The volatility effect and linear asset pricing models

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UNIVERSITY OF AMSTERDAM

The Volatility Effect and

Linear Asset Pricing Models

Master Thesis ofRonald Verhagen

Amsterdam Business School Master in International Finance Studentnumber: 10479503 Supervisor: dr. E. (Esther) Eiling June 2017

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ABSTRACT

This paper confirms earlier research findings that portfolios with high volatility have low mean returns. This effect cannot be explained by exposure to Market, Size, Book-to-Market or Momentum factors. In contrast to these findings, the volatility effect seems to be explained by the new Fama-French five-factor model which includes an Investment and Profitability factor. The predominant economic reason is the high profitability of low volatility firms. However, I have found that exposure to the market-beta in the cross-section adds to the volatility puzzle.

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INTRODUCTION

The volatility effect can be defined as low volatility stocks earning higher risk adjusted returns. Since volatility is used as a measurement of risk, this is viewed as a puzzle since this observed effect deviates from the risk-return trade off principle in the classical paradigm. For instance, Haugen and Heins point out in 1975 that the relationship between risk and return is inverted (Heins & Haugen, 1975). A more recent study by Ang, Hodrick, Xing, and Zhang (2006) renewed the attention to the analysis of low volatility stocks.

In the 1970s, Black, Jensen, and Scholes (1972) found that the relationship between risk and return is much flatter than the Capital Asset Pricing Model (CAPM) predicts. Over the past 45 years flaws in linear asset pricing models have been demonstrated on both empirical and theoretical grounds. Eugene Fama and Kenneth French (1992) designed a three-factor model to describe stock returns and improved upon the CAPM in order to explain over 90% of diversified portfolio returns. The three factors in the model are a size factor, Small minus Big (SMB), a value factor, High Minus Low (HML) and the CAPM market factor. This three-factor model is extended further with positive momentum in the Carhart four-factor model (Carhart, 1997). However, none of these models succeeded in explaining the volatility effect.

In 2013 Fama and French introduced a new linear asset pricing model: a five-factor asset pricing model (2014). They claim that this five-factor model is an improvement upon their earlier three-factor model by adding the factors profitability and investment. This paper will investigate the ability of the five-factor model in explaining the volatility effect. This is tested by time series regressions on variance and size sorted value-weighted portfolio’s. Finally, if the model is found to be sufficient in explaining the volatility effect, the most dominant factor explaining the effect is determined by a new series of regressions while omitting either the profitability or investment factor.

Given the recent large attention towards the low volatility investment strategy, this paper researches whether the low volatility premium persists in the three earlier described asset pricing models and if the new five-factor model explains cross-sectional variation of average stock returns in volatility sorted portfolio’s. Novy-Marx (2016) has concluded that high profitability is the single most significant predicator of low volatility returns. The five-factor model therefore should be able to explain the volatility effect since it includes the profitability factor. If the cross section of returns and the volatility effect is explained by the five-factor

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model, this model is further validated and suitable for future use by academics and practitioners. Fama and French have not tested or mentioned variance sorted portfolio’s in their paper introducing the five-factor model nor linked their new model to an explanation of the volatility effect.

This thesis finds similar results as in the paper of Ang et al. (2006), portfolios with high volatility or high exposure to systematic volatility have low average returns and this effect cannot be explained by exposure to aggregate volatility exposure and Size, Book-to-Market or Momentum factors. However, this anomaly seems to be explained by the newer five-factor model by Fama and French. This paper finds that there remains no significant pricing error when this model is tested on the variance and size sorted portfolios. This paper concludes that the low volatility anomaly no longer persists under the five-factor model. The Profitability factor is the most dominant factor in explaining the volatility effect. This is in line with the findings of the paper of Novy-Marx (2016) that concluded that high profitability is the single most significant predictor of low volatility returns.

The rest of the paper is organized as follows: Section I lays out the theoretical framework, Section II describes the research methods and techniques used, Section III provides the data and descriptive statistics, Section IV and V contains all the test findings and Section VI concludes.

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I. THEORETICAL FRAMEWORK

In this paper volatility sorted portfolios are empirically tested along four asset pricing models. This chapter starts with a discussion on the volatility effect. The second part of this chapter gives a background and in-depth view of each of the models used.

The Volatility Effect

The volatility effect can be defined as low volatility stocks earning higher risk adjusted returns. For instance, Haugen and Heins point out in 1975 that the relationship between risk and return is inverted (Heins & Haugen, 1975). A more recent study by Ang, Hodrick, Xing, and Zhang (2006) renewed the attention to the analysis of low volatility stocks and concludes that high-risk stocks have had “abysmally low average returns”. Blitz and van Vliet (2007) provided a detailed analysis of the effect and found that the low volatility anomaly was still in place after controlling for previous mentioned effects of size, value and momentum. The strategy of long low-beta, short high-beta stocks (adjusted to be beta-neutral) is as strong as value and momentum strategies in the 1968 – 2008 period (Baker, Bradley, & Wurgler, 2011).

Although the empirical evidence is compelling and has been since the work done in the 70s by Haugen and Heins, the investment community only started to pursue low volatility investing in the aftermath of the Global Financial Crisis of 2007-08. MSCI created Global Minimum Volatility Indices using an optimization approach in 2008. And in April 2011 the S&P 500 Index created a series of low volatility indexes, whose constituents are weighted in proportion to the inverse of their realized volatilities. Lastly, FTSE launched the Russell 1000 Low Volatility Focused Factor Index using a ranking methodology in September 2015.

A Low Volatility investment strategy is one of the categories in Smart Beta investing strategies. This type of investment strategy aims to outperform the capitalization weighted market in terms of risk-adjusted returns through alternative weighting methods that emphasize factors such as size (small cap), value, positive momentum, low volatility (beta), illiquid, profitability, equal weight, etc.

Since Ang et al. published their findings of the volatility effect in 2006 there have been studies that have challenged their conclusions, this paper shows whether the same conclusions hold for aggregate volatility as used in the publication of Ang et al. One of the papers challenging Ang et al. is the analysis of Bali and Cakici in 2008, they have found no robust significant

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(inverse) relation exists between idiosyncratic volatility (IVOL) and expected returns. Their paper indicates that i) different weighting schemes ii) data frequency to estimate idiosyncratic risk iii) the breakpoints utilized to sort stocks into quintile portfolio’s, and iv) using a screen for size, price, and liquidity play critical roles in determining the existence of the relationship between IVOL and the cross section of expected returns (Bali & Cakici, 2008).

More recently, Fu (2009) went even further and published an article finding a significantly positive relationship between the estimated conditional idiosyncratic volatilities and expected returns. He argues that IVOL is time varying and thus, the findings of Ang et al. should not be used to imply the relationship between idiosyncratic risk and expected returns. Fu used exponential GARCH models to estimate expected idiosyncratic volatilities and indicated that Ang et al.’s findings are largely explained by a subset of small stocks with high idiosyncratic volatilities. These illiquid stocks have a large bid-ask spread and thus skew the results of returns. Another bias from small firms is delisting, when they delist the returns were hugely negative, but this would be marked as 'N/A', greatly biasing returns.

The preference-for-gambling hypothesis argues that investors irrationally use high volatility stocks as lotteries; in this framework, investors are implicitly willing to accept lower expected returns by paying a premium to gamble with high volatility stocks (Baker, Bradley, & Wurgler, 2011). Recently Hou and Loh (2016) introduced a decomposition methodology to evaluate potential explanations for the negative relationship between IVOL and subsequent stock returns. Their examined explanations account for 29–54% of the puzzle, with explanations based on lottery preferences and market frictions making the biggest contributions. Together, the lottery preference proxies capture a good 10–25% of the puzzle.

Stambaugh, Yu and Yuan (2015) provide a theory for the negative relationship between IVOL and expected returns by starting with the principle that IVOL represents risk that deters arbitrage and the resulting reduction of mispricing. They combine this concept with their term arbitrage asymmetry: many investors who would buy a stock they see as underpriced are reluctant or unable to short a stock they see as overpriced. Higher IVOL, which translates into higher arbitrage risk, allows greater mispricing. As a result, expected return is negatively (positively) related to IVOL among overpriced (underpriced) securities.

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Another plausible behavioral hypothesis attributes the anomaly to analysts’ optimism about more volatile stocks. Hsu, Kudoh, and Yamada (2013) find that analysts tend to produce high growth forecasts for high-volatility stocks; this can push up their prices and correspondingly reduce future returns.

The delegated-agency model provides an explanation for why the low volatility premium could persist even when professional money managers have been aware of the anomaly. The theory contends that most portfolio managers are benchmarked against a common core equity index; they are simply unwilling to buy low volatility stocks, which would significantly increase their tracking error against the benchmark. (Brennan, Cheng, & Feifei, 2012); (Baker, Bradley, & Wurgler, 2011).

Novy-Marx shows that high volatility and high beta stocks tilt strongly to small, unprofitable, and growth firms. He concludes that these tilts can explain the poor absolute performance of the most aggressive stocks. In his working paper, he points out that the performance of a low volatility strategy is explained by properly controlling for size, profitability, and relative valuations. Furthermore, high profitability is the single most significant predictor of low volatility, exceeding the power even of market capitalization (Novy-Marx, Understanding defensive equity, 2016).

The papers of Fu (2009), and Bali and Cakici (2008) challenging Ang et al. look specifically at idiosyncratic volatility and this paper shows whether the same findings of Ang et al. hold for aggregate volatility as used in their publication. The three asset pricing models (I) CAPM, (II) Three-Factor model, (III) Four-Factor model, are used as an assessment of the volatility effect to contrast the Five-factor model. Finally, the new five factor model of Fama and French is tested on volatility based portfolio’s. With the introduction of this new model, the goal of this paper is to research if the low volatility effect can be explained by the five-factor model. Since this profitability factor is added to the five-factor model of Fama-French, my expectation is that the model will perform well in explaining the volatility effect. In the next section I will discuss the benchmark models used.

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The Capital Asset Pricing Model

This classical model is based upon earlier work of Markowitz (1952; 1959) and Tobin (1958) on the modern portfolio theory and diversification. An investor should select a diversified portfolio from the efficient part of the mean variance frontier according to Markowitz. The mean variance frontier represents a set of portfolios that for a given level of risk maximizes return, or for a given level of return minimizes risk. Dependent on the individual investors personal risk preference, the portfolio that is chosen maximizes their mean-variance utility. Tobin expanded upon the theory of Markowitz and described that each rational investor should hold a combination of the risky tangent portfolio T and a risk-free asset. The portfolio with the highest Sharpe Ratio, the mean portfolio return minus the risk-free rate divided by the standard deviation of the portfolio return, is defined as the tangent of the portfolio on the Efficient Frontier (Sharpe, 1966). It represents the classical risk-return trade-off and one of the most fundamental principles in finance.

The Capital Asset Pricing Model (CAPM) by Treynor (1962), Sharpe (1964), Linter (1965) and Mossin (1966) is presented according to the following relationship:

𝐸(𝑅𝑖) − 𝑅𝑓 = 𝑅𝑓+ 𝛽𝑖(𝑅𝑚− 𝑅𝑓) (1)

In the CAPM, the beta (𝛽𝑖 ) gives a measure of sensitivity of the return of the asset i in relation to the variation in the return of the market portfolio ((𝑅𝑚) − 𝑅𝑓). The risk-return trade-off is represented in this market beta (or systematic risk) and predicts that expected returns increase with an increase in risk. Low levels of risk, defined in 𝛽𝑖, are associated with low expected returns. The CAPM follows this framework under a set of assumptions. These assumptions are: (1) All investors are single period risk-averse utility maximizers who can choose among portfolio only on the basis of mean and variance, (2) investors can lend and borrow unlimited amounts under the risk-free rate, (3) there are no taxes or transaction costs, (4) financial claims on underlying assets have perfect divisibility and (5) all information is available at the same time to all investors.

Furthermore, the CAPM predicts that investors hold a combination of the tangent portfolio and the risk-free asset on the Capital Market Line (CML). Under CAPM the tangent portfolio is the market portfolio. The risk in an individual asset that is correlated with systemic risk cannot be reduced by diversification. All other risk (idiosyncratic risk) will be reduced to null through

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diversification by holding a frontier portfolio on the CML. As a result, idiosyncratic risk is not rewarded or related to returns according to CAPM.

For almost 30 years this asset pricing model has been the cornerstone of pricing assets, but as described in the first chapter, not without criticism. For example, Black, Jensen and Scholes (1972) found that the relationship between (market) risk and return was positive, but the relationship flatter than the CAPM predicts. Haugen and Heins (1975) concluded that "over the long run stock portfolios with lesser variance in monthly returns have experienced greater average returns than their ‘riskier’ counterparts" while studying the period from 1926 to 1971. The CAPM model is, despite the criticism and empirical evidence, still widely used today in practical applications and frequently cited in academic papers.

Fama – French Three-Factor Model

The size effect, market capitalization (a stock’s price times shares outstanding) adds to the explanation of the cross-section of average returns provided by market 𝛽 according to Banz (1981). Average return on small stocks (low market capitalization) are too high given their 𝛽 estimates, and average returns on large stocks are too low. The rationale behind the factor is that smaller companies are sensitive to more risk. They have limited financial means to endure negative impact events and are relatively undiversified. This results in a higher risk exposure than large firms so investors demand a premium. This “size effect” comes in the form of higher expected returns.

By not taking account of the size effect in the CAPM, the model underestimates the expected returns of small stocks and overestimates expected stock with a high market capitalization.

Stattman (1980) and Rosenberg, Reid and Lanstein (1985) found that average returns on stocks are positively related to the firm’s book-to-market ratio (BTM). This effect is called the “value effect”. This effect is explained by Fama and French according to a relative distress risk factor of Chan and Chen (1991). Firms that the market judges to have poor prospects is signaled by relative low stock prices and high ratios of book-to-market equity and thus have higher expected stock returns than firms with strong prospects. Since then, however, studies such as Dichev (1998), Griffin and Lemon (2002), and Campbell, Hilscher, and Szilagyi (2008) have shown that the direct relation between distress risk and return is actually negative.

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By not taking account of the value effect in the CAPM, the model underestimates the expected returns for high BTM stocks and overestimates the expected returns of low BTM stocks.

In 1992 Fama and French found that size and value effect can be incorporated in the CAPM model by two empirically determined variables: size (measured by market capitalization) and book-to-market ratio. The market beta alone does not seem to explain the cross-section of average stock returns for the 1963-1990 period. Fama and French (1992) designed the three-factor model to improve on the CAPM by adding two risk three-factors to the model, a size three-factor, Small minus Big (SMB) and a book-to-market (value) factor, High Minus Low (HML). These new factors are constructed to measure the size effect and the value effect. The factors are included into the model next to the original factor (market risk) to explain the cross-sectional variation of expected stock returns better than the CAPM.

The three-factor model is presented according the formula:

𝐸(𝑅𝑖) − 𝑅𝑓 = 𝛽𝑖(𝑅𝑚− 𝑅𝑓) + 𝛽𝑠∙ 𝑆𝑀𝐵 + 𝛽𝑣∙ 𝐻𝑀𝐿 (2)

SMB is constructed as a self-financing portfolio to capture the size effect. It consists of a long position in a portfolio of stocks with low market capitalization and a short position in a portfolio of stocks with high market capitalization. SMB is computed as the mean historic excess returns of stocks of the 30% smallest companies minus the mean historic excess returns of stocks of the 30% biggest companies. It measures the additional returns of small stocks (small market capitalization) over big stocks (large market capitalization).

HML is constructed as a self-financing portfolio to capture the value effect. It consists of a long position in a portfolio of high book-to-market stocks and a short position in a portfolio of low book-to-market stocks. HML is calculated as the mean historic excess returns of the 50% stocks with the highest book-to-market ratio minus the mean historic excess returns of the 50% with the lowest book-to-market ratio. It measures the additional returns of value stocks (high BTM stocks) over growth stocks (low BTM stocks).

Fama and French find that the factors SMB and HML positively explain the cross-section of variation in average stock returns. These results support the size effect of Banz (1981) and the value effect of Stattman (1980) and Rosenberg, Reid and Lanstein (1985). Where the

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traditional CAPM fails to account for these effects, the three-factor model of Fama and French is a successful extension to the CAPM in explaining the cross-section of variation in returns.

Carhart four-factor model

The Carhart four-factor model is an extension of the Fama-French three factor model and includes a momentum factor. Jegadeesh and Titman (1993) found that a momentum strategy that buys stocks that have performed well in the past and sell stocks that have performed poorly in the past generate significant positive returns over 3- to 12- month holding periods. This anomaly cannot be explained by systemic risks or delayed reactions to common factors. They show that stocks with strong past performance continue to outperform stocks with poor past performance in the next period with an average excess return of about 1% per month.

To account for the momentum effect of Jegadeesh and Titman, the additional momentum factor (MOM) is included by Carhart in the asset pricing model (1997). His four-factor model is presented according the following model:

𝐸(𝑅𝑖) − 𝑅𝑓 = 𝛽𝑖(𝑅𝑚− 𝑅𝑓) + 𝛽𝑠∙ 𝑆𝑀𝐵 + 𝛽𝑣∙ 𝐻𝑀𝐿 + 𝛽𝑚∙ 𝑀𝑂𝑀 (3)

The MOM factor is the additional return received by investing in a Momentum portfolio. This portfolio is a self-financing portfolio just as SML and HML. It is constructed monthly and is long the prior-moth winning stocks and short the prior month losing stocks. The selection of winners and losers is calculated from the equally weighted average return of the 30% firms with the highest eleven-month returns lagged one month minus the equally weighted average return of the 30% firms with the lowest eleven-month returns lagged one month.

The factors SMB and HML in this model are exactly the same as in the three-factor model of Fama and French (1992). These two factors have a clear economic explanation as a risk premium. The factor MOM does not have clear explanation as a risk premium. In his paper, Carhart (1997) points out that the model is only to explain expected returns and lets any risk clarifications open for interpretation.

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The Five-Factor model

In 2013 a new asset pricing model was introduced by Fama and French: The five-factor asset pricing model. They claim that this model is an improvement upon their earlier three-factor model: the factors Profitability (RMW) and Investment (CMA) have been added. Although the model was rejected on the GRS test, it provides an acceptable description of average returns for applied purposes according to their publication. The GRS statistic demonstrates whether

the estimated intercepts from a multiple regression model are jointly zero (Gibbons, Ross,

& Shanken, 1989). The five-factor model is explained according to the following formula:

𝐸(𝑅𝑖) − 𝑅𝑓 = 𝛽𝑖(𝑅𝑚− 𝑅𝑓) + 𝛽𝑠∙ 𝑆𝑀𝐵 + 𝛽𝑣∙ 𝐻𝑀𝐿 + 𝛽𝑟∙ 𝑅𝑀𝑊 + 𝛽𝑐∙ 𝐶𝑀𝐴 (4)

The factors Size (SMB) and Value (HML) of the earlier three-factor model are augmented. The details of the augmentation of these factors SMB and HML is clarified in the next chapter under the section ‘Factor Definitions’. The two new variables are related to the explanation of average returns via a rewritten dividend discount model. This discount model shows the role of expected profitability, expected investment and the book-to-market ratio as predictors of stock returns.

The dividend discount model shows that the value of a stock is the present value of the expected dividends per share as shown in the equation:

𝑀𝑡 = ∑ 𝐸(𝑅𝑡+𝜏)/(1 + 𝑟)𝜏 ∞

𝜏=1

(5)

In this formula 𝑀𝑡 is the stock price at time t and 𝐸(𝑅𝑡+𝜏) is the expected dividend per share in period 𝑡 + 𝜏 and the long-term average return on the stock is presented by 𝑟.

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By rewriting the formula, dividend payments 𝑅𝑡+𝜏 can be manipulated to total earnings minus the change in book market equity in the same period as shown by Miller and Modigliani (1961). The total value of the firm’s stock is:

𝑀𝑡 = ∑ 𝐸(𝑌𝑡+𝜏− 𝑑𝐵𝑡+τ)/(1 + 𝑟)𝜏 ∞

𝜏=1

(6)

In this formula 𝑌𝑡+𝜏 is the total equity earnings for period 𝑡 + 𝜏 and 𝑑𝐵𝑡+τ= 𝐵𝑡+τ− 𝐵𝑡+τ−1 is the change in total book equity. Then Fama and French divide formula (6) by time 𝑡 book equity (𝐵𝑡), this gives:

𝑀𝑡 𝐵𝑡 =

∑∞𝜏=1𝐸(𝑌𝑡+𝜏− 𝑑𝐵𝑡+τ)/(1 + 𝑟)𝜏

𝐵𝑡 (7)

Equation (7)gives us three implications about expected stock returns, corresponding with the addition of the value, profitability and investment factors to the model. First, when everything in the formula is fixed except the value of the stock 𝑀𝑡 and the expected stock return 𝑟 then a lower value of 𝑀𝑡 or a higher book-to-market ratio implies a higher expected return. This indicated the HML or value factor. Next, fix everything in the formula except all expected future earnings 𝑌𝑡+𝜏 and expected returns 𝑟 . The equation shows that higher expected future earnings imply a higher expected return. This is the reasoning behind the addition of the profitability factor. Finally, by fixing everything except the change in book equity 𝑑𝐵𝑡+τ and the expected returns, the equation tells us that a higher growth of book equity – investment – implies a lower expected rate of returns.

The size factor is not suggested by this dividend discount model, this is explained by the admission that the dividend discount model in equation (7) is incomplete and their empirical measures imperfect. The five-factor model does not include the momentum factor of Carhart. The authors claim that four variables may be the most they can control at the same time.

Fama and French do not provide any further rationale behind the added factors except the dividend discount model. It is not clear whether the expected returns are higher for firms with high profitability or low investment are due to higher risk or mispricing (behavioral reasons).

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To test the theory of the addition of the factors according to the rewritten dividend discount model a set of empirical proxies for expected future earnings and investments are necessary. A paper by Novy-Marx (2012) identifies a proxy for expected profitability that is strongly related to average returns. In his paper he concludes that profitability, measured by gross profits-to-assets, has roughly the same power as book-to-market predicting the cross-section of average returns.

RMW (Robust Minus Weak) is a self-financing portfolio similar to the other factors described earlier. It is calculated with an independent 2x3 sort, two size groups and three profitability groups. The profitability factor RMW is the average of the two Robust Profitability portfolio returns minus the average of the two Weak Profitability portfolio returns:

𝑅𝑀𝑊 =1

2(𝑆𝑚𝑎𝑙𝑙 𝑅𝑜𝑏𝑢𝑠𝑡 + 𝐵𝑖𝑔 𝑅𝑜𝑏𝑢𝑠𝑡) − 1

2(𝑆𝑚𝑎𝑙𝑙 𝑊𝑒𝑎𝑘 + 𝐵𝑖𝑔 𝑊𝑒𝑎𝑘) (8)

Aharoni, Grundy, and Zeng (2013) document a statistically reliable negative relationship between investment and average returns. CMA (Conservative Minus Aggressive) is a self-financing portfolio as the other factors described. Similar to the RMW factor, it is also computed with a 2x3 sort, two size groups and three investment groups. The Investment factor CMA is the average of the two Conservative Investment portfolio returns minus the average of the two Aggressive Investment returns:

𝐶𝑀𝐴 =1

2(𝑆𝑚a𝑙𝑙 𝐶𝑜𝑛𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑣𝑒 + 𝐵𝑖𝑔 𝐶𝑜𝑛𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑣𝑒) − 1

2(𝑆𝑚𝑎𝑙𝑙 𝐴𝑔𝑔𝑟𝑒𝑠𝑖𝑣𝑒 + 𝐵𝑖𝑔 𝐴𝑔𝑔𝑟𝑒𝑠𝑖𝑣𝑒) (9)

Investment is measured as the growth of total assets for the fiscal year ending in t-1 divided by total assets at the end of t-1. In the valuation equation (7),the investment variable is the expected growth of book equity, not the growth of total assets. Fama and French have replicated all tests using the growth of book equity, with results similar to those obtained with the growth of assets. The main difference, they point out, is that sorts on asset growth produce slightly larger spreads in average returns. The choice for growth of total assets becomes even more interesting when Fama and French (2007) concluded themselves that asset growth is not sufficiently robust. In that paper, they have found even better results for net share issuance, and since that variable also fits with the dividend discount model story, it appears to be a stronger candidate for the investment factor in the 5-factor model. Fama and French argue that

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their choice for this proxy does not change the robustness of their theory or conclusions. It remains unclear which variable is the best proxy for investment.

It is interesting to see whether the Fama French Five-Factor model is able to explain the volatility effect since they claim it is a significant improvement upon their earlier three-factor model. Since they claim it is a enhancement of their earlier model, this warrants an empirical thoroughly performed challenge, in this paper in the form of testing on volatility sorted portfolio’s. Their earlier three-factor model has been tested and couldn’t explain the volatility effect, this newer five-factor model has not yet been tested before on volatility sorted portfolio’s. The five-factor model is especially interesting since it includes a profitability factor, claimed to be the single most significant predictor of the volatility effect by Novy-Marx.

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II. RESEARCH METHODS AND TECHNIQUES

The goal of the analysis is to measure the effectiveness in explaining the cross-section of returns of four linear asset pricing models and their ability to explain the volatility effect. To measure their effectiveness, the models are tested in time-series regressions. The following four models are used:

The CAPM

𝐸(𝑅𝑖) − 𝑅𝑓 = 𝛽𝑎(𝑅𝑚− 𝑅𝑓) (10)

The three-factor model

𝐸(𝑅𝑖) − 𝑅𝑓 = 𝑅𝑓+ 𝛽𝑎(𝑅𝑚− 𝑅𝑓) + 𝛽𝑠∙ 𝑆𝑀𝐵 + 𝛽𝑣∙ 𝐻𝑀𝐿 (11)

The Carhart four-factor model

𝐸(𝑅𝑖) − 𝑅𝑓 = 𝑅𝑓+ 𝛽𝑎(𝑅𝑚− 𝑅𝑓) + 𝛽𝑠∙ 𝑆𝑀𝐵 + 𝛽𝑣∙ 𝐻𝑀𝐿 + 𝛽𝑚∙ 𝑀𝑂𝑀 (12)

The five-factor model

𝐸(𝑅𝑖) − 𝑅𝑓 = 𝑅𝑓+ 𝛽𝑎(𝑅𝑚− 𝑅𝑓) + 𝛽𝑠∙ 𝑆𝑀𝐵 + 𝛽𝑣∙ 𝐻𝑀𝐿 + 𝛽𝑟∙ 𝑅𝑀𝑊 + 𝛽𝑐∙ 𝐶𝑀𝐴 (13)

Portfolio Construction

The dependent variables of the regressions are excess returns of 25 monthly formed value weighted stock market portfolios. They are the intersections of 5 portfolios formed on size (market equity, ME) and 5 portfolios formed on the variance of daily returns (Var). Var is the variance of daily returns estimated using 60 days (minimum 20) of lagged returns. From these portfolio’s 5 size average portfolio’s, with equal weight (1/5) given to each size quintile and 5 volatility average portfolios with equal weight (1/5) given to each volatility quintile are constructed. One portfolio is constructed from the average of all of the 25 portfolios.

Finally, a separate series of (1-5) portfolios are constructed to indicate the volatility effect (premium). This is a long 1st_{quintile volatility (low) minus the 5}th_{quintile (high) portfolio returns.}

This exercise is for each of the 5 size quintiles and one 1-5 average portfolio. This last average portfolio is an average portfolio of each of the 5 long-short portfolio’s and will be used as the main indicator of the volatility premium.

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From these value-weighted portfolios excess returns are calculated by subtracting the risk-free rate. The one-month Treasury bill rate is used as the proxy for the risk-free rate.

The market risk premium(𝑅𝑚− 𝑅𝑓) is calculated as the monthly market return minus the risk-free rate.

Factor Definitions

The factors SMB, HML, MOM, RMW and CMA are constructed with methods explained in the previous chapter. The factors SMB and HML for the five-factor model are constructed with a different method compared to the three-factor model.

SMB for the three-factor model is computed as the mean historic excess returns of stocks of the 30% smallest companies minus the mean historic excess returns of stocks of the 30% biggest companies.

HML for the three-factor model is calculated as the mean historic excess returns of the 50% stocks with the highest book-to-market ratio minus the mean historic excess returns of the 50% with the lowest book-to-market ratio.

MOM is constructed from the equally weighted average return of the 30% firms with the highest eleven-month returns lagged one month minus the equally weighted average return of the 30% firms with the lowest eleven-month returns lagged one month.

The Five Factor Model uses a different approach to construct the factors. The SMB and HML factors use independent sorts of stocks in two Size groups and three B/M groups (independent 2x3 sorts). The Size breakpoint is the NYSE median market cap, and the B/M breakpoints are 30th_{and 70}th_{percentiles of B/M for NYSE stocks. The intersections of the sorts produce six}

portfolios. The Size factor, is the average of the three small stock portfolio returns minus the average of the three big stock portfolio returns. This results in the following formula:

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𝑆𝑀𝐵 =1

3(𝐿𝑜𝑤𝐵𝑀 𝑆𝑚𝑎𝑙𝑙 + 𝑀𝑖𝑑𝐵𝑀 𝑆𝑚𝑎𝑙𝑙 + HighBM Small) − 1

3(𝐿𝑜𝑤𝐵𝑀 𝐿𝑎𝑟𝑔𝑒 + 𝑀𝑖𝑑𝐵𝑀 𝐿𝑎𝑟𝑔𝑒 + HighBM 𝐿𝑎𝑟𝑔𝑒) (14)

The value factor HML in the five-factor model is the average of the two high B/M portfolio returns minus the average of the two low B/M portfolio returns. Equivalently, it is the average of small and big value factors constructed with portfolios of only small stocks and portfolios of only big stocks:

𝐻𝑀𝐿 =1

2(𝐻𝑖𝑔ℎ𝐵𝑀 𝑆𝑚𝑎𝑙𝑙 + 𝐻𝑖𝑔ℎ𝐵𝑀 𝐿𝑎𝑟𝑔𝑒) − 1

2(𝐿𝑜𝑤𝐵𝑀 𝑆𝑚𝑎𝑙𝑙 + HighBM Small) (15)

The profitability and investment factors of the 2x3 sorts, RMW and CMA, are calculated with the same method as HML except the second sort is either based on profitability (RMW) or investment (CMA). The measure for profitability is gross profits-to-assets. The measure for Investments is the growth of total assets divided by total assets.

Like HML, RMW and CMA can be interpreted as averages of profitability and investment factors for small and big stocks. The equations for these two factors are (8) and (9),provided in the previous chapter.

If the sensitivities to the factors (including the market risk factor) in these four models capture all the variation in the expected returns, I will not find any significant intercepts in the tested portfolio’s. If there are significant positive intercepts and thus pricing errors found in the regressions, this indicates that the model is not able to explain the returns. The volatility effect cannot be explained by the specific model if significant positive intercepts are found in the low volatility quintile, significant negative intercepts in the high volatility quintile or significant positive intercepts in the 1-5 quintile.

Besides the main research on aggregate volatility and size sorted value weighted portfolios, another two sets of portfolios are tested on the same asset pricing models to assure the quality of the findings in this paper. If the findings in all the sets of portfolios are the same or similar, the results are robust. The other two sets of portfolios are (i) aggregate volatility and size sorted equal weighted portfolios and (ii) residual volatility and size sorted value weighted portfolios. This means that there will be one robustness check on the measure of volatility and one check on the weighting scheme in the portfolio construction. The equal weighted portfolios give a 1/n

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weight to each of the individual stocks each month where n is the total number of stock in the given month. The impact of equal weighing the individual shares instead of value weighing is that there will be more weight on the smaller stocks and less weight on the larger stocks.

The residual variance and size sorted portfolios are formed monthly on the variance of the residuals from the Fama and French three-factor model. Residual Variance is estimated using 60 days (minimum 20) of lagged returns.

Expectation of the results

The expectation and hypothesis of this paper is that the CAPM, Fama-French Thee-Factor model and the Carhart four-factor model are not able to fully explain the cross sections of returns and a significant positive alpha (intercept) will be found in low volatility (1st_{quintile) and}

the average 1-5 volatility portfolio. This is in line with earlier literature referenced in this paper. The newest Fama-French five-factor model is an improvement on the earlier asset pricing models and my expectation is that this model is able to explain the low volatility anomaly since it includes the profitability factor. Novy-Marx (2016) has concluded that high profitability is the single most significant predicator of low volatility returns.

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III. DATA AND DESCRIPTIVE STATISTICS

The data used to construct the portfolio’s is based on the CRSP database and include NYSE, AMEX and NASDAQ stocks. These historical monthly values used for the portfolios and all the factors are accessible on the Kenneth French’ website1_.

The time series of portfolio returns and factor data is monthly data from July 1963 until the most recent data available, November 2016. This is the same time period used in the research of Ang et al (2006) and extended with the most recent period. 1963 is the first year used in this research because earlier years encounter selection bias and are tilted toward big historically successful firms (Fama & French, 1992). The end date is the most recent data available to maximize the amount of data available (after the publication of Ang et al.). There are 641 monthly observations per portfolio or factor.

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Mean Returns

Table 1. Mean Monthly Returns of Value-Weighted Portfolio’s.

These are the mean monthly returns of the portfolios that are the intersections of 5 portfolios formed on size (Small 1 to Big 5) and 5 portfolios formed on the variance of daily returns (Low Variance 1 to High Variance 5), Average portfolio’s and Long-Short portfolios (1-5).

Total Variance Low High 1 2 3 4 5 Average 1-5 Small Size 1 1,40% 1,55% 1,47% 1,16% 0,19% 1,15% 1,21% 2 1,30% 1,43% 1,43% 1,30% 0,67% 1,23% 0,76% 3 1,14% 1,23% 1,34% 1,27% 0,83% 1,16% 0,31% 4 1,08% 1,13% 1,17% 1,14% 0,87% 1,07% 0,21% Big Size 5 0,82% 0,92% 0,92% 0,86% 0,87% 0,88% -0,04% Avg. 1,15% 1,25% 1,27% 1,14% 0,68% 1,09% 0,49%

The mean monthly return of the average portfolio for the 1963-2016 period is 1,09%. There is a visible size effect in these mean returns as we observe that the average small portfolios 1 and 2 have a higher mean monthly return of 1,15% and 1,23% compared to the big portfolios 4 and 5 with a respective mean return of 1,07% and 0,88%. There is a visible volatility effect in these mean returns as we observe that low variance (1) portfolio has a mean return of 1,15% compared to the high variance portfolio mean return of 0,68%. This is also observable in the mean return of the long-short portfolios (1-5) and specifically the average portfolio. The average long low variance short high variance portfolio delivers a return of 0,49% over the 1963-2016 period.

The volatility effect is most strongly observable in the Small Stocks quintile (1), the return of that portfolio is 1,21%. The effect becomes less strong in the bigger size quintiles. Table 1 seems to indicate size and volatility effects. However, this needs to be controlled for systemic risk differences to confirm these effects.

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Variance of the monthly portfolio returns

Table 2. The Variance of the Monthly Portfolio Returns.

These are the mean variance of the portfolios that are the intersections of 5 portfolios formed on size (Small 1 to Big 5) and 5 portfolios formed on the variance of daily returns (Low Variance 1 to High Variance 5), Average portfolio’s and Long-Short portfolios (1-5).

Low Variance High Variance 1 2 3 4 5 Average 1-5 Small 1 16,29% 31,70% 41,80% 56,36% 85,02% 40,19% 43,44% 2 16,09% 27,34% 34,45% 45,98% 76,50% 34,32% 29,50% 3 13,37% 22,57% 28,72% 38,09% 64,55% 28,13% 36,68% 4 13,55% 19,78% 25,71% 32,99% 58,00% 24,71% 36,17% Big 5 11,55% 15,69% 19,66% 25,56% 44,40% 18,79% 28,55% Average 12,14% 20,52% 26,68% 35,32% 59,60% 26,64% 29,86%

The variance of the monthly returns of the average portfolio is 26,64%. The average low variance portfolio has a variance of 12,14% this is lower than the average high variance portfolio where the monthly returns have a variance of 59,60%. The almost equal portioned increase from the low variance quintile (1) to the high variance quintile (5) is explainable as a result of the portfolio construction. Stocks which are characterized as low variance by the past period seem to stay relative low variance in the subsequent period when it’s included in the specific low variance quintile portfolio. The same reasoning is applicable to high variance stocks and portfolios. The average portfolio of the size quintiles shows a similar but reverse effect, quintile 1, the small stocks quintile, has the highest average variance of monthly returns with 40,19%. The biggest stock portfolio measured by ME have the lowest variance of monthly returns with 18,89%.

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IV. MAIN RESULTS

In this chapter, the results of the regressions on the four asset pricing models will be presented and discussed. I have tested a total of 42 stock portfolios on 4 asset pricing models with a total of 168 time-series regressions. The resulting intercepts of these regressions are found in table 3 on the next page.

Intercepts

As discussed in an earlier chapter, if these asset pricing models are correctly specified, the intercept or pricing error is zero for all portfolio’s. As expected and in line with earlier research of other papers, the CAPM, Fama-French Thee-Factor model and the Carhart four-factor model are not able to fully capture cross sections of returns and the regression outcome of the tested models on the constructed portfolio’s result in significant pricing errors. The average low variance (1st_{quintile) and the average long-short (1-5) portfolio have positive intercepts, see}

table 3. The high variance (5th_{quintile) portfolio has a negative pricing error in all of these three}

models.

The intercept is the highest in the Capital Asset Pricing Model with 0,407 for the low variance quintile 1. This becomes gradually smaller when we look at subsequent quintiles and then minus 0,501 for the highest variance quintile. The average long-short portfolio (1-5) shows an intercept of 0,513 under CAPM. The intercepts for the low variance-, high variance- and the long-short portfolio are all significant with absolute t-statistics higher than 3. T-statistics higher than 1.96 are generally accepted as significant. That is the approximate value of the 97.5 percentile point of the normal distribution and thus the probability of obtaining a result equal to or "more extreme" than what was actually observed, when the null hypothesis is true is smaller than 5%. These results confirm the conclusions of the literature referenced in this paper.

Observing the results of the 3-factor model we notice a decrease in the intercept in the average long-short portfolio (0,385) compared to the CAPM model and an even further decrease when we look at the 4-factor model (0,204). However, the intercepts are significant with a t-statistic of 2,925 and 2,579 respectively. Book-to-Market, Size or Momentum factors are not able to explain the volatility effect.

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Table 3. Intercepts of the Four Asset Pricing Models

These intercepts are the result of the regressions performed on four asset pricing models, CAPM, 3-factor model, 4-factor model and the 5-factor model. The t-statistics are in brackets underneath each of the intercepts. The portfolios tested are the intersections of 5 portfolios formed on size (Small 1 to Big 5) and 5 portfolios formed on the variance of daily returns (Low Variance 1 to High Variance 5), Average portfolio’s and long short portfolios (1-5).

Intercept Low Variance High Variance Intercept Low Variance High Variance 𝒂𝒊 CAPM 1 2 3 4 5 Average 1-5 𝒂𝒊 3 Factor Model 1 2 3 4 5 Average 1-5 Small 1 0,65 0,631 0,458 0,056 -0,998 0,159 1,253 Small 1 0,368 0,287 0,114 -0,266 -1,316 -0,162 1,29 (6,325) (4,877) (3,173) (0,327) -(4,108) (1,114) (5,858) (5,513) (3,765) (1,398) -(2,681) -(8,026) -(2,203) (7,157) 2 0,529 0,53 0,451 0,215 -0,578 0,229 0,712 2 0,277 0,227 0,169 -0,03 -0,708 -0,013 0,541 (5,759) (4,808) (3,894) (1,685) -(3,140) (2,201) (3,974) (4,302) (3,160) (2,257) -(0,366) -(6,247) -(0,268) (3,579) 3 0,4 0,352 0,403 0,239 -0,378 0,203 0,383 3 0,193 0,114 0,174 0,019 -0,415 0,017 0,214 (4,918) (3,972) (4,141) (2,232) -(2,494) (2,592) (2,134) (2,864) (1,669) (2,265) (0,238) -(4,130) (0,351) (1,501) 4 0,344 0,283 0,239 0,136 -0,319 0,137 0,268 4 0,174 0,116 0,082 0,004 -0,3 0,015 0,08 (3,969) (3,457) (2,939) (1,594) -(2,472) (2,350) (1,479) (2,227) (1,572) (1,103) (0,054) -(2,936) (0,309) (0,527) Big 5 0,112 0,122 0,05 -0,09 -0,232 -0,007 -0,051 Big 5 0,067 0,124 0,053 -0,061 -0,129 0,011 -0,198 (1,436) (1,750) (0,794) -(1,439) -(2,220) -(0,268) _-(0,309) (0,987) (2,143) (0,941) -(1,028) -(1,281) (0,624) -(1,311) Average 0,407 0,384 0,32 0,111 -0,501 0,144 0,513 Average 0,216 0,174 0,119 -0,067 -0,573 -0,026 0,385 (5,950) (5,268) (4,213) (1,335) -(3,602) (2,167) (3,160) (4,013) (3,291) (2,229) -(1,215) -(6,393) -(0,780) (2,952) Intercept Low Variance High Variance Intercept Low Variance High Variance 𝒂_𝒊 4 Factor Model 1 2 3 4 5 Average 1-5 𝒂_𝒊 5 Factor Model 1 2 3 4 5 Average 1-5 Small 1 0,419 0,389 0,293 -0,026 -0,946 0,026 0,975 Small 1 0,27 0,166 0,067 -0,171 -0,915 -0,117 0,789 (6,183) (5,144) (3,855) -(0,285) -(6,217) (0,386) (5,590) (4,295) (2,465) (0,874) -(1,792) -(5,993) -(1,639) (4,903) 2 0,279 0,246 0,236 0,071 -0,487 0,069 0,343 2 0,143 0,057 0,034 -0,139 -0,433 -0,068 0,096 (4,238) (3,357) (3,120) (0,875) -(4,508) (1,464) (2,289) (2,444) (0,962) (0,523) -(1,962) -(4,113) -(1,687) (0,717) 3 0,178 0,16 0,201 0,085 -0,277 0,069 0,065 3 0,043 -0,023 0,011 -0,13 -0,181 -0,056 -0,171 (2,580) (2,319) (2,569) (1,027) -(2,784) (1,436) (0,451) (0,690) -(0,381) (0,163) -(1,804) -(1,889) -(1,338) -(1,304) 4 0,166 0,153 0,114 0,051 -0,161 0,064 -0,063 4 0,022 -0,053 -0,1 -0,107 -0,055 -0,058 -0,318 (2,079) (2,027) (1,491) (0,631) -(1,596) (1,286) _-(0,411) (0,295) -(0,773) -(1,504) -(1,386) -(0,570) -(1,228) -(2,289) Big 5 0,026 0,075 0,015 -0,059 -0,63 -0,001 -0,301 Big 5 -0,049 -0,048 -0,065 -0,098 0,129 -0,026 -0,573 (0,376) (1,274) (0,261) -(0,969) -(0,615) -(0,065) _-(1,962) -(0,731) -(0,926) -(1,206) -(1,609) (1,379) -(1,552) -(4,009) Average 0,213 0,205 0,172 0,025 -0,387 0,046 0,204 Average 0,086 0,02 -0,011 -0,129 -0,291 -0,065 -0,036 (3,880) (3,812) (3,215) (0,462) -(4,572) (1,428) (2,579) (1,762) (0,493) -(0,257) -(2,636) -(3,685) -(2,269) -(0,315)

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The newest Fama-French five-factor model is an improvement on these earlier asset pricing models as it seems that this model is able to explain the low volatility anomaly. The average long-short (1-5) portfolio intercept is not significant with a t-statistic of minus 0,315. The hypothesis cannot be rejected.

Market Beta’s

Observing the market beta’s in table 4 and comparing these with the mean monthly returns in table 1 it becomes clear that the exposure to the market-beta in the cross-section adds to the volatility puzzle. This observation can be concluded for each of the four asset pricing models. Each subsequent average volatility quintile has a higher market beta (𝛽𝑎), however, a higher market beta is not rewarded with a higher return. This is shown in the following figure, where the dotted line is the market beta of 5 average variance quintile portfolios in the five-factor asset pricing model with the mean monthly returns.

Figure 1: Comparison of the Market Beta in the Five Factor Asset Pricing Model with the Portfolio’s Mean Monthly Return.

In addition, when the volatility premium based on raw returns (table 1: 0.49%) is compared to the CAPM alpha (table 3: 0,513%) this indicates that the CAPM increases the volatility premium. This is not the case in the other asset pricing models tested.

0,40% 0,50% 0,60% 0,70% 0,80% 0,90% 1,00% 1,10% 1,20% 1,30% 1,40% 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 1 2 3 4 5 Me an M o n th ly r etu rn s Ma rk et Be ta Variance Quintiles

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Table 4. Market Beta’s of the Four Asset Pricing Models

These market Beta’s are the result of the regressions performed on four asset pricing models, CAPM, 3-factor model, 4-factor model and the 5-factor model. The t-statistics are in brackets underneath each of the intercepts. The portfolios tested are the intersections of 5 portfolios formed on size (Small 1 to Big 5) and 5 portfolios formed on the variance of daily returns (Low Variance 1 to High Variance 5), Average portfolio’s and long short portfolios (1-5).

βi Low High βi Low High

CAPM 1 2 3 4 5 Average 1-5 3 Factor Model 1 2 3 4 5 Average 1-5 Small 1 0,707 1,043 1,213 1,395 1,568 1,185 -0,856 Small 1 0,664 0,962 1,093 1,214 1,291 1,045 -0,624 (30,639) (35,901) (37,441) (36,315) (28,728) (36,909) -(17,828) (41,950) (53,269) (56,697) (51,650) (33,264) (59,807) -(14,605) 2 0,745 1,004 1,153 1,354 1,683 1,188 -0,674 2 0,721 0,959 1,079 1,235 1,419 1,083 -0,456 (36,092) (40,605) (44,278) (47,329) (40,704) (50,761) -(16,746) (47,349) (56,397) (60,709) (63,607) (52,876) (94,038) -(12,731) 3 0,688 0,949 1,079 1,258 1,603 1,116 -0,91 3 0,709 0,947 1,054 1,196 1,38 1,057 -0,667 (37,684) (47,629) (49,415) (52,222) (47,085) (63,316) -(22,594) (44,397) (58,738) (58,009) (61,860) (57,949) (92,442) -(19,722) 4 0,673 0,893 1,05 1,206 1,561 1,076 -0,883 4 0,732 0,934 1,081 1,198 1,394 1,068 -0,658 (34,553) (48,526) (57,405) (62,743) (53,775) (82,398) -(21,709) (39,647) (53,278) (61,010) (63,230) (57,627) (90,413) -(18,364) Big 5 0,628 0,805 0,937 1,088 1,387 0,969 -0,755 Big 5 0,717 0,885 0,996 1,128 1,318 1,009 -0,597 (35,680) (51,418) (66,102) (77,119) (59,200) (154,323) -(20,312) (44,766) (64,420) (74,264) (80,448) (55,426) (242,066) -(16,668) Average 0,688 0,939 1,087 1,26 1,56 1,107 -0,816 Average 0,708 0,938 1,061 1,194 1,36 1,052 -0,6 (44,779) (57,418) (63,635) (67,317) (49,940) (74,056) -(22,375) (55,661) (75,008) (84,220) (91,921) (64,031) (131,053) -(19,416)

βi Low High βi Low High

4 Factor Model 1 2 3 4 5 Average 1-5 5 Factor Model 1 2 3 4 5 Average 1-5 Small 1 0,653 0,941 1,055 1,163 1,213 1,005 -0,557 Small 1 0,685 0,984 1,093 1,17 1,171 1,021 -0,482 (40,922) (52,775) (58,894) (54,420) (33,797) (64,168) -(13,541) (44,309) (59,241) (57,937) (49,639) (31,146) (58,166) -(12,159) 2 0,72 0,955 1,065 1,214 1,372 1,065 -0,414 2 0,755 0,997 1,102 1,245 1,33 1,086 -0,329 (46,451) (55,194) (59,638) (63,049) (53,867) (95,688) -(11,703) (52,227) (67,878) (69,708) (71,430) (51,242) (110,016) -(10,024) 3 0,712 0,938 1,048 1,182 1,35 1,046 -0,636 3 0,75 0,973 1,084 1,223 1,305 1,067 -0,55 (43,819) (57,528) (56,756) (60,701) (57,600) (91,689) -(18,801) (48,749) (65,940) (67,293) (69,013) (55,335) (103,663) -(17,021) 4 0,734 0,927 1,074 1,188 1,364 1,057 -0,628 4 0,775 0,975 1,121 1,219 1,322 1,083 -0,544 (39,014) (52,086) (59,704) (61,925) (57,231) (89,402) -(17,472) (41,894) (58,234) (68,821) (64,054) (55,851) (92,454) -(15,880) Big 5 0,725 0,896 1,004 1,127 1,304 1,011 -0,576 Big 5 0,757 0,933 1,025 1,137 1,234 1,017 -0,473 (44,758) (64,816) (74,081) (78,940) (54,238) (240,212) -(15,893) (45,855) (73,419) (76,824) (75,875) (53,458) (245,337) -(13,441) Average 0,709 0,931 1,049 1,175 1,321 1,037 -0,562 Average 0,744 0,972 1,085 1,199 1,272 1,055 -0,476 (54,679) (73,556) (83,255) (93,364) (66,197) (138,015) -(18,470) (62,021) (97,016) (105,457) (99,444) (65,434) (149,573) -(17,133)

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Characteristics of Low Volatility Portfolios

The earlier three-factor model of Fama-French was not able to explain the volatility effect. But as shown earlier, when I tested the new five-factor on volatility and size sorted portfolios, no significant intercept in the average long-short portfolio was defined by the test result. The differences in these results must be explained by adding the two new factors RMW and CMA. In table 6 all the factor coefficients of the five-factor model are shown. As observed in table 6 the low volatility quintile and the average long-short portfolio show significantly positive factor coefficients. We observe significant negative coefficients for these two factors in the average high volatility (quintile 5) portfolio. We can conclude that the low volatility portfolios have characteristics in common with the self-financing portfolios of the two new factors. Low volatility stocks (and thus portfolios) have robust profitability records and/or a low (conservative) investment strategy. The most dominant factor of these two will be examined in the next section.

Dominant factor

In an analysis to find the dominant factor in explaining the volatility effect I have performed another set of regressions. The results are shown in table 5. These regressions are performed on the five-factor model omitting either the CMA factor or the RMW factor. By omitting one of the two variables in each regression and keeping everything else in the test the same, the significant factor with the highest coefficient is the most dominant in explaining the volatility effect.

Table 5. Factor coefficients of RMW and CMA

This table shows the coefficients of RMW and CMA and their corresponding t-statistic of the regressions performed on the five-factor model while either omitting the CMA or RMW factor. The portfolios are 5 volatility sorted quintiles averaged over the 5 size quintiles and one long-short portfolio (1-5).

RMW CMA

Quintile Coefficient _{T-Stat Coefficient} T-Stat

1 0,26 _(11,51) 0,09 (2,41) 2 0,37 _(19,23) 0,01 (0,39) 3 0,35 _(19,20) -0,05 (-1,29) 4 0,21 _(9,35) -0,08 (-2,31) 5 -0,49 _(-12,96) -0,25 (-4,04) 1-5 0,78 _(14,40) 0,34 (3,66)

In table 5 the coefficients for each factor are presented as well as the corresponding t-statistics in each test. The RMW factor has larger significant absolute coefficients in the first quintile portfolio, the fifth quintile portfolio and the long -short (1-5) portfolio. I can conclude that

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profitability is the most dominant factor in explaining the volatility effect. This is in line with the findings of the paper of Novy-Marx (2016) that concluded that high profitability is the single most significant predictor of low volatility returns.

The analyst optimism hypothesis of Hsu, Kudoh, and Yamada (2013) who attribute the volatility effect to analysts’ optimism about more volatile stocks can be linked to the factor coefficients of RMW in high volatility portfolio. They’ve hypothesized that analysts inflate earnings forecast more aggressively for volatile stocks. Because investors are known to overreact to analyst forecasts, this contributes to systemic overvaluation and low returns for high-volatility stocks. These high volatility stocks are characterized as having a relative weak profitability because they tilt strongly to small, unprofitable, and growth firms (Novy-Marx, 2016). It can be hypothesized that analysts are more comfortable to inflate the earnings forecasts of these type of firms because it is harder for clients to detect inflation in growth forecasts for stocks that have highly volatile growth. This is shown in the negative

coefficients of RMW in the high volatility portfolio with the five-factor model, these firms have a relativity low gross profits-to-assets ratio.

The average long-short (1-5) portfolio shows no significant pricing error in the time-series regression in contrast to several individual size quintile long short portfolios. The first size quintile has a positive significant pricing error of 0,789. The 4th_{and 5}th_{size quintiles show a}

contrary negative pricing error of -0,318 and -0,573 respectively. These contrasting findings are similar to the pricing errors shown in the other three models. It is not clear whether the volatility effect is explained by the five-factor model or it’s just because of the averaging of the specific errors results in an non-significant pricing error in the average long short portfolio.

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Table 6. Factor Coefficients 5-Factor model

These are four tables of the individual factor coefficients as a result of the regressions performed on the 5-factor model. The t-statistics are in brackets underneath each of the intercepts. The portfolios tested are the intersections of 5 portfolios formed on size (Small 1 to Big 5) and 5 portfolios formed on the variance of daily returns (Low Variance 1 to High Variance 5), Average portfolio’s and long short portfolios (1-5).

Low Variance High Variance Low Variance High Variance SMB 1 2 3 4 5 Average 1-5 HML 1 2 3 4 5 Average 1-5 Small 1 0,655 0,926 1,051 1,195 1,372 1,04 -0,714 Small 1 0,349 0,393 0,373 0,327 0,293 0,347 0,051 (30,482) (40,111) (40,078) (36,479) (26,259) _{(42,637) (12,956)} (11,747) (12,287) (10,272) (7,197) (4,054) (10,276) (0,669) 2 0,546 0,745 0,828 0,953 1,13 0,84 -0,382 2 0,296 0,344 0,309 0,217 -0,032 0,227 0,371 (27,150) (36,491) (37,667) (39,358) (31,310) _{(61,252) -(8,354)} (10,656) (12,173) (10,149) (6,466) -(0,637) (11,945) (5,863) 3 0,306 0,467 0,569 0,696 0,844 0,576 -0,535 3 0,272 0,353 0,299 0,221 -0,133 0,202 0,399 (14,286) (22,775) (25,389) (28,271) (25,743) _{(40,277) (11,896)} (9,182) (12,419) (9,646) (6,474) -(2,922) (10,218) (6,417) 4 0,095 0,192 0,239 0,318 0,511 0,271 -0,413 4 0,276 0,255 0,235 0,168 -0,158 0,155 0,429 (3,711) (8,249) (10,532) (12,013) (15,539) _{(16,653) -(8,670)} (7,763) (7,925) (7,493) (4,577) -(3,464) (6,891) (6,508) Big 5 -0,245 -0,226 -0,161 -0,178 0,011 -0,16 -0,253 Big 5 0,106 0,031 0,049 0,003 -0,036 0,031 0,137 (10,691) (12,808) -(8,673) -(8,530) (0,333) _{(27,740) -(5,160)} (3,347) (1,269) (1,892) (0,105) -(0,806) (3,839) (2,202) Average 0,271 0,421 0,505 0,597 0,774 0,514 -0,459 Average 0,26 0,275 0,253 0,187 -0,013 0,192 0,277 (16,263) (30,201) (35,308) (35,634) (28,619) _{(52,399) (11,894)} (11,261) (14,266) (12,770) (8,059) -(0,345) (14,182) (5,190) Low Variance High Variance Low Variance High Variance

RMW 1 2 3 4 5 Average 1-5 CMA 1 2 3 4 5 Average 1-5

Small 1 0,274 0,349 0,191 -0,108 -0,832 -0,025 1,109 Small 1 0,075 0,066 -0,029 -0,222 -0,493 -0,121 0,569 (9,254) (10,983) (5,300) -(2,401) (11,574) _{-(0,757) (14,625)} (1,693) (1,393) -(0,541) -(3,303) -(4,593) -(2,411) (5,029) 2 0,323 0,4334 0,393 0,366 -0,524 0,198 0,961 2 0,157 0,153 0,053 -0,038 -0,409 -0,017 0,563 (11,691) (15,436) (12,996) (10,993) (10,555) _{(10,513) (15,288)} (3,805) (3,645) (1,167) -(0,764) -(5,515) -(0,598) (6,003) 3 0,333 0,375 0,456 0,414 -0,436 0,229 0,772 3 0,207 0,061 0,06 0,069 -0,371 0,005 0,58 (11,311) (13,296) (14,802) (12,223) -(9,677) _{(11,605) (12,488)} (4,711) (1,456) (1,307) (1,373) -(5,520) (0,179) (6,282) 4 0,326 0,393 0,448 0,29 -0,513 0,189 0,842 4 0,208 0,175 0,145 0,068 -0,314 0,056 0,524 (9,210) (12,273) (14,389) (7,972) (11,331) (8,434) _(12,853) (3,950) (3,667) (3,114) (1,247) -(4,646) (1,690) (5,359) Big 5 0,186 0,372 0,282 0,079 -0,448 0,094 0,637 Big 5 0,247 0,205 0,089 0,031 -0,531 0,008 0,78 (5,906) (15,320) (11,065) (2,757) (10,144) (11,910) (9,461) (5,251) (5,660) (2,348) (0,720) -(8,056) (0,709) (7,754) Average 0,288 0,385 0,354 0,208 -0,551 0,137 0,864 Average 0,179 0,132 0,064 -0,018 -0,424 -0,013 0,603 (12,564) (20,064) (18,000) (9,036) -(14,807) _{(10,157) (16,271)} (5,222) (4,619) (2,164) -(0,536) -(7,631) -(0,670) (7,608)

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V. ROBUSTNESS CHECKS

As a robustness check to confirm the results found earlier, I have performed two sets of time-series regressions. One set of 42 equal weighted portfolios are tested on the same four asset-pricing models. Furthermore, a set of 42 Residual Volatility and Size sorted value-weighted portfolios are tested on the same models. The Residual Variance and Size sorted portfolios are formed monthly on the variance of the residuals from the Fama and French three-factor model.

Equal-Weighted Results

The equal weighted portfolios give a 1/n weight to each of the individual stocks each month where n is the total number of stock in the given month. The sorting method of the stocks in one of the 25 portfolios is exactly the same as in the earlier test. In table 7 the intercepts of each of the regressions are shown. The intercepts in this table are similar to the intercepts found in the main results, table 3. Specifically, significant intercepts are observed with the average long-short portfolio in the CAPM, three- and four-factor model. The five-factor model does not have a significant intercept with a t-stat of -1.378. The five-factor model is able to explain the variance of returns with equal-weighted, variance and size sorted portfolios.

Residual Volatility Results

Finally, regressions are performed on Residual Volatility and Size sorted Value-Weighted portfolios. These time-series regressions should also confirm the results find earlier. In table 8 the intercepts of these tests are presented. The results found in the main research are robust as these intercepts are also similar. Again, significant intercepts are observed with the average long-short portfolio in the CAPM, three- and four-factor model with t-stats of 3.225, 2.935 and 2.503 respectively. The five-factor does not have a significant intercept with a t-stat of -0.045.

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Table 7. Intercepts of the Four Asset Pricing Models - Equal Weighted Portfolio’s

These intercepts are the result of the regressions performed on four asset pricing models, CAPM, 3-factor model, 4-factor model and the 5-factor model. The t-statistics are in brackets underneath each of the intercepts. The portfolios tested are the intersections of 5 portfolios formed on size (Small 1 to Big 5) and 5 portfolios formed on the variance of daily returns (Low Variance 1 to High Variance 5), Average portfolio’s and long short portfolios (1-5). The portfolios are equal weighted. A 1/n weight is given to each of the individual stocks each month where n is the total number of stock in the given month.

Low Variance High Variance Low Variance High Variance 𝒂𝒊 CAPM 1 2 3 4 5 Average 1-5 𝒂𝒊 3 Factor Model 1 2 3 4 5 Average 1-5 Small 1 0,106 0,78 0,624 0,342 -0,118 0,46 0,396 Small 1 0,401 0,447 0,286 0,015 -0,446 0,141 0,453 (6,358) (5,820) (4,104) (1,918) -(0,433) (2,962) (1,726) (5,943) (5,155) (3,021) (0,135) -(2,204) (1,463) (2,298) 2 0,548 0,543 0,462 0,229 -0,561 0,244 0,713 2 0,294 0,237 0,175 -0,015 -0,698 -0,001 0,599 (5,936) (4,905) (3,974) (1,347) -(3,038) (2,325) (3,857) (4,582) (3,329) (2,369) -(0,189) -(6,243) -(0,028) (3,985) 3 0,413 0,375 0,394 0,261 -0,381 0,213 0,399 3 0,202 0,134 0,163 0,037 -0,424 0,022 0,233 (5,059) (4,190) (4,040) (2,407) -(2,497) (2,661) (2,226) (3,019) (1,972) (2,138) (0,453) -(4,278) (0,463) (1,641) 4 0,35 0,282 0,258 0,123 -0,33 0,136 0,285 4 0,181 0,11 0,094 -0,017 -0,319 0,01 0,107 (4,112) (3,426) (3,139) (1,420) -(2,553) (2,291) (1,584) (2,362) (1,490) (1,257) -(0,215) -(3,159) (0,193) (0,714) Big 5 0,208 0,151 0,091 0,003 -0,228 0,045 0,04 Big 5 0,106 0,087 0,024 -0,047 -0,148 0,004 -0,139 (2,542) (2,180) (1,394) (0,045) -(2,288) (1,303) (0,242) (1,479) (1,461) (0,421) -(0,830) -(1,640) (0,146) -(0,964) Average 0,439 0,426 0,366 0,191 -0,324 0,22 0,367 Average 0,237 0,203 0,148 -0,006 -0,407 0,035 0,25 (6,121) (5,513) (4,519) (2,158) -(2,259) (3,046) (3,225) (4,248) (3,645) (2,626) -(0,096) -(4,577) (0,920) (2,935) Low Variance High Variance Low Variance High Variance 𝒂_𝒊 4 Factor Model 1 2 3 4 5 Average 1-5 𝒂_𝒊 5 Factor Model 1 2 3 4 5 Average 1-5 Small 1 0,461 0,574 0,477 0,273 -0,053 0,346 0,124 Small 1 0,313 0,354 0,252 0,1 -0,083 0,187 0,001 (6,252) (6,725) (5,309) (2,589) -(0,274) (3,846) (0,646) (4,441) (4,290) (2,737) (0,885) -(0,418) (1,950) (0,005) 2 0,298 0,26 0,243 0,087 -0,477 0,082 0,385 2 0,161 0,068 0,045 -0,119 -0,431 -0,055 0,197 (4,529) (3,579) (3,263) (1,079) -(4,484) (1,770) (2,596) (2,743) (1,153) (0,719) -(1,686) -(4,136) -(1,386) (1,442) 3 0,189 0,178 0,19 0,101 -0,277 0,076 0,076 3 0,054 -0,005 -0,005 -0,113 -0,189 -0,051 -0,152 (2,759) (2,585) (2,455) (1,240) -(2,842) (1,596) (0,537) (0,874) -(0,077) -(0,079) -(1,606) -(2,007) -(1,257) -(1,168) 4 0,169 0,148 0,122 0,022 -0,177 0,057 -0,043 4 0,031 -0,066 -0,089 -0,142 -0,074 -0,068 -0,29 (2,161) (1,963) (1,598) (0,275) -(1,773) (1,126) -(0,289) (0,418) -(0,982) -(1,352) -(1,864) -(0,778) -(1,432) -(2,117) Big 5 0,084 0,063 0,026 -0,022 -0,085 0,013 -0,221 Big 5 -0,033 -0,095 -0,131 -0,155 0,104 -0,062 -0,532 (1,139) (1,047) (0,435) -(0,372) -(0,933) (0,423) -(1,500) -(0,464) -(1,814) -(2,467) -(2,792) (1,267) -(2,111) -(3,978) Average 0,24 0,245 0,212 0,092 -0,214 0,115 0,064 Average 0,105 0,051 0,015 -0,086 -0,134 -0,01 -0,155 (4,208) (4,339) (3,742) (1,631) -(2,569) (3,200) (2,503) (2,076) (1,138) (0,321) -(1,670) -(1,703) -(0,296) -(1,378)