8 References ... 40

9 Appendix ... 42

**1 ** **Introduction **

One of the main pillars on which financial asset pricing theory is built, is on the concept that investors want to be compensated for the risk they are taking on when investing in a particular asset. When a particular asset is riskier, the investor requires a higher expected return, and when a particular asset is less risky, the investor requires a lower expected return. In classical finance terms, this is called the risk-return trade-off (Bodie et al., 2018). One important implication of this notion is the expectation that investors who are willing to take on more risk, also gain the benefit of higher expected returns.

However, empirical studies like Ang et al. (2006); Clarke et al. (2006); Blitz and Van Vliet (2007); Ang et al. (2009); and Baker et al. (2011) all found evidence that opposes this concept of benefitting from higher returns when taking on more/additional risk. Ang et al. (2006) for example found that both stocks with high historical idiosyncratic volatility, and stocks with high exposure to systematic volatility risk, realize terribly low average returns. Clarke et al. (2006) additionally conclude that minimum- variance portfolios for the period 1968-2005 deliver comparable, or sometimes even higher average returns than the market portfolio. Blitz and Van Vliet (2007) even found that over a 1986-2006 period, a portfolio consisting of the 10% least volatile stocks globally outperform a portfolio of the 10% most volatile stocks globally by 12% (!) annually. They also found that this result cannot be explained by well-known risk factors such as value and size.

From the literature above, it can be seen that all of these papers use a unique methodology to, in the end, end up with some kind of similar conclusion: investing in riskier (more volatile) stocks doesn’t necessarily mean that one can expect to realize better returns on average than when one invests in less risky (less volatile) stocks. Both in absolute terms, as well as in relative (risk-adjusted) terms. The research paper of Blitz and Van Vliet (2007) even shows that low volatility stocks outperform high volatility stocks (both in absolute- and relative terms) by a wide margin over a long period of time.

This phenomenon – stocks with low volatility outperform stocks with high volatility – is nowadays known as the volatility effect (Blitz & Van Vliet, 2007) or the low volatility anomaly (Baker et al., 2011).

This thesis will first verify that the low volatility anomaly is (still) present in the US equity markets.

Thereafter, it will be examined if this phenomenon can be explained by new relevant academic insights with regards to additional commonly known risk factor exposure(s).

**1.1 ** **Research Question **

Therefore, the main research question of this paper can be formulated accordingly:

Is the volatility effect – low volatility stocks outperforming high volatility stocks – in the United States equity markets still robust after controlling for additional risk-factor exposure to new factors included in the Fama and French (2015) 5-factor model?

**1.2 ** **Motivation and Contribution to the Literature **

This thesis in general, but specifically the answer to the above stated research question is in multiple dimensions relevant. Most important on the academic side is the implication that, if this volatility effect is still robust after controlling for the risk factors proposed by Fama and French (2015), efficient market theory can (again, like Blitz and Van Vliet (2007) already mentioned) be seriously challenged.

This thesis will test the robustness of the low volatility anomaly after controlling for the risk factors included in the Fama-French (2015) 5-factor asset pricing model. These insights were simply not yet available at the time when papers like Ang et al. (2006), Clarke et al. (2006), Blitz and Van Vliet (2007), Ang et al. (2009) and Baker et al. (2011) were published, which makes that this thesis is also in this regard a contribution to the existing literature. The paper of Li et al. (2016) – published after the publication of Fama and French (2015) – also does not include the two additional risk factors included in the Fama-French 5-factor model, compared to the Fama-French 3-factor model (1993). This thesis will therefore fill the research gap arisen in the years after 2015.

Furthermore on the academic side, the research on the low volatility anomaly done by Blitz and Van Vliet (2007) includes only data from 1986 up until 2006. Li et al. (2016) published their paper on the volatility effect in the United States equity markets more recently, but they only include data from 1963 up until 2011. In that regard, this thesis will also be a real contribution to the literature because of a widened research window, namely by using a 1963-2020 time period. This also means that the COVID-19 crisis period starting March 2020 is included, which gives the unique opportunity to review if the volatility effect is severely impacted by this crisis.

On the practical side, the insights gained from this thesis can also benefit investors who are searching for alternative beta strategies (opposed to just passively holding the market portfolio) to generate alpha, or improved Sharpe ratios compared to the market portfolio. This is in both cases - if the low volatility appears to be robust after controlling for all risk factors included in the Fama-French 5-factor model (Fama & French, 2015), or not – useful, because in both cases investors know where (not) to look to achieve these objectives.

Additionally on the practical side, it can be concluded that from the academic research done in the past on anomalies in financial markets, most of the explanations for the existence of these anomalies are based on the notion that they arise from irrational investor behaviour (Blitz et al., 2014). However as Blitz et al. (2014) state in their paper, most of investor behaviour with regards to the low volatility anomaly comes from rational acting when taking institutional constraints into account. Therefore, extensive additional research on the low volatility anomaly can also serve as a starting point to better understand the implications of constraints in financial markets on the usefulness of models like the CAPM, in which one of the assumptions is that there are no financial constraints (Sharpe, 1964);

(Lintner, 1965); (Blitz et al., 2014).

**1.3 ** **Preview Methodology and Data **

To obtain the results needed to answer the research question, decile portfolios of stocks will be constructed based on their historic 36-month (average) total- or idiosyncratic volatility. Rebalancing of the portfolios will be done on a monthly basis. The excess return of the portfolios will be calculated for each month after the portfolios are rebalanced. Thereafter, the average-, standard deviation- and Sharpe ratio of the returns of each portfolio will be calculated. Furthermore, the Jobson-Korkie (1981) test with the Memmel (2003) adjustment will be used to test if the difference in the Sharpe ratios of the two outer decile portfolios is statistically significant, to verify the presence of the volatility effect.

To finally test if the volatility effect is robust after controlling for the relatively new risk factors included in the Fama-French 5-factor model, OLS regressions will be run to conclude if the low (high) volatility portfolios contain statistically significant positive (negative) alphas after controlling for these factors.

The methodology of using the Jobson-Korkie (1981) test with the Memmel (2003) adjustment is chosen because it is consistent with prior literature (Blitz & Van Vliet, 2007). Furthermore, the choice to use both total- and idiosyncratic volatility is made because both methods are used in the past academic literature (Blitz & Van Vliet, 2007); (Li et al., 2016). The choice to use monthly data is made because of consistency with already existing literature (Fama & French, 2015), besides the fact that a huge amount of computing power is already needed when using monthly data (each time-series regression for the idiosyncratic volatility calculations took ± 30 minutes). The time period of July 1963 – December 2020 is used because it is the longest time period for which the needed data is available and complete, which is also consistent with existing literature (Fama & French, 2015); (Li et al., 2016).

The stock price data and delisting return data used in this thesis is retrieved from CRSP. Data on the
Fama-French risk factors and the risk-free rate are retrieved from the Kenneth French data library.^{1}

**1.4 ** **Preview Most Important Findings **

Based on prior literature regarding the existence of the low volatility anomaly in financial markets all across the world, the expectation of the analysis was that the existence of the low volatility anomaly would be confirmed. However, based on new academic insights it was expected to find non- significant results after controlling for all Fama and French (2015) risk factors. The empirical analysis conducted in this thesis to research, validate, and test the volatility effect in the U.S. equity markets provides evidence that this is not entirely the case: the low volatility anomaly is still observable for the 1963-2020 time period, and is also still present after controlling for additional risk-factor exposure to new factors included in the Fama-French (2015) 5-factor model. I also found that the results are not sensitive to the use of different time periods (12 or 60 months) to construct volatility sorted portfolios, except when using average 12-month idiosyncratic volatility. This is likely caused by flawed average volatility calculations, resulting from using monthly- instead of daily data (not enough observations).

1 https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

**2 ** **Literature Review **

In this chapter, the history on the evolution of asset pricing models will be discussed first, to provide background knowledge on the research progress through the years in this academic field. This is done because the latest insights with regards to asset pricing models will be used in this thesis.

Thereafter, previous findings on the low volatility anomaly in the literature will be discussed, followed by the reasons given that could explain the existence of this anomaly. Based on these prior findings, the hypotheses for this thesis are formed. These are stated in the last paragraph of this chapter.

**2.1 ** **History of Asset Pricing Models **

In academic financial literature, the history of modern asset pricing models goes back to the year 1964, when William F. Sharpe published a framework on how to deal with uncertainty in financial markets: risk. Up until then, the models that were used had an assumption of certainty (Sharpe, 1964), which were therefore lacking a crucial part for pricing financial assets. Extending on this framework, Lintner (1965), Mossin (1966) and Merton (1973), among others, contributed to the (simple) Capital Asset Pricing Model (CAPM) as known in modern asset pricing theory today:

𝑟_{𝑖}− 𝑟_{𝑓}= 𝛼_{𝑖}+ 𝛽_{𝑖}(𝑟_{𝑚}− 𝑟_{𝑓}) + 𝜀_{𝑖} (1)

Here, 𝑟𝑖 represents the expected return of asset i, 𝑟𝑓 the risk-free rate, and 𝑟𝑚 the return of the market.

Furthermore, 𝛽_{𝑖} denotes the volatility (riskiness) of the return of asset i compared to the market, 𝛼_{𝑖}
represents a constant for the abnormal return of asset i, which is expected to be 0, and 𝜀𝑖 denotes
the error term. In this equation, the risk-return trade-off is clear: when 𝛽_{𝑖} (measure of risk) increases,
the expected return for asset i increases proportionally.

However after empirically testing the CAPM-model its ability to predict future stock returns, Black et al. (1972) contradictory found evidence that stocks with a low beta coefficient (𝛽𝑖) contained positive alpha on average, consistently. Something which cannot be possible, nor explained by the simple CAPM. From the paper of Fama and MacBeth (1973) this conclusion is supported, since they found that the risk-return relation of stocks is too flat, for which then can be argued that low-risk stocks are a better option to invest in.

After analysing the (flat) relationship between average return and beta (𝛽_{𝑖}), Fama and French (1993)
published a paper in which they described and added two additional common risk-factors in their
model – on top of the simple CAPM of Sharpe (1964) and Lintner (1965) – to estimate stock returns:

SMB and HML. SMB here is the risk-premium associated with small stocks relative to big stocks, and HML is the risk-premium associated with low value stocks relative to high value stocks:

𝑟𝑖− 𝑟𝑓 = 𝛼𝑖+ 𝛽𝑖(𝑟_{𝑚}− 𝑟𝑓) + 𝑠_{𝑖}𝑆𝑀𝐵 + 𝑣𝑖𝐻𝑀𝐿 + 𝜀𝑖 (2)

SMB is calculated by grouping the returns of the 30% smallest stocks, and subtracting (shorting) the returns of the 30% biggest stocks. In short small minus big, hence SMB. This results in a risk-premium which can be used in regression analysis to visualise exposure to this additional common risk factor.

HML, high minus low, is calculated in almost the same way, but opposed to grouping returns based on size, by grouping the returns of stocks relative to their book-to-market equity value. Stocks with high book-to-market equity ratios (known as value stocks) are assumed to be riskier (Fama & French, 1993), and are therefore expected to achieve higher average returns compared to stocks with low book-to-market equity ratios (known as growth stocks).

After controlling for the factors included in the Fama and French (1993) 3-factor asset pricing model, Carhart (1997) found that stocks that performed well in the previous 12 months also performed well in the next 12 months compared to stocks that performed not so well in the previous 12 months. For this reason, he added a momentum factor (UMD, up minus down) to the Fama and French (1993) 3- factor model, resulting in the Carhart (1997) 4-factor asset pricing model, displayed in equation 3:

𝑟_{𝑖}− 𝑟_{𝑓}= 𝛼_{𝑖}+ 𝛽_{𝑖}(𝑟_{𝑚}− 𝑟_{𝑓}) + 𝑠_{𝑖}𝑆𝑀𝐵 + 𝑣_{𝑖}𝐻𝑀𝐿 + 𝑚_{𝑖}𝑈𝑀𝐷 + 𝜀_{𝑖} (3)

Titman (2004) found that after controlling for the common risk factors included in the Fama and French (1993) 3-factor model and the Carhart (1997) momentum factor, that firms that invest conservatively outperform firms that invest aggressively. Therefore, he proposed an additional factor regarding firm investments (CMA, conservative minus aggressive) to be constructed and added to the asset pricing model in the same way as the size-, value- and momentum factor.

Novy-Marx (2013) found another anomaly when controlling for the common risk factors included in the Fama and French (1993) 3-factor model, namely that firms with more robust earnings outperform firms with weak earnings. In this paper, the anomaly is referred to as the gross profitability premium.

To also account for the anomalous returns associated with profitable firms, another risk factor (RMW, robust minus weak) constructed in a similar way as the size-, value-, momentum-, and investment factor needed to be added to the Fama and French (1993) 3-factor asset pricing model.

Two years after the publication of Novy-Marx (2013), Fama and French (2015) published a paper which combined the common risk factors included in their Fama-French 3-factor asset pricing model (1993), the insights of Titman (2004) with regards to the documented investment anomaly, and the insights of Novy-Marx (2013) with regards to the observed profitability anomaly. This resulted in the Fama-French 5-factor asset pricing model as known today:

𝑟_{𝑖}− 𝑟_{𝑓}= 𝛼_{𝑖}+ 𝛽_{𝑖}(𝑟_{𝑚}− 𝑟_{𝑓}) + 𝑠_{𝑖}𝑆𝑀𝐵 + 𝑣_{𝑖}𝐻𝑀𝐿 + 𝑝_{𝑖}𝑅𝑀𝑊 + 𝑖_{𝑖}𝐶𝑀𝐴 + 𝜀_{𝑖} (4)

The Carhart (1997) momentum factor is not included in this model because the regression slope of this factor became close to zero when the profitability factor and the investment factor were included, which therefore did not result in significant changes in model performance (Fama & French, 2015).

**2.2 ** **The Low Volatility Anomaly **

Stock returns and corresponding investment strategies that cannot be explained and/or predicted by the simple CAPM-model are called anomalies (Fama & French, 1996). Throughout time, numerous anomalies are found and documented in the academic financial literature. Examples of well-known anomalies are the aforementioned size- and value anomaly incorporated in the Fama-French 3-factor model (Fama & French, 1993), the momentum anomaly incorporated in the Carhart (1997) 4-factor model, and the investment- (Titman, 2004) and profitability anomaly (Novy-Marx, 2013), which are both incorporated in the Fama-French 5-factor asset pricing model as mentioned earlier.

Another anomaly which can be found in recent academic financial literature is the so-called volatility effect (Blitz & Van Vliet, 2007), or also spoken about as the low volatility anomaly (Baker et al., 2011).

This anomaly represents the phenomenon that stocks with low volatility tend to realize significantly better risk-adjusted returns in comparison with high volatility stocks (Ang et al., 2006); (Blitz & Van Vliet, 2007); (Baker et al., 2011); (Li et al., 2016). This is both a result of large negative alphas for stocks with high idiosyncratic volatility (Ang et al., 2006), and positive alphas for stocks with low total- and idiosyncratic volatility (Blitz & Van Vliet, 2007).

In their 2007 paper, Blitz and Van Vliet conclude that stocks with low average volatility (historically) present superior risk-adjusted returns compared to their peers. They found this result in terms of Sharpe ratios as well as with CAPM- and Fama and French (1993) 3-factor alphas. Additionally, they found that the volatility effect is similar in size compared to classic effects like the size, value and momentum risk-factors in the Fama and French (1993) 3-factor model and the Carhart (1997) 4-factor model. A benefit of this paper is that similar results are found as in Clarke et al. (2006) and Ang et al.

(2006), but that the methodology is much easier in terms of econometrics and statistical modelling.

More recently, Li et al. (2016) published their research where they first constructed portfolios of stocks based on their idiosyncratic volatility, using the Fama and French (1993) risk-factors market beta, size and value. Thereafter, the researchers constructed an idiosyncratic volatility factor (IVOL) in the same way as Fama and French (1993) constructed the factors for value and size. To measure the factor loadings of the constructed portfolios on the factors market beta, size, value and IVOL, Li et al. (2016) performed regression analyses on the portfolios sorted on idiosyncratic volatility. Their results show a significant difference in returns for the constructed value-weighted IVOL characteristic spread (High minus Low) portfolio of -1.09% monthly, which is an annualized difference in returns of 13.89% (Li et al., 2016) for an investment strategy that goes long low volatility stocks and short high volatility stocks.

Li et al. (2016) conclude their paper with the statement that there are no common systematic factors directly associated with low-volatility stocks when evaluating the sources and mechanisms (using a factor model to explain the returns by specific portfolio characteristics) of anomalous returns. Hence, the high returns of low volatility stocks are best characterized by market mispricing and consequently cannot be viewed as compensation for a hidden risk factor (Li et al., 2016).

**2.3 ** **Possible Explanations **

In the literature, there are a lot of empirical research papers published that find and show results of the existence of the low volatility in various international financial markets, both developed markets and emerging markets. Examples of this are Dutt and Humphery-Jenner (2013), Blitz et al. (2013), Nartea and Wu (2013) and Zaremba (2016). Most of which attribute the volatility effect to different factors. However, some conclusions of papers like Baker et al. (2011) are built upon and extended.

To start, Baker et al. (2011) attributed the existence of the volatility effect to dynamics of institutional money management mandates, which create limits to arbitrage. The theory proposed by Baker et al.

(2011) is that institutional money managers are evaluated against a benchmark which they have to outperform. However, these institutional money managers are subject to restrictions (like leverage and short selling constraints) and regulations with regards to risk. According to Baker et al. (2011) this discourages arbitrage activity in high-alpha, low-beta and low-alpha, high-beta stocks.

Blitz et al. (2013) additionally find that for emerging markets the volatility effect has strengthened over time. They argue that this – in line with Baker et al. (2011) – may be a result of increased institutionalization of emerging markets in combination with agency issues like Baker et al. (2011) already suggested to be present in institutional money- and portfolio management. Blitz et al. (2013) also point out that low correlations of volatility effects in emerging and developed equity markets argue against a common-factor explanation.

Dutt and Humphery-Jenner (2013) share the benchmarking argument of Baker et al. (2011) for the existence of the low volatility anomaly, but they additionally state that the effect can also partly be explained by company operating performance. Low volatility stocks have higher operating returns according to their research, which can be an explanation why the group of less volatile stocks outperforms the group of highly volatile stocks. Important to note here is that the Fama and French (2015) 5-factor asset pricing model, which incorporates a profitability factor, was not published yet.

Nartea and Wu (2013) studied the volatility effect in the Hong Kong equity market and also found the effect to be present. They argue that the effect can be attributed to population characteristics of investors in this market, namely individual investors with a preference for high volatility stocks (Nartea

& Wu, 2013). This argument to explain the low volatility anomaly is also used by Li et al. (2016). Nartea and Wu (2013) also repeat the argument of Blitz and Van Vliet (2007) that profit maximizing money managers value outperformance in up-markets more than outperformance in down-markets, which creates the possibility to grow preferences for high volatility stocks at the institutional level, additional to the individual level (Blitz & Van Vliet, 2007); (Nartea & Wu, 2013).

To round off, other explanations of the low volatility anomaly which can be found in the literature are the conclusions made in Zaremba (2016), who suggested besides the agreement for the benchmark- and limits to arbitrage explanations of Baker et al. (2011) that the volatility effect can be partly explained by liquidity factors, past maximum returns, short-term reversal and skewness. In the paper

of Blitz et al. (2014), where the authors extensively summarize explanations for the existence of the volatility effect, even more explanations can be found for the existence of the low volatility anomaly.

**2.4 ** **Alternative Explanation based on New Academic Insights **

In short, to summarize all the information in the previous paragraph, there are a lot of reasons stated in prior academic literature for the existence of the low volatility anomaly. However, because of newly available academic insights with respect to common risk factors in financial assets (Fama & French, 2015), some explanations are more interesting compared to others.

As mentioned in paragraph 2.1, there are currently two additional commonly used asset pricing models besides the Fama and French (1993) 3-factor model. Namely the Carhart (1997) 4-factor asset pricing model and the Fama and French (2015) 5-factor asset pricing model. However, Blitz and Van Vliet (2007) and Li et al. (2016) only performed their analyses with the common risk factors included in the Fama and French (1993) 3-factor model to analyse the low volatility anomaly.

Since both Blitz and Van Vliet (2007) and Li et al. (2016) in their analyses only control for the risk factors included in the Fama and French (1993) 3-factor model, one reason for the positive alphas (𝛼𝑖) of the portfolios consisting of low volatility stocks could be that these portfolios have additional positive exposure to common risk factors not included in this 3-factor model. In the same way, the negative alphas of the portfolios consisting of the high volatility stocks could maybe be explained by negative exposure to the additional common risk factors included in the Fama and French (2015) 5- factor model, but which were missing in the Fama and French (1993) 3-factor model.

Therefore, it can be possible that the lack of controlling for the two additional risk factors (profitability and investment) can be the cause for the existence of the observed low volatility anomaly, which combined could be responsible for this out- or underperformance of the low- and high volatility portfolios respectively. When this is indeed the case – Dutt and Humphery-Jenner (2013) already suggested a partial explanation of the volatility effect by company operating performance, which is related to profitability which Novy-Marx (2013) pointed out as a separate risk factor and included in the Fama-French 5-factor model – the expectation is that these positive (negative) alphas of the low (high) volatility portfolios disappear.

**2.5 ** **Hypotheses **

In the literature, the arguments for the existence of the low volatility anomaly (or anomalies in general) can be divided into two main categories, namely arguments based on behavioural biases of investors (both intentionally with measures in place like restrictions or regulations with regards to risk, as well as unintentionally) and arguments based on company characteristics or commonly known risk factors.

For this thesis, the objective is to test if the low volatility anomaly is still observable in the United States equity markets when controlling for the additional risk factors included in the Fama-French 5-factor model. In other words, it will be investigated if the portfolio consisting of low volatility stocks has exposure to factors such as market beta, size, value, profitability and investment.

The first test that will be performed is the Jobson and Korkie (1981) test with the Memmel (2003) adjustment, to test for statistically different Sharpe Ratios between the portfolios consisting of low- and high volatility stocks. For this test, the expectation is that there are significant differences in the Sharpe Ratios of both portfolios. The expectation here is based on the previously discussed empirical evidence as documented in the prior literature on the existence of the low volatility anomaly in various financial markets across the world. The hypotheses are therefore formulated as follows:

𝐻_{0}: there is a difference in the Sharpe Ratios of the portfolio consisting of low volatility stocks and the
portfolio consisting of high volatility stocks.

𝐻_{1}: there is no difference in the Sharpe Ratios of the portfolio consisting of low volatility stocks and
the portfolio consisting of high volatility stocks.

Secondly, it will be tested if the difference in realised returns between the low- and the high volatility portfolios can be explained by exposures to the common risk factors included in the Fama-French 5- factor asset pricing model, as already mentioned above. The expectation here is that the anomalous returns associated with volatility sorted portfolios can be explained by the common risk factors included in the Fama and French (2015) 5-factor model. The hypotheses are therefore formulated as follows:

𝐻_{0}: the outperformance of low volatility stocks relative to high volatility stocks can be explained by
the common risk factors included in the Fama-French 5-factor asset pricing model.

𝐻_{1}: the outperformance of low volatility stocks relative to high volatility stocks cannot be explained
by the common risk factors included in the Fama-French 5-factor asset pricing model.

In the next chapter, the methods used to empirically test the hypotheses above will be discussed.

**3 ** **Methodology and Data **

In this chapter, the process of empirically testing the hypotheses stated in the previous chapter will be thoroughly discussed. A brief summary of this chapter can be found in paragraph 1.3. First, the process of portfolio construction will be discussed, emphasizing on the reasons why certain choices are made. Second, the different ways in which statistical tests will be used to accept or reject the null hypotheses from will be explained. Third, the thoughts behind the robustness checks, combined with possible alternative methodologies which could have been used for the analysis, will be discussed.

Fourth and last, the data (sources) which will be used for the analysis will be mentioned.

**3.1 ** **Portfolio Construction **

The first step working towards a data structure on which statistical tests can be performed to answer the research question is to sort the dataset – consisting of all stocks on the various stock exchanges in the United States: NYSE, AMEX and NASDAQ – in groups of stocks based on their total- or idiosyncratic volatility for every time period (month) in the sample. In this thesis, both total volatility and idiosyncratic volatility will be used. This choice is made because Blitz and Van Vliet (2007) used total volatility in their analysis, and Li et al. (2016) used idiosyncratic volatility. Monthly rebalancing is implemented because of consistency with existing literature (Blitz & Van Vliet, 2007); (Li et al., 2016).

To sort the stocks in the dataset based on their total volatility for every month, a rolling standard deviation calculation is performed to get the average total volatility of the returns over the last 36 months. The choice to use 36 months is made because of consistency with the literature (Blitz & Van Vliet, 2007). First, the last 36 months of the monthly returns are calculated and selected. Second, for those 36 observations (t-1 up until t-36) the standard deviation is calculated for every stock for time period t. Third, for every time period t (every month in the sample) the stocks will be sorted based on their historic average 36-month total volatility. Last, equally weighted decile portfolios will be created for the sample period which are rebalanced monthly, to get the right data structure for the next steps.

To sort the stocks based on their idiosyncratic volatility, a different approach is needed. Since the total volatility (risk) of a stock can be split into two categories, systematic risk (undiversifiable risk, also called market risk) and unsystematic risk (diversifiable risk, also called idiosyncratic risk), systematic risk has to be subtracted from the total risk to end up with the idiosyncratic risk of a stock. To achieve this for all stocks for every time period, a rolling regression can be run for every sub time-period (the previous 36 months) to retrieve the residuals of the regression, i.e. the idiosyncratic risk of every stock over that time period. There are different ways to calculate idiosyncratic volatility, for example by using the CAPM model to retrieve the residuals, or by using the Fama- French 3-factor model, as Li et al. (2016) used in their analysis. Therefore, different ways of calculating idiosyncratic volatility will be used in the main analysis and the robustness checks of this thesis.

The rolling time series regressions (using data of the past 36 months for every time period in the sample) which will be used to calculate the idiosyncratic risk of every single stock in the sample, are displayed in Table 1. Depending on the different ways in which it is possible to calculate idiosyncratic risk, the residuals of the regression, the corresponding regression equation is used.

**Table 1: Time-series regression equations for idiosyncratic risk calculations **

Sharpe (1964) & Lintner

**(1965): CAPM model ** 𝑅_{𝑖,𝑡}= 𝛼_{𝑖}+ 𝛽_{𝑖}(𝑟_{𝑚,𝑡}− 𝑟_{𝑓,𝑡}) + 𝜀_{𝑖,𝑡}
Fama & French (1993):

3-factor model 𝑅_{𝑖,𝑡}= 𝛼_{𝑖}+ 𝛽_{𝑖}(𝑟_{𝑚,𝑡}− 𝑟_{𝑓,𝑡}) + 𝑠_{𝑖}𝑆𝑀𝐵_{𝑡}+ 𝑣_{𝑖}𝐻𝑀𝐿_{𝑡}+ 𝜀_{𝑖,𝑡}
Carhart (1997):

4-factor model 𝑅_{𝑖,𝑡}= 𝛼_{𝑖}+ 𝛽_{𝑖}(𝑟_{𝑚,𝑡}− 𝑟_{𝑓,𝑡}) + 𝑠_{𝑖}𝑆𝑀𝐵_{𝑡}+ 𝑣_{𝑖}𝐻𝑀𝐿_{𝑡}+ 𝑚_{𝑖}𝑈𝑀𝐷_{𝑡}+ 𝜀_{𝑖,𝑡}
Fama & French (2015):

5-factor model 𝑅_{𝑖,𝑡}= 𝛼_{𝑖}+ 𝛽_{𝑖}(𝑟_{𝑚,𝑡}− 𝑟_{𝑓,𝑡}) + 𝑠_{𝑖}𝑆𝑀𝐵_{𝑡}+ 𝑣_{𝑖}𝐻𝑀𝐿_{𝑡}+ 𝑝_{𝑖}𝑅𝑀𝑊_{𝑡}+ 𝑖_{𝑖}𝐶𝑀𝐴_{𝑡}+ 𝜀_{𝑖,𝑡}

Here, 𝑅_{𝑖,𝑡} is the excess return of the stock above the risk-free rate. This is the same as 𝑟_{𝑖,𝑡}− 𝑟_{𝑓,𝑡} which
is displayed in equations 1 till 4 in the previous chapter. When there was not enough data to perform
the time series regression, for example because there were not 36 historic months for the specific
company, the observation is dropped. Because of this, the first actual datapoint in the analysis is June
1966. Because the idiosyncratic volatilities (residuals) need to be averaged over the past 36 months
to end up with the 36 month measure for idiosyncratic volatility to sort the stocks on, another 3 years
of data will be lost, causing the time range on which the empirical analysis will be performed to be
May 1969 till December 2020. For this period, monthly rebalanced decile portfolios are constructed.

**3.2 ** **Methodology **

In this thesis, the methodology which will be used to test the low volatility anomaly is similar to the methodology used in the paper of Blitz and Van Vliet (2007). The first step is to sort all the stocks in the dataset, like already explained in paragraph 3.1. The time period which will be used for the analysis is July 1963 – which is the first month in which all required data for the analysis is present – up until December 2020, which was the last month of the previous year. This way of selecting a time range for the empirical analysis is consistent with existing literature (Fama & French, 2015); (Li et al., 2016). The choice to use monthly data is made because it is used in some of the most relevant academic financial papers out there currently (Fama & French, 2015), and because of the already large amount of observations (> 4.1 million) it contains after cleaning the dataset.

To also account for the returns of companies that got delisted in the dataset during the sample period, the return of this last month when listed was merged into the existing monthly dataset, where this last month of return data was missing. When all the right data is merged together in one dataset,

the equally weighted decile portfolios can be created. The choice to use equally weighted decile portfolios is because of consistency with the methodology to test the existence of the volatility effect (Blitz & Van Vliet, 2007). When analysing if the volatility effect is present inside groups of stocks with the same size, quintile groups are used to form a 5 by 5 matrix: 5 groups to sort the stocks by based on size, and 5 groups to sort on inside these size groups on (both total- and idiosyncratic) volatility.

After the portfolios are formed, the Jobson-Korkie (1981) test with the Memmel (2003) adjustment will be used to test if the low volatility anomaly is (still) present in the United States equity markets.

The choice to use this test is made because of consistency with the methodology used in prior empirical research (Blitz & Van Vliet, 2007). The test-statistic looks as follows:

𝑧 = 𝑆𝑅_{1}− 𝑆𝑅_{2}

√1𝑇[2(1 − 𝜌_{1,2}) +1

2 (𝑆𝑅^{1}^{2}+ 𝑆𝑅_{2}^{2}− 𝑆𝑅_{1}𝑆𝑅_{2}(1 + 𝜌_{1,2}^{2} ))]

~ 𝑁(0, 1) (5)

In this test, 𝑆𝑅_{𝑝} stands for the Sharpe Ratio of Portfolio p, 𝜌_{𝑝,𝑞} for the correlation between portfolio p
and portfolio q, and𝑇 for the number of observations, which is actually the number of time periods
in the sample over which the z-score will be calculated. To calculate the Sharpe Ratios of the portfolios
(𝑆𝑅𝑝), the standard way to calculate the realized Sharpe Ratio of a portfolio is used. The formula for
this calculation is displayed in equation 6:

𝑆𝑅_{𝑝}=𝑟_{𝑝}− 𝑟_{𝑓}

𝜎_{𝑝} (6)

Here, 𝑟_{𝑝} is the average annualized portfolio return, 𝑟_{𝑓} the average annualized risk-free rate of return
and 𝜎𝑝 the annualized portfolio standard deviation.

The Jobson-Korkie (1981) test is an adjusted and standardized t-test to analyse if the Sharpe Ratios of two portfolios are statistically different from each other. Therefore, conclusions can be drawn from this test to accept or reject the first set of formulated hypotheses in paragraph 2.5. For this test, the hypotheses formulated in paragraph 2.5 can be rewritten into testable hypotheses:

𝐻0: 𝑆𝑅1− 𝑆𝑅2= 0 𝐻1: 𝑆𝑅1− 𝑆𝑅2≠ 0

When the z-score resulting from the test is large – in the same way as how a normal t-test works – it can be concluded that the difference in Sharpe Ratios of the two portfolios is statistically significant.

This would lead to a rejection of the null hypothesis. In the case of this thesis, the difference of the
outer decile portfolios consisting of both the 10 percent least volatile stocks and the 10 percent most
volatile stocks will be substituted for 𝑆𝑅_{1} and 𝑆𝑅_{2} respectively. When this difference is statistically
significant, it can be concluded that the low volatility anomaly is still present in the U.S. equity markets
for the sample period used, which is May 1969 up until December 2020.

Subsequently, to analyse if the portfolio returns can be explained by exposure to common risk factors
such as systematic risk (𝑟𝑚− 𝑟𝑓), size (𝑆𝑀𝐵), value (𝐻𝑀𝐿), momentum (𝑈𝑀𝐷), profitability (𝑅𝑀𝑊) or
investment (𝐶𝑀𝐴), a regression of the portfolio returns (𝑅_{𝑝}) will be performed on these factors. The
models which will be used to do this analysis, using robust standard errors, are displayed in Table 2.

At first glance, the regression equations in Table 1 and Table 2 look similar, although there are a few
important differences between both. In Table 1, the time series regressions are displayed which are
used to calculate the idiosyncratic volatility of all the individual stocks for the different points in time,
based on the previous 36 months of data. This is represented by the residuals of the regressions (𝜀_{𝑖,𝑡}),
which is the part of the deviations in the excess returns of every individual stock (𝑅_{𝑖,𝑡}) that cannot be
explained by the independent variables: the common risk factors. In Table 2 however, the displayed
regression equations are for the total sample period, instead of only using the previous 36 months,
in which the portfolio excess return over the risk-free rate (𝑅𝑝) is used as the dependent variable, to
investigate the exposure of the different volatility sorted portfolios to the common risk factors.

**Table 2: Portfolio common risk factor exposure regression equations **

Sharpe (1964) & Lintner

**(1965): CAPM model ** 𝑅_{𝑝}= 𝛼_{𝑝}+ 𝛽_{𝑝}(𝑟_{𝑚}− 𝑟_{𝑓}) + 𝜀_{𝑝}
Fama & French (1993):

3-factor model 𝑅_{𝑝}= 𝛼_{𝑝}+ 𝛽_{𝑝}(𝑟_{𝑚}− 𝑟_{𝑓}) + 𝑠_{𝑝}𝑆𝑀𝐵 + 𝑣_{𝑝}𝐻𝑀𝐿 + 𝜀_{𝑝}
Carhart (1997):

4-factor model 𝑅_{𝑝}= 𝛼_{𝑝}+ 𝛽_{𝑝}(𝑟_{𝑚}− 𝑟_{𝑓}) + 𝑠_{𝑝}𝑆𝑀𝐵 + 𝑣_{𝑝}𝐻𝑀𝐿 + 𝑚_{𝑝}𝑈𝑀𝐷 + 𝜀_{𝑝}
Fama & French (2015):

5-factor model 𝑅_{𝑝}= 𝛼_{𝑝}+ 𝛽_{𝑝}(𝑟_{𝑚}− 𝑟_{𝑓}) + 𝑠_{𝑝}𝑆𝑀𝐵 + 𝑣_{𝑝}𝐻𝑀𝐿 + 𝑝_{𝑝}𝑅𝑀𝑊 + 𝑖_{𝑝}𝐶𝑀𝐴 + 𝜀_{𝑝}

From this regression analysis, the portfolio exposures to these common risk factors will be observable
in the factor coefficients. The exposure of the portfolio to systematic risk (market beta) can be drawn
from the coefficient 𝛽𝑝, the exposure to the size factor from the coefficient 𝑠𝑝, the exposure to the
value factor from the coefficient 𝑣_{𝑝}, the exposure to the momentum factor from the coefficient 𝑚_{𝑝},
the exposure to the profitability factor from the coefficient 𝑝𝑝 and the exposure to the investment
factor from the coefficient 𝑖_{𝑝}. For this thesis, the focus will be on the alphas of the portfolios using the
Fama-French 5-factor model, since this will yield the answer to the central research question.

To analyse the robustness of the volatility effect after controlling for the risk factors included in the Fama-French (2015) 5-factor model, the second set of hypotheses, as stated in paragraph 2.5, will be tested. Rewriting these into testable hypotheses yields the following results:

𝐻_{0}: 𝛼_{𝑝}= 0
𝐻1: 𝛼𝑝≠ 0

When the Fama and French (2015) risk factors could explain the discrepancies in returns between the portfolio consisting of the low- and the high volatility stocks, the alpha (𝛼𝑖) would be equal to zero. Therefore, the significant positive (negative) alphas found by Blitz and Van Vliet (2007) for the low (high) volatility portfolio when using the Fama-French 3-factor model, to control for exposure to, at that time, all common risk factors, would be expected to disappear. If it turns out however that this is not the case, i.e. the portfolio consisting of the low volatility stocks still contains positive alpha and the portfolio consisting of high volatility stocks still contains negative alpha, the conclusion that the low volatility anomaly is still robust after controlling for the common risk factors included in the Fama- French 5-factor model can be drawn. This would result in a rejection of the null hypothesis.

**3.3 ** **Checking Robustness and Alternative Methodologies **

For this subject, the low volatility anomaly, there are multiple approaches possible to empirically test the phenomenon. Even within one methodology, numerous choices have to be made. Therefore, to check if the volatility effect is also observable and robust when making slightly different choices within the chosen methodology, robustness checks will be performed.

As already mentioned in the paragraph on portfolio construction (3.1), both total- and idiosyncratic volatility will be used as a measure to group stocks in volatility sorted portfolios. Since both measures can be found in the literature (Blitz & Van Vliet, 2007); (Li et al., 2016), it will for both cases be analysed if the volatility effect is still robust after controlling for the Fama and French (2015) risk factors. For the measure of idiosyncratic volatility, there are also different possibilities to calculate this as explained in paragraph 3.1. For this reason, different ways of calculating the idiosyncratic volatilities will also be carried out to check the robustness of the results in the main analysis. Additionally, different time periods for calculating the idiosyncratic volatilities will be used. For example 12- or 60- month periods instead of the 36-month period used in the main analysis. Also, quintile instead of decile portfolios-, using a subsample of the dataset which only includes companies in the dataset when using the 36-month time period-, and double sorting on both size and volatility will be used as robustness checks for the results obtained in the main analysis.

The methodology of Blitz and Van Vliet (2007) is chosen to use for the analysis in this thesis because it is the most straightforward and best fitting way to derive clear conclusions to answer the research question. Another approach could have been to use (parts of) the methodology of Li et al. (2016).

These researchers used daily returns to calculate the monthly standard deviations of returns (volatility), after which they used the average of the last 36 months for every subsequent month. In this thesis, the average volatility of the monthly returns is used. Besides using equally weighted portfolios like Blitz and Van Vliet (2007) did, Li et al. (2016) additionally analysed the volatility effect with value weighted portfolios. In this thesis, equally weighted portfolios are used exclusively.

Opposed to Blitz and Van Vliet (2007), Li et al. (2016) created a factor which they added to the Fama-

French 3-factor model to test the low volatility anomaly. This approach was less straightforward for answering the research question in this thesis. Therefore, this alternative methodology is not used.

**3.4 ** **Data **

The data which will be used in this thesis is sourced from two high quality databases. The monthly
stock price, total return (including dividends) and, when applicable, delisting returns for all stocks on
the U.S. equity markets in the period 1963-2020 are retrieved from the CRSP database, accessed
through WRDS.^{2} The CRSP dataset is chosen because the most relevant literature in this field also
uses this data source, like Blitz and Van Vliet (2007), Fama and French (2015) and Li et al. (2016).

The return of the market 𝑟_{𝑚}, the risk-free rate 𝑟_{𝑓} and the common risk factor data, 𝑆𝑀𝐵, 𝐻𝑀𝐿, 𝑈𝑀𝐷,
𝑅𝑀𝑊 and 𝐶𝑀𝐴, are retrieved from the Kenneth French Data Library, as mentioned in paragraph 1.3.

To retrieve the excess return of the market over the risk free rate, a new variable “excess return” is created by subtracting the risk free rate from the market return (𝑟𝑚− 𝑟𝑓).

The risk free rate is the one-month U.S. Treasury bill rate. The choice to use this particular risk free rate is because most relevant past literature in this field also uses this rate (Fama & French, 2015); (Li et al., 2016). The original source of this risk free rate is the dataset of Ibbotson and Associates Inc., which is used by the Kenneth French Data Library to include this risk free rate in their dataset.

To account for possible outliers in the data caused by both corporate events (mergers, acquisitions,
spinoffs etc.) and the way in which the merging of the delisting returns in the dataset is done, the
monthly return data is winsorized at the 1^{st} and the 99^{th} percentile. In the literature, the way in which
the data was cleaned was not clearly stated. In this thesis, monthly return data is used instead of daily
or weekly because of the already large amount of data in the dataset, which already required a huge
amount of computing power. The time-series regressions for the idiosyncratic volatility calculations
took on average about 30 minutes for each way in which idiosyncratic volatility could be calculated.

For the return of the market, the value weighted excess return of the market is used as provided by the Kenneth French Data Library. This choice is made because after winsorizing, regression analyses showed a closer coefficient to 1.00 when using the value-weighted excess return of the market as the independent variable (0.9879, t = 630.66) opposed to using the equally-weighted excess return of the market (0.9221, t = 710.61). For this reason, this is a better proxy to use for the systematic risk factor in the regression analyses.

In the next chapter, the results of the main analysis will be displayed and discussed. In the chapter after that, different variations of the analysis will be displayed and discussed to check the robustness of the results from the main analysis.

2 https://wrds-www.wharton.upenn.edu/

**4 ** **Results **

In this chapter, the results of the empirical analysis conducted in this thesis will be discussed. This chapter is intentionally written in steps towards, in the end, answering the research question. In other words, this means that first the results will be discussed without incorporating the additional risk factors included in the Fama-French 5-factor asset pricing model, to verify results compared to findings in the existing literature. Thereafter, the additional risk factors will be included and it will be analysed if these are the causes of the existence of the low volatility anomaly found in prior literature.

This will first be done using total volatility to sort the stocks into decile portfolios. Thereafter, the same analysis will be conducted using idiosyncratic volatility. All regression results incorporate significance stars, where 1 star (*) refers to the significance of a coefficient at the 10% level, 2 stars (**) refers to significance at the 5% level and 3 stars (***) refers to significance at the 1% level.

**4.1 ** **Total Volatility **

To start off, the first logical step to begin with was to replicate the results of Blitz and Van Vliet (2007) including the additional years of data. For total volatility sorted portfolios, a time period of May 1965 until December 2020 is used. Here, factor data is not needed for the portfolio sorting. Blitz and Van Vliet (2007) used December 1986 – January 2006. The results of this analysis are shown in Table 3.

**Table 3: Portfolio characteristics: historic 36-month total volatility sorted **

Low 2 3 4 5 6 7 8 9 High

𝑅_{𝑝} 10.9% 13.3% 13.4% 13.0% 12.8% 12.1% 10.9% 9.5% 6.4% -0.3%

𝜎_{𝑝} 9.7% 13.3% 15.1% 16.8% 18.6% 20.2% 22.0% 23.6% 25.1% 27.4%

𝑅𝑝− 𝑟𝑓 6.3% 8.8% 8.8% 8.5% 8.2% 7.5% 6.3% 4.9% 1.9% -4.9%

𝑆𝑅𝑝 0.64 0.66 0.58 0.50 0.44 0.37 0.28 0.21 0.07 -0.18

𝑧 Jobson-Korkie test-statistic (z-score): 20.0

As can be observed in Table 3, the average returns of the volatility sorted decile portfolios almost perfectly decline from the portfolio consisting of the least volatile stocks to the one consisting of the most volatile stocks. Consequently, the corresponding portfolio Sharpe Ratios also almost perfectly decline. Both observations are consistent with findings in Blitz and Van Vliet (2007). When performing the Jobson-Korkie (1981) test with the Memmel (2003) correction, the z-score of 20.0 acknowledges a statistical difference in the Sharpe Ratios of portfolio 1 (Low) and portfolio 10 (High). Therefore, according to this test the low volatility anomaly is present and statistically significant for the extended time period. From Table 3 it can also be observed that there is a large difference between the average returns of portfolio 9 and portfolio 10 (High). This is in line with findings in Ang et al. (2006), in which the researchers find that the portfolio containing the most volatile stocks earn abnormally low returns.

Additionally, in Table 4 the exposure of the portfolios to the systematic risk factor 𝑟_{𝑀}− 𝑟_{𝑓}, also known
as market risk, are displayed by the coefficient 𝛽_{𝑝}. Here, it can be observed that the coefficient which
measures the exposure to the market risk factor steadily increases with the more volatile portfolios.

This is expected because the low(er) volatility portfolio(s) are expected to be less risky compared to the market, the portfolios in the middle are expected to be about as risky as the market, and the high(er) volatility portfolio(s) are expected to be riskier compared to the market. Since the coefficient for the low(er) volatility portfolio(s) is below 1, for the portfolios in the middle around 1 and for the high(er) volatility portfolios above 1, this is (as mentioned earlier) in line with expectations.

**Table 4: Portfolio factor exposure using CAPM: historic 36-month total volatility sorted **

Low 2 3 4 5 6 7 8 9 High

𝛼_{𝑝} 0.2%

(3.01)

***

0.3%

(3.50)

***

0.2%

(2.41)

**

0.1%

(1.36)

0.1%

(0.56)

-0.0%

(-0.31)

-0.2%

(-1.28)

-0.3%

(-2.01)

**

-0.6%

(-3.30)

***

-1.1%

(-5.67)

***

𝛽_{𝑝} 0.47
(18.1)

***

0.75 (29.5)

***

0.85 (30.7)

***

0.94 (32.0)

***

1.03 (33.1)

***

1.10 (32.3)

***

1.17 (32.3)

***

1.23 (30.1)

***

1.27 (29.0)

***

1.33 (26.4)

***

𝑅^{2} 0.599 0.799 0.803 0.794 0.770 0.747 0.717 0.681 0.644 0.591

Furthermore, it can be observed that the alphas of the portfolios, just like the returns and the Sharpe Ratios, almost perfectly decline for the more volatile portfolios. The alpha of the portfolio consisting of the least volatile stocks (Low) is 0.2% monthly, which is statistically significant at the 1% significance level (p-value = 0.003, t = 3.01). Contrary to the positive alpha found for the low volatility portfolio, the alpha of the portfolio consisting of the most volatile stocks (High) is minus 1.1% monthly, also significant at the 1% significance level (p-value = 0.000, t = -5.67). From the results in this table it can be concluded that the volatility effect is present and statistically significant after controlling for market risk. Annualized alpha values can be found in Table 5, where the alpha spread is better observable.

**Table 5: Annualized CAPM portfolio alphas: historic 36-month total volatility sorted **

Low 2 3 4 5 6 7 8 9 High

𝛼_{𝑝} 1.0% 0.4% -0.4% -0.9% -1.1% -1.6% -2.3% -3.0% -4.9% -9.7%

In Table 6, the regression results when additionally controlling for the size (𝑆𝑀𝐵) and value (𝐻𝑀𝐿) risk factors are displayed. The portfolio alphas are comparable to the alphas found when only controlling for market risk, as displayed in Table 4. To compare the results of the analysis in this thesis with the results in Blitz and Van Vliet (2007), where they found that the international low volatility portfolio outperformed the international high volatility portfolio by 12% annually, the alphas in Table 6 are annualized and displayed in Table 7.

**Table 6: Portfolio factor exposure using FF-3: historic 36-month total volatility sorted **

Low 2 3 4 5 6 7 8 9 High

𝛼_{𝑝} 0.2%

(2.20)

**

0.2%

(2.78)

***

0.1%

(1.61)

0.0%

(0.32)

-0.0%

(-0.65)

-0.1%

(-1.88)

*

-0.2%

(-3.10)

***

-0.4%

(-3.91)

***

-0.6%

(-5.37)

***

-1.1%

(-7.72)

***

𝛽_{𝑝} 0.48
(19.5)

***

0.74 (38.7)

***

0.82 (44.1)

***

0.88 (48.2)

***

0.93 (52.0)

***

0.98 (49.0)

***

1.01 (44.6)

***

1.04 (38.7)

***

1.05 (33.3)

***

1.08 (26.4)

***

𝑠_{𝑝} 0.14
(4.35)

***

0.26 (6.46)

***

0.39 (8.36)

***

0.54 (9.99)

***

0.70 (12.3)

***

0.83 (14.5)

***

0.95 (16.0)

***

1.07 (17.8)

***

1.15 (19.9)

***

1.20 (17.5)

***

𝑣𝑝 0.21 (5.20)

***

0.28 (8.57)

***

0.31 (9.61)

***

0.32 (9.45)

***

0.30 (8.94)

***

0.26 (7.41)

***

0.20 (5.04)

***

0.15 (3.14)

***

0.06 (1.18)

-0.05 (-0.65)

𝑅^{2} 0.660 0.873 0.901 0.920 0.924 0.923 0.908 0.887 0.858 0.790

Comparing the U.S. results with the international results of Blitz and Van Vliet (2007), an almost similar alpha spread can be observed between the low- and the high volatility portfolio. They however found a bigger positive alpha for the low volatility portfolio (+4.0%) and a less small negative alpha (-8.0%) for the high volatility portfolio. When comparing the results with the regional U.S. results of Blitz and Van Vliet (2007), the positive alpha found for the low volatility portfolio by Blitz and Van Vliet (2007) is also bigger (+3.3%). The negative alpha for the high volatility portfolio is again less small (-10.6%).

**Table 7: Annualized FF-3 portfolio alphas: historic 36-month total volatility sorted **

Low 2 3 4 5 6 7 8 9 High

𝛼_{𝑝} 1.9% 2.0% 1.1% 0.2% -0.5% -1.5% -2.8% -4.2% -6.9% -12.9%

A possible reason for the less small negative alpha found by Blitz and Van Vliet (2007) compared to
the results in this thesis can be that Blitz and Van Vliet (2007) only used the biggest 1000 companies
in their analysis (large cap). Because of this, it can be expected that the alpha in our analysis when
using all companies including mid-, small- and micro-cap, is more different from 0. Another possible
reason for the smaller positive alpha found for the low volatility portfolio compared to Blitz and Van
Vliet (2007) can be that the returns in this thesis are winsorized at the 1^{st} and the 99^{th} percentile.

When also controlling for the additional common risk factors included in the Fama-French 5-factor model, profitability (𝑅𝑀𝑊) and investment (𝐶𝑀𝐴), the alphas for both the low- and the high volatility portfolio move closer to zero. However, the annualized alpha spread between the low- and high volatility portfolio is still over 10%. These annualized alphas are displayed in Table 8.

**Table 8: Annualized FF-5 portfolio alphas: historic 36-month total volatility sorted **

Low 2 3 4 5 6 7 8 9 High

𝛼_{𝑝} 1.0% 0.4% -0.4% -0.9% -1.1% -1.6% -2.3% -3.0% -4.9% -9.7%

As can be seen in Table 8, when controlling for all common risk factors included in the Fama-French
5-factor asset pricing model, the annualized positive alpha for the portfolio consisting of the least
volatile stocks almost halved compared to only using the factors included in the Fama and French
(1993) 3-factor model. The alpha of the high volatility portfolio becomes more than three percentage
points less negative. From these results, it can be concluded that a (small) part of the low volatility
anomaly can be explained by the additional risk factors profitability and investment. More specifically,
as can be seen in Table 9, the majority of which is explained by the coefficient of the profitability
(𝑅𝑀𝑊) risk factor 𝑝_{𝑝}. For the low(er) volatility portfolio(s) a significant positive exposure to this risk
factor can be observed (p-value = 0.000, t = 4.01), and for the high(er) volatility portfolio(s) a
significant negative exposure to this risk factor can be observed (p-value = 0.000, t = -6.64).

**Table 9: Portfolio factor exposure using FF-5: historic 36-month total volatility sorted **

Low 2 3 4 5 6 7 8 9 High

𝛼_{𝑝} 0.1%

(1.11)

0.0%

(0.54)

-0.0%

(-0.53)

-0.1%

(-1.30)

-0.1%

(-1.37)

-0.1%

(-1.70)

*

-0.2%

(-2.22)

**

-0.3%

(-2.39)

**

-0.4%

(-3.50)

***

-0.8%

(-5.57)

***

𝛽_{𝑝} 0.49
(20.1)

***

0.76 (44.1)

***

0.85 (49.4)

***

0.90 (54.5)

***

0.93 (55.2)

***

0.97 (49.2)

***

1.00 (42.0)

***

1.01 (36.3)

***

1.02 (31.8)

***

1.03 (24.7)

***

𝑠_{𝑝} 0.18
(5.70)

***

0.34 (13.6)

***

0.48 (17.2)

***

0.62 (19.0)

***

0.76 (20.8)

***

0.86 (21.1)

***

0.95 (19.7)

***

1.01 (18.5)

***

1.03 (17.4)

***

0.98 (13.2)

***

𝑣_{𝑝} 0.18
(3.16)

***

0.23 (6.66)

***

0.28 (9.00)

***

0.31 (8.66)

***

0.33 (8.72)

***

0.32 (7.01)

***

0.25 (4.96)

***

0.22 (3.52)

***

0.11 (1.64)

0.01 (0.17)

𝑝_{𝑝} 0.16
(4.01)

***

0.31 (8.04)

***

0.32 (7.52)

***

0.28 (5.52)

***

0.21 (3.75)

***

0.11 (1.76)

*

-0.03 (-0.45)

-0.22 (-2.84)

***

-0.45 (-4.99)

***

-0.78 (-6.64)

***

𝑖_{𝑝} 0.08
(1.22)

0.11 (2.53)

**

0.07 (1.64)

0.01 (0.14)

-0.08 (-1.56)

-0.13 (-1.91)

*

-0.13 (-1.63)

-0.15 (-1.54)

-0.10 (-0.91)

-0.12 (-0.86)

𝑅^{2} 0.673 0.899 0.922 0.933 0.931 0.926 0.909 0.892 0.873 0.828

From this table, it can be observed that the monthly alpha of the low volatility portfolio decreases

significantly. The intercept coefficient (𝛼_{𝑝}) of the low volatility portfolio returns, controlling for all risk
factors included in the Fama-French 5-factor model, also becomes statistically insignificant (p-value

= 0.268, t = 1.11). When only controlling for the factors included in the Fama-French 3-factor model, the alpha for the low volatility portfolio (rounded to 0.2% monthly and 1.9% annually) was however observed to be significant at the 5% level, see Table 6. For the high volatility portfolios, the alpha becomes less negative and the t-value becomes smaller, but is still significant at the 1% level.

In conclusion, evidence is found that when using equally weighted decile portfolios sorted on total volatility, the low volatility anomaly is still robust after controlling for all five Fama-French risk factors.

**4.2 ** **Idiosyncratic Volatility **

Contrary to using total volatility when constructing the volatility sorted portfolios, another approach is to use idiosyncratic volatility as is done by Li et al. (2016) in their analysis. As already explained in chapter 3, different choices can be made to measure the idiosyncratic volatility of a stock. On the next pages, the results of using idiosyncratic volatility measured by only using the systematic risk factor (market beta), the Fama and French (1993) risk factors included in the 3-factor asset pricing model, and the Fama and French (2015) risk factors included in the 5-factor asset pricing model will be discussed. This approach – to use various measures for idiosyncratic volatility – is taken to be able to compare the results of portfolios sorted on these different measures of idiosyncratic volatility.

When CAPM is used to calculate the idiosyncratic risk of all individual stocks, to thereafter construct the sorted decile portfolios from, the following results as displayed in Table 10 can be found:

**Table 10: Portfolio characteristics: sorted on historic 36-month CAPM idiosyncratic volatility **

Low 2 3 4 5 6 7 8 9 High

𝑅_{𝑝} 10.8% 12.2% 12.7% 12.7% 12.6% 12.6% 13.1% 13.8% 13.9% 14.1%

𝜎_{𝑝} 10.3% 12.7% 14.4% 16.0% 17.6% 19.1% 20.9% 22.5% 25.0% 29.1%

𝑅_{𝑝}− 𝑟_{𝑓} 6.2% 7.6% 8.2% 8.1% 8.0% 8.0% 8.5% 9.2% 9.3% 9.5%

𝑆𝑅_{𝑝} 0.60 0.60 0.57 0.51 0.46 0.42 0.41 0.41 0.37 0.33

𝑧 Jobson-Korkie test-statistic (z-score): 7.35

In this results table, it can be observed that the average return of the decile portfolios almost perfectly increase when more risk (volatility) is taken. Contrary to the results obtained when using total volatility as the portfolio construction measure, also the average return of the highest (idiosyncratic) volatility portfolio here increases steadily. However, when accounting for the extra risk taken, the low(er) volatility portfolios are still the better risk-adjusted option when calculating idiosyncratic risk while only using the systematic risk factor (market beta), as can be observed by the portfolio Sharpe Ratios.