A decomposition methodology of hedge portfolio factor premia – the case for idiosyncratic volatility Student:

A decomposition methodology of hedge portfolio factor premia

– the case for idiosyncratic volatility

Student: Alexandru Robu

Student number: 10621628

Programme: MSc Finance (MSc FIN)

Track: Asset Management

Supervisor: Dr Liang Zou

Statement of Originality

This document is written by Student Alexandru Robu who declares to take full responsibility for the contents of this

document.

I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the

text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for

the supervision of completion of the work, not for the contents.

A decomposition methodology of hedge portfolio factor premia

– the case for idiosyncratic volatility

Abstract

The aim of this paper is to extend the methodology developed by Hou and Loh (2016) to the decomposition of hedge portfolio based factor premia in the spirit of Fama and French. We consider the case of idiosyncratic volatility as the candidate explanatory variable and decompose the size and value premia into an idiosyncratic volatility dependent component and a clean risk premium. The methodology developed has broad applicability to current issues in asset pricing literature such as disentangling the perplexing dynamics exhibited by multifactor models (FF Five Factor) as well as provide a generalized framework for decomposing market anomalies that have diverging underlying assumptions. Although the empirical results provide insufficient evidence to reject the risk-factor quality of the SMB and HML factors, we observe an increased significance of the role idiosyncratic volatility plays in determining the risk factor premia that coincides with increasing trade volumes.

1. Introduction

The concept of building a model that should describe (be that even retroactively) the movement of stock prices has been at the forefront of economic and financial research for decades. With a seminal paper in 1952, Markowitz establishes a key insight that stands until present day as the cornerstone of modern asset pricing theory. In his paper, Markowitz postulates that idiosyncratic (firm-specific) volatility (risk) is easily diversifiable and thus for assuming exposure to this type of risk, investors are not to be entitled to compensation. The concept has been widely agreed upon and adopted in most of the groundbreaking work that followed it. Sharpe (1964) and Lintner (1965) develop the ubiquitous Capital Asset Pricing Model that

proposes an asset pricing model in which investors are solely compensated for holding

systematic risk. Many attempts have been made ever since to improve upon this framework. All-encompassing studies ranging from consumption-based models to random walks have been proposed; and the more resounding the claim, the stronger its opposition. Through his 1970 paper, Eugene Fama postulated his Efficient Market Hypothesis, achieving a divide so great between its proponents and its critics that the debate still continues today.

The lack of consensus in many of these debates often stems from the lack of common assumptions and frameworks. Sharpe (1964) developed the CAPM, the critics brought light to the joint hypothesis issue (Roll’s Critique, 1977). Fama and French came up with the size and value premium, its critics pointed to the lack of theoretical backing of such findings, regarding them purely as anomalies. It is virtually impossible to have agreement on any empirical finding as long as the assumptions underlying the methodology diverge. Therefore we posit that any attempt at identifying the “first principles” of the field and establish a common methodology that can be universally agreed upon and generally employed by practitioners is a worthwhile

endeavor. We realize an all-encompassing common framework is a lofty goal that can only be achieved through a prolonged advancement of the scientific process. Nevertheless, incipient efforts have been made at developing generalized frameworks that could aide in reconciling much of the conflicting evidence present in literature. Aabo et al. (2017) show that the interaction between market volatility, idiosyncratic volatility and R-squared is important in understanding the mixed results of previous volatility related literature. Furthermore, Hou and Loh (HL, 2016) build the foundation of a methodological framework aimed at decomposing asset pricing anomalies. In their paper, they focus on evaluating the ability of a number of proposed market anomalies to explain the idiosyncratic volatility puzzle.

The main objective of this paper is to build upon their work and extend the applicability of the HL (2016) methodology to the decomposition of hedge portfolio factor premia in the spirit of Fama and French (1992, 1993, 1996 and 2015). To that end we propose the idiosyncratic volatility premium as the main explanatory variable behind Size and Value premia (Hou and Loh acknowledge the possibility of IVOL to be used as an explanatory variable). The economic intuition justifying this choice (please see Section 2.2 for a detailed rationale) is certainly not without merit however we stress that the main goal of this exercise is to develop a framework that can be universally employed by practitioners in decomposing asset pricing anomalies. Thus

the choice of idiosyncratic volatility as the explanatory variable is in part motivated by its “functional form1_{” when compared to the Fama French factors.}

As far as empirically driven asset pricing factors go, the idiosyncratic volatility premium has its origins right at the cornerstone of asset pricing research through the empirical rejection of Markowitz’s (1952) hypothesis of easily diversifiable firm-specific risk requiring no premium. Merton (1987) postulates that, provided incomplete information, a scenario where all investors will be fully diversified is unlikely, therefore investors will demand a premium for holding idiosyncratic risk. It is essential to note that the theoretical frameworks outlined above imply either a null or positive risk premium associated with idiosyncratic risk. This makes the empirical discovery of Ang et al. (2006) of a robust negative relation between IV and stock returns especially perplexing.

Recently the efforts to debunk the IV enigma have accelerated with many developments in both, empirical as well as theoretical and methodological studies as the idiosyncratic volatility (IV) puzzle has served as a confluence point for different asset pricing theory developments. The subject has received increased attention in the past few years as technological advances in

financial markets have allowed for testing various new hypotheses related to market efficiency (increased liquidity, lower transaction costs, high trading volume) and its links to mispricing (i.e. information vs. noise trading hypothesis). Hou and Loh (2016) provide a framework aimed at evaluating the numerous candidate explanations for the IVOL puzzle, however it is the paper of Aabo et al. that prompted us to consider the relation between idiosyncratic volatility and the size and value premia. Aabo et al. (2016) establish a strong association between absolute

idiosyncratic volatility and mispricing and give an intuitive rationale as to the role that noise trading may play in the perpetuation of mispricing and hence the idiosyncratic volatility puzzle.

If, as the criticism that holds that the Fama-French “anomalies” are indeed the result of suboptimal behavior by market participants, that is portfolio allocation decisions are made on the base of unrelated to fundamental information or what could be fairly categorized as informed trading in the spirit of Roll (1988) then we propose that with the increased trading volume we should see an increased role of idiosyncratic volatility (proxy for noise trading as instructed in Aabo et al., 2016) in the explanation of the cross section of returns. If the Fama-French factors

______________________________________________________________________________

1 _{We use the construction “functional form” loosely to refer to the difference between how the Fama and French}

hold the quality of stand-alone risk factors for which an investor will require compensation (FF, 1996), we would expect that the premium attributed to them remain largely unaffected by idiosyncratic volatility. Conversely if these can be regarded as pure anomalies we would expect their premium to be reduced significantly.

Thus our actual research questions can be framed as following: Will the size and value premia be deemed obsolete when idiosyncratic volatility is accounted for? Is the time-variance of such risk premia reduced when idiosyncratic volatility is controlled for?

The contribution of this paper to the existing academic literature is twofold. The first, and arguably most important, stems from the methodology developed which provides a common framework for evaluating asset pricing anomalies, even if some of the underlying assumptions differ. Furthermore with models in recent academic research containing an increasing number of factors, a new issue has come to the forefront through the overlapping effects of factors. A prime example of this issue is described in FF (2015) where through the introduction of the profitability (RMW) and investment (CMA) factors, the value factor has become redundant (insignificant in the five factor regression). Fama and French attempted to investigate further into what exactly caused this result by regressing the HML factor on the others. The coefficients were

counterintuitive which the authors deemed the result of interactions between the factors. They qualified this explanation unsatisfactory and suggested that further research is needed to

disentangle these effects. We propose that our model could be a good starting point for such an exercise. The second contribution stems from the actual empirical investigation employed in this paper. The answer to our research question should illuminate further on whether the Fama French proposed size and value factors are truly risk premia for which investors require compensation or purely anomalies.

The rest of the paper is structured as follows: the next section will provide the theoretical background necessary to understand the economic rationale and methodological choices made in employed in the paper. Section 3 will go into detail on the methodology employed, providing a detailed account into how our augmentations extend the applicability of the previous framework. The fourth section will outline the results and other observations that transpired in our testing. Section 5 will focus on discussing the methodological findings and empirical results, how they reconcile with existing literature as well as suggest possible paths for further research. Finally, section 6 will add our final thoughts and concluding remarks.

2. Theoretical Framework

This section aims to provide the theoretical background needed to justify the claims made in this paper. Furthermore, we hope to give some additional insight into the natural evolution of scientific discourse as it relates to the topics at hand and to the place in literature that this type of exercise strives to occupy.

We focus on understanding the purpose of asset pricing research as well as the causes that lead to such opposing views on some of the main issues of the field. As our paper addresses the disagreement concerning the quality of the Fama-French factors as either risk factors or anomalies, we investigate the narrative of their claims and the main dissenting views that

accompany them. Moreover, we aim to provide some theoretical background on the idiosyncratic volatility puzzle and its link to the noise trading hypothesis. Finally, we will conclude the section with an intuitive account of what we hope to achieve through this analysis given the established theoretical background.

2.1 Aim of asset pricing research

In order to better understand the reasons for discord and the prevalence of such polarizing answers to some of asset pricing’s most important questions, it is important we give a definition of what the discipline aims to achieve. In essence, it can be reduced to a theory of choice with incomplete information or, in the words of Kenneth Arrow (1951) “a realistic theory explaining

how individuals choose among alternate courses of action when the consequences of their actions are incompletely known to them... Risk and the human reactions to it have been called upon to explain everything...; according to Professor Frank Knight, even human consciousness itself would disappear in the absence of uncertainty."

Within that description, two types of applications can be defined: positive (attempting to understand the world we see and elaborate a model that accurately fits this information) or normative (elucidate inefficiencies within our world – i.e. identify where the real world deviates from theory thus resulting in mispricing). It is understandable then that if one were to subscribe to one of these goals in particular that may lead to disagreement on even the most fundamental

questions.

We define anomalies as empirical results that are inconsistent with a given asset pricing model’s predictions. Thus they suggest the presence of market inefficiency (consistent with the normative view) or rather an inadequacy in the specification of the asset pricing model.

2.2 Fama and French – main concepts and critique

This section describes the asset pricing framework outlined by Fama and French, its underlying principles and their motivation. It seeks to place in context what they want to achieve as well as their self-acknowledged weak points, the reliability of their performance measures and critiques.

In their 1992 paper, Fama and French assert that stock risks are multidimensional. One dimension is proxied by size, ME, while the other is proxied by the ratio of book value to market value, BE/ME. They (1993 and 1996) argue that the size and value of a stock are additional risk factors that together with the market premium explain the cross-sectional variation of returns.

Numerous dissenting voices have come to oppose the claims of Fama and French. The criticism ranges from “data snooping” and “survivorship bias” (Black, 1993) to questions about the validity of their methodology and model-testing techniques. Lewellen and Nagel (2006) claim that the level of fit achieved by the Fama-French model results from regarding the slopes in the second stage Fama-MacBeth regression as free parameters as well as statistically

unreliable evaluation methods. Even though data snooping and survivorship bias are valid concerns with regards to the validity of the model’s claims, they do not reject the fundamental way of approaching asset pricing. On the other hand when the econometric validity of the slope coefficients are put into question or the very method one uses to evaluate the capability of a model is subject to disagreement, it is no wonder then that we see a high incidence of conflicting results in finance research. This paper strives to aid the effort towards a set of generally agreed upon tools with which to conduct sound academic investigation in finance.

It is interesting to note that, even though Fama could argue that none of his critics has succeeded in elaborating a critique that is not at least as fickle as his own claims, the new factors that were introduced to complete the three-factor model have benefited from a much improved

level of theoretical backing. By their own admission, Fama and French (1993) lament the somewhat arbitrary nature of their first discoveries by noting that the chosen factors represent merely “variables that have no special standing in asset pricing theory show reliable power to explain the cross section of average returns”. What is more, Malagon et al. (2015) argue that profitability and investment growth are strongly linked with idiosyncratic risk. This we argue provides an added level of relevance to our investigation.

Notwithstanding the long battles fought to defend their discoveries, the doubts their equally esteemed colleagues expressed were not lost on them. If anything, it only goes to prove that the complexity of this question makes it so that either side can be defended ably. It is

through this that the importance of mending these concerns reveals itself to be an endeavor of the utmost virtue.

2.3 Idiosyncratic volatility puzzle and link to the Noise Trading Hypothesis

To understand the idiosyncratic volatility puzzle, we must take note of Merton’s (1987) claim that provided incomplete information a scenario where all investors will be fully

diversified is unlikely. This implies that investors will demand a premium for holding idiosyncratic risk, therefore effectively rejecting Markowitz’s (1952) hypothesis of easily

diversifiable firm-specific risk requiring no premium. The puzzle however only emerges with the empirical discovery of Ang et al. (2006) of a robust negative relation between IV and stock returns, the exact opposite of what was previously theorized. This is yet another example of conflicting ideas at the forefront of a field that without a common set of assumptions and tools may remain unresolved.

Previous research has established the IV effect is rather robust. The effect persists when controlling for numerous explanatory variables (such as liquidity risk, trading volume and transaction costs) as well as various firm characteristics that have been linked to stock returns. Furthermore, the negative IV-return relation has been confirmed in international markets (Ang et al., 2009; Pukthuanthong-Le and Visaltanachoti, 2009; Nartea et al., 2011).

Herskovic et al. (2016) find strong evidence for a factor structure among stocks’ idiosyncratic volatility. They note that a strong negative relation is present in the variation of cross-sectional returns which they attribute to the link between the IVOL factor and household

consumption risk. This view presents an alternative to our own approach in that it focuses on a market wide measure of idiosyncratic volatility rather than an individual stock approach.

Moreover, Herskovic et al. (2014) demonstrate that the comovement in idiosyncratic volatility can only be partially explained by missing factors. They show that even when

employing a 5-factor PCA (refer to Appendix B.b. for background), where 98% of the variation in raw return series has been explained, the residual variances of individual stocks remain highly correlated. Another interesting observation is that despite the impressive ability of the three factor model to explain the co-movement in returns, in a typical year only 9% of the average volatility is accounted for by the three factors, with idiosyncratic volatility capturing the

remaining 91% (Herskovic et al., 2016). This could be further reduced using the conclusions of this paper.

The concept of noise trading was first brought to the forefront through attempts at justifying one of the most puzzling empirical observations in financial research – the

inexplicably high trading volumes. Kyle (1985) and Roll (1986) discuss the issue through the lens of informed and uninformed investing. In the process, they define the notion of noise trading as the trading performed by “naïve” speculators that trade on pseudo information (or noise) as if it constituted fundamental information with a direct link to firm valuation (as echoed in Wang, 2009).

Since our aim is for a general decomposition methodology, it is important to note that idiosyncratic volatility itself has a joint-hypothesis concern embedded into its definition. You can only have idiosyncratic volatility if there is a clear distinction between that which is

systematic/market wide risk that cannot be diversified away and all the rest. In accordance with that concept, idiosyncratic volatility may be subject to a number of different definitions and interpretations based on the author’s belief of what constitutes un-diversifiable risk, i.e. to what asset pricing framework they subscribe. For that reason we must allow for the possibility that even the most generally agreed upon frameworks can still be subject to disagreement.

Nonetheless, for the purposes of this academic exercise we assume that a certain starting level of consensus is required to be able to build a common framework (A fitting anecdote would be “it is difficult to convince, using mathematical proof , the person who rejects multiplication”). Thus, since our proposed example discusses the concept of idiosyncratic risk, it is necessary that our framework allows for the concept to exist in the first place. Where the distinction comes into

effect is the definition of the IVOL measure. The core definition describes idiosyncratic volatility as the part of variation in returns that fails to be accounted for by the particular asset pricing model used. It is clear to see how this may result in disagreement as it inherits all those of the asset pricing model itself, as well as some of its own. Our proposition is to bring both

requirements down to their core principles: if a model for time-varying firm returns is required use the CAPM (deviates from Ang et al. (2006) who employ FF 3 Factor model).

2.4 Intuition for our investigation

This paper aims to leverage the newly acquired knowledge regarding the relation between idiosyncratic volatility and mispricing (and hence noise trading) to reevaluate the disagreement between the belief that factors such as size and book-to-market represent risk factors, and hence should require a positive return, and that they are purely “anomalies”, with no theoretical backing that have been identified solely as a result of “data snooping”.

The reason this debate is still ongoing is because of the degree of difficulty entailed by testing any of these claims definitively. A particular difficulty in disproving Fama and French’s claim is coming up with a valid explanation for the failure of CAPM, in that there may be a multitude of factors that work together to contribute to the flatness of the Security Market Line implied by the CAPM.

Intuitively, we propose that idiosyncratic volatility is a proxy for all the unknown variables that affect stock returns other than the market and use it to distinguish between the “true effect” of the candidate variable and what may be the effect of non-standard beliefs and behaviors. Thus our aim is not to propose new risk factors or to identify where the CAPM fails, rather we hope to provide a methodology that will help in deciding whether the existing

anomalies that have been as of yet identified in literature or any future ones that will be discovered are truly risk factors or purely anomalies that can be priced out of the market.

We posit that the idiosyncratic volatility premium is most likely anomalous as

diversification should price out any idiosyncratic risk. Therefore we propose that whatever risk premium idiosyncratic volatility seems to hold, it is the result of anomalous behavior (non-standard beliefs/ preferences, inattention, disagreement etc). Consequently the idiosyncratic

volatility risk premium should capture a large part of the anomalous effects of irrational

behaviors/ unknown variables and beliefs that go into determining stock returns, including those in hedge portfolios such as the HML and SMB factors. Thus we propose that by separating the part of each of these factors that is due to mispricing (irrational/non-standard behavior, distorted beliefs/preferences or any other deviations from theory), what will remain (if it will) will be a true risk factor.

3. Data and Methodology

For our main methodological framework we follow the general structure described by Hou and Loh (2016) for decomposing asset pricing anomalies. Although their paper focuses on idiosyncratic volatility as an anomaly and uses other candidate variables to explain it, they show that the methodology can be applied in the opposite direction as well. This is given as a

possibility but it is left largely unexplored, thus our paper aims to understand and extend the applicability of their framework to hedge portfolio factors and other types of asset pricing anomalies.

An obvious question regarding the added value of elaborating such a methodology is whether by just including an extra regression factor that proxies for the effect in question would not achieve comparable results – i.e. that a multifactor OLS yields equivalent results to our methodology. From this perspective, the decomposition adds considerable value in two ways. First it allows us to isolate the effect of one particular variable on another without interference from additional factors. This is particularly valuable as a need has been identified in various papers of further investigation into the inter-factor dynamics in asset pricing models (Fama and French, 2015). Moreover it provides a fairly precise, statistically testable value for the percentage of the anomaly that is explained by the candidate.

We make use of Kenneth French’s Database at Dartmouth to extract the Fama French factors, sorted portfolios, risk free rates and market proxies for both daily and monthly

frequencies. We retrieve daily returns from the CRSP database (share codes 10 and 11) from July 1963 to December 2012. This ensures comparability not only with Hou and Loh (2016) but also FF (2015), Herskovic et al. (2016) as well as Aabo et al. (2017). Since the methodology

small sample of stocks, namely the main constituents of the Dow Jones Industrial Average over the period in question. We realize this is not representative as the size and value effects seem to have a diminishing effect when the most extreme observations are eliminated (Knez and Ready, 1997). Nevertheless, since the empirical investigation constitutes only a secondary goal of this paper, we hold that a reduced sample preserves the relevant characteristics of the population but also allows us to have a clear picture of how every methodological choice plays out which gives immeasurable added value to the process of ensuring that all the steps are correctly implemented. Our expectations between the reduced (Dow-Jones stocks) and expanded sample (CRSP universe share code 10 and 11) are equivalent with the note that since the size and value effects are more pronounced in the expanded sample, we would expect the explanatory power of IVOL to be somewhat higher. Alternatively, the Dow Jones sample can be seen as an “acid test” for our hypothesis, if the size and value premiums are explained in the DJIA stocks sample by the IVOL puzzle, it is likely the effect will persist to the larger sample (with more extreme observations). To that end, henceforth our discussion will focus exclusively on the reduced sample to facilitate a clearer understanding of the steps involved.

We begin by constructing the idiosyncratic volatility measure. Hou and Loh (2016) follow Ang, Hodrick, Xing, and Zhang (2006) for constructing the IVOL measure in that they compute it as the as the standard deviation of the residuals from a regression of daily stock returns in month t −1 on the Fama and French (1993) factors. Since the purpose of this paper rests on not taking any sides on whether variables such as the Size and Value effect are truly risk factors or anomalies we must allow that everything that came after the CAPM to fall in either the risk factor category or the anomaly, therefore we will deviate slightly from Ang, Hodrick, Xing, and Zhang (2006) and use CAPM to build the IVOL measure.

Consequently, we perform a time series regression of daily excess returns (ri = Ri - rf) on the value-weighted market excess return (as computed on Ken French’s Database). Consistent with Ang et al. (2006) we compute per month idiosyncratic volatility (IVOL) as the standard deviation of the residuals from a regression of daily stock returns in month t-1 on the market excess return. We increase strictness and require at least 15 daily returns to compute the measure2.

______________________________________________________________________________

2 _{Hou and Loh (2016) show that there are negligible differences when only 10 observations are required or even}

(1)

(2)

Next, we calculate the size and value premiums using Fama MacBeth (1976) regressions. As the size and value factors are hedge portfolio factors, that are general for the entire market (as opposed to the security specific anomalies present in Hou and Loh such as Maxret, Skew, Momnetum etc), the methodology of Hou and Loh (2016) requires substantial adjustment to make it possible to accommodate the structure of the factors concerned. This extension and the observations that stem from it are the main contributions of this paper.

The methodology is aimed at accommodating two important characteristics: it is first and foremost aimed at decomposing factor premiums and it is attempting to do so by making use of a variable (IVOL) that loses its informational value in sorted portfolios3. We argue that this is largely the reason why the idiosyncratic volatility effect may be lost in sorted portfolio approach employed by FF and in much of the rest of the literature.

Consequently, we must begin by computing the size and value premiums that accompany individual stocks, rather than resort to the much more accessible approach of using sorted

portfolios. We note that by doing so concerns of errors in variables (Blume, 1970) and other possible measurement errors are left unaddressed. Finding an appropriate way to deal with these issues without using sorted portfolios is beyond the scope of this paper, however this reinforces the sentiment that using sorted portfolios is a limiting approach of analysis and the efforts aimed at discovering an alternative solution should not yet be abandoned.

To construct the value premium, Fama-MacBeth two-stage regression procedure is employed. Using monthly returns, we first perform a time-series regression of each individual stock’s return that, at first, solely includes the size/value factor. The inclusion of multiple variables involves additional complications which we postpone for now.

Cochrane (2001) offers two equally valid ways to perform Fama-MacBeth two-step

______________________________________________________________________________

3 _{Intuitively, idiosyncratic variance will cancel out to a certain extent when stocks are groped in portfolios.}

regression analysis: one that computes a full-period beta for the 1st step regression and one that computes rolling betas, using a 5-year moving regression window. For robustness purposes, both methods have been employed; both have yielded very similar result in terms of the premiums associated with the market, size and value risk factors. The important difference stems from the intermediary step of having a panel of first-pass betas associated with each security. Having this “functional form” is tremendously important in decomposing the risk premia, therefore we proceed with the first step of the Fama-MacBeth regression by performing time-series regressions of monthly individual stock returns on one of the Fama French factors.

(3)

(4)

We thus obtain a panel of _{values for all the individual stocks in our sample starting}

with July 1968 (the first 60 months are lost in estimating the rolling betas).

Next, we perform a cross-sectional regression for each point in time (t) on the betas obtained in the first step in order to compute the size premium. This is our second major

adjustment to the Hou-Loh methodology. In the case considered in their paper, they adjust Fama-MacBeth procedure so that they consider the estimation of the IVOL measure as the first pass in the Fama Macbeth regression. In our case, a standard Fama-MacBeth procedure is followed, so we proceed with the 2nd step – a cross-sectional regression on the betas obtained in the time-series.

(5)

(6)

At this point we have computed monthly values for the size and value premiums. The values are consistent with previous work (for the restricted sample, there are more significant differences, which is to be expected).

dependent variable (in panel form4) on the value of the idiosyncratic volatility measure computed previously. At this stage we only focus on univariate cross-sectional regressions where the dependent variable is one of the Fama-French factors and the explanatory variable is the idiosyncratic volatility measure (acting as a proxy for noise trading).

(7)

(8)

As prefaced above, this where the 3rd major adjustment to the Hou and Loh methodology comes into play. In their main analysis, the IVOL measure is a LHS variable and candidate explanations are proposed as explanatory variables. Naturally in our case, the puzzle at hand is no longer idiosyncratic volatility but rather the size and value premiums, whereas IVOL takes the role of the explanatory variable.

Finally we are able to decompose the 2nd pass regression coefficients associated with the size and value premia as follows:

Where _{represents the part of the SMB factor-return that is accounted for by}

idiosyncratic volatility and hence noise trading. Whereas represents the part of the value factor that remains and hence this should be able to give us the “cleaned-up” value premium.

The ratio _{measure the fraction of the size premium (or other considered}

puzzle) that is explained by the candidate explanatory variable, in our case – idiosyncratic volatility. Conversely _{represents the fraction of the puzzle that remains unexplained}

by the candidate variable.

______________________________________________________________________________

4 _{This is where the choice of using 5-year rolling regressions for the 1}st _{step in the Fama-MacBeth procedure}

comes into play. Without this a cross-sectional regression for the entire period is needed which is subject to loss of information due to time-varying volatility.

The means and standard errors of these ratios are unattainable in closed form. However, common practice (also employed by Hou and Loh, 2015) is to make use of a Taylor series expansion approach to find appropriate approximations. We present here only the final form equations for the mean and variance of the type of ratios presented above. Please refer to Casella and Berger (2001) for more background. For the clean coefficient ratio the following applies:

The estimated means and variances over the period in question (T months) are as follows:

To conclude our analysis we test the coefficients using Student-T tests for the ratios and analyze in more detailed the obtained coefficients by performing comparisons with the initial values. Compute the over-time variance to see whether our supposition stating that eliminating the effect of noise trading will lead to less time-varying risk premia. In the next section we go into more detail into the methodological and empirical observations that transpired as a result of this analysis.

This can be analogously applied to all the other FF factors: RMW, CMA as well as other anomalies documented in asset pricing literature: MOM (Carhart); SKD (Harvey and Siddique, 2000) etc. The tremendous advantage in doing so is that between the Hou and Loh (2016) and this current paper, the decomposition methodology has been expanded to accommodate a wide range of explanatory variables.

4. Results

4.1 Proposed empirical applications

Since the main concern of this paper was the methodology development, most of our robustness tests, due diligence and additional investigation were focused on methodological concerns. Consequently, due to time and space limitations we have chosen to constrict the empirical aspect of the paper to certain key findings and propose a extension to this paper which should include the following: an investigation of all the proposed Fama-French factors and other significant hedge portfolio derived factors in literature; multiple specifications of the models where the number of factors included is varied; several break-downs of the sample period to present a clearer picture of the time variability of the clean coefficient as opposed to the original; various sample selection methods should be investigated as well as country comparisons (to be able to see whether the effect intensifies where mispricing is a more prevalent issue). We consider that the topics mentioned in this paragraph should be included in the empirical extension of this paper in order to obtain a full picture of the performance of the framework as well as its implications for the existing asset pricing knowledge.

4.2 Actual results

This section aims to first provide a short overview of the results at each stage in our methodology as well answer our proposed research questions.

In the methodology we described our computational method for the idiosyncratic volatility measure. Consistent with previous literature, the idiosyncratic volatility is relatively constant in periods of financial stability with values hovering around 1.5% monthly. In Figure 1 we can easily observe that, as was expected, there are spikes in volatility that corresponding to periods of financial unrest. This was identified and discussed at length by Herskovic et al. (2016) as it is consistent with their claim regarding the co-movement of IV with general market

Figure 1 Idiosyncratic volatility July 1968 – December 2012

In relation to the other variables involved in our analysis we note that IVOL

(idiosyncratic volatility) has a negative correlation (Table 1) with the market and size premia (-0.17 and -0.09 respectively) and a positive, albeit weak, correlation with the value factor (0.07). This deviates only slightly in magnitude (not in sign) from previous studies however this is to be expected given the reduced sample employed.

Table 1

Correlations between factors: July 1968 – December 2012

IVOL Mkt-RF SMB HML

IVOL 1.00 -0.17 -0.09 0.07 Mkt-RF -0.17 1.00 0.27 -0.29 SMB -0.09 0.27 1.00 -0.10 HML 0.07 -0.29 -0.10 1.00

At the 1st step of the Fama-MacBeth procedure we decide that the more advantageous functional form is obtained by using rolling regressions (5-year window). We investigate whether the choice has impact on our results. The average size premium (as obtained in the second step of Fama MacBeth) equals 0.55% when the whole period is employed and only drops to 0.52% when rolling regressions are performed for the first step. As the rolling regression approach will reflect the beta of the firm more accurately at any given point in time we would expect that the volatility of the risk premium be reduced when the rolling window approach is

employed. This is indeed confirmed by our results as the variance drops by 26%. A similar dynamic is observed for the value premium. Therefore we conclude that the choice of using either the a full period estimation or a rolling-window regression in order to estimate the first-step coefficients in the Fama-MacBeth procedure has no real impact and thus the choice can be made based on the each individual case. At a later stage we will encounter a need for a cross-sectional regression of the time-series betas on the idiosyncratic volatility measure. For this purpose, the rolling window betas are better suited hence we choose to proceed as such.

For informative purposes we present descriptive statistics of the time-series regression coefficients for our reduced sample. This should give some background into the characteristics of the firms in our sample and help us establish baseline expectations for their behavior.

Table 2

First-pass coefficients descriptive statistics DJIA stocks

Beta_mkt Beta_smb Beta_hml

_{Mean 0.99 0.23 -0.35} Median 0.94 0.14 -0.34 Standard Deviation 0.323 0.399 0.456 Minimum 0.49 -0.51 -1.55 Maximum 1.75 0.95 0.61 Count 31 31 31

We direct the reader to return Table 2 if some additional mediation is needed in order to comprehend the later dynamics resulting from our analysis. For the moment we will only note the relative volatilities and range of the premia. As can be observed from Table 2 the volatility of the value premium is about 1.5 times the market beta volatility in our sample. We hold that this is relevant as a sample that would not exhibit a sufficient range will be unsuited for our purposes as it will likely understate our end results.

We continue with the cross-sectional regression, as described in Fama-Macbeth, to obtain the size and value premias over time. As mentioned earlier the values were consistent with previous research considering our reduced sample. As was previously observed (Cochrane, 2001), the size (Figure 2) and value (Figure 3) are highly volatile one month to month basis however the long-term effect remaining mostly constant around the mean (Table 4).

Table 3

Premium mean monthly value by first stage Fama-MacBeth computation method July 1968 - December 2012

Coefficient SMB HML

0.516% -0.064%

Percentage diff. -6.02% -26.25%

Figure 2. Size premium for period July 1968 – December 2012

-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

Size Premium

Figure 3. Value premium for period July 1968 – December 2012

As the next step of our analysis involves a cross-sectional regression of the time-series betas obtained in the earlier step on their corresponding idiosyncratic volatilities as part of our original contribution there are virtually no sources for comparison. We note that most

coefficients are significant5 however, at this stage, largely uninformative.

Finally, we compute the correlations, co-variances and variances necessary to compute the values to test our hypotheses (as described in equations 13 and 14).

Table 4

Premium mean July 1968 - December 2012

Coefficient SMB HML 0.458% -0.061%

Confirmed using an F-Test. -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

Value Premium

From Table 4 we observe the coefficient for the cleaned-up factor premiums which, for our reduced sample, stand at monthly values of 0.458% for the size premium and -0.061 for the value premium.

The percentage of the size premium that is explained by the idiosyncratic volatility puzzle is 11.32% when the entire period is taken into account. We consider this insufficient evidence to conclude that the size premium is the result of noise trading and thus entirely anomalous. Such a result was largely expected, given the overwhelming number of attempts made to disprove the validity of these factors. The evidence seems to be significantly weaker as far as the value premium is concerned, with only a 5.32% of the puzzle being explained by the candidate variable.

A more interesting observation arises however when we split the period in two periods: July 1968 – December 1997 and January 1998 – December 2012. The cutoff point represents a point where trading volumes began to increase dramatically as a result of technological

advancement and the rise of algorithmic trading. As can be seen from Table 5, for both

coefficients, the percentage that can be explained by the cross-sectional variation in idiosyncratic volatility increases significantly (to 27.7% and 13.4% for the size and value premiums

respectively).

Table 5

Percentage of premium explained by IVOL

Period SMB HML

1968-2012 11.32% 5.32% 1998-2012 27.73% 13.44%

This can also be observed when plotting the original and the clean versions of the size premium (Figure 4). Furthermore, from the same figure we can observe that as was proposed by our second hypothesis, the variance of the premiums was also reduced over the period. The exact values for the variance of the premia are mentioned in Table 6.

Figure 4

Size premium ( plotted against the clean size premium ( ) as computed using our methodology July 1968 - December 2012

Please see Appendix A for the Equivalent Value Premium graph.

Table 6

Premium variance July 1968 - December 2012

Coefficient SMB HML

0.145% 0.099%

Percentage diff. -29.10% -22.25%

With this observation, we conclude our results section. As we mentioned in Section 4.1, we constrict our results reporting to only those that are essential for demonstrating the

applicability of our methodology in our proposed setting. The next section will focus on the alternative methodological choices we have explored as part of our robustness checks.

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

Size Premium

4.3 Robustness checks

The methodology outlined in Section 3 has been the result of many different attempts to arrive at a valid decomposition of hedge portfolio derived factors. By exhausting all the other alternatives we can be fairly confident of the robustness of our methodology and results.

Apart from the investigating the alternative to every of the methodological choices outlined in Section 3, we have explored several alternatives to our chosen path.

First, although we have chosen to apply our analysis to individual stock returns (our main concern being the loss of information – consistent with Harvey and Siddique, 2000 and Hou and Loh, 2016) we have also entertained the idea of testing using sorted portfolios. We employed the 25 Size-B/M sorted portfolios and attempted a simple decomposition by following a

mathematical expansion of the each individual coefficient. The loss of information however was clearly visible as almost none of the 25 coefficients were significantly different than zero. A three-way sort using Size-B/M-IVOL was also attempted with unsatisfactory results.

To explore a base example that looks at a decomposition methodology that yields

equivalent results we have explored an orthogonal decomposition methodology of the FF factors (Appendix 1). This methodology does not propose any additional factors and in our research we prove empirically that the results of such a decomposition methodology are equivalent, it does provide a good benchmark to judge the methodological soundness of the model.

Moreover we considered the baseline of just including the factors in a multifactor

regression. Hou and Loh (2016) provide a more detailed account, the essence being of course the inability to separate between the direct effect and that which stems from the interaction of the effects of the other variables.

5. Discussion

This section reconciles the results with our expectations as well as previous research. It is essential that the insights we have acquired are thoroughly analyzed in relation to other similar or opposing results in literature.

5.1 Methodological findings

As we stressed from the beginning of the paper, the main purpose of this work was to extend the applicability of the Hou and Loh (2015) methodology to allow for its application in a wider range of settings and with anomalies of different origin and functional form. To that end several modifications have been made in order to accommodate the diverging characteristics of the variables employed.

We argue that, overall, this adjustment has been successful. We arrived at an end result without having to make use of any “unorthodox” assumptions. The steps follow a natural sequence of econometrically sound, generally accepted techniques. The loss of information is minimal, given that individual stock regressions have been used (with the exception of the first 5 years lost from the time-series regression which could not be avoided).

We propose that the end result is a slightly more generalized version of the Hou and Loh methodology. While their approach applied to one particular case, through this exercise we were able to identify the points in the methodology where adjustments are necessary in order to apply it in different settings. By exploring the feasibility of these choices we provide a roadmap that can be applied in further research as well as an additional particular case – the decomposition of a hedge portfolio based risk factor using a volatility-based measure.

A final observation that was left unaddressed in Hou and Loh is the possibility of having a negative percentage of the variable return explained, or a percentage of over 100%. We believe this was not addressed as it was not observed in the particular case of Hou and Loh. This is to be expected due to the more similar nature of the factors employed (i.e. both the dependent variable and the candidate explanations had similar origins in terms of underlying assumptions and computational procedure employed in order to obtain them). In our case we also don’t find this of particular great impact as it seems to be limited to only a few observations that on average balance out. Nevertheless a regression method that imposes a lower and upper bound may add value in future endeavors. This could also be developed in such a way that it complies with the instructions of Lewellen et al. (2010) of zero-beta restriction which they propose as a way of improving on the Fama-MacBeth framework.

5.2 Implications for Fama and French factor premia

We certainly cannot claim to have disproved any of the assertions made by Fama and French. However, from our observation of the increased explanatory power of idiosyncratic volatility in the second part of our timeframe we can conclude that mispricing and noise trading could increase both the absolute value and the volatility of the Fama French risk premia.

Apart from the methodology, possibly the most valuable insight for future research is the decreased volatility of the risk premia. As many recent studies have been concerned with the investigation of time-varying risk premia (e.g. Gagliardini et al., 2016) we hope that the new gained insights into the role that idiosyncratic volatility plays can help with further

developments.

5.3 Topics for further research

An appropriate way to address the measurement error, errors in variables issues that does not imply the use of sorted portfolios has been a topic discussion before. Even though, the sorted portfolio approach has become the standard, we argue that there is value in elaborating a solution that does not “diversify” away possible omitted effects. To have a decomposition methodology similar to the one outlined in this paper, it is imperative that a way to address EIV concerns without sorted portfolios is elaborated.

As proposed at the outset, an important application of this methodology is related to the actual work of Fama and French. With the introduction of their five-factor model (FF, 2015), the issue of factors with overlapping effects which result in one or multiple factors being deemed redundant has increased considerably. In their 2015 paper, FF discover that HML loses a lot of informational value once the RMW (profitability) and CMA (investment) factors are introduced. The way they evaluate this is by a series of regression where each of the RHS factors are moved to the LHS of the regression (as dependent variables) and regressed on the rest of the factors. Fama and French recognize that such a procedure leads to puzzling coefficients which they posit is due in large part to the interactions between the factors. They suggest that this explanation is unsatisfactory and propose that further research into the inter-factor dynamics is essential. We

propose that this model of decomposing factor premia is a great framework to investigate such puzzling observations (i.e. a study where the the five Fama French factors take turns as the puzzle and the candidate variables in our model, both individually and in a multifactor setting).

Finally in our investigation of the idiosyncratic volatility puzzle, we find the link to the noise trading hypothesis still fragile. A study that investigates this effect in markets where otherwise equal noise trading is less common could be very informative of whether the puzzle can truly be linked to noise trading.

6. Conclusion

The aim of this paper has been the development of a methodology that can be

successfully used in the decomposition of asset pricing risk-factors/anomalies that are commonly encountered in asset pricing models. The methodology described is an extension of the work of Hou and Loh (2016). To that end, we leverage their work to allow for a wider range of anomalies and explanatory variables to be adequately addressed within our framework. We consider the case for decomposing the Fama-French factors using a measure of idiosyncratic volatility in order to evaluate whether the increase in noise trading activity has considerably reduced the premium attributed to each factor and further elucidate on the “risk factor vs. anomaly” debate. The choice of the dependent and explanatory variables has been carefully considered to meet our requirements for the extension of the methodology as well as be backed by a strong economic rationale and address a gap in existing literature.

The paper contributes to the existing literature by providing a more generalized

methodology for decomposing asset pricing factors. This is of particular relevance as a need has been identified in recent literature for a framework for untangling the perplexing coefficient dynamics in newly developed multifactor models (Fama and French, 2015). Moreover the actual empirical investigation provides additional backing to the claims that the FF factors are not purely market anomalies. In light of our results, we cannot say conclusively whether size and value are stand-alone risk factors that justify a premium however, it is becoming increasingly clear that they capture an effect that persists in equity markets and should be accounted for.

The percentage of the premium explained by idiosyncratic volatility stand at 11.32% and 5.32% respectively for the size and value coefficients. This was insufficient to conclude that

either the size or value premia are the result of noise trading and thus entirely anomalous. Such a result was largely expected, given the overwhelming number of attempts made to disprove the validity of these factors.

The period split seems to indicate that as trading volume increases the effect of noise trading becomes significantly higher. This supports Black’s (1986) conjecture that “what is needed for a liquid market causes prices to be less efficient” and informs us that periods of trading predominantly on pseudo-information can lead to many of the well-established dynamics observed in financial markets to deviate from previous expectations.

We end by expressing our hope that the above described methodology will be used to elucidate other points of discord within finance research and contribute to the larger effort towards a unified set of first principles of asset pricing.

Appendix

A. Figure A1

Value premium ( plotted against the clean size premium ( ) as computed using our methodology July 1968 - December 2012

B. Alternative Decompositions a. Orthogonal decomposition

Let be , the HML risk that cannot be accounted for by the market. This is true insofar as: Then, -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

Value Premium

Where is the leftover effect of the SMB risk factor. This is orthogonalized across both the other 2 dimensions. This can be analogously redone for multiple factors. The scope of the paper will prompt us to also limit the analysis to the 3-Factor Model factors.

b. Principal component analysis

PCA is a method for constructing factors which are uncorrelated with each other and which allow us to maximize when running regressions on the target portfolios.

Following Jolliffe, PCA, the central idea of principal component analysis (PCA) is to reduce the dimensionality of a data set consisting of a large number of interrelated variables, while retaining as much as possible of the variation present in the data set. This is achieved by transforming to a new set of variables, the principal components (PCs), which are uncorrelated, and which are ordered so that the first few retain most of the variation present in all of the original variables.

Employing this methodology will provide a “simulated benchmark” against which to compare the performance of the other 2 decomposition methodologies as well as a parsimonious framework to guide the intuition of decomposing risk factors. PCA’s relatively simple

construction also allows us to reverse engineer the process by creating synthetic factors for the Mkt, SMB and HML. If we take linear combinations of the three PCA factors, we can preserve their ability to explain returns (the R^2and Alpha are unchanged) while removing the other PCA related constraints. This allows us to create nearly identical factors to the Fama-French factors and thus providing further insights into how the factors could be decomposed.

A short, naïve use of this methodology is used to confirm our initial prediction. The variance of the residuals remains informative even after accounting for over 90% of returns using principal components (the results were used purely for intuition purposes, for a more detailed exercise into the specifics of PCA in relation to the idiosyncratic volatility puzzle please see (Hu and Tsay, 2014).

References

Aabo, T., Pantzalis, C., Park, J., 2017. Idiosyncratic volatility: An indicator of noise trading? –– Journal of Banking and Finance

Ang, A. , Hodrick, R.J. , Xing, Y. , Zhang, X. , 2006. The cross-section of volatil- ity and expected returns. Journal of Finance 61, 259–299.

Black, F., 1986. Noise. Journal of Finance 41, 529–543.

Blume, E., 1973. Portfolio theory: a step toward its practical application. Journal of Business, 43, 152-73

Carhart, M., 1997. On persistence in mutual fund performance. Journal of Finance 52: 57–82. Cochrane, J.H., 2001. Asset Pricing (ISBN-13: 978-0-691-12137-6)

Daniel, K., Grinblatt, M., Titman, S., Wermers, R. , 1997. Measuring mutual fund performance with characteristic-based benchmarks. Journal of Finance 52, 1035–1058.

Fama, E.F., French, K.R., 1993. Common risk factors in the returns of stocks and bonds. Journal of Financial Economics 33, 3–56.

Fama, E.F., MacBeth, J.D., 1973. Risk,return and equilibrium: empirical tests. Journal of Political Economy 81, 607–636.

Fu, F., 2009. Idiosyncratic risk and the cross-section of expected stock returns. Journal of Financial Economics 91, 24–37.

Gagliardini, P., Ossola, E., Scailleti, O., 2016. Time-Varying Risk Premium In Large Cross-Sectional Equity Data Sets. Econometrica, Vol. 84, No. 3, 985–1046

Harvey, C.R., Siddique, A., 2000. Conditional skewness in asset pricing tests. Journal of Finance 55, 1263–1295.

Herskovic, B., Kelly, B., Lustig H., 2016. The common factor in idiosyncratic volatility: Quantitative asset pricing implications – Journal of Financial Economics

Hirshleifer, D., 1988. Residual risk, trading costs, and commodity futures risk premia. Review of Financial Studies 1, 173–193.

Hou and Loh, 2016. Have we solved the idiosyncratic volatility puzzle? – Journal of Financial Economics

Hu, Y.P., Tsay, R.S., “Principal Volatility Component Analysis”Journal of Business & Economic Statistics, 03 April 2014, Vol.32 (2), p.153-164

Hung, D., Shackleton, M., Xu, X., 2004. CAPM, higher co-moment and factor models of UK stock returns. Journal of Business Finance and Accounting 31: 87–112

Knez, P.J., and Ready, M.J., “On the Robustness of Size and Book-to-Market in Cross-Sectional Regressions,” The Journal of Finance, September 1997.

Kyle, A.S., 1985. Continuous auctions and insider trading. Econometrica 53, 1315–1336.

Lewellen, J., Nagel, S., 2006. The Conditional CAPM Does Not Explain Asset Pricing Anomalies. Journal of Financial Economics 82, 289-314.

Lewellen, J., Nagel, S., Shanken, J., 2010. A Skeptical Appraisal of Asset Pricing Tests. Journal of Financial Economics 96, 175-194.

Malagon, J., Moreno, D., Rodríguez, R., 2015. The idiosyncratic volatility anomaly: corporate investment or investor mispricing? J. Bank. Financ. 60, 224–238.

Merton, R.C., 1987. A simple model of capital market equilibrium with in- complete information. Journal of Finance 42, 483–510.

Nartea, G.V., Ward, B.D., Yao, L.J., 2011. Idiosyncratic volatility and cross-sectional stock returns in Southeast Asian stock markets. Account. Finance 51, 1031–1054. Pukthuanthong-Le, K., Visaltanachoti, N., 2009. Idiosyncratic volatility and stock returns: a

cross country analysis. Appl. Financ. Econ. 19, 1269–1281. Roll, R., 1988. . Journal of Finance 43, 541–566

Shanken, J., 1992. On the Estimation of Beta-Pricing Models. Review of Financial Studies 5, 1-33

Wang, F.A., 2010. Informed arbitrage with speculative noise trading. Journal of Banking & Finance 34, 304–313.