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On Drivers of Asset Pricing Factors

Vidojevic, M.

2019

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O N D R I V E R S O F A S S E T P R I C I N G FA C T O R S

milan vidojevic

VRIJE UNIVERSITEIT

ON DRIVERS OF ASSET PRICING FACTORS

ACADEMISCH PROEFSCHRIFT

ter verkrijging van de graad Doctor of Philosophy

aan de Vrije Universiteit Amsterdam,

op gezag van de rector magnificus

prof.dr. V. Subramaniam,

in het openbaar te verdedigen

ten overstaan van de promotiecommissie

van de School of Business and Economics

op maandag 8 april 2019 om 9.45 uur

in de aula van de universiteit,

De Boelelaan 1105

door

Milan Vidojević

geboren te Cacak, Joegoslavië

promotor: prof.dr. R.C.J. Zwinkels

copromotoren: prof.dr. P.A. Stork

prof.dr. T.B.M. Steenkamp

A C K N O W L E D G M E N T S

My Ph.D. trajectory was by no means conventional. I was very for-tunate to have been able to combine working at various universitieswith working in the industry, to receive guidance from academics andpractitioners that I admire greatly, and to meet numerous inspiringpeople who have left marks on my professional and personal devel-opment. The opportunity to acknowledge the contributions of othersto the feat of writing this dissertation that now lies in front of you isa great honor for me.

First and foremost, my gratitude goes to my supervisor, RemcoZwinkels. I vividly remember our meeting when you encouraged meto consider pursuing a Ph.D. Your belief in me in that moment andover these past five years has been humbling. Thank you for yourguidance throughout this journey, and for always finding time forme, even when that meant having meetings at inconvenient timesto accommodate the time difference between our locations. I wouldalso like to thank my co-supervisor, Philip Stork, for always challeng-ing me to think about the contributions of my work to the generalunderstanding of the subject matter. I am very proud that this disser-tation contains a paper that the three of us co-authored. My gratitudealso goes to my co-supervisor, Tom Steenkamp, without whose ap-proval and support I would not have been able to combine workingat Robeco Asset Management and doing a Ph.D. at the university.

I would like to thank the members of my Ph.D. committee, Prof. Al-bert Menkveld, Prof. Siem Jan Koopman, Prof. Mathijs van Dijk, Prof.Joop Huij, and Prof. Robert Hodrick for evaluating my dissertation.

I am thankful to Peter Ferket for his support in this endeavor. I amfortunate to have been able to work in the quantitative strategies de-partment at Robeco, and with some of the smartest and nicest peoplethat I know.

Simon, you are to blame for all of this. This journey started withyou hiring me back in 2014 and, together with Remco, supervisingmy master’s thesis that led to me developing an interest in empiricalfinance research. You encouraged me to pursue a Ph.D., supportedme in all big decisions that I have made over the years, and werealways there to give me advice. You are “my people.”

David, I admire your pragmatism and work ethic. You have playeda key role in my development as a researcher, and I am grateful foryour guidance over the years. Thank you for treating me as a peerand for giving me the space and freedom to express my opinions. Iam happy that two (and a half) of the five papers we have writtentogether thus far made their way into this dissertation.

vi

Joop, your drive and energy are unrivaled. You have been a rolemodel for me. Without your initiative to set up the Robeco Ph.D. pro-gram, I would not have been able to get the invaluable and uniqueexperience of the synergies between academia and practice. Thankyou. Georgi, my fellow comrade, it was great fun sharing this journeywith you. Viorel, thank you for being a great friend and listener, andfor always giving me your honest opinion and advice. Pim, your en-thusiasm for research is motivating. Laurens, thank you for showinggenuine interest in my work, for providing feedback, and for alwayschecking-in and offering advice. Matthias, thank you for being a co-author on one of the chapters in this thesis, and for sticking with methrough the numerous revisions.

I would like to thank my teammates in the selection research team,and the entire investment research group at Robeco for their supportand interest in my academic work, and for being colleagues one couldonly wish for. I would also like to thank my colleagues at Robeco NewYork for encouragement in the last phase of the doctoral program.Zoë, you yelling at me “Get it done!” seems to have worked.

I was fortunate to have been able to spend a significant part of myPh.D. at Columbia University in New York. I would like to thankProf. Gur Huberman for encouraging me to come to Columbia andProf. Kent Daniel for sponsoring my appointment at the universityand for providing comments that greatly helped improve this disser-tation. My deepest gratitude goes to Prof. Robert Hodrick for provid-ing invaluable feedback and guidance, and for being a part of myPh.D. committee. My experience at Columbia is something that I willcherish forever.

I would like to thank my friends for moral support and encour-agement. Renxuan, Kevin and Bert-Jan, it never gets boring with youguys, whether we are at Hamilton on the Upper West Side talkingabout superprocesses and market efficiency, or in a helicopter flyingover the Grand Canyon discussing the properties of the random walkin higher dimensions. Renxuan, thanks for being such a great hostat Columbia, and co-author on one of the papers. Sara and Roberto,the last three years would not have been nearly as much fun withoutyou. George and Alice, thank you for being great roommates over theyears. Davide and Giovanna, you are my Italian family. J.B., thankyou for being a great friend and colleague. Kristina, I am thankful forhaving your voice in my life.

Lastly, I would like to express gratitude to my family for support-ing me in all my personal and professional decisions, and for theencouragement and love they have unconditionally given to me.

Milan VidojevicNew York, 2019

C O N T E N T S

acknowledgments vi1 introduction 2

1.1 General introduction . . . . . . . . . . . . . . . . . . . . 2

1.2 Thesis chapters . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Practical relevance . . . . . . . . . . . . . . . . . . . . . 12

2 the profitability of low-volatility 14

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Significance tests . . . . . . . . . . . . . . . . . . . . . . 22

2.5 Robustness to the choice of profitability measure . . . 24

2.6 Robustness to measurement errors . . . . . . . . . . . . 26

2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 the idiosyncratic momentum anomaly 34

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Data and methodology . . . . . . . . . . . . . . . . . . . 42

3.3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3.2 Variable construction . . . . . . . . . . . . . . . . 44

3.3.3 Motivating results . . . . . . . . . . . . . . . . . 45

3.4 Time-series, cross-section, and factor-spanning tests . . 47

3.4.1 Empirical results . . . . . . . . . . . . . . . . . . 47

3.4.2 Relationship with idiosyncratic volatility . . . . 55

3.4.3 Liquidity and transactions costs . . . . . . . . . 57

3.4.4 Importance of factors in residualization . . . . . 58

3.4.5 Industry effects . . . . . . . . . . . . . . . . . . . 62

3.5 Explanations for the idiosyncratic momentum anomaly 66

3.5.1 Momentum crashes . . . . . . . . . . . . . . . . 68

3.5.2 Market states and dynamics . . . . . . . . . . . 70

3.5.3 Link with underreaction . . . . . . . . . . . . . . 73

3.6 International evidence . . . . . . . . . . . . . . . . . . . 78

3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4 macro drivers of low-volatility stock returns 89

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.2 Literature on low-risk anomaly . . . . . . . . . . . . . . 93

4.3 Motivating results . . . . . . . . . . . . . . . . . . . . . . 95

4.4 Methodology: firm-level VAR and portfolio dynamics . 100

4.4.1 VAR specification . . . . . . . . . . . . . . . . . . 101

4.4.2 Return variance decomposition . . . . . . . . . . 102

4.4.3 From single stock to portfolio-level shocks . . . 102

4.4.4 Impulse response function . . . . . . . . . . . . 104

viii

4.5 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.5.1 Clean surplus accounting earnings . . . . . . . . 106

4.6 Empirical results: VAR decomposition . . . . . . . . . . 107

4.6.1 Constant transition matrix . . . . . . . . . . . . . 107

4.6.2 Baseline results . . . . . . . . . . . . . . . . . . . 109

4.6.3 Other state variables . . . . . . . . . . . . . . . . 113

4.6.4 Return variance decomposition . . . . . . . . . . 117

4.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . 120

4.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.8.1 Relationship between dividend yield, return volatil-ity, and bond sensitivity . . . . . . . . . . . . . . 122

5 behavioral heterogeneity in return expectations

across equity style portfolios 128

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.3 Expected returns . . . . . . . . . . . . . . . . . . . . . . 137

5.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.5.1 Expected returns . . . . . . . . . . . . . . . . . . 141

5.5.2 Heterogeneous agent model estimation results . 147

5.5.3 Chartists’ expectation formation rules . . . . . . 151

5.5.4 Robustness to look-back for past performanceevaluation . . . . . . . . . . . . . . . . . . . . . . 153

5.5.5 Restricted models . . . . . . . . . . . . . . . . . . 153

5.5.6 Trading strategies . . . . . . . . . . . . . . . . . . 158

6 conclusion 161

references 165

summary 174

L I S T O F F I G U R E S

Figure 3.1 Idiosyncratic volatility and market capitaliza-tion across deciles . . . . . . . . . . . . . . . . . 59

Figure 3.2 Cumulative returns . . . . . . . . . . . . . . . . 71

Figure 3.3 Fama and MacBeth 1973 regressions with lagged(i)momentum signals with controls . . . . . . . 76

Figure 3.4 Fama and MacBeth 1973 regressions with lagged(i)momentum signals without controls . . . . . 77

Figure 3.5 Cumulative performance in restricted universe 79

Figure 3.6 Fama and MacBeth 1973 regressions with lagged(i)momentum signals with controls in interna-tional markets . . . . . . . . . . . . . . . . . . . 83

Figure 4.1 Variation in valuations . . . . . . . . . . . . . . 97

Figure 4.2 Sensitivity to bonds . . . . . . . . . . . . . . . . 98

Figure 4.3 Dividends . . . . . . . . . . . . . . . . . . . . . 99

Figure 4.4 Conditional return response to CP shock . . . 112

Figure 4.5 Conditional return response to slope shock . . 113

Figure 4.6 Conditional return response to inflation shock 115

Figure 4.7 Conditional return response to market varianceshock . . . . . . . . . . . . . . . . . . . . . . . . 117

Figure 5.1 10-year rolling Fama-MacBeth estimates of fac-tor premia . . . . . . . . . . . . . . . . . . . . . 143

Figure 5.2 Weight of chartists . . . . . . . . . . . . . . . . 150

Figure 5.3 Performance of trading strategies . . . . . . . . 159

L I S T O F TA B L E S

Table 2.1 Base-case Fama and MacBeth 1973 results . . . 20

Table 2.2 Regression of time-series of estimated returnsto beta in the cross-section on the market riskpremium . . . . . . . . . . . . . . . . . . . . . . 23

Table 2.3 Fama and MacBeth 1973 results using beta-adjusted returns on the left-hand side . . . . . 25

Table 2.4 Robustness to choice of profitability measure . 27

Table 2.5 Fama and MacBeth 1973 results with portfoliobeta . . . . . . . . . . . . . . . . . . . . . . . . . 29

Table 2.6 Fama MacBeth results on portfolio-level . . . . 30

x

Table 3.1 Performance of decile portfolios . . . . . . . . 46

Table 3.2 Performance in the large-cap universe 1963-2015 48

Table 3.3 Fama and MacBeth 1973 (1973) regressions . . 50

Table 3.4 Spanning tests . . . . . . . . . . . . . . . . . . . 52

Table 3.5 Spanning tests with other factor models . . . . 54

Table 3.6 Fama and MacBeth 1973 and Spanning Regres-sions with Idiosyncratic Volatility . . . . . . . . 56

Table 3.7 Importance of factors in residualization . . . . 61

Table 3.8 Performance of Within Industry and AcrossIndustry Strategies . . . . . . . . . . . . . . . . 64

Table 3.9 Spanning Regressions with Within Industry andAcross Industry Strategies . . . . . . . . . . . . 65

Table 3.10 Spanning Regressions with Within Industry andAcross Industry Strategies . . . . . . . . . . . . 67

Table 3.11 Momentum crashes - optionality in bear markets 69

Table 3.12 Market states . . . . . . . . . . . . . . . . . . . . 72

Table 3.13 Market dynamics . . . . . . . . . . . . . . . . . 74

Table 3.14 International results . . . . . . . . . . . . . . . . 81

Table 4.1 Performance characteristics . . . . . . . . . . . 96

Table 4.2 Dividend-price ratios of volatility-sorted port-folios . . . . . . . . . . . . . . . . . . . . . . . . 99

Table 4.3 Constant transition matrix with CP . . . . . . . 107

Table 4.4 Variance decomposition with constant transi-tion matrix . . . . . . . . . . . . . . . . . . . . . 109

Table 4.5 Transition matrix . . . . . . . . . . . . . . . . . 110

Table 4.6 VAR estimates with CP . . . . . . . . . . . . . . 111

Table 4.7 VAR estimates with slope . . . . . . . . . . . . 114

Table 4.8 VAR estimates with inflation . . . . . . . . . . 116

Table 4.9 VAR estimates with aggregate variance . . . . 118

Table 4.10 Beta-adjusted return variance decomposition . 119

Table 4.11 Spanning regressions . . . . . . . . . . . . . . . 121

Table 4.12 Bond betas within low-risk segment . . . . . . 123

Table 4.13 Bond betas within high-dividend yield segment 124

Table 5.1 Fama-MacBeth estimated premia . . . . . . . . 141

Table 5.2 Performance characteristics of styles . . . . . . 144

Table 5.3 Realized and expected returns . . . . . . . . . 145

Table 5.4 Characteristics . . . . . . . . . . . . . . . . . . . 146

Table 5.5 Output of HAM estimation . . . . . . . . . . . 148

Table 5.6 Weight of chartists . . . . . . . . . . . . . . . . 149

Table 5.7 Output of HAMs with different EWMA decayparameters . . . . . . . . . . . . . . . . . . . . . 152

Table 5.8 Output of HAM with longer profit look-back . 154

Table 5.9 Restricted models . . . . . . . . . . . . . . . . . 156

Table 5.10 LR-statistics of models with different EWMAdecay parameters . . . . . . . . . . . . . . . . . 157

"Market efficiency says that prices reflect all available information and thus pro-vide accurate signals for allocating resources to their most productive uses. This isthe fundamental principle of capitalism. To test market efficiency, however, we needa model that describes what the market is trying to do in setting prices. More specif-ically, we need to specify the equilibrium relation between risk and expected returnthat drives prices. The reverse is also true: Almost all asset pricing models assumethat markets are efficient. So, while some researchers talk about testing asset pricingmodels and others talk about testing market efficiency, both involve jointly testinga proposition about equilibrium risk pricing and market efficiency."

— Eugene F. Fama (Fama and Litterman 2012)

1I N T R O D U C T I O N

This chapter ispartly based on my

publication “FiveConcerns with the

Five Factor Model”in the Journal of

PortfolioManagement (seeBlitz et al. 2018).

1.1 general introduction

The core job of financial markets is to allocate resources toward theirproductive use with the ultimate goal of spurring economic growth.The fact that the markets have never been bigger than they are today,human productivity has never been at a higher level, and the worldin aggregate has never been wealthier can to a significant extent beattributed to their effectiveness at performing this function.

Yet it is important to differentiate between the concept of effective-ness and efficiency, as something that is done effectively does notnecessarily mean that it is executed in an efficient manner. The ques-tion of whether financial markets are also efficient is at the root of theasset pricing theory. In this context, the concept of efficiency refersto the ability of the market participants to efficiently process all avail-able information when forming expectations about future economicoutcomes. The theory of rational expectations, originally postulatedby Muth 1961, states that all economic agents are fully rational, mean-ing that they possess the knowledge of the models that determineeconomic outcomes, and have a cost-less access to information. Fama1965 further generalized the theory of rational expectation in the con-text of financial markets in his own Ph.D. dissertation, where he ar-gued that the agents in financial markets efficiently incorporate all

2

1.1 general introduction 3

available information in their assessment process, as they have thefull knowledge and understanding of the economic drivers of riskand return. In this world, all prices reflect the fundamental values ofassets, and one can only make high returns on their investments ifthey take on more risk. If all agents are assumed to be homogeneousand rational1, this means that they have to solve only two problems -understand what are the rewarded risk factors that drive investmentreturns, and then decide how much capital to allocate to these factors.

The fundamental questions of asset pricing can be brought back toone simple yet insightful equation that states that the price of an assetis equal to its expected discounted future payoff, pt = Et(mt+1xt+1),where pt is the price at time t, mt+1 is the applied discount factor,and xt+1 is the future payoff of the asset. The not-so-simple part thathas captivated the interests of researchers for decades is determiningthe right functional form of the discount factor2, and how investorsform expectations about future returns if the assumption of perfectrationality of the representative agent is relaxed.

In the linear factor model (beta) representation of the core pric-ing equation, expected asset returns are a function of the risk-freerate of return, and the regression coefficient β of asset returns on thestochastic discount factor, that is E(Ri) = Rf+ βi,mλm, where βi,m

is known as the quantity of risk, and λm as the price of risk3. Theprice of risk is the same for all securities in the market, while thequantity of risk varies (indicated by the subscript i). This means thatthe differences in expected returns across securities depend only ontheir co-variation (beta) with the stochastic discount factor. In com-plete markets, the concept of the linear discount factors, beta pricing,and the efficient frontier are all in fact equivalent (see Cochrane 2005;Ross 1978; Dybvig and Ingersoll 1982; Hansen and Richard 1987), im-plying that the mean-variance optimal portfolio contains all pricinginformation. Therefore, the problem of finding a parsimonious set offactors that explain the cross-section of stock returns is the same asfinding a set of factors that span the mean-variance efficient portfolio.This question remains the holy grail of empirical asset pricing.

1 The concept of the representative agent plays a prominent role in the rational expec-tations paradigm. For the prices to be efficient, agents can still have biased expecta-tions so long as the individual specific biases cancel out on the aggregate, and therepresentative (average) investor is rational.

2 This discount factor is also known as the stochastic (random) discount factor,marginal rate of substitution, or pricing kernel.

3 I suppress the time subscripts that are needed in case of conditional asset pricingmodels.

4 introduction

Sharpe 1964 proposed an equilibrium theory of asset pricing, knownas the Capital Asset Pricing Model (CAPM), where the mean-varianceefficient portfolio is spanned by the market portfolio, which is atraded portfolio of all assets in the economy. The predictions of theCAPM are straightforward: the expected return on an asset dependsonly on its beta on the market portfolio (also known as the mar-ket factor). Investors thus require higher expected returns in orderto hold assets with higher market betas. However, the theoreticallysound and normative implications of the CAPM turned out to lackempirical support. Early tests of the model indicated that the relationbetween beta and return is flatter than predicted; see, for instance,Black, Jensen, and Scholes 1972, Fama and MacBeth 1973, and Hau-gen and Heins 1975. In the years to come, researchers have identifiedmany patterns in the cross-section of stock return that could not beexplained by the CAPM. These are commonly referred to as the assetpricing anomalies. For instance, Basu 1977 showed that stocks withlow prices relative to their earnings generate returns higher than im-plied by their market betas. Banz 1981 documented that small-capstocks generate higher returns than large-cap stocks and that this em-pirical regularity also cannot be explained by the CAPM. De Bondtand Thaler 1985 showed that stocks with high past three-year returnsunderperform stocks with low past three-year returns, another pat-tern, entitled ’the long-term reversal’, that the CAPM could not ex-plain.

The empirical failure of the standard CAPM, as well as of its con-ditional variants, has motivated researchers to look for other modelswith the end goal of explaining why certain stocks, or more broadlyassets, have earned higher average returns than others.

Fama and French 1992 find strong evidence for the existence of sizeand value premiums4 in the cross-section of stock returns. Based onthese findings, Fama and French 1993 propose to extend the CAPMwith size (SMB, small minus big) and value (HML, high minus low)factors, resulting in a three-factor asset pricing model. As a motiva-tion for the inclusion of these factors, the authors propose a theorywhereby these so-called factor mimicking portfolios proxy for unob-servable state variables that carry information about the market-widedistress risk. In other words, the idea behind the three-factor model

4 Fama and French 1993 proxy for size using total market capitalization of stocks andfor value using the ratio of book value of equity to the market value of equity. Thesize and value effects were already known (see Banz 1981; Rosenberg, Reid, andLanstein 1985) at the time, however, the systematic treatment of these asset pricingissues is the main contribution of the seminal Fama and French 1992, 1993 papers.

1.1 general introduction 5

was that the CAPM is fundamentally right, in the sense that system-atically higher returns can only be obtained with higher systematicrisk, but apparently, size and value capture a dimension of systematicrisk that the plain CAPM market beta does not. This view was soonchallenged, e.g. by Lakonishok, Shleifer, and Vishny 1994, who ar-gued that value strategies are not particularly risky, and that, instead,their return seems to stem from behavioral biases of investors, in par-ticular extrapolation of past growth into the future. Another problemwith the distress risk argument is that studies which examine directindicators of distress risk find a negative relation with subsequentreturns, e.g. Dichev 1998, Griffin and Lemmon 2002, and Campbell,Hilscher, and Szilagyi 2008. These findings are in fact consistent withthe existence of a low-risk premium.

Following the publication of the three-factor model, it has becomecommon practice in the asset pricing literature to not only report one-factor (CAPM) alphas but also three-factor alphas. The three factor-model has been very influential for more than twenty years, andevery finance student is required to know and understand it. Withthis model, Fama and French 1993 brought some order into the as-set pricing chaos. For a while, it seemed that all the anomalies thathad popped up except for momentum and short-term reversal (seee.g. Fama and French 1996) could be brought back to the three fac-tors, and the risk-based interpretations of these factors meant thatthe CAPM did not need to abandon, but could be salvaged with a bitof modification.

However, many studies that followed the publication of the seminalpapers of Fama and French report three-factor alphas that are signif-icantly different from zero, which suggests that this model is alsoincomplete and that more factors are needed to accurately describethe cross-section of stock returns. Inspired by this mounting evidencethat the three factors do not suffice, Fama and French 2015 propose toaugment their model with two additional factors, namely profitabil-ity (RMW, robust minus weak) and investment (CMA, conservativeminus aggressive). This new five-factor model significantly raises thebar for new anomalies. Fama and French 2016 argue that it effectivelyaddresses the main shortcomings of the three-factor model, but theydo not proclaim their new model to be the last word on asset pricingor that it fully explains the cross-sectional variation in average stockreturns. Nevertheless, for practical purposes, the five-factor model is

6 introduction

likely to become the new benchmark in asset pricing in the years tocome.

Fama and French 2015 motivate their two new factors, profitabilityand investment, using a rewritten dividend discount model (DDM):for a given level of book-to-market and investments, higher futureprofitability implies higher expected returns, and for a given level ofbook-to-market and profitability, low investments also imply higherexpected returns. Interestingly, they no longer justify the additionof the two new factors in their five-factor model by providing anexplicit risk-based explanation. They do refer to the Merton 1974’sintertemporal CAPM (ICAPM) where the factors could proxy for un-observed state variables. In this rewritten DDM, the two additionalfactors directly imply expected returns, however, this model does notsay anything about the source of the factors, in particular, whetherthe observed premiums are compensations for systematic risks or be-havioral anomalies. The fact that the authors make no explicit state-ment about the source of these factor premiums makes the five-factormodel a paradigm shift compared to the CAPM and the three-factormodel, which both had risk foundations.

1.2 thesis chapters

Chapter 2 addresses the internal inconsistency of the five-factor modelthat results from the retention of the fundamental CAPM relation be-tween the market beta and expected returns. Using the CAPM as astarting point for an asset pricing model is appealing for various rea-sons. First, the CAPM has strong theoretical underpinnings. Second,it helps to capture the equity risk premium, i.e. why stocks on aver-age have a return that is higher than the risk-free rate of return. Thisargument was also used by Fama and French 1993 in the context oftheir three-factor model. Third, the CAPM is effective at explainingthe time-series variation in stock returns, because when the marketgoes up (down), high-beta stocks tend to go up (down) more, whilelow-beta stocks tend to go up (down) less.

Crucially, however, the CAPM also implies that a higher marketbeta should be rewarded with a higher expected return in the cross-section of stocks. This assumption denies the existence of a low-betaanomaly. The first empirical tests of the CAPM by Black, Jensen, andScholes 1972 and Fama and MacBeth 1973 already observed a flatterrelation between the market betas and average returns than predicted

1.2 thesis chapters 7

by the model, while Haugen and Heins 1975 even find a negative re-lation. Two decades later, using Fama and MacBeth 1973 regressions,Fama and French 1992 themselves conclude that the market beta isnot a priced variable. Blitz and van Vliet 2007 show that the low-betaeffect has not just persisted, but it has become more pronounced overtime and that the effect is even stronger when volatility is used in-stead of beta. Ang et al. (2006, 2009) document a strong idiosyncraticvolatility anomaly in the cross-section of the US and international eq-uities. More recent studies, such as Baker, Bradley, and Wurgler 2011,Baker and Wurgler 2012, Frazzini and Pedersen 2014, Hong and Sraer2016, confirm the low-volatility and (or) low-beta effects.

The five-factor model postulates a positive, linear relationship be-tween factor loadings (betas) and expected stock returns. This meansthat if one properly accounts for the size, value, profitability, andinvestment factors, long-term average returns should increase withmarket betas. Essentially, the original CAPM is a nested version ofthe five-factor model, where additional factors are added to aid themarket beta in explaining the cross-section. The market factor is, afterall, the most widely accepted factor with the highest variance, that isnot spanned by the other factors. That said, its inclusion in the five-factor model, which aims to explain the cross-section of returns, isquestionable given the lack of empirical support for the claim thatreturns increase with market betas. Fama and French 2016 justify theCAPM basis of their model by showing that the low-beta anomaly islargely explained by their five-factor model. This result is in line withNovy-Marx 2014, who finds that the low-beta and low-volatility ef-fects are explained by the three-factor model augmented with a prof-itability factor. Both studies use time-series regressions to come tothese conclusions.

Chapter 2 takes a closer look at these results5. We observe that di-rect evidence for a linear, positive relation between the market betaand average returns, which is assumed in the models of Fama andFrench and Novy-Marx, is still lacking, and therefore, it is prema-ture to conclude that the low-risk anomaly is explained. More specif-ically, if the Fama and French 2015 asset pricing model were correct,it should be possible to construct portfolios which show that the pre-dicted linear relationship between the market beta and returns holdsin practice, provided one controls appropriately for the other factorsin the model. In this chapter, we test whether this premise is sup-

5 This chapter appears in the Journal of Empirical Finance as Blitz and Vidojevic 2017.

8 introduction

ported by the data using Fama and MacBeth 1973 regressions, wherethe estimated coefficients can be interpreted as returns on portfolioswhich have unit exposure (ex-ante) to one specific factor, controllingfor the exposures (ex-ante) to all other factors included in the regres-sion. We find that all factors in the five-factor model are rewardedwith significant premia, except the market. In other words, a unitexposure to the market beta in the cross-section does not result in sig-nificantly higher returns, regardless of whether one controls for otherfactors in the five-factor model. We further modify the testing pro-cedure and go on to show that the magnitude of the deviation fromthe theoretical relationship is significant. Taken together, these resultsimply that the relationship between these risk measures and returnin the cross-section is flat instead of positive, i.e. there still exists amajor low-risk anomaly.

This, however, does not mean that we advocate the addition of alow-versus-high beta factor to the asset pricing models that are builton the CAPM basis, because a model which starts by assuming theCAPM relation and then adds a factor with the sole purpose to alterthat relation would be internally inconsistent. Instead, we questionwhether the CAPM should be used as the basis for an asset pricingmodel in the first place. Ideally, an asset pricing model should be ableto explain the existence of the equity risk premium, but also allowfor the absence of a return premium to market beta exposure in thecross-section that is observed in practice. The asset pricing models ofBlitz 2014 and Clarke, de Silva, and Thorley 2014 take a step in thatdirection.

Chapter 3 seeks to uncover the sources of the anomalously highprofits of the idiosyncratic momentum anomaly. Although the empiri-cal evidence for the (conventional) momentum premium documentedby Jegadeesh and Titman 1993 is as strong as that for the size andvalue premiums, Fama and French 1993 did not include it in theirthree-factor model. This might be because the three-factor model wasdeveloped around the same time as the momentum phenomenon be-came known. A more fundamental problem with adding momentumto the three-factor model is that it is hard to argue that it can be seenas a priced risk factor. However, the momentum premium has turnedout to be too “pervasive” (Fama and French 2008, p. 1653) and strongto simply ignore, and is by now an established factor. The four-factormodel, i.e. the three-factor model augmented with a momentum fac-


tor, is as popular in the asset pricing literature as the three-factormodel.

Interestingly, the momentum factor is still conspicuously absentfrom the five-factor model, despite the clear opportunity this pre-sented to include it once and for all. Fama and French 2016 acknowl-edge that similar to the three-factor model, the five-factor model isunable to explain the momentum effect. They also mention that thefocus of the model is on explaining long-term expected returns ratherthan short-term variation in returns. Many recent studies prefer touse a six-factor model, i.e. the five-factor model augmented with thesame momentum factor that is commonly used to transform the three-factor model into a four-factor model.

The idiosyncratic momentum phenomenon raises more asset pric-ing questions. Gutierrez and Pirinsky 2007 consider a momentumstrategy in which stocks are sorted on their idiosyncratic returns, i.e.the stock-specific, residual returns that follow from regressions of to-tal stock returns on the three Fama and French 1993 factors. Blitz,Huij, and Martens 2011 further show that the risk-adjusted returnof this idiosyncratic momentum strategy is double that of the con-ventional momentum strategy over a sample that spans around eightdecades. Yet neither one of these studies addresses the fundamentalquestion of whether the idiosyncratic momentum expands the effi-cient frontier comprised of the already established factors, or whatcould be the source of this anomaly. Chapter 3 examines these ques-tions.

We show that the idiosyncratic momentum is a distinct phenomenonthat exists next to conventional momentum, and is not explainedby it; the idiosyncratic momentum is priced in the cross-section ofstock returns after controlling for the established and recently pro-posed asset pricing factors, including the ones that explain a host ofmomentum-related anomalies; some of the prominent explanationsfor the momentum premium, such as the crash risk, and investor over-confidence and overreaction linked to the market states and dynamicscannot explain the idiosyncratic momentum profits; long-term returndynamics of the idiosyncratic momentum support the underreactionhypothesis for its existence; and lastly, the idiosyncratic momentumgenerates robust returns across a range of developed and emergingmarkets. The fact that we cannot conclusively reject one momentumfactor in favor of the other, and a lack of evidence that links thesetwo momentum strategies to the same underlying mechanisms, leads

10 introduction

us to conclude that they behave more like complements rather thansubstitutes.

Chapter 4 aims to answer the question of what drives the anoma-lously high (low) returns of low (high) volatility portfolios. In thischapter, we propose a novel approach to decompose returns of re-balanced portfolios into the discount rate and cash-flow news on asingle-stock level that explicitly takes into account stock’s dynamicexposures to aggregate shocks. We impose the present value identityof Campbell and Shiller (1988), Campbell (1991), and Vuolteenaho(2002) on the log book-to-market ratio, decompose stock returns on afirm-level, and aggregate them into portfolios. Differently from priorwork, we include additional indicator variable interactions with thestock-level state variables in our vector autoregression model (VAR),where indicators tracks in which volatility-sorted portfolio a partic-ular firm is at any point in time. That is, our approach takes intoaccount the fact that, over time, stocks migrate from one portfolio toanother; in other words, the fact that a stock that could have startedoff as a high-vol stock could have become a mid or a low-vol stock,or the other way around.

We study the dynamics of the volatility-sorted portfolios, and doc-ument that on a long-short beta-adjusted anomaly level, cash-flownews drive around four times more of the return variance than thediscount rate news, however, we also find that the two news com-ponents are highly negatively correlated, making the decompositionhard to interpret. These results are consistent with those of Lochstoerand Tetlock 2016 in the case of other asset pricing anomalies that theyexamine using a related, but different model, and have implicationsfor our understanding of what theories are more likely to explainthese anomalies.

Most importantly, by incorporating a rich set of macroeconomicvariables, we show that low-volatility stocks underperform followingperiods of increasing bond yields, inflation, and aggregate marketvariance. High-vol stocks hedge against these macroeconomic states.Provided that these macro state variables are proxies for investors’long-term investment opportunities, our results imply investors’ in-centives for hedging against these risks as potential drivers of thelow-volatility anomaly. This paper thus establishes a fundamentallink between the low-volatility anomaly returns and the macroeco-nomic state variables.


The final chapter of the thesis (chapter 5) belongs in the asset pric-ing literature that studies how agents form expectations in financialmarkets. However, just like the previous chapters, this one also aimsto uncover the sources of various asset pricing anomalies.

People’s decisions are guided by their expectations of future out-comes. Therefore, understanding how people process informationand form expectations is at the forefront of finance. The theory ofrational expectations says that agents in the market are fully rational,meaning that they know the correct economic model that generatesa certain economic outcome and that they have cost-less access to allrelevant information. This theory has been the dominant paradigmbehind much of the research in the field of economics ever since itwas postulated by Muth 1961.

The last couple of decades have seen a proliferation in the the-oretical and empirical work that produced a considerable amountof evidence for the existence of agents with boundedly rational andheterogeneous beliefs about future asset prices, who trade on theseexpectations, thus causing prices to deviate from their fundamentalvalues. Depending on which group of agents prevails in the market,different price dynamics are generated.

In this chapter, we estimate a heterogeneous agent model (HAM),which features two groups of boundedly rational investors on fiveprominent equity investment styles - value, size, profitability, invest-ment, and momentum - four of which are a part of the Fama andFrench 2015 five-factor model. One group of agents is known as thefundamentalists, and they are assumed to trade based on the devia-tions between the fundamental and the observed market price. Theother group of agents is known as the chartists or technical traders,and they form expectations by extrapolating past trends in prices.In the absence of chartists, fundamentalists would be the agents withfully rational expectations, however, in a market in which the chartistsare also present in a significant number, the fundamentalists are bound-edly rational because they do not take into account the fact that thechartists are also present when they form their expectations aboutfuture prices.

The contribution of our paper is two-fold. First, to the best of ourknowledge, we are the first to estimate a heterogeneous agent modelwithin the equity market, on a level more granular than that of the eq-uity market index. Our test assets are the characteristics-sorted port-folios, whose anomalously high returns present some of the biggest

12 introduction

puzzles in the empirical asset pricing literature. We find evidencefor existence of heterogeneous agents in the market for these assets.Second, we use the insights from the vast literature on the role ofstock-level characteristics in explaining the cross-section of stock re-turns and link it with the segregated literature on heterogeneousagent models. In our model, the two groups of boundedly rationalagents have demand functions for the investment styles that dependon their respective expected style return forecasts. In order to empir-ically estimate the fundamentalists’ expected returns on stocks, ourpaper departs from the prior papers that apply HAMs on an equityindex-level, where agents estimate fundamental values based on avariant of the Gordon growth model or a simple moving average ofpast prices. We build on the findings from a rich literature on em-pirical equity pricing and formulate a characteristics-based expectedreturn model on a single-stock level, where the variation in expectedreturns across stocks is driven by the differences in their characteris-tics, such the market capitalization, book-to-market ratio, operatingprofitability, investment, and past momentum. Furthermore, the vari-ation in expected returns over time is driven by the variation in thestock-level characteristics and also the variation in the premia associ-ated with these characteristics.

The fact that we are able to identify both groups of agents in themarket with economically and statistically significant effects on mar-ket prices casts doubt on the theories that assume perfect rationalityof the representative agent in financial markets, and give support tothe behavioral theories with heterogeneous agents.

1.3 practical relevance

The importance of understanding the factor structure of expectedstock returns has far-reaching implication. The last decade has seena rise of an investment paradigm that directly leverages on the find-ings of the empirical asset pricing literature, that is often referred toas factor investing6. The idea behind this investment approach is toexplicitly allocate to the factor premiums that are rewarded with highreturns, and to manage the factor exposures in a top-down way, there-fore thinking about returns and risks from a holistic portfolio factorperspective, as opposed to a more traditional asset class risk framing.

6 The terms style investing and smart beta investing are commonly used among prac-titioners, and refer to the similar underlying ideas.

1.3 practical relevance 13

Some examples of such strategies are value, momentum, quality, andlow-volatility. Today, more than $1 trillion is reported to be investedin factor-based portfolios7.

The finding of my dissertations have some prescriptive suggestionsfor the design of real-world factor investing strategies. For instance,chapter 2 shows that the low-volatility anomaly is not redundant inan allocation that includes value and profitability factor premiums,as earlier studies may have suggested. Chapter 3 shows that investorscan allocate to the momentum premium in a more efficient way thanby just holding stocks with high past total returns, in particular, bytaking stocks’ idiosyncratic momentum into account. Chapter 4 for-mally shows that the low-volatility strategies are exposed to certainmacroeconomic factors, which investors should be aware of when de-ciding how to allocate across asset classes. Lastly, chapter 5 providesfurther evidence for the pervasiveness of these factors, and how be-havioral finance models can help explain the existence of factor pre-mia.

It is only by understanding what drives these factors that investorscan have a conviction as to whether or not they will persist in thefuture.

7 https://www.economist.com/finance-and-economics/2018/02/01/factor-investing-gains-popularity

https://www.economist.com/finance-and-economics/2018/02/01/factor-investing-gains-popularity

https://www.economist.com/finance-and-economics/2018/02/01/factor-investing-gains-popularity

2T H E P R O F I TA B I L I T Y O F L O W- V O L AT I L I T Y

This chapter is jointwork with David

Blitz, and it ispublished in the

Journal of EmpiricalFinance (see Blitz

and Vidojevic 2017).

2.1 introduction

Vast empirical evidence shows that the unconditional Capital AssetPricing Model fails to explain cross-sectional differences in averagestock returns. The early tests of the model already indicated that therelation between beta and return is flatter than predicted; see, forinstance, Black, Jensen, and Scholes 1972, Fama and MacBeth 1973,and Haugen and Heins 1975. Two decades later, Fama and French1992 conclude that, if one controls for size effects, market beta is un-priced in the cross-section of stock returns, implying that firms withhigher market sensitivity are not rewarded with higher average re-turns. Closely related to the low-beta anomaly is the low-volatilityeffect of Blitz and van Vliet 2007 and Blitz, Pang, and van Vliet 2013,who document that the relation between past stock volatilities andsubsequent stock returns is not merely flat, but even negative in allmajor stock markets over recent decades. The low-volatility effect isalso related to studies which report superior risk-adjusted returns forminimum-variance portfolios, such as Haugen and Baker 1991 andClarke, de Silva, and Thorley 2010, and to the work of Ang et al. (2006,2009), who find a similar anomaly for very short-term idiosyncraticvolatility. More recent studies such as Baker, Bradley, and Wurgler2011, Baker and Haugen 2012, and Frazzini and Pedersen 2014 con-firm the low-volatility and/or low-beta effects.

Various studies show that the three- and four-factor models failto explain the low-risk anomaly. For instance, Blitz 2016 finds thatthe three-factor model is unable to explain anomalously high returnsof low-volatility stocks, and Frazzini and Pedersen 2014 report thatthe low-beta anomaly is not subsumed by the three- and four-factormodels. However, Novy-Marx 2014 argues that the low-beta and low-volatility anomalies are explained by a three-factor model augmented

14

2.1 introduction 15

with a profitability factor. Fama and French 2016 also find that theirFama and French 2015 five-factor model, which adds profitability andinvestment factors to their original three-factor model, is able to ex-plain returns on beta-sorted portfolios. Both papers use time-seriesregressions to come to these conclusions. This means that they firstcreate beta- or volatility-sorted portfolios, and next regress the result-ing time series of portfolio returns on the time series of the factorsthat comprise their proposed asset pricing models. The absence ofeconomically large and statistically significant alphas in these regres-sions is interpreted as evidence that the low-beta and low-volatilityanomalies are explained. This paper does not question the empiricalresults of Fama and French 2016 and Novy-Marx 2014 but argues thatdirect evidence for a linear, positive relation between market beta andreturns, which is assumed in their models, is still lacking. If the Famaand French 2015 and Novy-Marx 2014 asset pricing models were cor-rect, it should be possible to construct portfolios which show thatthe positive, linear relation between beta and returns holds in prac-tice, provided one controls appropriately for the other factors in theirmodels. This can be tested by the use of Fama and MacBeth 1973

regressions, as the estimated coefficients in these regressions can beinterpreted as returns on portfolios which have unit exposure (ex-ante) to factors, controlling for exposures (ex-ante) to all other factorsincluded in the regression. Fama 2015 also argues for considering notjust one, but multiple asset pricing tests, including Fama and Mac-Beth 1973 cross-section regressions. However, the rejections of thelow-beta anomaly by Novy-Marx 2014 and Fama and French 2016

are solely supported by time-series spanning tests.Using Fama and MacBeth 1973 regressions we test whether the

factors in the five-factor model are rewarded with significant premia,and find that all factors are, except market beta. In other words, aunit exposure to market beta in the cross-section does not result insignificantly higher returns, regardless of whether one controls forthe additional factors proposed by Fama and French 2015. At thesame time, the constant in the regressions, which ought to be zeroaccording to this asset pricing model (if returns in excess of the risk-free return are used), is large and significant. Taken together, theseresults imply that the relation between market beta and return in thecross-section is flat instead of positive, which is consistent with theasset pricing models of Blitz 2014 and Clarke, de Silva, and Thorley2014. Simply put, we are unable to construct high-beta portfolios with

16 the profitability of low-volatility

high returns and low-beta portfolios with low returns by controllingfor factors such as profitability, while it should be possible to do so ifthe low-beta anomaly is fully explained by such factors.

We also find more pronounced mispricing for volatility than forbeta. This suggests that the low-volatility anomaly is stronger thanthe low-beta anomaly, and, given that the two are closely related, thatthe low-volatility anomaly is the dominant phenomenon. These re-sults are consistent with the earlier findings of Blitz and van Vliet2007, who find higher alpha spreads for volatility-sorted portfoliosthan for beta-sorted portfolios. Lastly, we show that our results arerobust to the choice of profitability measure, and also, using two dis-tinct methodologies, to the well-known errors-in-variables problem.

2.2 data

We consider all common stocks (share codes 10 and 11) in the CRSPdatabase traded on NYSE, AMEX, and NASDAQ exchanges, exceptthose with a share price below 1 dollar. Following Frazzini and Ped-ersen 2014, we estimate stock and market return volatilities over thepast year (minimum 120 days) and correlations with the market port-folio over the preceding five year period (minimum three years). Forvolatilities, we use log returns (log(1+r)), whereas for correlations weuse the average of past three log returns to control for non-synchronoustrading. If daily data are not available, we use past twelve monthly re-turns to calculate volatilities and sixty (minimum 36) for correlations.These estimates are used to calculate market betas, which we shrinktowards one using the commonly employed shrinkage factor of 1/3,as proposed originally by Blume (1971, 1975).

For the calculation of size, value and momentum characteristics wefollow the standard Fama-French methodology. The market capital-ization (ME) of a stock is its price times the number of shares out-standing, and size is defined as the natural logarithm of market capi-talization at the end of the previous month. The balance sheet andincome statement information stem from Compustat North Amer-ica annual files. Book value is the book value of shareholders’ eq-uity, plus balance sheet deferred taxes and investment tax credit, ifavailable, minus the book value of preferred stock (calculated usingthe redemption, liquidation, or par value, in that order). If available,we use shareholders’ equity from either Compustat or Moody’s In-dustrial manuals, otherwise, we measure stockholders’ equity as the

2.2 data 17

book value of common equity plus the par value of preferred stock,or the book value of assets minus total liabilities. The valuation ofa stock is defined as its book-to-market ratio, i.e. BE/ME, calculatedas the book value of common equity at the previous calendar year’sfiscal year-end divided by the market value of equity at the end ofthe previous calendar year, updated at the end of June each year. Themomentum of a stock is defined as its total return over the precedingtwelve months excluding the most recent month.

For profitability, Fama and French (2015, 2016) use an operatingprofitability ratio which is defined for all stocks, while Novy-Marx2014 uses a gross profitability ratio which is undefined for financials.As their results suggest that for explaining the alpha of low-risk strate-gies it does not matter which definition is used, we use the Famaand French (2015, 2016) measure as our base-case definition of prof-itability. This means that we calculate operating profitability as an-nual revenues minus cost of goods sold, interest expense, and selling,general, and administrative expenses divided by book equity. In thepenultimate section of the paper, we challenge the robustness of ourresults to the choice of profitability measure. Next to Novy-Marx 2014

gross profitability we also consider cash-based operating profitability,based on Ball et al. 2016, and return on equity based on Hou, Xue,and Zhang 2015. Gross profitability is defined as annual revenues mi-nus cost of goods sold divided by total assets. Cash-based operatingprofitability is defined as revenues minus costs of goods sold minusreported sales, general, and administrative expenses minus change inaccount receivables, inventory, and prepaid expenses, plus change indeferred revenue, trade account payable, and accrued expenses. Allchanges are calculated on the year-to-year bases. Return on equity(ROE) is defined as income before extraordinary items divided bybook equity, but we deviate slightly from the definition of Hou, Xue,and Zhang 2015 by using annual instead of quarterly earnings data.In this way, we ensure consistency with the frequency of the other ac-counting factors that we consider. Another advantage of annual ROEis that it is available for the full span of our sample, while the quar-terly measure of Hou, Xue, and Zhang 2015 is only available from1972. Moreover, Novy-Marx 2015 analyzes the profitability measureof Hou, Xue, and Zhang 2015 and concludes that their ROE factor isa convoluted proxy for profitability, as it derives a significant portionof its pricing power from past earnings surprises that it incorporatesthrough the use of quarterly earnings data. This enables their ROE


factor to price momentum portfolios. He goes on to argue that grossprofitability is a superior measure of a firm’s profitability once this isaccounted for. This also supports the use of annual data.

Investment (asset growth) is the percentage change in firms’ totalassets from year t-2 to t-1. Stocks for which one or more data items aremissing are dropped from the sample for that month. The start dateof the sample period is July 1963, as this is the earliest date for whichthe new Fama-French factors are available, and the sample periodends in December 2015. The average number of stocks per month is2,972.

2.3 main results

According to the asset pricing models of Fama and French 2015 andNovy-Marx 2014, expected stock returns should be linearly propor-tional to beta, as in the CAPM, after accounting for the other pricedfactors in their respective models. A popular way to analyze the sig-nificance of a factor while controlling for other factors is to conductdouble or triple sorts. However, that approach becomes practically in-feasible when one wants to control for more than three factors. Thispaper uses firm-level Fama and MacBeth 1973 regressions instead. Ev-ery month, we run cross-section regressions of stock returns in excessof the risk-free return on a set of characteristics. The crucial realiza-tion here is that the coefficients estimated in such a regression canbe interpreted as returns on portfolios with a unit exposure (ex-ante)to a factor in the cross-section while controlling for all other factorsincluded in the regression. The next step consists of calculating theaverage premium associated with each factor over all months andthe corresponding t-statistics (using Newey-West corrected standarderrors) and verifying whether these levels are consistent with the pre-dictions of the asset pricing models under investigation. We includemomentum as one of the control factors in our analyses even thoughit is not part of the Fama and French 2015 five-factor model, becauseit is widely recognized as an important driver of stock returns in thecross-section. We note that our conclusions do not materially changeif momentum is excluded from the analysis. Our results are also ro-bust to using weighted least squares (WLS) instead of OLS regres-sions, using either market capitalization or price to weigh observa-tions.

2.3 main results 19

All explanatory variables are winsorized at the 1% and 99% levelsin order to avoid a large potential influence of outliers, and cross-sectional z-score normalizations are applied to all variables, exceptmarket beta. As a result, the intercept of the regression can be inter-preted as the expected excess return on a stock with average factorcharacteristics, adjusted for the part of the return that can be em-pirically attributed to its market beta. Based on the CAPM and itsextensions, the intercept of the regressions should not be statisticallydifferent from zero, and the reward to a unit of beta exposure shouldbe equal to the equity risk premium, which amounts to 0.50 percentper month over our sample period.

Table 2.1 reports results for various Fama and MacBeth 1973 regres-sions. Regression I tests the CAPM, as next to the intercept it onlyincludes market beta. Every month, we run a cross-section regressionof stock returns in excess of the risk-free rate on their market be-tas. This procedure yields a time-series of estimated coefficients, forwhich we report the averages and the corresponding t-statistics. If theCAPM holds and beta explains the cross-sectional differences in av-erage stock returns, the intercept should not be statistically differentfrom zero, while the slope, i.e. the reward to beta exposure, should bepositive, matching the equity risk premium over our sample period.The results in the table show that the reverse is true in reality: theintercept is large and highly statistically significant, while the returnassociated with beta exposure is indistinguishable from zero. This im-plies a flat, instead of a positive, relation between risk and return inthe cross-section.

In regression II, the size, value, and momentum characteristics areadded to the regression. The sign and significance of the estimatedpremia for these factors are fully consistent with the existing litera-ture; however, they do not help to save the market beta. Controllingfor the influence of size, value and momentum, the average return toa unit of beta exposure in the cross-section remains indistinguishablefrom zero, while the intercept remains positive and significant. Theinterpretation of these results is that the base-case expected returnis positive and the same for every stock, regardless of whether it co-varies strongly with the market or not, and that only the size, valueand momentum characteristics of a stock add or detract from thatbase-case expected return. These results are consistent with the as-set pricing models proposed by Blitz 2014 and Clarke, de Silva, andThorley 2014, in which market beta also does not carry a premium


Table 2.1: Base-case Fama and MacBeth 1973 results

This table reports the results of Fama and MacBeth 1973 regressions. We in-clude all common stocks traded on NYSE, AMEX and NASDAQ exchangesfrom July 1963 to December 2015 with share price above $1. Beta is estimatedusing past five years of daily stock returns – correlations and volatilities areseparately estimated over the five (min three) and one (min 120 days) yearwindows, respectively, and beta is calculated as the ratio of volatilities mul-tiplied by correlation; if daily data is not available, we use past sixty (min36) and twelve monthly returns to calculate correlations and volatilities, re-spectively. We shrink beta towards one using shrinkage parameter of 1/3.Size is the natural logarithm of firm’s market capitalization at the end ofmonth t, value is the natural logarithm of the ratio of firms book equity forthe fiscal year ending in t-1 and market cap at the end of December of t-1;momentum is the total stock return from t-12 to t-2; profitability is the ratioof operating profits and book equity at the fiscal year ending in t-1, andinvestment is growth of total assets for the fiscal year ending in t-1. All vari-ables are winsorized at 1% and 99%, and we normalize all variables exceptbeta. Reported are the average coefficients and t-statistics calculated usingNewey-West corrected standard errors.

I II III

intercept 1.03 0.85 0.74

(5.69) (4.83) (4.35)beta -0.20 -0.01 0.11

(-0.71) (-0.02) (0.40)size -0.16 -0.22

(-1.80) (-2.68)value 0.18 0.15

(3.89) (3.48)momentum 0.34 0.32

(6.52) (6.42)profitability 0.21

(6.07)investment -0.20

(-9.91)

2.3 main results 21

in the long run, but is only included to help explain the short-termcross-sectional variation in stock returns.

Regression III shows the results when adding the two new vari-ables of Fama and French 2015, profitability and investment. In thisregression, market beta should clearly show up as a priced factor, ifit were true that with the new factors the low-beta anomaly is re-solved. Consistent with the predictions of the five-factor model, bothnew factors are rewarded with significant premia. However, the ta-ble also shows that, again, the return to beta is insignificant, whilethe intercept of the regressions remains positive and significant. Inother words, contrary to the predictions of the asset pricing modelsof Fama and French 2015 and Novy-Marx 2014, we are unable to findclear evidence of a positive relation between beta and return whencontrolling for the factors they argue explain the anomaly.

In the analysis above, each stock’s covariation with the market port-folio, i.e. its market beta, was estimated in a univariate framework,that is, our estimate of the market beta is closely related to the slopeestimate from a simple regression of stock returns on the market port-folio (with the difference that we estimate correlations and volatilitiesover different time periods). An alternative to this would be to esti-mate the sensitivity to the market portfolio using the multivariate re-gression, that is, accounting for the effects of the other four factors inthe five-factor model. We estimate this beta over past 60 months (min-imum 24), shrink it towards one using the same shrinkage parameteras before (1/3), and run the Fama and MacBeth 1973 regressions us-ing this measure instead of the univariate one. These results can befound in Table 2.7 in the appendix. Note that the sample starts in July1968, as the Fama and French 2015 factor returns are available fromJuly 1963. Our conclusions remain unchanged: the estimated marketpremium remains insignificant indicating that high beta stocks do notearn higher returns than their low beta counterparts.

We further examine the robustness of our results by splitting theuniverse at two size breakpoints and repeat the analysis within thesegroups. Following academic convention, we define large-caps as stockswith a market capitalization above NYSE median, and small-caps asthose below this threshold. We also examine whether this relationshipholds amongst micro-caps, defined as stock with a market capitaliza-tion below the 20th percentile of NYSE listed stocks. These results arereported in Table 2.8 in the appendix. Once again, the sample startsin July 1968 to ensure significant coverage on all variables, especially


amongst micro-caps (our conclusions do not change if we start in1963). We observe that the results are robust across all size groups.

2.4 significance tests

Our results unambiguously indicate that none of the examined mod-els manages to establish a positive relationship between market betaand average returns; however, they do not say anything about themagnitude of the deviation between the expected (theory-based) andthe realized premium for the market beta. In order to determine thesignificance of our results, we conduct two follow-up tests. For thefirst test, we note that under the null hypothesis that there is a linearrelationship between market beta and stock returns, a unit of beta ex-posure in the cross-section should be rewarded with the market riskpremium. We test whether this premise is supported by the data byregressing the time series of the estimated market premiums (i.e. coef-ficients for market beta) on realized market returns (Mkt-Rf). Accord-ing to the examined linear pricing models, the estimated premium tobeta in the cross section should be explained entirely by exposure tothe market factor, so the intercept of this regression should be zero. Ifinstead, the intercept is positive (negative), then the return to a unitof beta is higher (lower) than predicted.

Table 2.2 shows that the ex-post estimated slope coefficients areclose to one and highly significant, which is what we would expectfor portfolios with unit beta exposure ex-ante; but, more importantly,the estimated coefficient on the constant term is significantly belowzero in all three cases. In other words, we observed before that beta inthe cross-section is not rewarded with a significantly positive return,and this additional test shows that this deviation from the models’prediction is statistically significant.

Our second test is also aimed at assessing the significance of thedeviation from the predicted relation between market beta and re-turn. We do so by re-running the Fama and MacBeth 1973 regres-sions, only this time using beta-adjusted returns on the left-hand sideas the dependent variable. In other words, for each stock return inthe regression, we not only subtract the risk-free return but also itsbeta times the market excess return in that period. The main differ-ence with the previous test is that the beta adjustment executed inthis way is applied ex-ante to each stock in the universe, while inthe prior test, the adjustment was done using the time-series of ex-

2.4 significance tests 23

Table 2.2: Regression of time-series of estimated returns to beta in the cross-section on the market risk premium

This table reports results of time-series regressions of the estimated returnsto market beta in the cross-section on the realized market premium (Mkt-Rf). To estimate returns to market beta in the cross-section, we use FamaMacBeth (1973) regressions. We include all common stocks traded on NYSE,AMEX and NASDAQ exchanges from July 1963 to December 2015 withshare price above $1. For specification I, we use market beta as the explana-tory variable in Fama and MacBeth 1973 regressions; in specification II, weuse beta, size, value and momentum; in specification III we use beta, size,value, momentum, profitability, and investment. Beta is estimated using pastfive years of daily stock returns – correlations and volatilities are separatelyestimated over the five (min three) and one (min 120 days) year windows,respectively, and beta is calculated as the ratio of volatilities multiplied bycorrelation; if daily data is not available, we use past sixty (min 36) andtwelve monthly returns to calculate correlations and volatilities, respectively.We shrink beta towards one using shrinkage parameter of 1/3. Size is thenatural logarithm of firm’s market capitalization at the end of month t, valueis the natural logarithm of the ratio of firms book equity for the fiscal yearending in t-1 and market cap at the end of December of t-1; momentumis the total stock return from t-12 to t-2; profitability is the ratio of operat-ing profits and book equity at the fiscal year ending in t-1, and investmentis growth of total assets for the fiscal year ending in t-1. All variables arewinsorized at 1% and 99%, and we normalize all variables except beta.

I II III

alpha -0.74 -0.53 -0.40

(-3.86) (-2.68) (-2.11)slope 1.09 1.05 1.04

(25.34) (23.70) (24.22)


post estimated returns to market beta. If the linear relation betweenbeta and return holds, using beta-adjusted returns on the left-handside of the regressions should result in a zero coefficient for beta onthe right-hand side of the regressions; if instead, the coefficients arepositive (negative), the return to a unit of beta is higher (lower) thanpredicted. The results are shown in the first three columns of Table2.3. We find statistically significant, negative premiums for beta onthe right-hand side of the regressions, contrary to the predictions ofthe CAPM and its extensions, which is another confirmation of thelow-beta anomaly. We do note that the statistical significance weakensas we add more control factors, and even drops below the commonlyused 5% threshold level in the regression which includes profitabilityand investment factors. This last bit of ambiguity disappears whenwe consider not only beta but also volatility.

The use of beta-adjusted returns on the left-hand side also allowsus to let beta compete head-on with volatility, by simply addingvolatility to the set of explanatory variables on the right-hand side ofthe regressions. If any of the considered models hold, both beta andvolatility should be unpriced on the right-hand side. These results arereported in the last three columns of Table 3. Controlling for only betaand volatility (specification IV), beta remains the dominant source ofmispricing. Interestingly, this changes when we add other control fac-tors. In these specifications (V and VI) we observe that the negativealpha shifts entirely from beta to volatility, and that the statistical sig-nificance of this negative alpha of volatility is much larger, in absoluteterms, than the levels observed previously for the beta. Consequently,the addition of volatility renders the estimated beta mispricing com-pletely insignificant, suggesting that the low-volatility anomaly is thedominant phenomenon. This is consistent with the earlier findingsof Blitz and van Vliet 2007, who come to the same conclusion basedon the observation that alpha spreads are higher for volatility-sortedportfolios than for those sorted on beta.

2.5 robustness to the choice of profitability measure

The Fama and French 2015 five-factor model is motivated by the val-uation theory of asset pricing. One of the implications of this theoryis that high profitability, for a given level of investment and book-to-market, implies a high internal rate of return on expected dividends,and consequently, high expected returns. As their main measure of

2.5 robustness to the choice of profitability measure 25

Table 2.3: Fama and MacBeth 1973 results using beta-adjusted returns onthe left-hand side

This table reports the results of modified Fama and MacBeth 1973 regres-sions. Each month, we regress beta-adjusted stock returns on a set of charac-teristics and reported are the average coefficients and t-statistics calculatedusing Newey-West corrected standard errors. We include all common stockstraded on NYSE, AMEX and NASDAQ exchanges from July 1963 to Decem-ber 2015 with share price above $1. Beta is estimated using past five yearsof daily stock returns – correlations and volatilities are separately estimatedover the five (min three) and one (min 120 days) year windows, respectively,and beta is calculated as the ratio of volatilities multiplied by correlation; ifdaily data is not available, we use past sixty (min 36) and twelve monthlyreturns to calculate correlations and volatilities, respectively. We shrink betatowards one using shrinkage parameter of 1/3. Size is the natural logarithmof firm’s market capitalization at the end of month t, value is the naturallogarithm of the ratio of firms book equity for the fiscal year ending in t-1and market cap at the end of December of t-1; momentum is the total stockreturn from t-12 to t-2; profitability is the ratio of operating profits and bookequity at the fiscal year ending in t-1, and investment is growth of total as-sets for the fiscal year ending in t-1. All variables are winsorized at 1% and99%, and we normalize all variables except beta.

I II III IV V VI

intercept 1.04 0.86 0.74 1.07 0.32 0.30

(5.70) (4.85) (4.37) (4.62) (1.64) (1.54)beta -0.70 -0.51 -0.39 -0.75 0.06 0.08

(-3.46) (-2.35) (-1.90) (-3.91) (0.29) (0.41)volatility 0.01 -0.32 -0.27

(0.07) (-4.44) (-4.04)size -0.16 -0.22 -0.36 -0.38

(-1.80) (-2.68) (-6.01) (-6.39)value 0.18 0.15 0.15 0.12

(3.88) (3.47) (3.34) (2.85)momentum 0.34 0.32 0.36 0.34

(6.52) (6.42) (7.27) (6.98)profitability 0.21 0.17

(6.07) (5.73)investment -0.20 -0.20

(-9.89) (-9.69)R-square (%) 1.36 4.25 4.61 3.14 4.76 5.05


profitability, they opt for operating profitability, but this choice hasbeen challenged in some studies. For instance, Ball et al. 2016 showthat operating profitability is a poor measure of profitability as it in-cludes accounting accruals, that have been shown to predict negativefuture returns. They propose a cash-based profitability measure thatis purged of accruals and find that it dominates the Fama-Frenchvariable. Hou, Xue, and Zhang 2015 put forward the q-factor modelmotivated by the neoclassical q theory of investment. Their modelconsists of four factors: market, size, investment, and profitability,with their measure of profitability being the return on equity. In theinvestment model, the value factor is not necessary to explain thecross-section of stock returns, whereas the dividend discount modeldirectly implies its existence. Novy-Marx 2013, on the other hand, hashis own measure of profitability, gross-profits-to-assets. In this sec-tion, we run another robustness test using the profitability measuresthat have been proposed by Ball et al. 2016, Novy-Marx (2013, 2014),and Hou, Xue, and Zhang 2015 together with the respective controlsthat they have in their models. As Ball et al. 2016 do not explicitlypropose another asset pricing model, we use the Fama and French2015 five-factor model and replace operating profitability with cash-based operating profitability. Novy-Marx considers models with andwithout the momentum factor, so we also include the two alterna-tives. The market, size, and investment (asset growth) characteristicsof Fama and French 2015 and of Hou, Xue, and Zhang 2015 are de-fined in the same way. As all three of the above-mentioned papersexclude financial firms, we do so as well by removing all stocks withfour-digit SIC codes starting with 6. Results are presented in Table2.4. We also include the Fama and French 2015 model for comparisonpurposes, as the stock universe is restricted to non-financials.

In all model specifications, market beta remains unpriced in thecross-section of stock returns, while all other characteristics are eco-nomically and statistically significant. Consequently, all of these em-pirical models appear to be challenged by the same problem: the pre-dicted positive, linear relationship between market beta and returnsis not found in the cross-section.

2.6 robustness to measurement errors

Thus far, the analysis in this paper was conducted on an individual-firm level, which may raise the question of whether our results are

2.6 robustness to measurement errors 27

Table 2.4: Robustness to choice of profitability measure

This table reports the results of Fama and MacBeth 1973 regressions. Weinclude non-financial stocks traded on NYSE, AMEX and NASDAQ ex-changes from July 1963 to December 2015 with share price above $1. Betais estimated using past five years of daily stock returns – correlations andvolatilities are separately estimated over the five (min three) and one (min120 days) year windows, respectively, and beta is calculated as the ratio ofvolatilities multiplied by correlation; if daily data is not available, we usepast sixty (min 36) and twelve monthly returns to calculate correlations andvolatilities, respectively. We shrink beta towards one using shrinkage param-eter of 1/3. Size is the natural logarithm of firm’s market capitalization atthe end of month t, value is the natural logarithm of the ratio of firms bookequity for the fiscal year ending in t-1 and market cap at the end of Decem-ber of t-1; momentum is the total stock return from t-12 to t-2; operatingprofitability is the ratio of operating profits and book equity at the fiscalyear ending in t-1, cash-based operating profitability is operating profitabil-ity (of Fama-French) minus change in account receivables, inventory, andprepaid expenses, plus change in deferred revenue, account payable, andaccrued expenses. All changes are calculated on the year-to-year bases. Re-turn on equity as income before extraordinary items divided by book equity.Investment is growth of total assets for the fiscal year ending in t-1. All vari-ables are winsorized at 1% and 99%, and we normalize all variables exceptbeta. Reported are the average coefficients and t-statistics calculated usingNewey-West corrected standard errors.

FF BALL HXZ NM1 NM2

intercept 0.76 0.80 0.82 0.85 0.87

(4.27) (4.35) (4.42) (4.70) (5.04)beta 0.12 0.07 0.05 0.02 0.00

(0.40) (0.23) (0.15) (0.06) (0.00)size -0.19 -0.20 -0.21 -0.12 -0.15

(-2.27) (-2.23) (-2.58) (-1.28) (-1.71)value 0.16 0.19 0.23 0.23

(3.61) (3.96) (4.63) (4.69)investment -0.23 -0.16 -0.26

(-10.17) (-6.52) (-9.90)momentum 0.31

(5.87)operating profitability 0.23

(5.80)cash-based oper. prof. 0.28

(10.67)return on equity 0.18

(4.79)gross profitability 0.18 0.16

(5.20) (4.86)R-square (%) 4.92 4.84 4.52 4.83 5.51


affected by the errors-in-variables problem. If our independent vari-ables are measured with systematic errors, this may lead to biases inour estimates and incorrect inferences. In order to explicitly addressthis problem, we test the robustness of our findings using two differ-ent methodologies.

We already showed that our results are robust to two different waysof estimating market beta; however, here we explicitly address theconcern that our beta estimate is unpriced simply because it is riddledwith estimation errors. To this end, we form 50 portfolios by sortingstocks on their past market betas, calculate the beta of the overallportfolio, and assign this value to each stock in the correspondingportfolio. This approach is similar to that used in Fama and French1992 where they assign betas of size-beta portfolios to each stock inthe portfolio. Assuming that the errors in the estimation of single-stock betas are not perfectly correlated, this should lead to a reductionin estimation noise. Results are presented in Table 2.5 and show thatmarket beta remains unpriced in the cross-section in all three modelspecifications. We note that this conclusion does not depend on thenumber of portfolios used for stock grouping.

Second, instead of using individual-firm data, we form sets of port-folios and conduct our analysis on a portfolio level. We have tworequirements for stock grouping: (i) the resulting portfolios shouldexhibit significant variation in average (expected) returns; (ii) theyshould exhibit significant variation in characteristics. In the asset pric-ing literature, studies often use the 5x5 size-book-to-market sortedportfolios; however, not only do we consider these portfolios, butalso 5x5 portfolios sorted on size and each of the other control vari-ables included in our study. That is, each month we form 25 size-beta,25 size-value, 25 size-momentum, 25 size-profitability, and 25 size-investment sorted portfolios, and calculate the corresponding portfo-lio characteristics. Stocks are equal-weighted and the investment uni-verse and sample period are the same as in our base case. For eachof the portfolio sets, we run Fama and MacBeth 1973 cross-section re-gressions just as we do for single stocks. Results are reported in Table2.6.

Our conclusions do not change regardless of which set of test port-folios is used: beta is unpriced in the cross-section of stock returns.All other characteristics have the correct sign, although they are notalways significant. As a possible reason, we contend that some of ourdouble sorts do not produce enough variation in expected returns (in

2.6 robustness to measurement errors 29

Table 2.5: Fama and MacBeth 1973 results with portfolio beta

This table reports the results of Fama and MacBeth 1973 regressions. We in-clude all common stocks traded on NYSE, AMEX and NASDAQ exchangesfrom July 1963 to December 2015 with share price above $1. Beta is estimatedusing past five years of daily stock returns – correlations and volatilities areseparately estimated over the five (min three) and one (min 120 days) yearwindows, respectively, and beta is calculated as the ratio of volatilities mul-tiplied by correlation; if daily data is not available, we use past sixty (min36) and twelve monthly returns to calculate correlations and volatilities, re-spectively. This beta is shrunk towards one using shrinkage parameter of1/3. We form 50 portfolios by sorting stocks on their past market betas, cal-culate the beta of the overall portfolio, and assign this value to each stock inthe corresponding portfolio. Size is the natural logarithm of firm’s marketcapitalization at the end of month t, value is the natural logarithm of theratio of firms book equity for the fiscal year ending in t-1 and market capat the end of December of t-1; momentum is the total stock return from t-12

to t-2; profitability is the ratio of operating profits and book equity at thefiscal year ending in t-1, and investment is growth of total assets for thefiscal year ending in t-1. All variables are winsorized at 1% and 99%, andwe normalize all variables except beta. Reported are the average coefficientsand t-statistics calculated using Newey-West corrected standard errors.

I II III

intercept 1.02 0.84 0.74

(5.70) (4.87) (4.38)beta -0.19 0.00 0.11

(-0.70) (-0.01) (0.41)size -0.16 -0.21

(-1.80) (-2.68)value 0.18 0.15

(3.90) (3.50)momentum 0.34 0.32



(-9.90)R-square (%) 2.58 5.41 5.76


Table 2.6: Fama MacBeth results on portfolio-level

This table reports the results of Fama and MacBeth 1973 regressions con-ducted on the portfolio level. We include all common stocks traded onNYSE, AMEX and NASDAQ exchanges from July 1963 to December 2015

with share price above $1, and non-missing values for considered character-istics. Each month we form 25 size-beta, 25 size-value, 25 size-momentum,25 size-profitability, and 25 size-investment sorted portfolios, and calcu-late the corresponding portfolio characteristics. Portfolio returns are equal-weighted and the investment universe and sample period are the same asin our base case. Beta is estimated using past five years of daily stock re-turns – correlations and volatilities are separately estimated over the five(min three) and one (min 120 days) year windows, respectively, and beta iscalculated as the ratio of volatilities multiplied by correlation; if daily datais not available, we use past sixty (min 36) and twelve monthly returns tocalculate correlations and volatilities, respectively. We shrink beta towardsone using shrinkage parameter of 1/3. Size is the natural logarithm of firm’smarket capitalization at the end of month t, value is the natural logarithmof the ratio of firms book equity for the fiscal year ending in t-1 and marketcap at the end of December of t-1; momentum is the total stock return fromt-12 to t-2; profitability is the ratio of operating profits and book equity atthe fiscal year ending in t-1, and investment is growth of total assets for thefiscal year ending in t-1. All variables are winsorized at 1% and 99%, andwe normalize all variables except beta. Reported are the average coefficientsand t-statistics.

size- size- size- size- size-beta value momentum profitability investment

intercept 0.93 0.86 1.12 0.64 0.57

(4.31) (2.58) (3.59) (1.65) (1.53)beta -0.12 -0.10 -0.30 0.13 0.24

(-0.43) (-0.25) (-0.75) (0.27) (0.55)size -0.16 -0.24 -0.18 -0.22 -0.15

(-1.94) (-3.08) (-2.06) (-2.41) (-1.88)value 0.03 0.18 0.14 0.13 0.07

(0.44) (2.75) (2.17) (2.45) (1.36)momentum 0.12 0.07 0.34 0.06 0.05

(2.21) (1.62) (6.49) (1.44) (1.27)profitability 0.05 0.21 0.17 0.18 -0.00

(0.76) (3.77) (2.88) (3.28) (-0.07)investment -0.10 -0.09 -0.06 -0.04 -0.15

(-3.57) (-2.83) (-2.10) (-1.60) (-4.85)R-square (%) 60.12 54.94 58.28 53.89 55.01

2.7 conclusion 31

the case of size-beta), or not enough dispersion in certain portfoliocharacteristics. Nevertheless, regardless of how we group stocks, ourconclusions do not change. In unreported tests, we also use industry-sorted portfolios, as suggested by Lewellen, Nagel, and Shanken 2010,as well portfolios sorted on value and profitability, the two factors thatarguably explain the low-risk anomaly, and also find our conclusionsunchanged.

2.7 conclusion

We find that exposure to market beta in the cross-section is not re-warded with significantly higher returns, regardless of whether onecontrols for the additional factors proposed by Fama and French(2015, 2016). At the same time, the constant in the regressions, whichought to be zero according to their asset pricing model, is large andsignificant. Taken together these results imply that the relation be-tween risk and return in the cross-section is flat instead of positive.We also find that the mispricing is even more pronounced for volatil-ity than for beta. This suggests that the low-volatility anomaly isstronger than the low-beta anomaly, and, given that the two are closelyrelated, that the low-volatility anomaly is the dominant one. We chal-lenge the robustness of our results using different profitability mea-sures that have been discussed in the literature and find that none ofthem is able to establish a positive relationship between market betaand stock returns. We also find that our results are robust to usingcharacteristics-sorted portfolios instead of individual stocks, whichaddresses the concern that our findings might be affected by theerrors-in-variables problem.

Of course, the results in this paper represent just one attempt atobtaining a positive risk-return relation by controlling for the factorsthat supposedly explain the low-risk anomaly. The fact that this at-tempt is unsuccessful does not rule out that portfolios constructed ina different manner do exhibit a clear positive risk-return relation con-sistent with the predictions of the Fama and French 2015 and Novy-Marx 2015 models. For instance, the market betas or factor exposuresused in this paper might not be appropriate, and it is possible thata different methodology would lead to different conclusions. But aslong as the data indicates that portfolios with higher risk do not gen-erate higher returns, it is premature to conclude that the low-riskanomaly has been resolved.


2.8 appendix

Table 2.7: Fama and MacBeth 1973 results with multivariate beta

This table reports the results of Fama and MacBeth 1973 regressions. We in-clude all common stocks traded on NYSE, AMEX and NASDAQ exchangesfrom July 1968 to December 2015 with share price above $1. Beta is the slopecoefficient on the market factor estimated using multivariate regressions ofstock excess returns on the five factor model from t-60 to t-1 (min t-24 tot-1). We shrink beta towards one using shrinkage parameter of 1/3. Size isthe natural logarithm of firm’s market capitalization at the end of montht, value is the natural logarithm of the ratio of firms book equity for thefiscal year ending in t-1 and market cap at the end of December of t-1; mo-mentum is the total stock return from t-12 to t-2; profitability is the ratioof operating profits and book equity at the fiscal year ending in t-1, andinvestment is growth of total assets for the fiscal year ending in t-1. All vari-ables are winsorized at 1% and 99%, and we normalize all variables exceptbeta. Reported are the average coefficients and t-statistics calculated usingNewey-West corrected standard errors.

I II III

intercept 0.71 0.62 0.59

(3.14) (2.89) (2.71)beta -0.02 0.08 0.11

(-0.15) (0.79) (1.15)size -0.09 -0.15

(-1.07) (-2.06)value 0.26 0.20

(4.09) (3.64)momentum 0.35 0.33



(-9.77)R-square (%) 0.58 3.58 4.05

2.8 appendix 33

Table2.

8:Fama

andM

acBeth1

97

3in

differentsize

market

segments

Thistable

reportsthe

resultsofFam

aand

MacBeth

19

73

regressionsin

threesize

marketsegm

ents:Large,Smalland

Micro

caps.We

includeallcom

mon

stockstraded

onN

YSE,A

MEX

andN

ASD

AQ

exchangesfrom

July1

96

8to

Decem

ber2

01

5w

ithshare

priceabove

$1.Large

stocksare

thosew

ithm

arketcapitalizationabove

median

capitalizationof

NY

SElisted

stocks;Smallstocks

arethose

belowthis

threshold,andM

icrostocks

arethose

with

capitalizationbelow

20th

percentileof

capitalizationof

NY

SEstocks.Beta

isestim

atedusing

pastfive

yearsof

dailystock

returns–

correlationsand

volatilitiesare

separatelyestim

atedover

thefive

(min

three)and

one(m

in1

20

days)year

window

s,respectively,andbeta

iscalculated

asthe

ratioof

volatilitiesm

ultipliedby

correlation;ifdaily

datais

notavailable,w

euse

pastsixty

(min

36)

andtw

elvem

onthlyreturns

tocalculate

correlationsand

volatilities,respectively.We

shrinkbeta

towards

oneusing

shrinkageparam

eterof

1/3.Size

isthe

naturallogarithmof

firm’s

market

capitalizationat

theend

ofm

ontht,value

isthe

naturallogarithmof

theratio

offirm

sbook

equityfor

thefiscalyear

endingin

t-1

andm

arketcap

atthe

endof

Decem

berof

t-1;m

omentum

isthe

totalstockreturn

fromt-

12

tot-

2;profitabilityis

theratio

ofoperating

profitsand

bookequity

atthe

fiscalyear

endingin

t-1,and

investment

isgrow

thof

totalassets

forthe

fiscalyear

endingin

t-1.A

llvariables

arew

insorizedat

1%and

99%

,andw

enorm

alizeall

variablesexcept

beta.Reported

arethe

averagecoefficients

andt-statistics

calculatedusing

New

ey-West

correctedstandard

errors.

Above

NY

SEm

edianBelow

NY

SEm

edianBelow

NY

SE20

percentile

III

IIII

IIIII

III

III

intercept0.

96

1.04

0.91

1.15

0.92

0.80

1.17

0.83

0.73

(3.

85)

(4.

81)

(4.

36)

(5.

69)

(4.

67)

(4.

22)

(5.

59)

(4.

11)

(3.

70)

beta-0.

29

-0.

38

-0.

25

-0.

33

-0.

08

0.05

-0.

36

0.02

0.14

(-0.

84)

(-1.

22)

(-0.

83)

(-1.

01)

(-0.

23)

(0.

15)

(-1.

01)

(0.

06)

(0.

42)

size-0.

07

-0.

08

-0.

08

-0.

13

-0.

16

-0.

20

(-1.

67)

(-2.

07)

(-1.

03)

(-1.

83)

(-2.

29)

(-3.

07)

value0.

08

0.13

0.23

0.16

0.24

0.16

(1.

81)

(2.

33)

(4.

26)

(3.

38)

(4.

31)

(3.

12)

mom

entum0.

21

0.20

0.36

0.34

0.41

0.39

(3.

18)

(3.

24)

(6.

31)

(6.

21)

(7.

66)

(7.

52)

profitability0.

13

0.24

0.26

(3.

51)

(6.

13)

(5.

61)

investment

-0.

10

-0.

25

-0.

27

(-4.

37)

(-10.

85)

(-10.

37)

R-square

(%)

5.06

8.81

9.44

2.28

4.25

4.57

2.07

3.62

3.94

3T H E I D I O S Y N C R AT I C M O M E N T U M A N O M A LY

This chapter is jointwork with David

Blitz and MatthiasX. Hanauer.

3.1 introduction

The momentum effect is one of the most pervasive asset pricing anoma-lies documented in the financial literature: stocks with the highestreturns over the past six to twelve months continue to deliver above-average returns in the subsequent period (see Jegadeesh and Titman1993, 2001). Momentum strategies are known to exhibit significantdynamic exposures to systematic risk factors (styles). For instance, inbull markets high-beta stocks tend to, on average, outperform low-beta stocks, and a zero-investment momentum factor has a net pos-itive exposure to the market factor. The opposite happens in bearmarkets. Such exposures can be particularly hurtful during style re-versals: the Fama-French momentum factor returned -83% in 2009

when stocks that had suffered the largest losses during the financialcrisis made a strong recovery.

Grundy and Martin 2001 show that dynamic hedging of the mo-mentum strategy’s market and size exposures substantially reducesthe volatility of the strategy without a loss in return, but Daniel andMoskowitz 2016 show that the superior performance of their strategyis crucially dependent on the fact that they use ex-post factor betasto hedge these exposures. A hedging strategy based on ex-ante be-tas does not generate the same improvement. Gutierrez and Pirinsky2007 propose an alternative method to reduce these systematic styletilts by making individual stock returns in the ranking period orthog-onal to the three factors that explain a major part of the variation inaverage returns - the market, size, and value factors. Using this ap-proach, the authors document that, after a similar performance in thefirst year after formation, this idiosyncratic1 momentum strategy con-tinues to generate abnormal returns for years, while the total return

1 Gutierrez and Pirinsky 2007 refer to this effect as abnormal return momentum, andBlitz, Huij, and Martens 2011 refer to it as residual momentum.

34

3.1 introduction 35

momentum strategy reverses strongly. Although their results suggestthat the performance difference in the first year after formation isnegligible, Blitz, Huij, and Martens 2011 observe that the idiosyn-cratic momentum strategy exhibits only half of the volatility of theconventional momentum strategy without a significant reduction inreturn, thus doubling the Sharpe ratio of the strategy. However, nei-ther Gutierrez and Pirinsky 2007 nor Blitz, Huij, and Martens 2011

address one of the fundamental asset pricing questions, namely, ifthe idiosyncratic momentum is a distinct factor that expands the effi-cient frontier comprised of already documented factors. The inclusionof the total return momentum to the set of control variables in assetpricing tests is of paramount importance that was overlooked by pre-vious studies, as it is not a priori clear if idiosyncratic momentumcontains information about expected returns that is not contained intotal return momentum or a linear combination of factors that formthe basis of the established asset pricing models (such as the Famaand French 2015, five-factor model) augmented with the total returnmomentum factor2. This is also in line with the arguments in Barillasand Shanken 2017, 2018 that all factors should be considered jointly.This paper provides strong evidence that the idiosyncratic momen-tum is a distinct phenomenon from the conventional momentum.

Using a set of time-series, cross-section, and factor-spanning tests,we show that the idiosyncratic momentum cannot be explained byany of the established asset pricing factors, such as market, size, value,operating profitability, and investment, even if the total return mo-mentum factor is included. In fact, the idiosyncratic momentum sub-sumes the total return momentum in some tests, while the converse isnever the case. The recently proposed alternative asset pricing mod-els of Hou, Xue, and Zhang 2015 and Stambaugh and Yuan 2017,which have been shown to explain the total return momentum, fail toexplain its idiosyncratic counterpart.

Furthermore, we examine the links between the idiosyncratic mo-mentum and its conceptually related idiosyncratic volatility anomalyand find a much weaker relationship between these two effects thanbetween the idiosyncratic volatility and the total return momentum.We also find that, relative to the idiosyncratic momentum, the totalreturn momentum tends to be invested in stocks of smaller marketcapitalization with higher levels of Amihud 2002 illiquidity where

2 Gutierrez and Pirinsky 2007 and Blitz, Huij, and Martens 2011 apply only the Famaand French 1993 three factor model that was still the “industry standard” Subrah-manyam 2009, p. 45 back then.

36 the idiosyncratic momentum anomaly

the limits to arbitrage likely play a larger role in preserving the pre-mium. While the idiosyncratic momentum is associated with a some-what higher level of turnover, we show that the break-even transac-tions costs necessary to render it insignificant are 15% higher than forthe conventional momentum. The idiosyncratic momentum stocks ex-hibit a lower average level of the idiosyncratic volatility and highermarket capitalization relative to the conventional momentum stocks,which have been shown to be, respectively, positively and negativelyrelated to the transactions costs of a strategy by, for instance, Novy-Marx and Velikov 2016 and Frazzini, Israel, and Moskowitz 2018. Weconjecture that the transactions costs of trading the idiosyncratic mo-mentum are likely to be lower than those of trading the conventionalmomentum. Therefore, we conclude that the total costs of trading theidiosyncratic momentum are lower or at least no higher than those oftrading the convectional momentum.

When examining the importance of factors with respect to whichstock excess returns are orthogonalized in order to obtain idiosyn-cratic momentum scores, we find that the market is by far the mostimportant one. This should not come as a surprise as it is the factorwith the highest risk premium, volatility, and power in explaining thevariation in returns in the time series. Adding size (SMB) and value(HML) factors further enhances risk-adjusted returns of the strategy,but RMW and CMA add value only marginally.

Moskowitz and Grinblatt 1999 show that industry portfolios ex-hibit significant momentum that is not subsumed by the individualfirm momentum and other standard asset pricing factors. We testthe strength of the idiosyncratic momentum on an industry level andfind that, while present, it is less important than in the case of the con-ventional momentum. Consistent with this, we find that adding theindustry portfolios to the set of factors against which we orthogonal-ize stock returns to lower the return of the idiosyncratic momentumstrategy, but to lower the risk (volatility) even more thus leading to ahigher Sharpe ratio. This indicates that the industry exposures in theidiosyncratic momentum do contribute to the risk of the strategy, butthat this risk is not fully compensated by the market.

We also provide a fresh perspective on the various explanationsfor the momentum phenomenon that have been put forwarded inthe literature. These include investor overconfidence, investor over-and underreaction, as well as risk-based explanations. Gutierrez andPirinsky 2007 argue that idiosyncratic momentum is an underreac-

3.1 introduction 37

tion phenomenon, caused by gradual diffusion of information, giventheir finding that abnormal returns do not reverse over multi-yearholding periods. Prior research has established links between conven-tional momentum profits and investors’ overconfidence, overreaction,and risk-based explanations. If idiosyncratic momentum is a distinctphenomenon that is driven by something else, such as investor un-derreaction, one would expect such links to be absent, or at leastmuch less pronounced. If, on the other hand, idiosyncratic momen-tum and total return momentum are driven by the same underlyingmarket mechanisms, the superiority of idiosyncratic might simply bedue to more extreme exposures to these sources. We empirically testthis and reject the latter hypothesis, i.e. we find that the strong linkbetween conventional momentum and investors’ overconfidence oroverreaction, as well as risk-based explanations, is much weaker foridiosyncratic momentum.

Our results support the underreaction hypothesis. First, we itera-tively run sixty Fama and MacBeth 1973 regressions with varyinglags of the two momentum signals, controlling for the other knownpredictors of stock returns in the cross-section, and find that idiosyn-cratic momentum forecasts high short and long-term excess returns,while conventional momentum forecasts high short-term, and nega-tive long-term excess returns. Second, as conventional and idiosyn-cratic momentum strategies are positively correlated, we argue thatone can use idiosyncratic momentum as a signal to distinguish be-tween momentum stocks with high future returns, that are morelikely to be caused by underreaction, and those whose returns re-verse, consistent with initial overreaction and long-term reversal. Wefind evidence in support of this argument and show that a portfoliothat is long idiosyncratic momentum winners and short losers withinpast conventional momentum winners generates high, non-revertingreturns over the next five years. On the other hand, a portfolio that islong conventional momentum winners and short losers within pastidiosyncratic momentum winners generates negative long-term re-turns.

The final contribution of this paper is to document that idiosyn-cratic momentum shows robust out-of-sample performance in inter-national developed and emerging equity markets. Our results are con-sistent with Chaves 2016, who finds strong results for a simplifieddefinition of idiosyncratic momentum in 21 developed countries, in


addition to the U.S.3 In our study, we use the original definition andapply it uniformly in all considered regions, including emerging mar-kets, which have not been examined before. The work of Chaves 2016

also shows that the effect is robust to the methodological choices.Conventional momentum is known to be ineffective in Japan, at

least unconditionally, therefore giving rise to data mining results.In line with Chaves 2016 and the recent work of Chang et al. 2018

we find that idiosyncratic momentum does work in Japan. Chang etal. 2018 specifically examine idiosyncratic momentum in Japan andlink it to explanations based on investor underreaction. We add tothis existing literature by providing additional evidence for the un-derreaction explanation. Specifically, we show that similar to the U.S.,idiosyncratic momentum profits in international markets remain pos-itive up to five years after portfolio formation, while conventionalmomentum profits already start reversing after less than one year.

This paper is organized as follows: In section 3.2, we discuss vari-ous explanations for momentum that have been proposed in the lit-erature and their links to idiosyncratic momentum. In section 3.3, wedescribe the data and methodology used to construct idiosyncraticmomentum. In section 3.4, we present results of asset pricing tests,and analyze the importance of factors in residualization. In section3.5, we show that none of the explanations for sources of the momen-tum premium hold for idiosyncratic momentum, and discuss the linkbetween idiosyncratic momentum profits and underreaction. Section3.6 presents international results, and section 3.7 concludes the paper.

3.2 discussion

While momentum is one of the most pervasive asset pricing anoma-lies, it is also one of the least systematic, in the sense that the compo-sition of the momentum portfolio is solely determined by the recentperformance of the stocks in the investment universe. Kothari andShanken 1992 show that past return sorted portfolios have significanttime-varying exposure to systematic factors. Consequently, the longleg of momentum has positive exposures to the styles that performedwell in the recent past, and the short leg is exposed to those that un-derperformed. An intuitive example is the relative outperformanceof high-beta stocks in bull markets, and a contemporaneous under-

3 Chaves 2016 considers one-factor (market) unscaled residuals estimated over thepast 12-2 month window, whereas we use three-factor model volatility-scaled 12-2month residuals estimated over the past 36 months.

3.2 discussion 39

performance of low-beta stocks. As a direct consequence of this, mo-mentum can exhibit negative returns if the market experiences sharpturns.

Daniel and Moskowitz 2016 demonstrate that not only does themarket beta of the momentum portfolio differ depending on the pastmarket performance, but also that after bear markets, the beta of themomentum portfolio becomes even more negative when the marketsubsequently reverses. The authors assert that, in bear markets, themomentum portfolio behaves like a short call option on the market,meaning that it gains little if the market further declines, but if themarket rises, the portfolio loses a lot. Moreover, they document thatthe loser portfolio is the predominant source of this optionality, andargue that this evidence is consistent with the theory of Merton 1974

in which common equity is viewed as a call option on the value ofthe firm. Especially after a bear market environment, stocks of theloser portfolio are not as deep in-the-money as stocks of the winnerportfolio and consequently have a stronger option-like behavior.

If this momentum crash risk is the driver of momentum returns,and idiosyncratic momentum is simply more exposed to this risk, thesuperiority of idiosyncratic momentum could be explained. However,since idiosyncratic momentum explicitly attempts to eliminate thetime-varying style exposures that seem to be the source of the crashrisk, it may also turn out to be less prone to crashes. We empiricallytest this hypothesis and find that idiosyncratic momentum is signifi-cantly less exposed to the crash risk. Consequently, this cannot be anexplanation for its superiority over conventional momentum.

Cooper, Gutierrez, and Hameed 2004 find that momentum returnsare positive following periods of positive market returns, and neg-ative after periods of negative market returns, and argue that thisbehavior is consistent with the overreaction hypothesis for the exis-tence of the momentum premium. They further link the overreactionhypothesis to cognitive biases described in Daniel, Hirshleifer, andSubrahmanyam 1998 and Hong and Stein 1999.

Daniel, Hirshleifer, and Subrahmanyam 1998 assume that investorsare overconfident about their private information, and this overconfi-dence, together with self-attribution bias, leads to an overreaction thatdrives momentum returns. As overconfidence tends to be greater af-ter bull than after bear markets, as argued in Gervais and Odean 2001,overreaction and, therefore, momentum returns, are higher after bullmarkets. In the behavioral model of Hong and Stein 1999 an initial


underreaction to information, and subsequent overreaction, driven bymomentum traders, leads to momentum returns. According to theirmodel, overreaction, and momentum returns, are negatively corre-lated with the risk aversion of momentum traders. As risk aversiondecreases with wealth (e.g Campbell and Cochrane 1999), momentumreturns should be higher after market increases.

We replicate and confirm the findings of Cooper, Gutierrez, andHameed 2004 for conventional momentum; however, we find that inthe case of idiosyncratic momentum, returns are still positive, albeitinsignificant, after periods of negative market returns, and not sta-tistically different from the returns after periods of positive marketreturns. Therefore, this overreaction explanation also does not applyto idiosyncratic momentum.

Asem and Tian 2010 further investigate the asymmetric momentumprofits following bull and bear markets. They show that momentumreturns for different market states (as shown in Cooper, Gutierrez,and Hameed 2004) are dominated by returns for different marketdynamics, where the subsequent market return is also taken intoaccount. They document that momentum returns are significantlyhigher when the market stays in the same condition than when it tran-sitions to another state. Asem and Tian 2010 argue that this pattern isconsistent with the model of Daniel, Hirshleifer, and Subrahmanyam1998, but not with the competing models of Hong and Stein 1999 orSagi and Seasholes 2007 that predict high momentum returns in caseof a market reversal after a bear market.

In the model of Daniel, Hirshleifer, and Subrahmanyam 1998 tradersreceive public signals after trading a stock based on a private signal. Ifthe public signal confirms their private signal, the investors attributethe success to their skills, however, they attribute non-confirming sig-nals to bad luck. Because of this self-attribution bias, traders becomeoverconfident about their stock selection skills, and this overconfi-dence drives momentum. Asem and Tian 2010 argue that investors,on average, traded more based on positive (negative) private signalswhen the past market was positive (negative). Consequently, subse-quent positive months should drive overconfidence more than subse-quent negative months and vice versa. Therefore, momentum returnsshould also be higher for market continuations than for market rever-sals.

We validate the findings of Asem and Tian 2010 that results fordifferent market states are dominated by results for different mar-

3.2 discussion 41

ket dynamics, but again we show that idiosyncratic momentum isless affected by these market dynamics than its total return counter-part, which unequivocally indicates that overconfidence if capturedby these metrics, cannot explain it.

Gutierrez and Pirinsky 2007 argue that incentives of delegated wealthmanagers may lead them to underreact to firm-specific news andoverreact to relative (total) past returns. Using data on institutionalownership, they show that institutional investors buy/sell total re-turn winners/losers significantly more than any other stock in theuniverse, a pattern that is largely absent in the case of their idiosyn-cratic momentum counterparts. This empirical observation is in linewith the fact that institutional investors largely neglect firm-specificreturns. The authors find that high momentum stocks with the small-est change in institutional ownership during the formation periodare precisely those that have the highest long-term returns, and con-versely, high momentum stocks that are bought the most by institu-tions during the formation period are those that exhibit the strongestreversals. Thus, long-term returns of momentum strategies dependon the level at which they are held by institutions4. As total returnmomentum stocks are subject to substantially higher changes in insti-tutional ownership than idiosyncratic momentum stocks, the authorscontend that this effect is driven by investor’s overreaction, while thelatter one can be attributed to underreaction.

We confirm the findings of Gutierrez and Pirinsky 2007 that whileconventional momentum forecast high short to medium-term returns,its significance drops to zero fairly quickly and, consistent with Je-gadeesh and Titman 2001, turns into a long-term reversal after aroundone year following portfolio formation. In contrast, idiosyncratic mo-mentum forecasts high (or at least non-negative) short and long-termreturns. Differently from Gutierrez and Pirinsky 2007, we use Famaand MacBeth 1973 regressions with lagged conventional and idiosyn-cratic momentum signals (up to 60 lags). This approach enables us tocontrol for other known predictors of stock returns, such as marketbeta, book-to-market, size, profitability, and investment. These resultsare consistent with the underreaction and overreaction hypothesis foridiosyncratic and conventional momentum, respectively.

Related to our work is that of Haesen, Houweling, and Zundert2015 who document idiosyncratic momentum spillover effects fromthe equity to the credit market. In particular, they find that stocks

4 This assumes that institutions are marginal investors in the market.


with the highest past idiosyncratic returns are also future winnersin the credit market. The momentum spillover effect, whereby com-panies whose stocks are past total return winners have high creditreturns going forward, has been documented by Gebhardt, Hvidk-jaer, and Swaminathan 2005, however, Haesen, Houweling, and Zun-dert 2015 show that this effect has a structural bias towards low-riskcredits and is, consequently, dependent on the performance of thecredit market during the portfolio holding period. They also showthat the default risk exposure of momentum spillover depends onthe equity market return during the formation period. On the otherhand, the idiosyncratic momentum spillover is substantially less ex-posed to these systematic biases, and cannot be explained by any ofthe other credit market factors, such as default and term, nor theFama-French-Carhart5 factors. Their results indicate that the idiosyn-cratic momentum spillover effect also stems from underreaction.

3.3 data and methodology

3.3.1 Data

The data used in this study come from multiple sources. For the U.S.sample, we obtain security level information from the Center for Re-search in Security Prices (CRSP) from the end of December of 1925 tillthe end of December of 2015. We include all common shares (sharecodes 10 and 11) that are traded on the NYSE/AMEX and NASDAQexchanges (exchange codes 1, 2, and 3) except those with the begin-ning of month share price below $1. For the bulk of the analysis,we also exclude microcaps6, defined as stocks with market capitaliza-tion below the 20th percentile market capitalization of NYSE-tradedstocks, in order to dismiss concerns that our results are driven bytiny stocks that are out of reach for institutional investors or marketmicrostructure issues.

We estimate market betas using univariate regressions of excessstock returns on the market factor over the most recent sixty months(minimum twenty-four). Size is defined as the natural logarithm ofa firm’s market capitalization (shares outstanding times price per

5 Carhart 1997 introduces the momentum factor.6 The only exception is when we construct the value-weighted idiosyncratic momen-

tum factor, as, for consistency, we follow the standard Fama and French 1993 factorconstruction methodology, and they also do not exclude microcaps. The portfoliosare, however, value-weighted and that ensures that micro-caps do not dominate theportfolio returns.

3.3 data and methodology 43

share). The balance sheet and income statement information usedin the cross-section tests stems from Compustat’s annual files. Bookvalue is the sum of book value of stockholders’ equity, balance sheetdeferred taxes and investment tax credit (if available), minus the bookvalue of preferred stock. If available, we use the redemption, liquida-tion, or par value to calculate the book value of preferred stock. Stock-holders’ equity is obtained either from Moody’s industrial manualsor Compustat. If it is not available, we measure stockholders’ equitypreferably as the sum of book value of common equity and the parvalue of preferred stock, or the book value of assets minus total li-abilities if the first one is not available. The book value of equity isthen divided by the market capitalization calculated at the end of theprevious calendar year to obtain the book-to-market ratio, which isfurther log-transformed. The 12-2 month total return momentum isthe total return from month t-12 to t-2. Operating profitability is de-fined as annual revenues minus cost of goods sold, interest expense,and selling, general, and administrative expenses divided by book eq-uity for the last fiscal year end in t-1, and investment (asset growth)is the percentage change in firms’ total assets from year t-2 to t-1. Ac-counting data for a given fiscal year are updated once a year at theend of June of the following calendar year. Idiosyncratic volatility isthe standard deviation of residuals from a regression of stock excessreturns on the three Fama and French 1993 factors over the last month(twenty-two, minimum sixteen trading days). Amihud 2002 illiquid-ity measure is defined as the ratio of absolute stock return to its dollarvolume averaged over the last month (twenty-two, minimum sixteentrading days).

As a proxy for the risk-free rate, we use the one-month U.S. Trea-sury bill rate that, together with the Fama-French factor returns thatare used to construct idiosyncratic momentum in the United States,is obtained from the website of Professor Kenneth French7. The Hou,Xue, and Zhang 2015 Q-factor model return series come from Profes-sor Lu Zhang8, and returns of the mispricing factors of Stambaughand Yuan 2017 come from the website of Professor Yu Yuan9.

7 http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html8 We thank Professor Lu Zhang for providing these return series.9 http://www.saif.sjtu.edu.cn/facultylist/yyuan/

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


http://www.saif.sjtu.edu.cn/facultylist/yyuan/

http://www.saif.sjtu.edu.cn/facultylist/yyuan/


3.3.2 Variable construction

We calculate idiosyncratic momentum in multiple stages, followingthe methodology of Gutierrez and Pirinsky 2007 and Blitz, Huij, andMartens 2011. First, each month, we estimate model (1) over the past36-month window for all stocks in the investment universe. We re-quire the full 36-month return history to estimate the model.

Ri,t − Rf,t = αi +βmkt,i · (Rmkt,t − Rf,t)+

βhml,i · Rhml,t +βsmb,i · Rsmb,t + εi,t(3.1)

In the second step, we calculate idiosyncratic returns as:

ei,t = Ri,t − Rf,t − αiι− βmkt,i · (Rmkt,t − Rf,t)−

βhml,i · Rhml,t − βsmb,i · Rsmb,t(3.2)

Finally, the idiosyncratic momentum score is the last 12-2 monthvolatility-adjusted mean idiosyncratic return10:

IdiosyncraticMomentumi,t =

∑t−2t−12 ei,t√∑t−2

t−12(ei,t − ei)2(3.3)

Our results are robust to a host of commonly used portfolio con-struction methodologies. For the base case, we form equal-weightedportfolios, and address some of the concerns associated with equal-weighting by excluding micro-caps, as defined in subsection 3.3.1,from the investment universe. Fama and French 2008 note that theseare stocks that represent around 60% of the universe, but account foronly 3% of total market capitalization. Equal-weighting ensures thatportfolios have enough breadth, as returns on value-weighted portfo-lios can be heavily dominated by returns on a small number of verylarge stocks.

Furthermore, we validate our findings by constructing an idiosyn-cratic momentum factor following the portfolio construction method-ology that Fama and French use to the construct the conventionalmomentum (WML) factor. Thus, the factor is based on the six size -idiosyncratic momentum sorted value-weighted portfolios, and it isa zero-investment portfolio that is long small and big (idiosyncratic)

10 We follow Blitz, Huij, and Martens 2011 and scale residuals with their volatilities.This adjustment leads to a slight improvement in the signal, especially for the topportfolio, but our results do not hinge on it. In fact, the unscaled top-bottom id-iosyncratic momentum portfolio generates absolute and risk-adjusted returns thatare comparable in magnitude to those of the scaled version.

3.3 data and methodology 45

winners, and short small and big (idiosyncratic) losers. Portfolios arereformed monthly and stocks are held for one month.

iMOMt =1

2(BigidioWinnerst + SmallidioWinnerst)

−1

2(BigidioLoserst + SmallidioLoserst)

(3.4)

We confirm that all conclusions in our paper remain qualitative un-changed if we use the value-weighted iMom factor portfolio, insteadof the decile spread portfolio. The reason we opt to use the latteras a base case is that most papers that we reference and replicate inthis paper use decile portfolios, and we do not want to depart fromthis choice. We do, however, challenge the robustness of this choiceall-throughout.

We provide yet another robustness check, where we show that ourresults hold if we consider only large stocks - commonly defined asstocks with a beginning-of-month market capitalization above the me-dian market capitalization of NYSE-traded stocks.

Hou, Xue, and Zhang 2017 find that many anomalies documentedover the past decades do not survive these robustness tests and thatthe strong results reported in the original studies are oftentimes drivenby micro-caps. We establish that idiosyncratic momentum is not amicro-cap phenomenon, but a robust and pervasive effect that consis-tently shows up throughout the entire cross-section of US equities.

3.3.3 Motivating results

In Table 3.1, we report descriptive statistics of the decile portfoliosformed as univariate sorts on idiosyncratic, as well as total returnmomentum, over the 1963-2015 period. Results for the long sample(1929-2015) can be found in Table 3.15 in the appendix, and they areconsistent with what we observe in the shorter sample.

We observe a monotonically increasing pattern in excess returns,Sharpe ratios, and factor-adjusted returns (i.e. alphas) going fromlow (D1) to high (D10) idiosyncratic and total return momentumportfolios. The self-financing D10-D1 idiosyncratic momentum port-folio generates a monthly return of 0.98%, which is somewhat lowerthan that of total return momentum (1.07%), however, with substan-tially lower volatility. The Sharpe ratio of the idiosyncratic momen-tum strategy is 0.29 per month, 77% higher than that of conventionalmomentum (0.17). The CAPM and three-factor alphas are higher for


Table3.

1:Performance

ofdecile

portfolios

Thistable

reportsperform

ancecharacteristics

ofdecileportfolios

constructedas

univariatesorts

onidiosyncratic

andtotalreturn

mom

entum,respectively.W

einclude

allcom

mon

stockstraded

onN

YSE,A

MEX

,andN

ASD

AQ

exchangesfrom

July1

96

3to

Decem

ber2

01

5above

20th

percentileof

market

capof

NY

SEtraded

stocksand

with

shareprice

above$

1,with

validtotalreturn

andidiosyncratic

mom

entumscores.Totalreturn

mom

entumis

definedas

the1

2-2

month

totalstockreturn

andidiosyncratic

mom

entumis

the1

2-2

month

volatility-scaledidiosyncratic

returnestim

atedover

past3

6m

onthsusing

theFam

aand

French1

99

3three-factor

model.

Foreach

portfolio,we

reportreturns

inexcess

ofthe

risk-freerate,volatility,ex-post

Sharperatios,C

APM

-,three-factor-,andfive-factor

alphasand

correspondingt-

statistics.Also

reportedare

theG

RS

teststatisticsfor

eachofthe

correspondingassetpricing

models,w

herethe

testassetsare

totalreturnand

idiosyncraticm

omentum

sorteddeciles.Portfolios

areequal-w

eightedand

reformed

monthly.

ExcessR

eturnVol

SharpeR

atioA

lphaC

APM

tstatA

lpha3FM

tstatA

lpha5FM

tstat

IdiosyncraticM

omentum

D1

0.22

5.92

0.04

-0.

38

(-3.

48)

-0.

53

(-5.

81)

-0.

47

(-5.

04)

D2

0.43

5.54

0.08

-0.

13

(-1.

47)

-0.

30

(-4.

26)

-0.

32

(-4.

43)

D3

0.52

5.29

0.10

-0.

03

(-0.

41)

-0.

20

(-3.

58)

-0.

24

(-4.

17)

D4

0.64

5.20

0.12

0.10

(1.

28)

-0.

08

(-1.

62)

-0.

11

(-2.

14)

D5

0.73

5.12

0.14

0.20

(2.

66)

0.02

(0.

39)

-0.

03

(-0.

72)

D6

0.77

5.10

0.15

0.24

(3.

21)

0.07

(1.

69)

0.02

(0.

42)

D7

0.82

5.09

0.16

0.28

(3.

97)

0.12

(2.

8)0.

06

(1.

49)

D8

0.87

5.09

0.17

0.34

(4.

46)

0.18

(3.

94)

0.14

(2.

99)

D9

1.01

5.26

0.19

0.46

(5.

74)

0.31

(5.

66)

0.27

(4.

74)

D10

1.19

5.57

0.21

0.63

(6.

7)0.

52

(7.

32)

0.51

(6.

86)

D10-D

10.

98

3.33

0.29

1.00

(7.

51)

1.05

(7.

78)

0.98

(7.

02)

GR

S7.

72

0.00

7.12

0.00

5.70

0.00

TotalReturn

Mom

entum

D1

0.13

7.74

0.02

-0.

58

(-3.

29)

-0.

81

(-5.

12)

-0.

59

(-3.

72)

D2

0.48

5.97

0.08

-0.

11

(-0.

92)

-0.

32

(-3.

17)

-0.

30

(-2.

89)

D3

0.63

5.26

0.12

0.10

(1.

03)

-0.

12

(-1.

59)

-0.

18

(-2.

33)

D4

0.64

4.86

0.13

0.15

(1.

85)

-0.

06

(-1.

01)

-0.

13

(-2.

32)

D5

0.67

4.71

0.14

0.18

(2.

47)

-0.

01

(-0.

21)

-0.

10

(-1.

96)

D6

0.73

4.65

0.16

0.24

(3.

58)

0.06

(1.

28)

-0.

05

(-1.

19)

D7

0.83

4.69

0.18

0.34

(4.

97)

0.18

(3.

76)

0.07

(1.

58)

D8

0.86

4.94

0.17

0.35

(4.

53)

0.21

(3.

77)

0.10

(1.

85)

D9

1.04

5.60

0.19

0.48

(4.

58)

0.41

(5.

19)

0.37

(4.

57)

D10

1.20

7.09

0.17

0.55

(3.

33)

0.57

(4.

6)0.

64

(5.

07)

D10-D

11.

07

6.42

0.17

1.14

(4.

42)

1.37

(5.

47)

1.23

(4.

77)

GR

S5.

14

0.00

4.60

0.00

3.66

0.00

3.4 time-series , cross-section, and factor-spanning tests 47

total return momentum, but the standard errors associated with theseestimates are also substantially higher, resulting in lower t-statistics.Thus, idiosyncratic momentum generates more stable alphas. TheFama and French 2015 five-factor model is also unable to explain theextreme decile return spreads of the two strategies, and the patternthat we observe for other models is also present here: the t-statistic issubstantially higher for idiosyncratic than for conventional momen-tum despite the lower abnormal return.

In order to ensure that our results are not driven by small, illiquidstocks, we also form decile portfolios that exclude stocks with mar-ket capitalization below NYSE median (i.e. if we only consider large-caps). Results reported in Table 3.2 show that our conclusions are notqualitatively affected by this alteration, although both strategies gen-erate somewhat smaller returns in the large-cap universe. Based onall these results, we conclude that, on a stand-alone basis, idiosyn-cratic momentum is a much stronger phenomenon that conventionalmomentum.

3.4 time-series , cross-section, and factor-spanning tests

3.4.1 Empirical results

We conduct three tests to examine whether idiosyncratic momen-tum is a separate factor that expands the efficient frontier, i.e. thatit cannot be subsumed by other asset pricing factors: the time-seriesGRS11, cross-section Fama and MacBeth 1973, and factor-spanningtests. Fama 2015 argues that time-series and cross-section asset pric-ing tests should be examined jointly as they provide unique perspec-tives that complement each other.

We first turn to the time-series GRS test where we test whether theidiosyncratic momentum decile portfolios have a joint alpha of zero.Thus, under the null hypothesis, the asset pricing model is able to per-fectly explain returns of the test portfolios. All GRS test statistics arepresented in the bottom rows of Table 3.1. With the CAPM and three-factor model GRS statistics of 7.72 and 7.12, respectively, we rejectboth asset pricing models at the most conservative significance levels,and conclude that they cannot explain the idiosyncratic momentumanomaly. The addition of the two new Fama French factors to theoriginal 1993 three-factor set does not lead to a change in conclusion

11 Gibbons, Ross, and Shanken 1989


Table3.

2:Performance

inthe

large-capuniverse

19

63-

20

15

Thistable

reportsperform

ancecharacteristics

ofdecile

portfoliosconstructed

asunivariate

sortson

idiosyncraticand

totalreturn

mom

entum,

respectively,in

theuniverse

thatexcludes

stocksw

ithm

arketcapitalization

belowN

YSE

median.

We

includeall

large-capcom

mon

stockstraded

onN

YSE,

AM

EX,

andN

ASD

AQ

exchangesfrom

July1

96

3to

Decem

ber2

01

5,except

thosew

ithshare

pricebelow

$1,

with

validtotal

returnand

idiosyncraticm

omentum

scores.Total

returnm

omentum

isdefined

asthe

12-

2m

onthtotal

stockreturn

andidiosyncratic

mom

entumis

the1

2-2

month


returnestim

atedover

past3

6m

onthsusing

theFam

aand

French1

99

3three

factorm

odel.Foreach

portfolio,we

reportreturns

inexcess

ofthe


Sharperatios,

five-factoralphas

andcorresponding

t-stats,num

berof

observationsin

eachportfolio,

andpercent

median

market

capitalizationof

eachportfolio.

Portfoliosare

equal-weighted

andreform

edm

onthly.ExcessR

eturnVolatility

SharpeR

atioA

lphaC

APM

t-statA

lpha3FM

t-statA

lpha5FM

t-stat

IdiosyncraticM

omentum

D1

0.20

5.55

0.04

-0.

37

(-3.

82)

-0.

46

(-4.

89)

-0.

40

(-4.

19)

D2

0.39

5.16

0.08

-0.

15

(-1.

93)

-0.

25

(-3.

43)

-0.

28

(-3.

7)D

30.

52

5.05

0.10

-0.

02

(-0.

27)

-0.

13

(-2.

22)

-0.

17

(-2.

81)

D4

0.56

4.82

0.12

0.05

(0.

82)

-0.

06

(-1.

12)

-0.

10

(-1.

77)

D5

0.65

4.77

0.14

0.14

(2.

3)0.

02

(0.

38)

-0.

04

(-0.

77)

D6

0.70

4.82

0.15

0.19

(3.

11)

0.06

(1.

22)

0.00

(-0.

03)

D7

0.68

4.76

0.14

0.17

(3.

16)

0.07

(1.

39)

0.02

(0.

46)

D8

0.78

4.71

0.17

0.28

(4.

56)

0.18

(3.

29)

0.11

(2.

01)

D9

0.85

4.93

0.17

0.33

(4.

99)

0.23

(3.

84)

0.20

(3.

21)

D10

1.05

5.26

0.20

0.51

(5.

81)

0.45

(5.

38)

0.44

(5.

04)

D10-D

10.

85

3.59

0.24

0.88

(6.

11)

0.91

(6.

27)

0.84

(5.

63)

TotalReturn

Mom

entum

D1

0.16

7.05

0.02

-0.

49

(-3.

14)

-0.

65

(-4.

18)

-0.

45

(-2.

87)

D2

0.51

5.50

0.09

-0.

03

(-0.

33)

-0.

19

(-1.

99)

-0.

19

(-1.

93)

D3

0.56

5.03

0.11

0.05

(0.

59)

-0.

10

(-1.

31)

-0.

15

(-1.

88)

D4

0.55

4.68

0.12

0.07

(0.

92)

-0.

09

(-1.

5)-0.

17

(-2.

63)

D5

0.58

4.53

0.13

0.10

(1.

57)

-0.

04

(-0.

65)

-0.

12

(-2.

14)

D6

0.66

4.43

0.15

0.19

(3.

16)

0.06

(1.

2)-0.

05

(-1.

02)

D7

0.66

4.51

0.15

0.18

(3.

02)

0.07

(1.

29)

-0.

05

(-0.

92)

D8

0.69

4.65

0.15

0.21

(3.

12)

0.13

(2.

02)

0.02

(0.

25)

D9

0.85

5.18

0.16

0.32

(3.

56)

0.30

(3.

64)

0.26

(3.

02)

D10

1.14

6.64

0.17

0.53

(3.

43)

0.60

(4.

45)

0.65

(4.

74)

D10-D

10.

97

6.56

0.15

1.03

(3.

9)1.

24

(4.

86)

1.10

(4.

2)


(GRS statistic of 5.70): the pattern in average returns generated byunivariate idiosyncratic momentum sorts cannot be explained by anyof the leading asset pricing models.12

We repeat the analysis with total return momentum sorted portfo-lios as test assets. The GRS statistics for each of the models are sub-stantially smaller than for idiosyncratic momentum. For instance, theCAPM GRS test statistic for momentum deciles is 5.14, which is 33%lower than that of idiosyncratic momentum, and for the five-factormodel, the GRS test statistic is 3.66, which is 36% lower.

The second test we consider is the Fama and MacBeth 1973 cross-section test, whereby, each month, we regress stock returns on a set ofcharacteristics to obtain a time-series of coefficients, and subsequentlycalculate averages and corresponding t-statistics of the resulting time-series. The estimated slope coefficient can be interpreted as premiaassociated with a unit exposure to a factor (characteristic), holding allother factors constant. We include the following controls: market beta,natural logarithms of size and ratio of book to market equity, operat-ing profitability, investment (asset growth), and the main variables ofinterest - idiosyncratic and total return momentum. All variables arewinsorized at 1% and 99% levels, and t-statistics are calculated usingNewey and West 1987 adjusted standard errors with a maximum lagof 3 months. Once again, we remove micro-caps from our analysis.Results are shown in Table 3.3.

We note that all control characteristics have signs and magnitudesconsistent with those reported in the literature. Stand-alone, both mo-mentum strategies are highly economically and statistically signifi-cant regardless of the model specification. If we include both charac-teristics at the same time, idiosyncratic momentum emerges strongerwith a higher t-stat, however, total return momentum remains signifi-cant. This suggests that there is information about average returns inidiosyncratic momentum that is not contained in total return momen-tum and vice versa. In Table 3.16 in the appendix, we show that theseresults also hold over the long (1929-2015) sample.

Panel A of Table 3.4 presents results of a series of spanning testswhere we regress idiosyncratic momentum factor returns on (i) threeFama and French 1993 factors; (ii) five Fama and French 2015 factors;(iii) five Fama-French factors and Carhart’s momentum factor. In thenext step, we reverse the position of total return and idiosyncratic

12 In unreported test, we augmented the five-factor model with the conventional mo-mentum factor and we still reject this model when we test it on idiosyncratic mo-mentum portfolios.


Table3.

3:Fama

andM

acBeth1

97

3(1

97

3)regressions

This

tablereports

theresults

ofFama

andM

acBeth1

97

3regressions.W

einclude

allcomm

onstocks

tradedon

NY

SE,AM

EX,and

NA

SDA

Qexchanges

fromJuly

19

63

toD

ecember

20

15

abovethe

20th

percentileof

market

capof

NY

SEtraded

stocksand

with

shareprice

above$1,w

ithnon-m

issingcharacteristics.Beta

isthe

slopecoefficienton

them

arketfactorestim

atedusing

univariateregressions

ofstockexcess

returnson

theone-factor

model

(CA

PM)

fromt-

60

tot-

1(m

int-

24

tot-

1).Size

isthe

naturallogarithm

offirm

’sm

arketcapitalization

atthe

endof

month

t,value

isthe

naturallogarithm

ofthe

ratioof

firms

bookequity

forthe

fiscalyear

endingin

t-1

andm

arketcap

atthe

endof

Decem

berof

t-1;

profitabilityis

theratio

ofoperating

profitsand

bookequity

atthe

fiscalyear

endingin

t-1,and

investment

isgrow

thin

totalassets

forthe

fiscalyear

endingin

t-1.Total

returnm

omentum

isdefined

asthe

12-

2m

onthtotalstock

returnand

idiosyncraticm

omentum

isthe

12-

2m

onthvolatility-scaled

idiosyncraticreturn

estimated

overpast

36

months

usingthe

Fama

andFrench

19

93

three-factorm

odel.A

llvariables

arew

insorizedat

1%and

99%

.R

eportedare

theaverage

coefficientsand

t-statisticscalculated

usingN

ewey-W

estcorrected

standarderrors

with

am

aximum

of3

lags.

InterceptBeta

ln(ME)

ln(BtM)

OP

INV

iMO

MM

OM

R2

N

coeff1.

66

-0.

05

-0.

08

0.17

0.78

7.12

1447

t-stat(3.

50)

(-0.

36)

(-2.

45)

(2.

61)

(4.

76)

coeff1.

75

0.03

-0.

08

0.09

0.91

6.33

1447

t-stat(3.

61)

(0.

18)

(-2.

42)

(1.

25)

(5.

85)

coeff1.

78

-0.

08

-0.

09

0.14

0.45

0.59

7.48

1447

t-stat(3.

75)

(-0.

58)

(-2.

68)

(2.

19)

(3.

39)

(3.

14)

coeff1.

63

0.05

-0.

10

0.20

0.84

-0.

44

0.79

8.06

1417

t-stat(3.

53)

(0.

39)

(-3.

06)

(2.

77)

(4.

88)

(-5.

59)

(4.

69)

coeff1.

71

0.13

-0.

10

0.13

0.87

-0.

38

0.89

7.28

1417

t-stat(3.

65)

(0.

87)

(-3.

05)

-1.

60

(4.

73)

(-4.

73)

(5.

82)

coeff1.

74

0.01

-0.

10

0.18

0.84

-0.

42

0.44

0.59

8.39

1417

t-stat(3.

76)

(0.

11)

(-3.

25)

(2.

44)

(4.

96)

(-5.

4)(3.

38)

(3.

09)


momentum. For this analysis, we use the formal, value-weighted id-iosyncratic momentum factor, as defined in equation (3.4), to ensurethat our results are not driven by different portfolio concentrations,weighting schemes, or rebalancing frequencies.

Unlike cross-section tests that do not provide robust evidence forone factor over the other, spanning tests strongly show that conven-tional momentum is redundant when we control for idiosyncraticmomentum. While both factors have significant three- and five-factoralphas, the addition of idiosyncratic momentum to the five Fama-French factors renders total return momentum insignificant, whilethe converse is not true. Over the 1963-2015 sample, the five-factormodel augmented with the total return momentum factor brings thealpha of idiosyncratic momentum to 0.35% per month (t-statistic of5.55), while the five-factor model augmented with the idiosyncraticmomentum factor brings the alpha of the total return momentum to-0.13% (t-statistic of -1.09). In Table 3.17 in the appendix, we showthat these results also hold up in our long sample (1929-2015): thefour factor model that includes the idiosyncratic momentum factor isfully able to capture average returns on the conventional momentumfactor, while the converse is not the case.

Panels B and C show results of spanning tests where we considerthe big and small legs of the two momentum factor separately, basedon 2x3 portfolio sorts. For each of the factors, we construct Big fac-tors by going long the big (above NYSE) portfolio with a favorablefactor exposure (idiosyncratic and total return winners, value, highprofitability, and conservative investment, receptively), and shortingthe big unfavorable factor legs. The Big-RF portfolio is the value-weighted portfolio of stocks with a market capitalization above me-dian capitalization of NYSE-traded stocks, financed by shorting therisk-free portfolio. Similarly for Small factors, we go long the smallfavorable factor legs and short the small unfavorable ones. The Small-RF is the value-weighted portfolio of stocks with market capitaliza-tion below median capitalization of NYSE-traded stocks, financed byshorting the risk-free portfolio. In these tests, we find that the value-weighted idiosyncratic momentum factor subsumes the total returnmomentum factor holds for both big, as well as small-cap factors,while total return momentum does not subsume idiosyncratic mo-mentum in either sub-universe.

In Table 3.5 we examine whether the Q-factor model of Hou, Xue,and Zhang 2015, or the model based on the mispricing factors of

52 the idiosyncratic momentum anomalyTable

3.4:Spanning

tests

This

tablepresents

theresults

ofthe

time-series

spanningtests.The

idiosyncraticm

omentum

(iMO

M)

factoris

constructedusing

independentsorts

ofstocks

intotw

osize

andthree

idiosyncraticm

omentum

groups,where

thesize

breakpointis

theN

YSE

median

market

capitalization,andthe

idiosyncraticm

omentum

breakpointsare

the3

0thand

70th

percentilesof

idiosyncraticm

omentum

forN

YSE

stocks.Thisprocess

yieldssix

value-weighted

portfolios.Thefinal

idiosyncraticm

omentum

factoris

azero-investm

ent,equal-weighted

portfoliothat

islong

small

andbig

(idiosyncratic)w

inners,andshort

small

andbig

(idiosyncratic)losers.Portfolios

arereform

edm

onthly.Allother

factorsare

obtainedfrom

thew

ebsiteof

ProfessorK

ennethFrench.The

sample

periodruns

fromJuly

19

63

toD

ecember

20

15.

PanelA:A

llStocksA

lphaM

kt-Rf

SMB

HM

LR

MW

CM

AM

OM

iMO

M

IdiosyncraticM

omentum

(i)0.

68

-0.

06

0.0

2-0.

03

(7.

71)

(-2.

81)

(0.

65)

(-0.

90)

(ii)0.

64

-0.

04

0.0

3-0.

10

0.0

40.

16

(7.

02)

(-1.

92)

(0.

80)

(-2.

36)

(0.

78)

(2.

50)

(iii)0.

35

0.0

10.

00

0.1

0-0.

07

-0.

01

0.3

9

(5.

55)

(0.

60)

(-0.

06)

(3.

25)

(-2.

10)

(-0.

12)

(2

6.4

7)

TotalReturn

Mom

entum

(iv)0.

91

-0.

19

0.0

1-0.

33

(5.

44)

(-4.

66)

(0.

14)

(-5.

44)

(v)0.

74

-0.

13

0.0

7-0.

52

0.2

70.

43

(4.

32)

(-3.

22)

(1.

16)

(-6.

41)

(3.

08)

(3.

55)

(vi)-0.

13

-0.

08

0.0

3-0.

38

0.2

20.

21

1.3

6

(-1.

09)

(-2.

64)

(0.

83)

(-6.

81)

(3.

65)

(2.

51)

(2

6.4

7)

PanelB:Only

BigStocks

Alpha

Big-RF

HM

L_BR

MW

_BC

MA

_BM

om_B

iMom

_B

iMom

_B(vii)

0.2

60.

00

0.0

0-0.

11

0.1

70.

38

(3.

31)

(0.

00)

(0.

13)

(-3.

16)

(4.

53)

(2

2.3

4)M

om_B

(viii)0.

00

-0.

06

-0.

18

0.2

5-0.

17

1.1

6

(-0.

01)

(-1.

82)

(-3.

16)

(4.

30)

(-2.

64)

(2

2.3

4)

PanelC:O

nlySm

allStocksA

lphaSm

all-RF

HM

L_SR

MW

_SC

MA

_SM

om_S

iMom

_S

iMom

_S(ix)

0.3

90.

03

0.0

80.

00

-0.

01

0.4

1

(5.

84)

(2.

51)

(2.

67)

(-0.

16)

(-0.

31)

(2

7.5

0)M

om_S

(x)-0.

06

-0.

10

-0.

28

0.0

70.

20

1.3

3

(-0.

52)

(-4.

63)

(-5.

06)

(1.

27)

(2.

31)

(2

7.5

0)


Stambaugh and Yuan 2017 are able to explain idiosyncratic momen-tum profits, by considering spanning tests based on these alternativeasset pricing models. The Q-factor model consists of four factors - themarket, size (ME), investment (IA), and profitability (ROE), and hasbeen shown to explain returns of the conventional return momentumfactor. Novy-Marx 2015 shows that the ROE factor that Hou, Xue, andZhang 2015 use is a convoluted proxy for profitability, as it mechan-ically incorporates earnings surprises through the use of quarterlyearnings data, and consequently explains returns of the momentumfactor.

In Table 3.5, we confirm these results and also show that the Q-factor model is unable to explain returns of the idiosyncratic momen-tum factor. Over the January 1967 to December 2015 sample13, the id-iosyncratic momentum factor has a Q-factor alpha of 0.39% a month,with a t-statistic of 4.23

14.We also find that the mispricing factors of Stambaugh and Yuan

2017 are unable to explain returns of the idiosyncratic momentumfactor, while they have no issues with the conventional momentumfactor. The model of Stambaugh and Yuan 2017 consists of the mar-ket, size, and two mispricing factors constructed from a set of 11 assetpricing anomalies. The first factor consists of six anomalies related tofirms’ management and it is labeled as MGMT. The second factor isconstructed from anomalies that are less related to firms’ manage-ment, one of which is total return momentum, and it is labeled asPERF. Naturally, the PERF factor plays a prominent role in explain-ing returns on the conventional momentum factor. Nevertheless, inthe case of idiosyncratic momentum, this model leaves an alpha of0.38% a month with a t-statistic of 4.15.

We conclude that results of the asset pricing tests do not conclu-sively reject one factor in favor of the other, however, idiosyncratic

13 The sample starts in January of 1967 as the Q-factor model returns are available fromthat date.

14 Hou, Xue, and Zhang 2017 show that the Q-factor model leaves an insignificant re-turn spread between the top and the bottom idiosyncratic momentum portfolios,although the intercept remains economically significant at 0.32% a month (t-statisticof 1.46). The reason these findings differ from ours is a difference in portfolio con-struction. Our idiosyncratic momentum factor is constructed following the standardFama-French factor construction methodology based on 2x3 portfolio sorts, whichis also used to construct the factors in the Q-factor model. Hou, Xue, and Zhang2017 use an idiosyncratic momentum portfolio based on value-weighted decile sortswith NYSE break-points, that excludes micro-caps, and consequently, their left handside (idiosyncratic momentum) portfolio is significantly more large-cap tilted thanthe factors in the Q-factor model, on the right hand side of the spanning regression.Our comparison, on the other hand, is done using portfolios constructed in the sameway.


Table3.

5:Spanningtests

with

otherfactor

models

Thistable

presentsthe

resultsof

thetim

e-seriesspanning

testsw

ithQ

-factorm

odelandStam

baugh-Yuanm

ispricingfactor

model.The

idiosyncraticm

omentum

(iMO

M)

factoris

constructedusing

independentsorts

ofstocks

intotw

osize

andthree

idiosyncraticm

omentum

groups,where

thesize

breakpointis

theN

YSE

median

market

capitalization,andthe

idiosyncraticm

omentum

breakpointsare

the3

0thand

70th

percentilesof

idiosyncraticm

omentum

forN

YSE

stocks.Thisprocess

yieldssix

value-weighted

portfolios.Thefinalidiosyncratic

mom

entumfactor

isa

zero-investment,equal-

weighted

portfoliothat

islong

smalland

big(idiosyncratic)

winners,and

shortsm

allandbig


arereform

edm

onthly.T

heQ

factorm

odelfactorsare

obtainedfrom

LuZ

hangand

Stambaugh-Yuan

mispricing

factorsare

obtainedfrom

thew

ebsiteofYu

Yuan.Thesam

pleperiod

runsfrom

January1

96

7to

Decem

ber2

01

5,limited

bythe

availabilityof

theQ

-factorm

odelreturnseries.

Alpha

Mkt-R

fSM

B/ME

IAR

OE

MG

MT

PERF

IdiosyncraticM

omentum

(i)0.

39

-0.

01

0.09

0.12

0.33

(4.

23)

(-0.

69)

(3.

01)

(2.

43)

(9.

35)

TotalReturn

Mom

entum(ii)

0.13

-0.

07

0.24

0.01

0.92

(0.

79)

(-1.

92)

(4.

54)

(0.

13)

(14.

32)

IdiosyncraticM

omentum

(iii)0.

38

0.02

0.04

0.08

0.27

(4.

15)

(0.

97)

(1.

44)

(2.

31)

(12.

14)

TotalReturn

Mom

entum(iv)

-0.

11

0.09

0.11

0.16

0.85

(-0.

80)

(2.

57)

(2.

49)

(3.

14)

(25.

37)


momentum seems to pose an even bigger challenge to the standardasset pricing models than conventional momentum.

3.4.2 Relationship with idiosyncratic volatility

The idiosyncratic volatility anomaly of Ang et al. 2006 is conceptuallyrelated to the idiosyncratic momentum anomaly. Idiosyncratic volatil-ity is defined as the standard deviation of residuals from a regressionof stock excess returns on the three Fama and French 1993 factorsover the last month, and it has been shown to be negatively relatedto future stock returns. In a well-specified expected return model,residuals should be pure noise, but given that the three-factor modeldoes not fully explain stock returns, Ang et al. 2006 postulate thatthe omitted information will be captured in the regression residuals.The intuition behind the idiosyncratic momentum is similar, however,the signal is calculated based on the average value of residuals overthe last 12-2 months, scaled by their volatility over the congruent win-dow, while idiosyncratic volatility is based on the volatility of residu-als over a much shorter time-span. Since the idiosyncratic momentumcorrects for the impact of the (longer term) idiosyncratic volatility, wetest whether the difference between the idiosyncratic and the total re-turn momentum comes from different exposures to the idiosyncraticvolatility anomaly.

In order to address this, we calculate stocks’ idiosyncratic volatili-ties following the methodology of Ang et al. 2006. Panel A of Table3.6 shows the output of the Fama and MacBeth 1973 regression wherewe add the idiosyncratic volatility to the set of return predictors. Weconfirm the results of the prior studies that the idiosyncratic volatilityis rewarded in the cross-section of stock returns with a negative andhighly statistically significant premium (t-statistic of -5.44). However,the addition of this factor does not change the conclusions regardingthe two momentum factors - both remain priced and similar in magni-tude to the estimates from the specification without the idiosyncraticvolatility.

We next construct an idiosyncratic volatility factor15 following thesame methodology used to construct the standard Fama-French fac-tors, and our idiosyncratic momentum factor, and add it to the setof the other explanatory factors that we used in the factor spanning

15 The portfolio is long low idiosyncratic volatility small and big portfolio and shorthigh idiosyncratic counterparts.

56 the idiosyncratic momentum anomalyTable

3.6:Fam

aand

MacBeth

19

73

andSpanning

Regressions

with

IdiosyncraticVolatility

Thistable

presentsresults

ofthe

Fama

andM

acBeth1

97

3tim

e-seriesspanning

testsw

ithidiosyncratic

volatility.PanelA

:W

einclude

allcom

mon

stockstraded

onN

YSE,A

MEX

,andN

ASD

AQ

exchangesfrom

July1

96

3to

Decem

ber2

01

5above

the2

0thpercentile

ofm

arketcap

ofN

YSE

tradedstocks

andw

ithshare

priceabove

$1,w

ithnon-m


isthe

slopecoefficient

onthe

market

factorestim

atedusing


ofstock

excessreturns

onthe

one-factorm

odel(C

APM

)from

t-6

0to

t-1

(min

t-2

4to

t-1).Size

isthe

naturallogarithm

offirm

’sm

arketcapitalization

atthe

endof

month

t,valueis

thenaturallogarithm

ofthe

ratioof

firms

bookequity

forthe

fiscalyearending

int-

1and

market

capat

theend

ofD

ecember

oft-

1;profitabilityis

theratio

ofoperating

profitsand

bookequity

atthe

fiscalyearending

int-

1,andinvestm

entis

growth

intotal

assetsfor

thefiscal

yearending

int-

1.Totalreturn

mom

entumis

definedas

the1

2-2

month

totalstock

returnand

idiosyncraticm

omentum

isthe

12-

2m


idiosyncraticreturn

estimated

overpast

36

months

usingthe

Fama

andFrench

19

93

three-factorm

odel.Idiosyncraticvolatility

isdefined

asthe

volatilityof

residualsfrom

regressionsof

stockreturns

onthe

threeFam

aand

French1

99

3factors

overthe

last2

2(m

in1

6)days.

All

variablesare

winsorized

at1%

and9

9%.

Reported

arethe

averagecoefficients

andt-statistics

calculatedusing

New

ey-West

correctedstandard

errorsw

itha

maxim

umof

3lags.Panel

B:The

idiosyncraticm

omentum

andvolatility

factorsare

constructedusing

independentsorts

ofstocks

intotw

osize

andthree

idiosyncraticm

omentum

/volatilitygroups,w

herethe

sizebreakpointis

theN

YSE

median

marketcapitalization,and

theidiosyncratic

mom

entum/volatility

breakpointsare

the3

0thand

70th

percentilesofidiosyncratic

mom

entum/volatility

forN

YSE

stocks.Thisprocess

yieldssix

value-weighted

portfolios.The

finalidiosyncratic

mom

entum/volatility

factoris

azero-investm

ent,equal-w

eightedportfolio

thatis

longsm

allandbig

(idiosyncratic)winners-volatility,and

shortsmalland

big(idiosyncratic)losers-volatility

portfolios.Portfoliosare

reformed

monthly.A

llother

factorsare

obtainedfrom

thew

ebsiteof

ProfessorK

ennethFrench.The

sample

periodruns

fromJuly

19

63

toD

ecember

20

15.

PanelA

:Fama

MacB

ethR

egressions

InterceptBeta

ln(ME)

ln(BtM)

OP

INV

iMom

Mom

iVolR

sqN

coeff2.

41

0.1

-0.

13

0.14

0.72

-0.

39

0.42

0.61

-0.

22

8.97

1417

t-stat(5.

71)

(0.

85)

(-4.

72)

(1.

96)

(4.

4)(-

5.08)

(3.

33)

(3.

32)

(-5.

44)

PanelB

:SpanningR

egressions

Alpha

Mkt-R

FSM

BH

ML

RM

WC

MA

iMom

Mom

iVolR

sqN

IdiosyncraticM

omentum

0.35

0.01

00.

1-0.

07

-0.

01

0.39

54

630

-5.

55

-0.

6(-

0.06)

-3.

25

(-2.

10)

(-0.

12)

-26.

47

0.36

-0.

01

-0.

03

0.11

-0.

04

0.01

0.39

-0.

05

54.

2630

(5.

64)

(-0.

36)

(-0.

96)

(3.

49)

(-1.

06)

(0.

32)

(26.

11)

(-1.

55)

TotalReturn

Mom

entum

-0.

13

-0.

08

0.03

-0.

38

0.22

0.21

1.36

57

630

(-1.

09)

(-2.

64)

-0.

83

(-6.

81)

-3.

65

-2.

51

-26.

47

-0.

17

0.02

0.18

-0.

43

0.03

0.08

1.32

0.29

58.

9630

(-1.

44)

(0.

73)

(3.

77)

(-7.

65)

(0.

51)

(0.

89)

(26.

11)

(5.

32)


tests. In panel B of Table 6 we show that there is no significant rela-tionship between the idiosyncratic momentum and volatility and thattherefore, the idiosyncratic momentum continues to exhibit a statis-tically significant alpha of 36 bps a month (t-stat of 5.64). We repeatthe same analysis for the total return momentum and find a positiveand significant relationship with the idiosyncratic volatility, whichfurther lowers its alpha to -0.17 bps a month (t-stat -1.44) when wealso control for the idiosyncratic momentum and other characteris-tics that form the foundation of the Fama and French 2015 model.We conclude that the relationship between the idiosyncratic volatil-ity and the idiosyncratic momentum is in fact much weaker than itsrelationship with the total return momentum and that consequently,differences in idiosyncratic volatility cannot explain the superiorityof the idiosyncratic momentum over its conventional counterpart.

3.4.3 Liquidity and transactions costs

Novy-Marx and Velikov 2016 show that momentum has higher trad-ing costs compared to lower turnover strategies based on variablessuch as value or size. However, the after-trading-cost performance ofmomentum remains statistically different from zero even for naivestrategy implementation and even more so for the more sophisti-cated trading cost mitigation strategies. If trading costs for imple-menting idiosyncratic momentum are higher than for conventionalmomentum, the superiority of the idiosyncratic momentum couldbe explained by higher limits to arbitrage. Trading costs for a strat-egy depend on (i) the average transaction costs per trade and (ii) theturnover of the strategy. Below we address each of these two compo-nents.

Novy-Marx and Velikov 2016 and Frazzini, Israel, and Moskowitz2018 argue that transaction costs increase with the idiosyncratic volatil-ity and decrease with the market capitalization of the stocks beingtraded. The left panel of Figure 3.1 shows the average idiosyncraticvolatility of stocks across the total and idiosyncratic momentum deciles,and the right panel shows the average market capitalization. Thereis a distinct (inverse) U-shaped pattern where the extreme deciles,winners and losers, exhibit higher (lower) levels of the idiosyncraticvolatility (market capitalization), however, the effect is more pronouncedfor the total return, than for the idiosyncratic momentum. Further-more, in unreported tests, we have found that the total return mo-


mentum on average also has a significantly higher level of Amihud2002 illiquidity16, which is considered to be a measure of price im-pact, and consequently stock-level liquidity and implicit trading costs.These results indicate that the average costs of trading the idiosyn-cratic momentum stocks should be lower, or at most as high as thoseof trading the conventional momentum stocks.

While the idiosyncratic momentum tends to be invested in stockswith, on average, lower transactions costs per trade than the con-stituents of the total return momentum strategy, it does come with ahigher level of turnover. We find that the turnover of the conventionalmomentum strategy is 64% per month, and that of the idiosyncraticmomentum is 87% per month17. Similar to in Grundy and Martin2001 and Barroso and Santa-Clara 2015, we calculate the round-tripcutoff costs that would render the profits of both strategies insignif-icant at the 1% level. We find these break-even costs to be 63 basispoints for the conventional momentum and 73 basis points for the id-iosyncratic momentum. Therefore, the hypothetical transactions coststhat would remove the significance of the profits of the idiosyncraticmomentum would be 15% higher than for the conventional momen-tum despite its higher turnover. Having established above that thetransactions costs of the average idiosyncratic momentum stock donot seem to exceed the ones of the conventional momentum, we con-clude that higher trading costs cannot explain the superiority of theidiosyncratic momentum.

3.4.4 Importance of factors in residualization

Fama and French (1993, 1996) show that their empirically motivatedthree-factor model spans a wide range of equity portfolio returns, andclaim that the market, size, and value factors represent systematic riskfactors. With affirmations coming from more than two decades of newdata, and evidence from international markets, no one can disputethat the three-factor model successfully manages to capture much ofthe variation in average returns. In order to isolate the stock specificmomentum from the common style momentum, following Gutierrezand Pirinsky 2007 and Blitz, Huij, and Martens 2011, we opt to or-thogonalize stock returns against these three factors. In fact, any well-

16 Amihud’s illiquidity measure is defined as the absolute daily return scaled by thedollar trading volume.

17 We calculate turnover as the sum of the average one-way turnover of the short andthe long leg.


Figu

re3.1

:Idi

osyn

crat

icvo

lati

lity

and

mar

ket

capi

taliz

atio

nac

ross

deci

les

This

figur

esh

ows

the

aver

age

leve

lof

the

idio

sync

rati

cvo

lati

lity

and

mar

ket

capi

taliz

atio

nac

ross

deci

les

port

folio

sso

rted

onth

eto

tal

retu

rnan

did

iosy

ncra

tic

mom

entu

m.

Tota

lre

turn

mom

entu

mis

defin

edas

the

12

-2m

onth

tota

lst

ock

retu

rnan

did

iosy

ncra

tic

mom

entu

mis

the

12

-2m

onth

vola

tilit

y-sc

aled

idio

sync

rati

cre

turn

esti

mat

edov

erth

epa

st3

6m

onth

sus

ing

the

Fam

aan

dFr

ench

19

93

thre

e-fa

ctor

mod

el.W

ein

clud

eal

lcom

mon

stoc

kstr

aded

onN

YSE

,AM

EX,a

ndN

ASD

AQ

exch

ange

sfr

omJu

ly1

96

3to

Dec

embe

r2

01

5ab

ove

20

thpe

rcen

tile

ofm

arke

tcap

ofN

YSE

trad

edst

ocks

and

wit

hsh

are

pric

eab

ove

$1,w

ith

valid

tota

lret

urn

and

idio

sync

rati

cm

omen

tum

scor

es.P

ortf

olio

sar

eeq

ual-

wei

ghte

dan

dre

form

edm

onth

ly.


defined model that captures commonalities in stock returns is eligible,including variance decomposition models, such as those based on theprincipal component analysis. Factors such as market, size, and valueimpose a structure that is accepted in the financial literature, and thismotivates our decision to prefer them over others. We do, however,recognize that not all factors contribute equally in this process. Forinstance, the market factor may be the strongest driver of systematicreturns, which is reflected in its significance in the time-series regres-sions. In this subsection, we consider alternatives to the three-factormodel. We recalculate idiosyncratic momentum using:

o market factor

o market, size, and value factors

o market, size, value, operating profitability, and investment fac-tors

For each variable specification, we calculate equal-weighted, monthly-rebalanced decile portfolios, excluding micro-caps, and report the per-formance characteristics of the top, bottom, and top-bottom deciles inTable 3.7. As three years of factor returns are necessary to calculateidiosyncratic momentum, and operating profitability and investmentfactors are available from 1963, we start the analysis in July, 1966 forall three specifications. This enables us to make a fair comparison ofthe models.

We note that most of the effect comes from the market factor. Theinclusion of the two additional Fama-French factors leads to furtherimprovement as more of the stock-specific momentum is isolated;however, the incremental value is greatly diminished. In fact, bothreturn and volatility are reduced as we add more factors, but the riskreduction is much larger, resulting in a higher risk-adjusted return.This is further evidence that dynamic style exposures are not fully re-warded, or put differently, stock specific momentum dominates stylemomentum. The model that includes the profitability and investmentfactors generates a monthly Sharpe ratio that is 20% higher than ifonly the market factor is used.

The pattern in five-factor alphas is similar to the one we observefor raw returns: alpha decreases as we add more factors, however,t-statistics increase substantially. The D10-D1 portfolio alpha of thespecification that only includes the market factor is 10% higher thanif the five-factor model is used, but the t-statistic is 19% lower. Thus,with the five-factor model, we are able to distill a stronger signal than


Tabl

e3

.7:I

mpo

rtan

ceof

fact

ors

inre

sidu

aliz

atio

n

Thi

sta

ble

repo

rts

perf

orm

ance

char

acte

rist

ics

ofth

eto

p,bo

ttom

,an

dto

p-bo

ttom

deci

lepo

rtfo

lios

cons

truc

ted

asun

ivar

iate

sort

son

12

-2m

onth

idio

sync

rati

cre

turn

esti

mat

edus

ing

(i)o

ne-f

acto

rm

odel

(CA

PM),

(ii)

Fam

aan

dFr

ench

19

93

thre

e-fa

ctor

mod

el,(

iii)F

ama

and

Fren

ch2

01

5fiv

e-fa

ctor

mod

elov

erth

epa

st3

6m

onth

s,(i

v)Fa

ma

and

Fren

ch1

99

3th

ree-

fact

orm

odel

augm

ente

dw

ith

10

indu

stry

port

folio

s.W

ein

clud

eal

lcom

mon

stoc

kstr

aded

onN

YSE

,AM

EX,a

ndN

ASD

AQ

exch

ange

sfr

omJu

ly1

96

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Dec

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sth

eR

MW

and

CM

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ctor

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rns

are

avai

labl

efr

omJu

ly1

96

3on

war

dsan

dw

ene

edth

ree

year

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data

toca

lcul

ate

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rati

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omen

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ove

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rcen

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SEtr

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ksan

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ith

shar

epr

ice

abov

e$1

,wit

hva

lidto

talr

etur

nan

did

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ncra

tic

mom

entu

msc

ores

.For

each

port

folio

,we

repo

rtth

ere

turn

inex

cess

ofth

eri

sk-f

ree

rate

,vol

atili

ty,S

harp

era

tio,

and

five-

fact

orm

odel

alph

aan

dsl

ope

coef

ficie

nts

wit

hit

sas

soci

ated

t-st

ats.

Port

folio

sar

eeq

ual-

wei

ghte

dan

dre

form

edm

onth

ly.

CA

PM3

FM5FM

3FM

+IN

D

D1

D10

D10

-D1

D1

D10

D10

-D1

D1

D10

D10

-D1

D1

D10

D10

-D1

Exce

ssR

etur

n0.1

31

.23

1.0

90.2

01

.15

0.9

50.2

21

.15

0.9

30.3

21

.04

0.7

1

Vol

6.2

85

.89

4.3

56.0

65

.65

3.3

75.9

85

.61

3.0

95.7

35

.41

2.2

8

Shar

peR

atio

0.0

20

.21

0.2

50.0

30

.20

0.2

80.0

40

.20

0.3

00.0

60

.19

0.3

1

Alp

haC

APM

-0.4

70.6

61

.13

-0.3

90.5

90

.98

-0.3

70.5

90

.96

-0.2

50.4

90

.74

tsta

t(-

3.7

7)

(5.6

7)

(6.3

1)

(-3

.38)

(6.1

5)

(7.0

1)

(-3

.33)

(6.3

2)

(7.4

9)

(-2.6

5)

(5.9

2)

(7.8

6)

Alp

ha3FM

-0.5

90.5

71

.17

-0.5

30.5

01

.03

-0.5

10.4

91

.00

-0.4

00.3

80

.78

tsta

t(-

5.2

3)

(6.5

0)

(6.5

1)

(-5

.58)

(6.8

2)

(7.3

6)

(-5

.79)

(7.1

2)

(7.7

5)

(-5.6

4)

(7.4

5)

(8.3

4)

Alp

ha5FM

-0.5

00.5

61

.06

-0.4

60.4

80

.94

-0.4

70.4

90

.96

-0.3

90.3

70

.76

tsta

t(-

4.3

1)

(6.1

8)

(5.7

8)

(-4

.74)

(6.2

6)

(6.5

0)

(-5

.16)

(6.8

3)

(7.1

7)

(-5.2

6)

(6.9

2)

(7.7

7)


with the market factor alone, which comes with the cost of lowerreturn that is due to the elimination of some rewarded style momen-tum.

Lastly, we consider the impact of adding industry portfolio returnsto the three Fama and French 1993 factors that form our base case.Industry portfolios are generally considered to be important from aportfolio risk management perspective, as stocks within the same in-dustry tend to co-move together, however, unlike other factors, thereis no evidence that industry exposures are priced in the cross-sectionof stocks. We obtain the returns of the ten industry portfolios18 fromthe website of Professor Kenneth French and add them to the threeFama and French 1993 factors. The reason for using ten industryportfolios is that a finer classification would result in too many in-dependent variables in our regression, which is performed using 36

monthly return observations.We find that the addition of the industry factors leads to a 25% drop

in excess return of the idiosyncratic momentum on a long-short level,but an even bigger reduction in volatility, resulting in a marginal in-crease in the Sharpe ratio. We observe the same pattern for the CAPM,three- and five-factor alphas and the associated standard errors andconsequently t-statistics. The significant drop in return seems to indi-cate that industry exposures of the strategy do matter, however, theyare also not fully compensated by the market, given that the Sharperatio increases once they are removed. In the following subsection,we dive deeper into the relationship between the total return and id-iosyncratic momentum anomalies and their industry exposures.

3.4.5 Industry effects

Moskowitz and Grinblatt 1999 document that industry portfolios ex-hibit significant momentum that is not subsumed by the individualfirm momentum and other standard asset pricing factors, such assize and value. They conjecture that this finding implies that con-ventional momentum strategies are not well diversified, as winning(losing) stocks tend to come from the same industries.

In order to test the importance of industry effects when it comesto the idiosyncratic momentum, we split the signal into two: the al-location signal that is the average idiosyncratic momentum of the

18 The ten industries are consumer non-durables, consumer durables, manufacturing,energy, high-tech, telecom, shops, healthcare, utilities, and other.


industry19 assigned to each stock in that industry, and the selectionsignal that is equal to stock’s idiosyncratic momentum in excess ofthat of its corresponding industry. The same signals are calculatedfor the conventional momentum. We then use the selection signal toform decile portfolios of within-industry momentum strategies, andthe allocation signal to construct the across-industry strategy wherethe top portfolio consists of all stocks in the industry with the highestallocation signal, and the bottom portfolio consists of all stock in theindustry with the lowest signal20. Table 3.8 shows the performancecharacteristics of these portfolios, as well as the portfolios based onthe total signal for comparison.

In the case of the conventional momentum, we observe a drop of13% in excess return, in relative terms, going from the total to selec-tion (within-industry) signal, most of it coming from the short sideof the portfolio. However, we also observe an even larger drop involatility of 16%, thus leading to a marginally higher Shape ratio ofthe within-industry strategy compared to the one based on the to-tal signal. The across-industry strategy, in fact, performs significantlyworse, with a Sharpe ratio of 0.11, which is 39% smaller than that ofthe based-case strategy21.

When it comes to the idiosyncratic momentum we observe similarpatterns - a drop in return when going from the total to the selectionsignal, which is more than offset by the drop in volatility, and pooroverall performance of the across-industry strategy.

We next run spanning regressions of the total and idiosyncraticmomentum strategy returns on the across and within-industry cor-responding long-short portfolios. Table 3.9 shows that in the case ofthe idiosyncratic momentum, both signals are significant, albeit theselection strategy gets a much higher coefficient (1.03 versus 0.14). Infact, the within-industry strategy alone fully spans the normal strat-egy, and this is not the case when only the across-industry strategy isused.

When it comes to the total return momentum, we observe the samepattern, however, the importance of the across-industry effect appearsto be larger. This finding is supported by the results of the Fama andMacBeth 1973 regressions where we include both the selection, as

19 We use the standard 10 industry classification based on SIC codes.20 The across-industry strategy buys and sells entire industries. Consequently, the decile

portfolios do not have the same number of stocks.21 We obtain quantitatively similar results when constructing a 12-1M industry momen-

tum strategy using 10 value-weighed industry return portfolios from Professor KenFrench’s website.


Table 3.8: Performance of Within Industry and Across Industry Strategies

This table reports performance characteristics of the top, bottom, and top-bottom decile portfolios constructed as univariate sorts on idiosyncratic andtotal return momentum signals within and across industries. We include allcommon stocks traded on NYSE, AMEX, and NASDAQ exchanges fromJuly 1963 to December 2015 above 20th percentile of market cap of NYSEtraded stocks and with share price above $1, with valid total return andidiosyncratic momentum scores. For each portfolio, we report the return inexcess of the risk-free rate, volatility and Sharpe ratio. Portfolios are equal-weighted and reformed monthly. Industry classification is based on firstdigit of the SIC codes provided by CRSP.

ExcessReturn

Volatility SharpeRatio

IdiosyncraticMomentum

Total

D1 0.22 5.91 0.04

D10 1.19 5.57 0.21

D10-D1 0.98 3.33 0.29

WithinIndustry

D1 0.24 5.88 0.04

D10 1.13 5.5 0.21

D10-D1 0.89 2.69 0.33

AcrossIndustry

D1 0.26 6.12 0.04

D10 1.02 6.23 0.16

D10-D1 0.76 5.81 0.13

Total ReturnMomentum

TotalD1 0.13 7.73 0.02

D10 1.2 7.09 0.17

D10-D1 1.07 6.43 0.17

WithinIndustry

D1 0.23 7.47 0.03

D10 1.15 6.88 0.17

D10-D1 0.93 5.39 0.17

AcrossIndustry

D1 0.36 6.22 0.06

D10 1.02 6.49 0.16

D10-D1 0.66 6.27 0.11


Tabl

e3

.9:S

pann

ing

Reg

ress

ions

wit

hW

ithi

nIn

dust

ryan

dA

cros

sIn

dust

rySt

rate

gies

This

tabl

epr

esen

tsre

sult

sof

the

tim

e-se

ries

span

ning

test

sba

sed

onto

p-bo

ttom

deci

lepo

rtfo

lios

cons

truc

ted

asun

ivar

iate

sort

son

idio

sync

rati

can

dto

talr

etur

nm

omen

tum

sign

als

wit

hin

and

acro

ssin

dust

ries

.We

incl

ude

allc

omm

onst

ocks

trad

edon

NY

SE,A

MEX

,and

NA

SDA

Qex

chan

ges

from

July

19

63

toD

ecem

ber

20

15

abov

e2

0th

perc

enti

leof

mar

ket

cap

ofN

YSE

trad

edst

ocks

and

wit

hsh

are

pric

eab

ove

$1,w

ith

valid

tota

lre

turn

and

idio

sync

rati

cm

omen

tum

scor

es.P

ortf

olio

sar

eeq

ual-

wei

ghte

dan

dre

form

edm

onth

ly.I

ndus

try

clas

sific

atio

nis

base

don

first

digi

tof

the

SIC

code

spr

ovid

edby

CR

SP.

Alp

haA

cros

sIn

dust

ryW

ithi

nIn

dust

ryR

sqN

Idio

sync

rati

cM

omen

tum

coef

f-0

.05

0.1

41.0

30

.90

630

t-st

at(-

1.0

3)

(18.6

8)

(61

.79)

coef

f0

.06

0.7

90

.85

630

t-st

at(0

.26)

(9.7

8)

coef

f0

.76

0.1

70

.31

630

t-st

at(7

.57)

(9.7

8)

Tota

lRet

urn

Mom

entu

m

coef

f-0

.01

0.1

91.0

30

.95

630

t-st

at(-

0.2

2)

(18.0

6)

(83

.50)

coef

f0

.08

0.6

20

.93

630

t-st

at(0

.38)

(15.8

9)

coef

f0

.62

0.4

60

.42

630

t-st

at(3

.42)

(15.8

9)


well as the allocation variables. Table 3.10 shows that in the case of theidiosyncratic momentum, only the selection part remains priced inthe cross-section, while in the case of the total return momentum bothdo. In the regression where we include all four variables, only theallocation component of the idiosyncratic momentum is insignificant.

We should note that our analysis differs from that of Moskowitzand Grinblatt 1999 in a number of ways. First, we consider the 12-2 month look-back window for both the idiosyncratic and total re-turn momentum strategies to be consistent with the rest of our paper,whereas Moskowitz and Grinblatt 1999 find that industry momentumis stronger when a shorter look-back is used and the last month is notskipped. Second, we consider 10 industry portfolio classification to beconsistent with the previous section where we add 10 industry port-folios to the set of factors against we residualize stock returns. Giventhat there are 36-monthly return observations that are used for theseregressions, we cannot use finer industry grouping. Moskowitz andGrinblatt 1999, on the other hand, use a finer industry classificationbased on 20 industries, which leads to a higher breadth of their in-dustry momentum strategy. Nevertheless, we are able to confirm theimportance of industry effects in the total return momentum strategy,however, we also find that in the case of idiosyncratic momentumthese effects seem to be of lesser importance.

3.5 explanations for the idiosyncratic momentum anomaly

In this section, we discuss and empirically test whether the mostprominent explanations for the momentum phenomenon, which in-clude momentum crashes, investor overconfidence, and over- andunderreaction, also hold for idiosyncratic momentum. We conjecturethat if both strategies are driven by the same underlying mechanisms,the observed performance differences could simply be due to differ-ent degrees of exposure to these sources. On the other hand, if thelinks between these explanations and idiosyncratic momentum areabsent, the two strategies are more likely to be separate phenomena.

3.5 explanations for the idiosyncratic momentum anomaly 67

Tabl

e3.1

0:S

pann

ing

Reg

ress

ions

wit

hW

ithi

nIn

dust

ryan

dA

cros

sIn

dust

rySt

rate

gies

This

tabl

epr

esen

tsre

sult

sof

the

Fam

aan

dM

acBe

th1

97

3re

gres

sion

sba

sed

onid

iosy

ncra

tic

and

tota

lre

turn

mom

entu

mse

lect

ion

and

allo

cati

onsi

gnal

s.Th

eal

loca

tion

sign

alth

atis

the

aver

age

valu

eof

the

indu

stry

assi

gned

toea

chst

ock

inth

atin

dust

ry,a

ndth

ese

lect

ion

sign

alth

atis

equa

lto

stoc

k’s

sign

alin

exce

ssof

that

ofit

sco

rres

pond

ing

indu

stry

.Ind

ustr

ycl

assi

ficat

ion

isba

sed

onfir

stdi

git

ofth

eSI

Cco

des

prov

ided

byC

RSP

.We

incl

ude

all

com

mon

stoc

kstr

aded

onN

YSE

,AM

EX,a

ndN

ASD

AQ

exch

ange

sfr

omJu

ly1

96

3to

Dec

embe

r2

01

5ab

ove

the

20

thpe

rcen

tile

ofm

arke

tca

pof

NY

SEtr

aded

stoc

ksan

dw

ith

shar

epr

ice

abov

e$1

,wit

hno

n-m

issi

ngch

arac

teri

stic

s.Be

tais

the

slop

eco

effic

ient

onth

em

arke

tfac

tor

esti

mat

edus

ing

univ

aria

tere

gres

sion

sof

stoc

kex

cess

retu

rns

onth

eon

e-fa

ctor

mod

el(C

APM

)fr

omt-

60

tot-

1(m

int-

24

tot-

1).

Size

isth

ena

tura

llog

arit

hmof

firm

’sm

arke

tca

pita

lizat

ion

atth

een

dof

mon

tht,

valu

eis

the

natu

rall

ogar

ithm

ofth

era

tio

offir

ms

book

equi

tyfo

rth

efis

caly

ear

endi

ngin

t-1

and

mar

ket

cap

atth

een

dof

Dec

embe

rof

t-1

;pro

fitab

ility

isth

era

tio

ofop

erat

ing

profi

tsan

dbo

okeq

uity

atth

efis

caly

ear

endi

ngin

t-1

,and

inve

stm

ent

isgr

owth

into

tal

asse

tsfo

rth

efis

cal

year

endi

ngin

t-1

.To

tal

retu

rnm

omen

tum

isde

fined

asth

e1

2-2

mon

thto

tal

stoc

kre

turn

and

idio

sync

rati

cm

omen

tum

isth

e1

2-2

mon

thvo

lati

lity-

scal

edid

iosy

ncra

tic

retu

rnes

tim

ated

over

past

36

mon

ths

usin

gth

eFa

ma-

Fren

ch(1

99

3)

thre

e-fa

ctor

mod

el.

All

vari

able

sar

ew

inso

rize

dat

1%

and

99

%.

Rep

orte

dar

eth

eav

erag

eco

effic

ient

san

dt-

stat

isti

csca

lcul

ated

usin

gN

ewey

-Wes

tco

rrec

ted

stan

dard

erro

rsw

ith

am

axim

umof

3la

gs.

Inte

rcep

tBe

taln

(ME)

ln(B

tM)

OP

INV

iMom

_SiM

om_A

Mom

_SM

om_A

Rsq

N

coef

f1.7

40.1

2-0

.10

0.1

20.8

5-0

.40

0.8

41.1

88.0

41417

t-st

at(3

.71)

(0.8

8)

(-3

.11)

(1.5

9(4

.72)

(-5

.07)

(6.1

7)

(1.7

1)

coef

f1.5

40.0

4-0

.10

0.1

90.7

9-0

.47

0.7

21.5

98.8

31417

t-st

at(3

.33)

(0.3

4)

(-3

.10)

(2.6

6)

(4.6

3)

(-6

.01)

(4.5

3)

(3.3

9)

coef

f1.7

70.0

1-0

.10

0.1

60.8

0-0

.44

0.4

40.5

50.5

21.3

39.7

01417

t-st

at(3

.77)

(0.0

4)

(-3

.39)

(2.2

9)

(4.7

8)

(-5

.91)

(3.9

6)

(0.6

4)

(2.8

4)

(2.2

8)


3.5.1 Momentum crashes

In Table 3.11, we replicate results of Daniel and Moskowitz 201622 us-

ing conventional momentum decile portfolios by estimating the fol-lowing regression:

Rt = α0 +αBIB,t + [β0 + IB,t(βB + IU,tβB,U)]·

(Rmkt,t − Rf,t) + εt(3.5)

where IB and IU are dummy variables that indicate whether the pastcumulative twelve-month return of the market portfolio is negative(IB) and whether the subsequent month is non-negative (IU). βB indi-cates whether the market-beta differs after past bear markets, whileβB,U indicates the extent to which the subsequent up- and down-market betas differ after such market.

Consistent with their results, we find that the market beta of the mo-mentum strategy is significantly lower (-0.58 with a t-stat of -5.28) af-ter a bear market than after a bull market. If the market subsequentlyfurther declines, the point estimate for beta is -0.51 (= β0 + βB), butif the market reverses, the beta is additional -0.85 (t-stat -5.83) lower,resulting in overall beta of -1.36 (=β0 + βB + βB,U). The predomi-nant source of this optionality comes from the loser portfolio with adown-market beta of 1.52 (=1.31+0.21), and an up-market beta of 2.18

(=1.31+0.21+0.66). As the momentum strategy is short these losers, itexhibits the most negative market exposure precisely when the mar-ket recovers after bear markets. In unreported tests, we also find thatthe optionality is asymmetric; i.e. the market beta difference for sub-sequent up- and down-markets following bull markets is more thantwo times smaller in magnitude.

We apply the same methodology to idiosyncratic momentum port-folios and find that the beta differences are much smaller than in thecase of conventional momentum, however, still statistically significant.While the difference in market betas following bull and bear mar-kets is only -0.18 (t-stat -3.10) the differences between up- and down-market betas following bear markets is -0.28 (t-stat -3.58) comparedto -0.85 for conventional momentum. If the market further declinesfollowing a bear market, the point estimate of beta is -0.14 (=0.04-

22 In their study, Daniel and Moskowitz 2016 use value-weighted portfolios. For con-sistency with the rest of the paper, we use equal-weighted portfolios, but note thatour conclusions do not differ if we value-weight stocks.


Tabl

e3.1

1:M

omen

tum

cras

hes

-op

tion

alit

yin

bear

mar

kets

This

tabl

epr

esen

tsth

ese

resu

lts

ofco

ndit

iona

lreg

ress

ionsRt=α0+αBI B

,t+[β

0+I B

,t(β

B+I U

,tβB

,U)]·(Rm

kt

,t−Rf

,t),

whe

reth

ede

pend

entv

aria

bles

are

pric

ean

did

iosy

ncra

tic

long

-sho

rt(d

ecile

)po

rtfo

lios;

(IB

,t)

andI U

,tar

edu

mm

ies

indi

cati

ngw

heth

erth

epa

stcu

mul

ativ

etw

elve

-mon

thre

turn

ofth

em

arke

tis

nega

tive

((I B

,t))

and

whe

ther

the

subs

eque

ntm

onth

isno

n-ne

gati

ve((I U

,t))

.βB

indi

cate

sw

heth

erth

em

arke

tbe

tadi

ffer

saf

ter

past

bear

mar

kets

,w

hileβU

indi

cate

sth

eex

tent

tow

hich

the

subs

eque

ntup

-and

dow

n-m

arke

tbet

asdi

ffer

afte

rsu

chm

arke

t.W

ein

clud

eal

lcom

mon

stoc

kstr

aded

onN

YSE

,AM

EX,a

ndN

ASD

AQ

exch

ange

sfr

omJu

ly1

92

9to

Dec

embe

r2

01

5ab

ove

20th

perc

enti

leof

mar

ketc

apof

NY

SEtr

aded

stoc

ksan

dw

ith

shar

epr

ice

abov

e$1

,wit

hva

lidto

talr

etur

nan

did

iosy

ncra

tic

mom

entu

msc

ores

.Tot

alre

turn

mom

entu

mis

defin

edas

the

12

-2m

onth

tota

lst

ock

retu

rnan

did

iosy

ncra

tic

mom

entu

mis

the

12-2

mon

thvo

lati

lity-

scal

edid

iosy

ncra

tic

retu

rnes

tim

ated

over

past

36

mon

ths

usin

gth

eFa

ma

and

Fren

ch1

99

3th

ree-

fact

orm

odel

.Por

tfol

ios

are

equa

l-w

eigh

ted

and

refo

rmed

mon

thly

.

Tota

lRet

urn

Mom

entu

m

D1

D2

D3

D4

D5

D6

D7

D8

D9

D1

0D

10-D

1

alph

a_0

-0.9

1-0

.37

-0.1

20.0

30.0

50.1

90.2

30.2

60.3

70.4

11.3

2

(-5

.76)

(-3

.47)

(-1

.41

)(0

.4)

(0.7

7)

(2.7

7)

(3.4

0)

(3.7

1)

(4.2

5)

(3.1

3)

(6.3

3)

alph

a_B

-0.5

3-0

.91

-0.6

6-0

.51

-0.3

6-0

.27

0.1

00.0

10.3

80.8

81.4

1

(-1

.04)

(-2

.63)

(-2

.36)

(-2

.08

)(-

1.6

)(-

1.2

7)

(0.4

4)

(0.0

6)

(1.3

3)

(2.0

6)

(2.0

9)

beta

_01.3

11.1

31.0

31.0

11.0

01.0

21.0

41.1

01.1

91.3

80.0

7

(36.3

)(4

6.1

5)

(52.6

2)

(57.9

5)

(62

.75)

(66.6

5)

(67

.12)

(67.1

5)

(59

.64)

(45.5

9)

(1.4

8)

beta

_B0.2

10.2

00.1

80.1

90.1

10.0

60.0

2-0

.08

-0.1

8-0

.37

-0.5

8

(2.4

9)

(3.5

3)

(4.0

8)

(4.7

1)

(3.1

1)

(1.5

6)

(0.6

1)

(-2

.2)

(-3.9

8)

(-5.3

7)

(-5

.28)

beta

_B,U

0.6

60.6

10.4

40.2

80.2

50.1

20.0

40.0

5-0

.09

-0.1

9-0

.85

(6.0

2)

(8.1

5)

(7.3

1)

(5.3

)(5

.1)

(2.6

3)

(0.9

2)

(1.0

8)

(-1.4

8)

(-2.0

3)

(-5

.83

)

Idio

sync

rati

cM

omen

tum

D1

D2

D3

D4

D5

D6

D7

D8

D9

D1

0D

10-D

1

alph

a_0

-0.6

0-0

.30

-0.1

5-0

.04

0.0

60.0

90.1

60.1

70.3

20.4

41.0

4

(-6

.18)

(-3

.74)

(-2

.07)

(-0

.53

)(0

.81)

(1.2

9)

(2.0

5)

(2.2

)(4

.32)

(5.3

2)

(9.2

5)

alph

a_B

-0.2

8-0

.18

0.0

1-0

.17

-0.2

2-0

.47

-0.2

6-0

.25

-0.3

10.2

80.5

6

(-0.9

)(-

0.6

8)

(0.0

2)

(-0

.69

)(-

0.8

9)

(-2

.13

)(-

1.0

5)

(-0

.98

)(-

1.2

7)

(1.0

5)

(1.5

5)

beta

_01.1

61.1

21.0

91.0

91.0

81.1

01.1

11.1

21.1

41.1

90.0

4

(52.4

3)

(60.4

5)

(63.7

6)

(61.8

5)

(62

.6)

(69.9

6)

(63

.05)

(62.0

2)

(67

.57)

(62.7

9)

(1.4

6)

beta

_B0.0

90.0

80.1

60.0

70.0

90.0

30.0

1-0

.04

-0.0

6-0

.09

-0.1

8

(1.7

4)

(1.9

6)

(3.9

6)

(1.8

4)

(2.3

6)

(0.6

9)

(0.2

2)

(-1

.02)

(-1

.6)

(-2.1

6)

(-3.1

)be

ta_B

,U0.3

20.2

10.1

10.2

30.2

10.2

50.2

80.3

30.2

20.0

4-0

.28

(4.6

9)

(3.7

3)

(2.0

4)

(4.1

9)

(3.9

8)

(5.1

3)

(5.1

1)

(5.9

3)

(4.2

9)

(0.6

1)

(-3

.58

)


0.18), and if the market recovers, the beta is -0.42 (=-0.14-0.28). Thus,the (negative) beta adjustment for idiosyncratic momentum is abouta third of what we found for conventional momentum. In unreportedtests, we find that idiosyncratic momentum also exhibits less time-varying beta following bull markets than conventional momentum.

In Figure 3.2, we show the cumulative returns of the two momen-tum strategies from July 1929 till December 2015. While the residual-ization process nearly eliminated the crash in the early 1930s, it wasnot fully effective in 2009 when momentum strategies exhibited oneof their largest drawdowns in history. Nevertheless, the drawdownof idiosyncratic momentum was much smaller than that of conven-tional momentum. Taken together, our results indicate that crash riskcannot explain the superior performance of idiosyncratic momentum.

3.5.2 Market states and dynamics

We now turn to the overreaction argument, as proposed in Cooper,Gutierrez, and Hameed 2004. We calculate average returns of the twomomentum strategies23 following bull (bear) markets, which are de-fined as periods with positive (negative) 36-month market returns24,and present results in Table 3.12.

In line with the results in Cooper, Gutierrez, and Hameed 2004,momentum returns are positive (1.32%) and significant following bullmarkets, and negative (-0.48%), though insignificant, following bearmarkets. In contrast, idiosyncratic momentum returns are positive inboth cases (1.08% and 0.39%, respectively), although also insignificantafter bear markets. Furthermore, the dispersion in returns followingbull and bear markets is marginally significant for total return butinsignificant for idiosyncratic momentum. In term of economic signif-icance, we observe that the average difference between bull and bearmarkets is almost three times higher to total return than for idiosyn-cratic momentum. This indicates that this overreaction argument isless strong for idiosyncratic momentum.

Asem and Tian 2010 show that results for different market states(as shown in Cooper, Gutierrez, and Hameed 2004) are dominated byresults for different market dynamics, where the subsequent market

23 We use decile spread portfolios following Cooper, Gutierrez, and Hameed 2004.24 The choice of 36-month lookback window also follows Cooper, Gutierrez, and

Hameed 2004.


Figu

re3

.2:C

umul

ativ

ere

turn

s

Thi

sfig

ure

show

scu

mul

ativ

eou

tper

form

ance

ofth

eto

pov

erth

ebo

ttom

tota

lret

urn

and

idio

sync

rati

cm

omen

tum

port

folio

s.To

talr

etur

nm

omen

tum

isde

fined

asth

e1

2-2

mon

thto

tal

stoc

kre

turn

and

idio

sync

rati

cm

omen

tum

isth

e1

2-2

mon

thvo

lati

lity-

scal

edid

iosy

ncra

tic

retu

rnes

tim

ated

over

past

36

mon

ths

usin

gth

eFa

ma

and

Fren

ch1

99

3th

ree-

fact

orm

odel

.We

incl

ude

allc

omm

onst

ocks

trad

edon

NY

SE,A

MEX

,and

NA

SDA

Qex

chan

ges

from

July

19

63

toD

ecem

ber

20

15

abov

e2

0th

perc

enti

leof

mar

ket

cap

ofN

YSE

trad

edst

ocks

and

wit

hsh

are

pric

eab

ove

$1,w

ith

valid

tota

lre

turn

and

idio

sync

rati

cm

omen

tum

scor

es.P

ortf

olio

sar

eeq

ual-

wei

ghte

dan

dre

form

edm

onth

ly.


Table 3.12: Market states

This table reports average returns and associated t-statistics of the idiosyn-cratic momentum (D10-D1), as well as total return momentum strategy fol-lowing bull and bear markets, defined as in Cooper, Gutierrez, and Hameed2004 as periods with positive (negative) 36-month market returns. Total re-turn momentum is defined as the 12-2 month total stock return and idiosyn-cratic momentum is the 12-2 month volatility-scaled idiosyncratic returnestimated over past 36 months using the Fama and French 1993 three-factormodel. We include all common stocks traded on NYSE, AMEX, and NAS-DAQ exchanges from July 1929 to December 2015 above 20th percentileof market cap of NYSE traded stocks and with share price above $1, withvalid total return and idiosyncratic momentum scores. Portfolios are equal-weighted and reformed monthly.

Past Bull MKT Past Bear MKT Diff

Total ReturnMomentum

1.32 -0.48 1.80

(6.88) (-0.53) (1.93)

IdiosyncraticMomentum

1.08 0.39 0.69

(11.09) (0.98) (1.67)

return is also taken into account.25 We now empirically test whethertheir findings apply to idiosyncratic momentum. Following the method-ology proposed in Asem and Tian 2010, we define bull markets asperiods in which the cumulative 12-month market return is positive,and bear markets in which it is negative. If the return during thesubsequent month is positive, it is classified as an ‘up’ month, and ifit is negative, it is a ‘down’ month. In Table 3.13, we show that theaverage return of momentum following market downturns (a downmonth following a bull market) is 0.09% (t-stat 0.33), and followingmarket upturns (an up month following a bear market) is -4.55% (t-stat -4.26). On the other hand, the average return in an up monthfollowing bull markets is 2.03% (t-stat 9.31), and in a down monthfollowing bear markets is 5.41% (t-stat 9.34). This clearly illustratesthat momentum delivers high returns in trending markets, and un-derperforms if markets reverse.

We test the sensitivity of idiosyncratic momentum to market dy-namics and find that it delivers a positive return in trending marketsand also in market downturns. The return in down months followingbull markets is 0.84% (t-stat 4.87), but, more importantly, return in

25 Hanauer 2014 finds similar results for Japan, Korea, Taiwan, and Turkey. In contrast,Cheema and Nartea 2017 report positive momentum returns exclusively after bearmarkets for Chinese A-shares. Furthermore, only in down months following bearmarkets, momentum returns are significantly different from zero.


up months following bear markets, which is particularly hurtful forconventional momentum, is -0.66% and statistically indistinguishablefrom zero (t-stat -1.49).

We further test for differences in average returns of idiosyncraticand total return momentum in different market states and find thataverage returns of the conventional momentum significantly differsdepending on the subsequent month, however, in case of idiosyn-cratic momentum the difference is only significant following bearmarkets, and still one third of that observed in the base of total returnmomentum. We also test the difference between the two strategies ineach of the four cases and find that total return momentum deliv-ers significantly higher returns in trending markets, while idiosyn-cratic momentum significantly outperforms when market dynamicschange.

Our results show that idiosyncratic momentum is substantially lessaffected by market dynamics (market continuations versus market re-versals), which is not surprising given that idiosyncratic momentumis designed to exhibit smaller time-varying style exposures. Asemand Tian 2010 claim that the documented patterns for conventionalmomentum are in line with the investor overconfidence hypothesis,but this explanation also seems less applicable to idiosyncratic mo-mentum.

3.5.3 Link with underreaction

Gutierrez and Pirinsky 2007 argue that idiosyncratic momentum isgrounded in behavioral, rather than risk-based explanations. Theylink idiosyncratic momentum profits to investors’ underreaction tonews (slow diffusion of information) given their observation that id-iosyncratic momentum profits are sustained even 60 months follow-ing portfolio formation. However, the authors do not control for theimpact of other stock characteristics that could also be related to fu-ture returns. Conventional and idiosyncratic momentum strategiescould be different in other dimensions, that is, they could be relatedto other stock level characteristics, that may be causing their returnsto be sustained over longer periods.

In order to address this concern we run Fama and MacBeth 1973

regression of next month’s stock excess returns on their lagged con-ventional and idiosyncratic momentum scores, and also other knownpredictors of stock returns, such as market beta, book-to-market, size,


Table3.

13:M

arketdynam

ics

Thistable

reportsaverage

returnsand

associatedt-statistics

ofthe

idiosyncratic(D

10-D

1),as

well

astotal

returnm

omentum

strategyfollow

ingm

arketdow

nturns(an

down

month

following

abull

market),m

arketupturns

(anup

month

following

abear

market),and

market

continuations(an

upm

onthfollow

inga

bullm

arketand

adow

nm

onthfollow

inga

bearm

arket).Follow

ingA

semand

Tian(2

01

0),w

edefine

bullm

arketas

periodsin

which

thecum

ulative1

2-month

market

returnis

positive,andbear

market

inw

hichit

isnegative.Total

returnm

omentum

isdefined

asthe

12-

2m

onthtotal

stockreturn

andidiosyncratic

mom

entumis

the1

2-2

month


returnestim

atedover

past3

6m

onthsusing

theFam

aand

French1

99

3three-factor

model.W

einclude

allcomm

onstocks

tradedon

NY

SE,AM

EX,and

NA

SDA

Qexchanges

fromJuly

19

29

toD

ecember

20

15

above2

0thpercentile

ofm

arketcap

ofN

YSE

tradedstocks

andw

ithshare

priceabove

$1,w

ithvalid

totalreturnand

idiosyncraticm

omentum

scores.Portfoliosare

equal-weighted

andreform

edm

onthly.

TotalReturn

Mom

entum

Up

Month

Dow

nM

onthD

iffPast

BullMK

Tm

ean2.

03

0.09

1.94

tstat(9.

31)

(0.

33)

(5.

49)

PastBear

MK

Tm

ean-4.

55

5.41

-9.

97

tstat(-

4.26)

(9.

34)

(8.

10)

IdiosyncraticM

omentum

Up

Month

Dow

nM

onthD

iffPast

BullMK

Tm

ean1.

18

0.84

0.34

tstat(9.

32)

(4.

87)

(1.

60)

PastBear

MK

Tm

ean-0.

66

2.27

-2.

94

tstat(-

1.49)

(7.

57)

(5.

46)

Difference

(iMom

-Mom

)U

pM

onthD

own

Month

PastBullM

KT

mean

-0.

85

0.75

tstat(-

4.96)

(3.

58)

PastBear

MK

Tm

ean3.

89

-3.

14

tstat(5.

24)

(-6.

65)


profitability, and investment. We iteratively run 60 such regressions,in each using lags of momentum signals ranging from 1 to 60 months.This approach is similar to that used in Ball et al. 2016 with the differ-ence that we are focusing on the ability of lagged momentum signals,as opposed to lagged profitability, to predict future returns. In Fig-ure 3.3, we show the point estimates of the slope coefficients (LEFT)and the corresponding t-statistics (RIGHT) of the conventional andidiosyncratic momentum characteristics at lags t-1 to t-60. All inde-pendent variables are winsorized at 1% and 99% levels, and standarderrors are calculated using Newey and West 1987 correction with amaximum lag of 3 months. Results for the 1929-2015 sample are con-sistent with the ones from the shorter sample and can be found inFigure 3.7 in the appendix26.

While conventional momentum forecasts high short to mediumterm returns, the power of conventional momentum drops to zerofairly quickly and, consistent with Jegadeesh and Titman 2001, turnsinto a long-term reversal after around one year following portfolioformation. In contrast, idiosyncratic momentum forecasts high, or atleast non-negative, returns up to 40 months following formation. Fig-ure 3.4 shows slope estimates for the two momentum signals whenwe do not control for other characteristics. In this scenario, we findmuch more significant results, clearly highlighting the importance ofcontrolling for other characteristics, in particular value, that is closelyrelated to the long-term reversal factor of De Bondt and Thaler 1985,which plays an important role for conventional momentum.

To further gauge the link between idiosyncratic momentum andunderreaction, we hypothesize the following: although on the margin,total return momentum is caused by overreaction, as the two momen-tum strategies are positively correlated, there is a subset of firms withhigh momentum (henceforth: Mom) scores that are caused by under-reaction, and a subset of high idiosyncratic momentum (henceforth:iMom) firms that are caused by overreaction. Using iMom as a proxyfor momentum that is caused by underreaction, we construct portfo-lios by first restricting the stock universe to 20% of stocks with thehighest total return momentum scores, and then within this group,we sort stocks into five portfolios on their idiosyncratic momentumscores. We then construct an equal-weighted portfolio that is longiMom winners and short iMom losers within this high total returnmomentum universe and track the performance of this portfolio up

26 For the longer sample, we do not control for profitability and investment character-istics as they are not available in the COMPUSTAT database.


Figure3.

3:Fama

andM

acBeth1

97

3regressions

with

lagged(i)m

omentum

signalsw

ithcontrols

Thisfigure

shows

theFam

aand

MacBeth

19

73

coefficients,and

correspondingt-statistics

onthe

laggedtotal

returnand

idiosyncraticm

omentum

characteristics.W

eiteratively

run6

0such

regressionsw

ithone

lag-pairin

each,w

iththe

following

controlsfixed

attim

et-

1:beta,

size,value,

profitability,and

investment.

We

includeall

comm

onstocks

tradedon

NY

SE,A

MEX

,and

NA

SDA

Qexchanges

fromJuly

19

63

toD

ecember

20

15

above2

0thpercentile

ofmarketcap

ofNY

SEtraded

stocksand

with

shareprice

above$1,w

ithnon-m


isthe

slopecoefficient

onthe

market

factorestim

atedusing


ofstock

excessreturns

onthe

one-factorm

odel(CA

PM)

fromt-

60

tot-

1(m

int-

24

tot-

1).Size

isthe

naturallogarithmof

firm’s

market

capitalizationat

theend

ofm

ontht,value

isthe

naturallogarithmof

theratio

offirm

sbook

equityfor

thefiscal

yearending

int-

1and

market

capat

theend

ofD

ecember

oft-

1;profitabilityis

theratio

ofoperating

profitsand

bookequity

atthe

fiscalyear

endingin

t-1,and

investment

isgrow

thin

totalassetsfor

thefiscalyear

endingin

t-1.Totalreturn

mom

entumis

definedas

the1

2-2

month

totalstock

returnand

idiosyncraticm

omentum

isthe

12-

2m


idiosyncraticreturn

estimated

overpast

36

months

usingthe

Fama

andFrench

19

93

three-factorm

odel.A

llvariables

arew

insorizedat

1%and

99%

.R

eportedare

theaverage

coefficientsand


usingN

ewey-W

estcorrected

standarderrors

with

am

aximum

of3

lags.


Figu

re3

.4:F

ama

and

Mac

Beth

19

73

regr

essi

ons

wit

hla

gged

(i)m

omen

tum

sign

als

wit

hout

cont

rols

Thi

sfig

ure

show

sth

eFa

ma

and

Mac

Beth

19

73

coef

ficie

nts,

and

corr

espo

ndin

gt-

stat

isti

cson

the

lagg

edto

tal

retu

rnan

did

iosy

ncra

tic

mom

entu

mch

arac

teri

stic

s.W

eit

erat

ivel

yru

n6

0su

chre

gres

sion

sw

ith

one

lag

pair

inea

ch,w

ith

noot

her

cont

rols

.We

incl

ude

allc

omm

onst

ocks

trad

edon

NY

SE,

AM

EX,a

ndN

ASD

AQ

exch

ange

sfr

omJu

ly1

96

3to

Dec

embe

r2

01

5ab

ove

20

thpe

rcen

tile

ofm

arke

tca

pof

NY

SEtr

aded

stoc

ksan

dw

ith

shar

epr

ice

abov

e$1

,wit

hno

n-m

issi

ngch

arac

teri

stic

s.To

tal

retu

rnm

omen

tum

isde

fined

asth

e1

2-2

mon

thto

tal

stoc

kre

turn

and

idio

sync

rati

cm

omen

tum

isth

e1

2-2

mon

thvo

lati

lity-

scal

edid

iosy

ncra

tic

retu

rnes

tim

ated

over

past

36

mon

ths

usin

gth

eFa

ma

and

Fren

ch1

99

3th

ree-

fact

orm

odel

.All

vari

able

sar

ew

inso

rize

dat

1%

and

99%

.R

epor

ted

are

the

aver

age

coef

ficie

nts

and

t-st

atis

tics

calc

ulat

edus

ing

New

ey-W

est

corr

ecte

dst

anda

rder

rors

wit

ha

max

imum

of3

lags

.


to 60 months following portfolio formation27. Micro-caps, as definedin the data section, are excluded from the analysis. Similarly, we useMom as a proxy for momentum that is caused by overreaction andconstruct a long-short total return momentum portfolio in a universethat is restricted to 20% of stocks with the highest idiosyncratic mo-mentum scores. Results are presented in Figure 3.5.

Consistent with our hypothesis, we observe that the gap in out-performance of the top over the bottom iMom portfolio in the highMom universe widens over time, and reaches 6% five years after ini-tial portfolio formation (LEFT figure). Conversely, the gap betweenthe top and the bottom Mom portfolio in the iMom universe quicklybecomes negative (RIGHT figure), and continues to widen reaching-5.5% five years into the future. These results support the hypothesisthat idiosyncratic return momentum is more likely to be attributableto investors’ underreaction to firm-specific returns, while in the caseof total return momentum, overreaction seems to be a more plausibleexplanation.

3.6 international evidence

We further test the robustness of our results in four broad regions:Europe, Japan, Asia Pacific (excluding Japan), and emerging markets.The investment universe consists of all constituents of the FTSE WorldDeveloped Index or S&P Developed BMI for Europe, Japan, and AsiaPacific, and for emerging markets, we use S&P/IFC Global EmergingMarkets Index constituents28. The resulting universe consists of ap-proximately 1600, 1200, 450, and 1200 stocks, on average, for Europe,Japan, Asia Pacific, and emerging markets, respectively.

We gather monthly gross stock returns in local currencies, as wellas in U.S. dollars, taking into account dividends, stock splits, andother capital adjustments. Our stock return data sources are Inter-active Data Exshare, MSCI, and S&P/IFS, in that order. Monthly re-turns are truncated at 500%. The free-float adjusted market capitaliza-tion data come from FTSE and S&P/IFC, and accounting data, thatwe need to construct the HML factor, are obtained from Worldscope,MSCI, and SP/IFC.

27 We apply the overlapping portfolio approach, as used in Jegadeesh and Titman 1993.28 Stocks that are included in the broad S&P Developed BMI index are usually also

included in the FTSE World Developed Index. However, the S&P Developed BMIstarts only in 1989. As we aim to obtain the longest possible time series we alsoinclude FTSE World Developed Index constituents that are available from December1985 onwards.

3.6 international evidence 79

Figu

re3

.5:C

umul

ativ

epe

rfor

man

cein

rest

rict

edun

iver

se

The

left

pane

lsho

ws

cum

ulat

ive

retu

rnof

the

top-

bott

om(q

unit

ile)

idio

sync

rati

cm

omen

tum

port

folio

cons

truc

ted

inth

ere

stri

cted

univ

erse

ofst

ocks

wit

h2

0%

high

estt

otal

retu

rnm

omen

tum

scor

es;T

heri

ghtp

anel

show

scu

mul

ativ

ere

turn

ofth

eto

p-bo

ttom

(qun

itile

)tot

alre

turn

mom

entu

mpo

rtfo

lioco

nstr

ucte

din

the

rest

rict

edun

iver

seof

stoc

ksw

ith

20

%hi

ghes

tid

iosy

ncra

tic

retu

rnm

omen

tum

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es.W

eke

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ack

ofth

ese

port

folio

sup

to6

0

mon

ths

follo

win

gpo

rtfo

liofo

rmat

ion.

Tota

lre

turn

mom

entu

mis

defin

edas

the

12

-2m

onth

tota

lst

ock

retu

rnan

did

iosy

ncra

tic

mom

entu

mis

the

12-2

mon

thvo

lati

lity-

scal

edid

iosy

ncra

tic

retu

rnes

tim

ated

over

past

36

mon

ths

usin

gth

eFa

ma

and

Fren

ch1

99

3th

ree-

fact

orm

odel

.We

incl

ude

all

com

mon

stoc

kstr

aded

onN

YSE

,AM

EX,a

ndN

ASD

AQ

exch

ange

sfr

omJu

ly1

96

3to

Dec

embe

r2

01

5ab

ove

20

thpe

rcen

tile

ofm

arke

tca

pof

NY

SEtr

aded

stoc

ksan

dw

ith

shar

epr

ice

abov

e$1

,wit

hva

lidto

talr

etur

nan

did

iosy

ncra

tic

mom

entu

msc

ores

.Por

tfol

ios

are

equa

l-w

eigh

ted.

We

appl

yth

eov

erla

ppin

gpo

rtfo

lioap

proa

chof

Jega

dees

han

dTi

tman

19

93

.


Within each region, we construct equal-weighted quintile portfo-lios by ranking stocks on idiosyncratic and total return momentum,respectively, in a country-neutral manner. This ensures that we donot have strong structural biases to any given country within the cor-responding region. Both momentum measures are defined as for theU.S. stocks, and we use local returns to avoid introducing noise thatis due to currency effects29. Portfolio returns are in U.S. dollars inexcess of the one-month Treasury bill rate from January 1989 for Eu-rope, Asia Pacific, and Japan, and January 1992 for emerging markets,till the end of December 2015. These results are reported in Table 10.

Similar to our earlier results for the U.S., idiosyncratic momentumgenerates superior risk-adjusted top-minus-bottom returns relative toconventional momentum in all markets that we consider. While thissample period was very favorable for total return momentum, withreturns in excess of 80 basis points per month in Europe, Asia Pacific,and emerging markets, idiosyncratic momentum still delivered highreturns with a substantial reduction in volatility, resulting in substan-tially higher Sharpe ratios and t-statistics for CAPM and three-factoralphas. Our results are consistent with Chaves 2016, who finds strongresults for a simplified definition of idiosyncratic momentum in inter-national developed equity markets. We stick to the original definitionand also consider emerging equity markets, which have not been ex-amined before.

When we regress the conventional (idiosyncratic) momentum re-turns on the three-factor model augmented with an idiosyncratic(conventional) momentum factor, both strategies exhibit highly signif-icant four-factor model alphas for Europe, Asia Pacific, and emergingmarkets. Our results, therefore, do not favor one momentum factorover the other for these regions and indicate that they behave likecomplements, as opposed to substitutes.

Our results are even more compelling for Japan. Similar to otherstudies (e.g. Griffin, Ji, and Martin 2003; Fama and French 2012), wefind momentum returns that are close to zero. Hanauer 2014 arguesthat different market dynamics in Japan, i.e. the prevalence of peri-ods with market reversals compared to periods with market continua-tions, cause these overall low returns of momentum. He further docu-ments that momentum returns are also significantly positive in Japan

29 We construct regional Fama-French equivalent hedge factors by ranking stocks ontheir market capitalization and book-to-market ratio, respectively. Thereby, we definethe size (small-minus-big, SMB) and value (high-minus-low, HML) factors as thereturn difference between the value-weighted top and bottom tercile.

3.6 international evidence 81

Tabl

e3

.14:I

nter

nati

onal

resu

lts

This

tabl

esh

ows

perf

orm

ance

ofto

tal

retu

rnan

did

iosy

ncra

tic

mom

entu

min

inte

rnat

iona

lm

arke

ts.

We

focu

son

four

broa

dre

gion

s:Eu

rope

,Ja

pan,

Asi

aPa

cific

(exc

ludi

ngJa

pan)

and

emer

ging

mar

kets

.Our

univ

erse

cons

ists

ofal

lco

nsti

tuen

tsof

the

FTSE

Wor

ldD

evel

oped

Inde

xor

SPD

evel

oped

BMI

for

Euro

pe,J

apan

,and

Asi

aPa

cific

,and

for

emer

ging

mar

kets

,we

use

S&P/

IFC

Glo

balE

mer

ging

Mar

kets

Inde

xco

nsti

tuen

ts.O

ursa

mpl

eco

vers

the

peri

odfr

omth

een

dof

Dec

embe

r1

98

9

toth

een

dof

Dec

embe

r2

01

5fo

rEu

rope

,Jap

an,a

ndA

sia

Paci

fic,a

ndfr

omth

een

dof

Dec

embe

r1

99

2to

the

end

ofD

ecem

ber

20

15

for

emer

ging

mar

kets

.Wit

hin

each

regi

on,w

eco

nstr

uct

equa

lly-w

eigh

ted

quin

tile

port

folio

sby

rank

ing

stoc

kson

idio

sync

rati

can

dto

tal

retu

rnm

omen

tum

,res

pect

ivel

y,in

aco

untr

yne

utra

lm

anne

r.To

talr

etur

nm

omen

tum

isde

fined

asth

e1

2-2

mon

thto

tals

tock

retu

rnan

did

iosy

ncra

tic

mom

entu

mis

the

12-2

mon

thvo

lati

lity-

scal

edid

iosy

ncra

tic

retu

rnes

tim

ated

over

past

36

mon

ths

usin

gth

ere

gion

alm

arke

t,si

ze,a

ndva

lue

fact

ors.

Port

folio

retu

rns

are

inU

.S.d

olla

rsin

exce

ssof

the

one-

mon

thTr

easu

rybi

llra

te.

Exce

ssR

etur

nVo

lSh

arpe

Rat

ioA

lpha

CA

PMt-

stat

Alp

ha3FM

t-st

atA

lpha

4FM

*t-

stat

Idio

sync

rati

cM

omen

tum

EUR

Q1

1.0

05.2

00

.19

0.5

2(5

.19

)0

.54

(8.7

1)

0.4

2(7

.89

)Q

50.1

35.9

10

.02

-0.4

1(-

3.2

1)

-0.3

6(-

5.6

5)

-0.2

5(-

4.3

9)

Q1

-Q5

0.8

72.1

60

.40

0.9

2(7

.91

)0

.90

(8.6

1)

0.6

7(7

.94

)

JAP

Q1

0.2

86.5

90

.04

0.3

9(2

.52

)0

.21

(2.4

2)

0.1

3(1

.82

)Q

5-0

.16

7.1

8-0

.02

-0.0

4(-

0.2

0)

-0.3

2(-

4.0

2)

-0.2

4(-

3.8

4)

Q1

-Q5

0.4

42.6

90

.16

0.4

3(2

.92

)0

.53

(3.6

1)

0.3

7(3

.40

)

PCFx

JPQ

11.0

96.4

10

.17

0.4

0(3

.32

)0

.65

(6.7

9)

0.6

3(6

.61

)Q

50.0

57.1

30

.01

-0.7

0(-

4.5

2)

-0.3

1(-

3.0

7)

-0.2

4(-

2.5

5)

Q1

-Q5

1.0

42.7

30

.38

1.1

0(7

.35

)0

.96

(6.5

6)

0.8

7(6

.25

)

EMQ

11.1

46.6

20

.17

0.6

0(4

.36

)0

.58

(4.6

9)

0.4

1(3

.41

)Q

50.3

97.1

60

.05

-0.1

8(-

1.0

2)

-0.2

1(-

1.4

4)

-0.1

1(-

0.7

6)

Q1

-Q5

0.7

52.3

00

.33

0.7

8(5

.77

)0

.79

(5.8

8)

0.5

3(4

.24

)

Tota

lRet

urn

Mom

entu

m

EUR

Q1

1.1

05.1

60

.21

0.6

3(5

.30

)0

.64

(6.7

2)

0.4

1(6

.80

)Q

5-0

.07

7.3

6-0

.01

-0.7

0(-

3.4

4)

-0.6

3(-

5.3

9)

-0.3

7(-

4.4

7)

Q1

-Q5

1.1

74.6

10

.25

1.3

4(5

.57

)1

.27

(6.5

9)

0.7

8(6

.97

)

JAP

Q1

0.0

76.4

00

.01

0.1

8(1

.14

)0

.12

(0.9

1)

0.0

0(-

0.0

1)

Q5

0.0

18.6

40

.00

0.1

5(0

.58

)-0

.26

(-1

.67

)-0

.12

(-1

.21

)Q

1-Q

50

.06

5.5

90

.01

0.0

3(0

.10

)0

.38

(1.3

9)

0.1

1(0

.76

)

PCFx

JPQ

11.0

46.3

10

.17

0.3

6(2

.89

)0

.57

(5.0

9)

0.4

5(4

.54

)Q

5-0

.35

8.5

0-0

.04

-1.2

1(-

5.2

9)

-0.6

2(-

4.3

7)

-0.4

5(-

3.7

2)

Q1

-Q5

1.3

94.7

00

.30

1.5

7(6

.32

)1

.19

(5.5

2)

0.9

0(5

.19

)

EMQ

11.1

36.4

70

.18

0.6

1(4

.43

)0

.61

(4.7

2)

0.3

3(2

.81

)Q

50.3

28.1

70

.04

-0.3

0(-

1.2

9)

-0.3

6(-

1.8

5)

-0.0

9(-

0.4

5)

Q1

-Q5

0.8

14.1

80

.19

0.9

1(3

.82

)0

.97

(4.3

2)

0.4

2(2

.14

)


when the market stays in the same condition, but significantly neg-ative when it reverses. As these market reversals occurred more fre-quently compared to the U.S. or European markets during our sampleperiod, and conventional momentum tends to underperform idiosyn-cratic momentum during these periods (as shown in section 3.5.2), id-iosyncratic momentum should show better performance. Indeed, id-iosyncratic momentum generates a return of 0.44% per month whichis statistically significant at the most conservative levels. Also, theCAPM, three-factor, and four-factor model regressions show highlysignificant alphas for idiosyncratic momentum while the ones for con-ventional momentum are insignificant. These results indicate that thereduced time-varying exposures to systematic risk factors of idiosyn-cratic momentum significantly enhance the effectiveness of the strat-egy to such an extent that it even becomes a successful momentum-related strategy in Japan. Our finding that idiosyncratic momentumis also effective in Japan, contrary to conventional momentum, is con-sistent with Chaves 2016 and Chang et al. 2018.

Figure 3.6 shows the results of Fama and MacBeth 1973 regressionswith the lagged conventional and idiosyncratic momentum scores,controlling for the market beta, size, and value characteristics. Inorder to increase the breadth of the sample, we pull all regions to-gether and include country-level dummy variables to absorb country-specific effects. Similar to our findings for the US market, lagged val-ues of idiosyncratic momentum forecast positive returns up to fiveyears into the future, while those of total return momentum forecastnegative returns already one year following portfolio formation, con-sistent with the under-reaction hypothesis for idiosyncratic momen-tum and overreaction hypothesis for conventional momentum. Also,without applying lags for either of the momentum signals, that is us-ing standard Fama and MacBeth 1973 regressions, idiosyncratic mo-mentum emerges stronger with a t-statistic of 4.20 while total returnmomentum is not priced with a t-statistic of 1.27. Our results areconsistent with Chang et al. 2018, who link idiosyncratic momentumprofits in Japan to investor under-reaction using different arguments.

3.7 conclusion

Portfolios formed on idiosyncratic, as opposed to total past returnsgenerate comparable average returns, with half the volatility of theconventional momentum strategy. We provide evidence that idiosyn-

3.7 conclusion 83

Figu

re3.6

:Fam

aan

dM

acBe

th1

97

3re

gres

sion

sw

ith

lagg

ed(i

)mom

entu

msi

gnal

sw

ith

cont

rols

inin

tern

atio

nalm

arke

ts

This

figur

esh

ows

the

Fam

aan

dM

acBe

th1

97

3co

effic

ient

s,an

dco

rres

pond

ing

t-st

atis

tics

onth

ela

gged

tota

lre

turn

and

idio

sync

rati

cm

omen

tum

char

acte

rist

ics.

We

iter

ativ

ely

run

60

such

regr

essi

ons

wit

hon

ela

g-pa

irin

each

,w

ith

the

follo

win

gco

ntro

lsfix

edat

tim

et-

1:

beta

,siz

e,va

lue,

and

coun

try

dum

mie

s.W

efo

cus

onfo

urbr

oad

regi

ons:

Euro

pe,

Japa

n,A

sia

Paci

fic(e

xclu

ding

Japa

n)an

dem

ergi

ngm

arke

ts.

Our

univ

erse

cons

ists

ofal

lco

nsti

tuen

tsof

the

FTSE

Wor

ldD

evel

oped

Inde

xor

S&P

Dev

elop

edBM

Ifo

rEu

rope

,Jap

an,a

ndA

sia

Paci

fic,a

ndfo

rem

ergi

ngm

arke

ts,w

eus

eS&

P/IF

CG

loba

lEm

ergi

ngM

arke

tsIn

dex

cons

titu

ents

.Our

sam

ple

cove

rsth

epe

riod

from

the

end

ofD

ecem

ber

19

89

toth

een

dof

Dec

embe

r2

01

5fo

rEu

rope

,Jap

an,a

ndA

sia

Paci

fic,a

ndfr

omth

een

dof

Dec

embe

r1

99

2to

the

end

ofD

ecem

ber

20

15

for

emer

ging

mar

kets

.Bet

ais

the

slop

eco

effic

ient

onth

em

arke

tfa

ctor

esti

mat

edus

ing

univ

aria

tere

gres

sion

sof

stoc

kex

cess

retu

rns

onth

eon

e-fa

ctor

mod

el(C

APM

)fr

omt-

36

tot-

1(m

int-

12

tot-

1).

Size

isth

ena

tura

llog

arit

hmof

firm

’sm

arke

tca

pita

lizat

ion

atth

een

dof

mon

tht,

valu

eis

the

natu

rall

ogar

ithm

ofth

era

tio

offir

ms

book

equi

tyfo

rth

efis

caly

ear

lagg

edby

atle

ast

6m

onth

san

dm

arke

tca

pal

sola

gged

by6

mon

ths.

Tota

lret

urn

mom

entu

mis

defin

edas

the

12

-2m

onth

tota

lsto

ckre

turn

and

idio

sync

rati

cm

omen

tum

isth

e1

2-2

mon

thvo

lati

lity-

scal

edid

iosy

ncra

tic

retu

rnes

tim

ated

over

past

36

mon

ths

usin

gth

eFa

ma

and

Fren

ch1

99

3th

ree-

fact

orm

odel

.All

vari

able

sar

ew

inso

rize

dat

1%

and

99

%.R

epor

ted

are

the

aver

age

coef

ficie

nts

and

t-st

atis

tics

calc

ulat

edus

ing

New

ey-W

est

corr

ecte

dst

anda

rder

rors

wit

ha

max

imum

of3

lags

.


cratic momentum is a separate factor that expands the efficient fron-tier comprised of already established asset pricing factors, even ifone accounts for conventional momentum. We further discuss andtest some of the most prominent explanations that have been put for-ward for conventional momentum, and find that none of them holdfor idiosyncratic momentum. In particular, we show that, unlike con-ventional momentum, idiosyncratic momentum profits are positivefollowing bull, as well as bear markets, albeit insignificant in the lattercase, and that they are substantially less affected by market dynam-ics, where the return in the month following a bull or bear market istaken into account. These findings go against the overconfidence andoverreaction explanations for the anomaly. We also find substantiallylower non-linear crash risk exposure embedded in idiosyncratic mo-mentum than in its total return counterpart, which leads us to rejectthe hypothesis that superior performance of idiosyncratic over totalreturn momentum can be explained by this risk source. Our empiricalresults support the underreaction hypothesis for the existence of theidiosyncratic momentum premium. We find that controlling for otherknown predictors of stock returns in the cross-section, conventionalmomentum forecasts high short term returns, but becomes insignif-icant quickly, and turns negative around one year following portfo-lio formation. On the other hand, idiosyncratic momentum forecastshigh short and long-term returns. We also show that one can use id-iosyncratic momentum to distinguish between past total return win-ners that are prone to long term reversal and those that are not.

The fact that we cannot conclusively reject one factor in favor ofthe other, and our inability to link these two momentum phenomenato the same underlying mechanisms, leads us to conclude that theybehave more like complements than like substitutes. Finally, we docu-ment significant idiosyncratic momentum profits in international eq-uity markets, including the one market where conventional momen-tum is known to be ineffective - Japan. We conclude that idiosyncraticmomentum presents an even bigger challenge to the asset pricingliterature and that the underreaction explanation for the premiumseems more likely than the various risk-based and behavioral expla-nations that have been proposed for conventional momentum.

3.8 appendix

3.8 appendix 85

Table3.

15:Perform

anceof

decileportfolios

19

29-

20

15

Thistable

reportsperform

ancecharacteristics

ofequal-w

eighteddecile

portfoliosconstructed

asunivariate

sortson

idiosyncraticand

totalreturn

mom

entum,

re-spectively.W

einclude

allcom

mon

stockstraded

onN

YSE,A

MEX

,andN

ASD

AQ

exchangesfrom

July1

92

9to

Decem

ber2

01

5above

20th

percentileof

market

capof

NY

SEtraded

stocksand

with

shareprice

above$1,

with

validtotal

returnand

idiosyncraticm

omentum

scores.Total

returnm

omentum

isdefined

asthe

12-

2

month

totalstock

returnand

idiosyncraticm

omentum

isthe

12-

2m


idiosyncraticreturn

estimated

overpast

36

months

usingthe

Fama

andFrench

19

93

three-factorm

odel.Foreach

portfolio,we

reportreturns

inexcess

ofthe


Sharperatios,C

APM

-,three-factorm

odelalphas,and

correspondingt-statistics.A

lsoreported

arethe

GR

Stest

statisticsfor

eachof

thecorresponding

assetpricing

models,w

herethe

testassets

aretotal

returnand

idiosyncraticm

omentum

sorteddeciles.Portfolios

arereform

edm

onthly.

ExcessR

eturnVol

SharpeR

atioA

lphaC

APM

tstatA

lpha3FM

tstat

IdiosyncraticM

omentum

D1

0.31

0.08

0.04

-0.

47

-5.

15

-0.

61

-8.

96

D2

0.52

0.07

0.07

-0.

21

-2.

86

-0.

34

-6.

10

D3

0.61

0.07

0.09

-0.

10

-1.

49

-0.

24

-4.

79

D4

0.76

0.07

0.11

0.05

0.67

-0.

10

-2.

28

D5

0.84

0.07

0.12

0.13

1.85

-0.

01

-0.

25

D6

0.86

0.07

0.13

0.15

2.29

0.01

0.35

D7

0.98

0.07

0.14

0.27

3.78

0.12

2.85

D8

1.04

0.07

0.15

0.32

4.41

0.18

3.87

D9

1.13

0.07

0.17

0.42

6.28

0.30

6.58

D10

1.27

0.07

0.19

0.58

7.75

0.50

8.66

D10-D

10.

96

3.44

0.28

1.04

9.94

1.11

10.

80

GR

S12.

85

0.00

12.

04

0.00

TotalReturn

Mom

entum

D1

0.32

0.10

0.03

-0.

63

-4.

14

-0.

89

-7.

19

D2

0.60

0.08

0.07

-0.

24

-2.

16

-0.

46

-5.

54

D3

0.70

0.07

0.10

-0.

05

-0.

56

-0.

23

-3.

72

D4

0.75

0.07

0.11

0.05

0.61

-0.

12

-2.

37

D5

0.76

0.06

0.12

0.09

1.32

-0.

05

-1.

15

D6

0.84

0.06

0.14

0.19

3.15

0.08

1.68

D7

0.91

0.06

0.15

0.27

4.54

0.17

3.83

D8

0.97

0.06

0.16

0.33

5.21

0.25

4.92

D9

1.13

0.06

0.18

0.48

5.94

0.43

6.60

D10

1.33

0.07

0.18

0.65

5.17

0.63

6.48

D10-D

11.

01

7.21

0.14

1.28

6.05

1.52

7.80

GR

S6.

46

0.00

6.88

0.00


Table3.

16:

19

29-

20

15

Fama

andM

acBeth1

97

3regressions

Thistable

reportsthe

resultsofFam

aand

MacBeth

19

73

regressions.We

includeallcom

mon

stockstraded

onN

YSE,A

MEX

,andN

ASD

AQ

exchangesfrom

July1

92

9to

Decem

ber2

01

5above

20th

percentileof

market

capof

NY

SEtraded

stocksand

with

shareprice

above$1,

with

non-missing

characteristics.Betais

theslope

coefficientonthe

marketfactor

estimated

usingunivariate

regressionsofstock

excessreturns

onthe

onefactor

model

(CA

PM)

fromt-

60

tot-

1(m

int-

24

tot-

1).Size

isthe

naturallogarithm

offirm

’sm

arketcapitalization

atthe

endof

month

t,value

isthe

naturallogarithm

ofthe

ratioof

firms

bookequity

forthe

fiscalyearending

int-

1and

market

capat

theend

ofD

ecember

oft-

1.Totalreturnm

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Table3.

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4M A C R O D R I V E R S O F L O W- V O L AT I L I T Y S T O C KR E T U R N S

This chapter is jointwork with RenxuanWang.

4.1 introduction

A portfolio long stocks with low past return volatility and short theirhigh-volatility counterparts generates significant abnormal returns(alphas) with respect to the commonly used asset pricing benchmarks,yet the source of these seemingly risk-less profits remains a puzzle inthe asset pricing literature1. For long-term investors who hold diver-sified portfolios of stocks and bonds, and aim for high long-term risk-adjusted returns, hedges against deteriorating investment opportuni-ties should come at the expense of expected returns. In this paper, weprovide evidence that low and high-vol stock beta-adjusted returnsreact in opposite ways in response to shocks to a host of macroeco-nomic variables stemming from bond yields, forward rates, inflation,and aggregate volatility.

We examine the links between low-volatility anomaly returns andthese macroeconomic state variables through the lens of the presentvalue identity and propose a novel methodology to decompose re-turns of dynamic (rebalanced) portfolios into the discount rate andcash-flow news components on a single stock-level. Differently fromprior papers, our approach takes into account the fact that, over time,stocks migrate from one portfolio to another. Thus, we account forthe fact that a stock that could have started off as a high-vol stockcould have become a mid or a low-vol stock, or the other way around.While in the short-run return volatility is quite persistent, over theentire half-life of a stock, it is not unlikely that a stocks transitions

1 The concept of low-risk is not uniformly defined; in general, it refers to the empiricalobservation that stocks that appear to be less risky under commonly used statisticalrisk measures such as beta, past or implied return volatility, or idiosyncratic volatil-ity, deliver higher risk-adjusted returns than their more risky peers. To avoid anyambiguity, throughout this study, we use low past realized volatility as a proxy forlow-risk.

89

90 macro drivers of low-volatility stock returns

in or out of high or low-volatility portfolio, just as a small cap canbecome a large-cap, or a growth stock reaches its steady state growthand becomes a value stock. We show that failing to incorporate thesedynamics within the VAR framework would lead to a conclusion thataggregate variables do not predict stock returns, and thus that aggre-gate macro series do not have any impact on the low and high-volstock alphas.

Our VAR allows us to incorporate a rich set of aggregate state vari-ables, and using impulse responses, we can properly control for thecovariance structure of their shocks and shocks to stock-level statevariables, and quantitatively describe the long-term return dynamicsof returns on these portfolios. Furthermore, our choice to conductanalysis on a single stock-level, as opposed to portfolio-level, can bemotivated by Lewellen, Nagel, and Shanken 2010 who raise a numberof concerns about asset pricing tests that use as test assets portfolioswith strong factor structures.

Our main results hold strong implications: controlling for their mar-ket betas, high-vol stocks hedge against states of the world in whichbond returns/yields are falling/rising, inflation is rising, and mar-ket variance spikes. Low-vol stocks, on the contrary, perform poorlyunder these conditions. Using the Cochrane and Piazzesi 2005 tent-shaped return forecasting factor, or the slope of the term structure ofinterest rates as the aggregate state variables, we show that a positiveshock to these series has a negative impact on the beta-adjusted re-turns of low-vol stocks, and a positive impact on returns of high-volstocks. These effects are highly economically and statistically signifi-cant, and very persistent over time.

Within the same framework, we also explore the relationship be-tween low and high-vol stock returns and the level of the term struc-ture of interest rates and the short rate. Consistent with other results,we find a strong relationship between these state variables and re-turns on low and high-vol stocks.

Another state variable we analyze is inflation. As inflation is an im-portant driver of nominal yields, we naturally ask if inflation shockshave an impact on beta-adjusted returns on low and high-vol stocks.We find results in support of this hypothesis.

High-vol stocks also hedge against periods when market volatilityspikes. Contrary to this, low-vol stocks have negative beta-adjusted re-turns in these states. These results are consistent with those of Camp-bell et al. 2017 who find that high beta stocks, related to high-volatility

4.1 introduction 91

stocks, also hedge against sharp increases in market volatility. Barinov2013 finds that stocks with many growth options and high idiosyn-cratic volatility serve as a natural hedge against increases in aggre-gate volatility and proposes a two-factor Merton 1973 ICAPM modelto capture the idiosyncratic volatility anomaly. He argues that stockswith the same levels of market exposure, i.e. beta, but different levelsof idiosyncratic volatility will react differently when the aggregatevolatility spikes. While the whole equity market tends to collapseduring these periods, high-ivol stocks win by losing less than low-ivolstocks, adjusted for their market betas.

From a point of view of a mean-variance investor who allocatestheir wealth to stocks and bonds, the high, positive covariance be-tween low-vol stocks and bonds implies that their hedging utility issignificantly reduced if they opt to add low-vol stocks into their port-folio. Consequently, this investor needs to be compensated for thisrisk and requires a higher expected return on low-vol stocks thanwhat is justified by their market betas. Our hypothetical investor isnot dissimilar to their real-world analog that subscribes to the tradi-tional 60-40 allocation scheme.

Boons 2016 also examines the risk premiums for exposure to vari-ous macroeconomics state variables, such as term, default, short rate,dividend yield, price-earnings ratio, value spread, and Cochrane andPiazzesi 2005 factor in the cross-section of individual stocks. He findsthat risk premiums of these state variables are consistent in sign withhow they forecast macroeconomic activity in the time-series. His re-sults are consistent with ICAPM and suggest that investors do careabout macroeconomic news. Related to this, Koijen, Lusting, and vanNieuwerburgh 2017 show that a parsimonious three-factor modelcomprised of the market factor, shocks to the level of the term struc-ture, and shocks to the CP factor explains the anomalous returns ofbook-to-market sorted portfolios, maturity sorted government bondportfolios and credit portfolios of different ratings.

Our work builds on a large literature that exploits the present valuereturn identity. Campbell and Shiller 1988 approximated the log re-turn on a dividend-paying asset around the mean log dividend-priceratio using a first-order Taylor series expansion. They show that thelog price-dividend ratio is high when either the expected dividendgrowth is high, or expected stock returns are low (i.e expected dis-count rates are low).


Campbell 1991 further showed that one can (approximately) de-compose returns into news about future cash flows, and news aboutfuture discount rates. An increase in expected dividend growth is as-sociated with a price appreciation today, and an increase in discountrates is associated with a price depreciation today. Campbell andVuolteenaho 2004 decompose market returns into cash-flow and dis-count rate news, and estimate sensitivities of various anomaly port-folios to these news shocks. They show that the price of risk for theCF news is much higher than that of DR news, and discuss how vari-ous asset pricing anomalies can be resolved through the lens of theirtwo-beta model.

Vuolteenaho 2002 extends the methodology of Campbell and Shiller1988 to impose the present value identity and decompose firm’s logbook-to-market ratio into the expected return and cash-flow compo-nents. He finds that firm-level returns are mostly driven by chang-ing expectations about cash-flows, with the variance of the cash-flowcomponent being two times larger than that of the discount rate (i.e.expected return) component, but that cash-flow components can belargely diversified away in aggregate portfolios.

On the other hand, Lochstoer and Tetlock 2016 find that, on ananomaly portfolio-level (i.e. long-short), cash-flow shocks explain moreof the return volatility than the discount rate shocks. They focus onvalue, size, profitability, investment, share issuance, and price mo-mentum anomalies and find consistent results for all of them. Further,they show that cash-flow and discount rate components across theseanomalies are not strongly related, suggesting that aggregate shocksare unlikely to be the main drivers of anomaly returns.

We study the dynamics of the volatility-sorted portfolios and doc-ument that on a long-short beta-adjusted anomaly-level, cash-flownews drive around four times more of return variance than the dis-count rate news, however, we also find that the two news series arehighly negatively correlated, making the decomposition hard to inter-pret.

Our main methodological contribution to the literature is to incor-porate a transition matrix, that governs in which portfolios stocksbelong, into the VAR dynamics. Just like Lochstoer and Tetlock 2016,we impose the present value identity on the log book-to-market ratio,decompose stock returns on a firm-level, and aggregate them intoportfolios; but differently from prior work, we include additional in-dicator variable interactions with the stock-level state variables in our

4.2 literature on low-risk anomaly 93

VAR, where indicators track in which volatility-sorted portfolio a par-ticular firm is in at any point in time. We show that there is significantheterogeneity in the dynamics across these portfolios that impacts theconclusions substantially.

What makes low-vol stocks so sensitive to bond risk premia? Ourresults indicate that factors that drive the term structure of inter-est rates also drive returns on volatility-sorted portfolios. However,our paper does not have much to say on where these factors stemfrom; instead, we refer readers to the vast literature on the termstructure (see, Ang and Piazzesi 2003, Litterman and Scheinkman1991). However, we do consider one important driver of the termstructure, inflation, and find it to be a significant driver of the low-minus-high return spread. Baker and Wurgler 2012 also examine thelinks between returns on government bonds and the cross-section ofbond-like stocks, that is, stocks of large, mature, profitable, high div-idend, low-volatility firms. They find strong and stable relationshipsbetween these assets and suggest that more research in this area isneeded to pin down the exact drivers of these patterns.

This paper is organized as follows. Section 4.2 provides a summaryof the literature on the low-risk anomaly. Section 4.3 presents the mo-tivating results on the relationship between volatility-sorted portfolioreturns and bond yields. Section 4.4 presents the methodology forreturn variance decomposition. In section 4.5, we describe the data,and section 4.6 presents the main empirical results. Section 4.7 linksour findings to the established Fama and French 1993 and Fama andFrench 2015 asset pricing models and concludes the paper.

4.2 literature on low-risk anomaly

Black, Jensen, and Scholes 1972 were among the first to show thatthe security market line is much flatter than one would expect underthe general equilibrium market model, where expected stock returnsare a positive, linear function of exposures to only one factor - themarket. Almost half a century later, the failure of the unconditionalCAPM to explain the cross-section of stock returns is taken as anaxiom. More recently, Frazzini and Pedersen 2014 present a modelwith leverage and margin constraints that generates empirically con-sistent results, whereby stocks with high betas have low alphas, andconversely, stocks with low betas have high alphas. Hong and Sraer2016 argue that high beta stocks are more speculative than low beta


stocks, and due to short-sale constraints, they are more prone to over-pricing. Their model also generates the low-beta (low-risk) anomaly.More recently, Liu, Stambaugh, and Yuan 2017 argue that the low-beta anomaly arises due to a positive correlation between beta andidiosyncratic volatility, which makes beta a convoluted proxy for thetrue misprinting.

The low-risk literature has also evolved in other directions. Blitzand van Vliet 2007, Baker, Bradley, and Wurgler 2011, Baker andHaugen 2012, show that portfolio sorted on past return volatility alsogenerate anomalously high risk-adjusted returns. This anomaly oftenreferred to as the low-volatility effect, is closely related to the low-beta effect. On the other hand, Ang et al. 2006 show that stocks withlow idiosyncratic volatility with regard to the Fama and French 1993

model have high average returns. They go on to argue that residualsin the regression of stock returns on the Fama-French factors containinformation about other factors that are not included in this model,and as such could be priced in the cross-section. Their empirical find-ings strongly support this hypothesis.

The robustness of all these effects across different markets and timeperiods has been confirmed in a number of papers. Unsurprisingly,many researchers have tried to develop models, mostly conditional innature, that attempt to capture these patterns. Proponents of the (ra-tional) risk-based asset pricing have argued that low-risk stocks earnhigh average returns because they are riskier, and the CAPM failsto capture some aspects of their risk. For instance, both Novy-Marx2014 and Fama and French 2015 try to attribute abnormal returns oflow-risk stocks to their value and profitability slopes, and while it istrue that these stocks load positively on these factors, their evidenceremains inconclusive (see Blitz and Vidojevic 2017 for a detailed dis-cussion). Also, it remains unclear whether factors in their proposedmodels capture systematic risks in the first place. Schneider, Wagner,and Zechner 2016 provide another potential explanation for the low-beta and low-volatility anomalies: these strategies are profitable dueto high skew risk that causes CAPM to overestimate the required eq-uity returns relative to a skew-adjusted model. The mechanism thatgenerates skewness of returns in their model is linked to the defaultrisk.

On the other hand, institutional frictions, such as regulatory, lever-age, and short-selling constraints (Black 1972, Frazzini and Pedersen2014), incentives of delegated wealth managers (Chistoffersen and

4.3 motivating results 95

Simutin 2017, Sirri and Tufano 1998), propensity of people to play lot-teries (Barberis and Huang 2007) also provide explanations for whylow-risk stocks end up being neglected and generate high abnormalreturns, despite their low levels of risk. At present, the literature hasnot settled on the explanation for this effect.

4.3 motivating results

Table 4.1 shows the performance characteristics of five value-weighted,volatility-sorted portfolios from 1952 till 2015. Low-vol stocks haveaverage excess returns of 0.59% a month, 0.15% (t-stat of 3.04) and0.10% (t-stat of 2.51) of which cannot be explained by the CAPM andthe Fama and French 1993 three factor models, respectively. The high-vol portfolio has an average excess return of 0.47% a month, which islower than that of the low-vol portfolio, and both the CAPM, as wellas the three factor model alphas are negative and significant: - 0.49%(t-stat of -2.99) and -0.42% (t-stat of -3.63). On a long-short portfolio-level, these alphas amount to 0.64% month (t-stat of 3.19) and 0.53%(t-stat of 3.73) in the case of CAPM and three factor model, confirm-ing the presence of the low-vol effect in our sample.

Figure 4.1 plots the change in the dividend-price ratio of the low-minus-high risk portfolio over the past 36 months, together with thechange in the level of the term structure of interest rates over the sameperiod2. The pattern is pervasive: changes in the dividend-price ratioof low-minus-high risk portfolio move closely with the changes inthe level. The correlations between these series is 48%, with a p-valuepractically equal to 0.

The relationship between low-vol stocks and bonds is also visi-ble from their ex-post returns. In Figure 4.2 we plot the slope coef-ficients from a regression of portfolio excess returns on the (negativeof) shocks to the level of the term structure3, controlling for the mar-ket factor4. The level of the term structure is widely recognized as amajor factor that drives bond yields, and shocks to level are definedas residuals from an AR(1) model.

2 Level is the first principal component extracted from 30 to 1 year maturity-sortedgovernment bond portfolios.

3 This factor has also been used by Koijen, Lusting, and van Nieuwerburgh 2017.4 A similar picture emerges if we regress on a simple bond index.


Table 4.1: Performance characteristics

This tables contains the performance characteristics (excess return, volatil-ity, ex-post Sharpe ratio, CAPM and Fama-French three factor model alphasand slope coefficients with respective t-stats) of the volatility-sorted port-folios . The sample consists of all common stocks traded on NYSE/AMEXand NASDAQ exchanges from January 1952 to December 2015, except thosewith prices below $1. Volatility is defined as the standard deviation of pre-vious 36-monthly return observations. Portfolios are value-weighted andrebalanced monthly.

LowVol Q2 Q3 Q4 HighVol Low-High

ex.ret 0.59 0.66 0.63 0.70 0.47 0.13

vol 3.53 4.68 5.67 6.89 8.34 6.70

sharpe 0.17 0.14 0.11 0.10 0.06 0.02

α 0.15 0.04 -0.10 -0.14 -0.49 0.64

(3.04) (0.97) (-1.64) (-1.32) (-2.99) (3.19)βmkt 0.76 1.06 1.26 1.45 1.64 -0.88

(64.84) (111.13) (85.70) (58.21) (43.66) (-19.11)r.sq 0.85 0.94 0.91 0.82 0.71 0.32

α 0.10 0.00 -0.09 -0.09 -0.42 0.53

(2.51) (-0.09) (-1.54) (-1.14) (-3.63) (3.73)βmkt 0.82 1.09 1.20 1.30 1.40 -0.58

(82.24) (112.35) (85.08) (65.17) (49.66) (-16.86)βsmb -0.24 -0.05 0.25 0.64 1.06 -1.29

(-16.11) (-3.78) (12.11) (22.02) (25.60) (-25.85)βhml 0.15 0.11 -0.07 -0.21 -0.31 0.46

(9.71) (7.26) (-3.27) (-6.71) (-6.97) (8.60)r.sq 0.90 0.95 0.92 0.90 0.86 0.67


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Figure 4.2: Sensitivity to bonds

This figure plots the point estimates of the regression of excess returns on volatility-sorted portfolios on the excess returns on the (inverse) of the shocks to the level ofthe term structure of interest rates, controlling for the equity market factor. The sam-ple consists of all common stocks traded on NYSE/AMEX and NASDAQ exchangesfrom February 1952 to December 2015. Level is the first principal component ex-tracted from 30 to 1 year maturity-sorted government bond portfolios. The sampleconsists of all common stocks traded on NYSE/AMEX and NASDAQ exchangesfrom January 1952 to December 2015. Shocks are defined as residuals from an AR1

model. Volatility is defined as the standard deviation of previous 36-monthly returnobservations. Portfolios are value-weighted and rebalanced monthly.

Returns of low-vol stocks co-vary positively/negatively and signifi-cantly with bond returns/yields (t-stat of 6.10), whereas in the case ofhigh-risk stocks, the opposite is true (t-stat of -4.50). In fact, the sen-sitivity of these portfolios to bond returns (yields) appears to mono-tonically decrease (increase) as we move from the low to the high-riskportfolio.

Low-vol stocks also have higher dividend yields than their high-volcounterparts, a feature that can be particularly appealing to investorswho like regular coupon payouts - the fixed-income investors. Table4.2 shows the average annual dividend-price ratios of these five port-folios together with the standard deviation of their AR1 innovations5.

Figure 4.3 shows the dynamics of the dividend-price ratios of lowand high-vol portfolios over time. Two interesting observations standout: (i) low-vol stocks have a persistently higher dividend-price ra-tio than high-vol stocks; (ii) dividend-price ratios appear to be non-stationary, especially in the case of high-vol stocks, that tend to be

5 Due to a high level of autocorrelation in the d-p series, we calculate standard devia-tions of the AR(1) residuals.


Table 4.2: Dividend-price ratios of volatility-sorted portfolios

This table shows the full sample mean 12-month dividend yield of thevolatility-sorted portfolios and the volatility of their AR1 innovations. Divi-dend yield is the difference between 12-month log gross and net portfolio-level returns. The sample consists of all common stocks traded on NY-SE/AMEX and NASDAQ exchanges from January 1952 to December 2015,except those with prices below $1. Volatility is defined as the standard de-viation of previous 36-monthly return observations. Portfolios are value-weighted and rebalanced monthly.

LowVol Q2 Q3 Q4 HighVol

mean 3.76% 2.85% 2.24% 1.58% 0.95%vol of innov 0.41% 0.49% 0.61% 0.43% 0.34%

small, growth-like stocks that often times do not pay dividends. Forthis reason, a decomposition of the portfolio-level log dividend-priceratio would not be possible - an approach that is commonly used onthe index-level series. We leave a more detailed discussion of the rela-tionship between dividend-price ratio, volatility, and bond sensitivityfor the appendix.

Figure 4.3: Dividends

This figure plots the 12-month dividend yield of the low and high-risk portfoliosover time . Dividend yield is the difference between 12-month log gross and netportfolio returns. The sample consists of all common stocks traded on NYSE/AMEXand NASDAQ exchanges from January 1952 to December 2015. Volatility is definedas the standard deviation of previous 36-monthly return observations. Portfolios arevalue-weighted and rebalanced monthly.


4.4 methodology : firm-level var and portfolio dynam-ics

Portfolios constructed as sorts on the past return volatility have dif-ferent exposures to aggregate state variables. For instance, Figure 2

shows that low-vol stocks, on average, co-vary positively with the re-turns on government bonds (shocks to the level of the term structure),whereas the opposite is true in the case of high-vol stocks. In orderto investigate what governs these dynamics in a systematic way, thatis, to identify the channels through which aggregate shocks affect re-turns of single stocks, and furthermore, portfolios of stocks groupedby a particular characteristic, we embed each stock’s return, earnings,and book-to-market ratio in a present value identity whose parame-ters we estimate using a vector autoregression model.

Since our analysis concerns returns of an anomaly strategy for whichperiodical rebalancing is needed in order to maintain desired portfo-lio properties (characteristics), it is of crucial importance that we takeinto account stock migrations. We explicitly acknowledge that over itslifetime, a stock can transition in or out of low/high volatility portfo-lio, and emphasize the importance of capturing these dynamics whenanalyzing returns of rebalanced portfolios. In order to do so, we useindicator variables to track in which portfolio a given stock is at eachportfolio rebalancing occurrence. By interacting these indicators withthe VAR stock-level state variables (i.e. return, book-to-market, andprofitability), we are able to capture the differential effects of aggre-gate shocks on these portfolios. Such a system also allows us to useimpulse response functions to analyze and quantify the long-termaverage impact of aggregate shocks on stock returns.

Previous research, such as Vuolteenaho 2002 and Lochstoer andTetlock 2016 focused on estimating return variance decomposition ofan average single stock by assuming a constant transition matrix ina VAR system. Although their approach is sufficient in answeringhow much cash-flow news versus discount rate news contribute tosingle stock return variance, it is not sufficient for our purpose. Weare interested in quantifying the differences between low and high-vol stocks with regard to their exposures to aggregate shocks. Sincea stock can transition from one portfolio to another, for instance, ifa small, volatile stock grows and becomes a large, stable stock, itsexposure to aggregate shocks can change. For comparison, we also

4.4 methodology : firm-level var and portfolio dynamics 101

estimate a constant transition matrix and show that this model failsto capture this heterogeneity.

In order to analyze the return variance decomposition on portfolio-level returns, we follow the approach in Lochstoer and Tetlock 2016

to aggregate portfolio-level shocks from the single stock-level shocks.Compared to Campbell and Vuolteenaho 2004, this approach avoidsthe problem associated with the estimation of VAR-based only onmarket-level time-series, which has been shown to be unstable. Fur-thermore, we also avoid decomposing the price-dividend ratio on aportfolio-level directly due to stationarity issues associated with someof these portfolios and the fact that we are analyzing returns of rebal-anced portfolios. Lochstoer and Tetlock 2016 also argue that resultsof Campbell, Polk, and Vuolteenaho 2010 where they decompose re-turns of book-to-market sorted portfolios on a portfolio-level, overes-timate the importance of the cash-flow component relative to whenthe decomposition is done on a single stock-level.

4.4.1 VAR specification

We consider a VAR with three stock-level state variables that definethe present value identity:

zi,t = (ri,t,bmi,t, ei,t)′ (4.1)

where ri,t, bmi,t and ei,t are stock i’s log return, log book-to-marketratio and log book earnings, respectively.

The nuance of our approach is that we allow the stock to migrateacross J different portfolios, and we capture this using the indicatorvariable:

Ii,t = (I(1)i,t , I(2)i,t , I(3)i,t , . . . , I(J)i,t )

′ (4.2)

where I(j)i,t , j = 1, 2, 3, . . . , J equals one if stock i belongs to portfolio jat time t and otherwise equals zero.

We augment the state vector with K aggregate variables, xt:

Zi,t = (z ′i,t ⊗ Ii,t, xt) ′ (4.3)

and model Zi,t as a first order autoregressive process:

Zi,t = µ+ ΓZi,t−1 + ui,t, Σ = E(ui,tu′i,t), (4.4)


where µ is a vector of constants capturing the mean of Zi,t.The transition matrix Γ is constrained in that the aggregate state

variables follow independent dynamics that influence individual stocks,but there is no feedback from single firms to aggregate variables, thatis, aggregate shocks are assumed to be exogenous:

Γ =

Γ1 01×K

03J×1 Γs

where Γ1: 3J× 3J is a full matrix; Γs : K×K the transition matrix thatgoverns the dynamics of the aggregate state variable.

4.4.2 Return variance decomposition

Based on (4.4), we can construct discount rate shocks and cash-flowshocks for each stock in the same way as in Campbell and Vuolteenaho2004:

λ = ρΓ(I− ρΓ)−1

where ρ = 0.96. As a result, the discount rate shocks (DRit) and cashflow shocks (CFit) are defined as:

DRshocki,t = e1 ′λui,t (4.5)

where e1 = (1, 0, . . . , 0) ′ and

CFshocki,t = (e1 ′ + e1 ′λ)ui,t (4.6)

Furthermore, we can conveniently extract DR and CF shocks for anaverage stock in portfolio j as

DRj,shocki,t = e ′(j−1)∗4+1λui,t (4.7)

andCF

j,shocki,t = (e ′(j−1)∗4+1 + e

′(j−1)∗4+1λ)ui,t (4.8)

4.4.3 From single stock to portfolio-level shocks

We follow Lochstoer and Tetlock 2016 to aggregate stock-level intoportfolio-level VAR estimates. As the return decomposition was doneon log returns, we need to approximate level returns using a second-order Taylor series expansion around stock’s current log return.

4.4 methodology : firm-level var and portfolio dynamics 103

Ri,t+1 ≡ exp(ri,t+1) (4.9)

Ri,t+1 = exp(Etri,t+1)exp(CFshocki,t+1 −DRshock

i,t+1 ) (4.10)

where exp(Etri,t+1) is the fitted value from the firm-level VAR.A second order expansion for cash-flow and discount rate shocksaround zero gives:

Ri,t+1 = exp(Etri,t+1){1+CFshocki,t+1 +

1

2(CFshock

i,t+1 )2

−DRshocki,t+1 +

1

2(DRshock

i,t+1 )2 +CFshocki,t+1 DR

shocki,t+1 }

(4.11)

CFlevel_shocki,t+1 ≡ exp(Etri,t+1){CF

shocki,t+1 +

1

2(CFshock

i,t+1 )2} (4.12)

DRlevel_shocki,t+1 ≡ exp(Etri,t+1){DR

shocki,t+1 −

1

2(DRshock

i,t+1 )2} (4.13)

CFDRcrossi,t+1 ≡ exp(Etri,t+1)CFshocki,t+1 DF

shocki,t+1 (4.14)

Portfolio returns in levels can be approximated as:

Rp,t+1 −

n∑i=1

ωpi,texp(Etri,t+1) ≈

CFlevel_shockp,t+1 −DRlevel_shock

p,t+1 +CFDRcrossp,t+1

(4.15)

where

CFlevel_shockp,t+1 =

n∑i=1

ωpi,tCF

level_shocki,t+1 (4.16)

DRlevel_shockp,t+1 =

n∑i=1

ωpi,tDR

level_shocki,t+1 (4.17)

CFDRcrossp,t+1 =

n∑i=1

ωpi,tCFDR

crossi,t+1 (4.18)

The return variance is given by:


var(Rp,t+1) ≈var(CFlevel_shockp,t+1 ) + var(DRlevel_shock

p,t+1 )

− 2cov(CFlevel_shockp,t+1 ,DRlevel_shock

p,t+1 ) + var(CFDRcrossp,t+1)

(4.19)where

Rp,t+1 ≡ Rp,t+1 −

n∑i=1

ωpi,texp(Etri,t+1). (4.20)

4.4.4 Impulse response function

The impulse responses are a convenient way to quantify the long-term impact of shocks in the VAR system. For our application, we areinterested in analyzing the impact of shocks to aggregate variables onstock returns, depending on the portfolio in which they belong. Theseresponses are implied directly from the system in (4.4). We employthe Cholesky factorization of the covariance matrix of VAR residuals,whereby we assume exogeneity of the aggregate shocks. The signifi-cance bounds around impulse responses are obtained using the jack-knife method, whereby we remove one cross-section at a time andre-estimate the model parameters. As we have 50 (T) independentcross-sections, the VAR is estimated once using all T cross-sections,and 49 times using T-1 cross-sections, where at each iteration, a dif-ferent cross-section is dropped from the sample. Just as it is the casewith the Γ coefficients, the interpretation of parameters is conditionaland should be multiplied by the probability measure in order to getthe unconditional responses.

4.5 data

We consider common stocks (share codes 10 and 11) in the CRSPdatabase traded on NYSE, AMEX, and NASDAQ exchanges exclud-ing penny stocks6. Our measure of total return volatility is definedas the standard deviation of the past 36-month return observations,and our results are robust to alternative measures of the ex-ante stockvolatility measured at either daily or monthly frequency. The ex-anteestimates of market betas are obtained using univariate regressionsof excess stock returns on the excess return on market portfolio over

6 Penny stocks are commonly defined as those with the beginning-of-month pricesabove $1.

4.5 data 105

the past 60 (minimum 24) months7. Dividend-price ratio is the dif-ference between gross and net portfolio (log) returns. Returns grossand net of dividends are obtained from CRSP and are adjusted forstock delistings. The market capitalization (ME) of a stock is its pricetimes the number of shares outstanding. Book value is the sum ofbook value of stockholders’ equity, balance sheet deferred taxes andinvestment tax credit (if available), minus the book value of preferredstock. If available, we use the redemption, liquidation, or par valueto calculate the book value of preferred stock. Stockholders’ equityis obtained either from Moody’s industrial manuals or Compustat. Ifit is not available, we measure stockholders’ equity preferably as thesum of book value of common equity and the par value of preferredstock, or the book value of assets minus total liabilities, if the first oneis not available. For the fiscal years ending in 1993 or later, we do notadd deferred taxes to book equity due to changes in their treatment(FASB 109)8. The book value of equity is then divided by the marketcapitalization calculated at the end of the previous calendar year toobtain the book-to-market ratio.

Bond returns and yields are obtained from the CRSP bond databases.Equity factors that are used in this study are downloaded from thewebsite of Professor Kenneth French. The level and slope of the gov-ernment bond yield curve are the first two principal components es-timated using 30, 20, 10, 7, 5, 2, and 1-year fixed-maturity treasurybond yields from January 1952 till December 2015.

We follow Cochrane and Piazzesi 2005 and define bond risk pre-mium as the single return forecasting (CP) factor constructed as alinear combination of one to five year forward rates. We use monthlyFama-Bliss 1 to 5-year zero-coupon bond yields to construct one tofive-year forward rates and regress the equal-weighted average of theone-year excess returns on bonds of maturities 2, 3, 4 and 5 years onthe one-year yield and 2 through 5-year forward rates. The CP factoris the resulting fitted value. As suggested in Cochrane and Piazzesi2005, we discard the data before 1964 due to reliability concerns.

The length of our sample is governed by the availability of the data.For the analysis of dividend yields and the long-term portfolio-levelperformance, the start date of the sample period is January 1952 andthe end is in December 2015. We form value-weighted portfolios thatwe rebalance each month.

7 Results are robust to other measures of beta estimated using monthly or daily data.8 As explained onmba.tuck.dartmouth.edu/pages/faculty/ken.french/datalibrary.html


The VAR-based analysis is done at an annual frequency and start in1964 and end in 2014, due to limited availability of COMPUSTAT datathat we need to construct accounting variables and the CP factor. Thisresults in 50 independent cross-sections. Furthermore, for the VARdecomposition, we exclude stocks below the 20th percentile marketcapitalization of NYSE listed stocks, to alleviate the concerns that ourresults might be driven by micro-caps. We subtract beta times themarket return from the single stock returns to obtain the (market)beta-adjusted returns.

We consider the following set of state variables: short rate, past mar-ket return, level and slope of the term structure, CP, aggregate marketvariance, and inflation. Short rate is the log risk-free rate taken fromthe website of professor Ken French; level and slope are the first andsecond principal components as explained above; CP is the Cochraneand Piazzesi 2005 factor; aggregate market variance is the variance ofdaily returns on the value-weighted market portfolio over the previ-ous year obtained from the website of Amit Goyal9; inflation is thelog annual change in CPI obtained from the FRED database. Table4.14 in the appendix shows the full sample mean, median, and stan-dard deviation for each of the state variables at an annual frequency(June-June).

4.5.1 Clean surplus accounting earnings

We use the clean surplus accounting principles to deduce firm’s earn-ings. The log of earnings is the log stock return minus the change inlog book-to-market ratio:

ln(ROE)i,t+1 = ri,i+1 + ρbmi,t+1 − bmi,t (4.21)

where ρ is set to 0.96. As firm-level profitability is neither fully areal nor a nominal quantity, and our VAR is specified in terms of realvalues, we subtract 0.4*inflation from this variable. Our results arerobust to this choice, which is guided by the evidence presented inCampbell, Polk, and Vuolteenaho 2010.

Further, we consider a more direct measure of company earningstaken directly from Compustat files defined as net income (NI) di-vided by lagged book equity and find our baseline results robust tothis alteration.

9 This variable was used in Goyal and Welch 2008.

4.6 empirical results : var decomposition 107

Table 4.3: Constant transition matrix with CP

This table shows the VAR transition Γ matrix estimated using ordinary leastsquares with CP as the aggregate state variance. Standard errors are clus-tered by time and firm. The sample consists of all common stocks tradedon NYSE/AMEX and NASDAQ exchanges from January 1964 to Decem-ber 2014, except those with prices below $1 and market cap below the 20thNYSE percentile. CP is the Cochrane and Piazzesi (2005) tent-shaped returnforecasting factor.

Rt bmt et CPt

Rt+1 -0.03 0.06 -0.04 -0.12

(-0.60) (4.49) (-1.73) (-1.57)bmt+1 -0.37 0.84 0.13 0.09

(-7.17) (50.84) (5.05) (0.95)et+1 0.15 -0.08 0.07 0.20

(12.90) (-12.90) (4.25) (3.26)CPt+1 0.41

(3.10)

4.6 empirical results : var decomposition

4.6.1 Constant transition matrix

Prior empirical applications of the present value identity on a singlestock-level involved estimation of a single, constant VAR transitionmatrix, Γ , that governs the model dynamics. This approach ignoresthe fact that, over time, stock characteristics change, and they co-varydifferently with macroeconomic indicators. We start by estimatingthis simple VAR model using pooled least squares and a single macrostate variable, the CP factor. Table 4.3 shows the estimated VAR coeffi-cients together with their t-statistics, calculated using standard errorsthat are clustered by year and firm, as explained in Petersen 2009.

Consistent with other papers, we find that past beta-adjusted re-turns are a poor predictor of future beta-adjusted returns on a singlestock-level, but there is a high degree of persistence in firm funda-mentals: past book-to-market predicts future book-to-market with acoefficient of 0.84 (t-stat of 50.84) and past real earnings predict futurereal earnings with a coefficient of 0.07 (t-stat of 4.25). Contrary to this,the persistence of year on year earnings defined as net incomes overbook equity is around 4 to 5 times higher10, caused by the fact that

10 When we define earning as net income over lagged book value, this result obtains.


companies engage in earnings smoothing. We, on the other hand, useclean-surplus earnings that is necessary for the present value identityto hold.

High past book-to-market ratios are also significantly associatedwith high future beta-adjusted returns and low future real profitabil-ity, and past real profitability forecast high future book-to-market ra-tios, and negative beta-adjusted return, albeit the latter is insignifi-cant.

CP is also highly persistent, with a coefficient of 0.41 on its own lag,but the constant gamma approach would lead us to conclude that CPpredicts single stock beta-adjusted returns with a negative, but statis-tically insignificant coefficient. Impulse responses are also unable totrace out the difference between low and high-vol portfolio reactionsto changes in the aggregate variables since there is no differentiationin the impact of CP on returns of different stocks, i.e. all stocks followdynamics governed by the same transition matrix Γ .

The constant Γ approach assumes the same transition matrix forevery stock in the universe, and thus ignores the fact that stocks withdifferent characteristics can co-vary with different state variables inopposite ways. Our approach enables us to trace out these differ-ences, and, as we discuss below, we find that CP predicts low andhigh-vol stock-level returns with statistically significant coefficientsof opposite signs.

When it comes to return variance decomposition (Table 4.4), wefind that on an anomaly (long-short) level, 62% of the beta-adjustedreturn variance comes from the cash-flow news, 11% from discountrate news, but with a substantial, negative correlation between thetwo news component: the −2Cov term explains 26% of the returnvariance, implying a correlation coefficient between the news termsof -49%. These results are in line with those reported by Lochstoerand Tetlock 2016 for other anomaly portfolios. Empirically, one couldorthogonalize one news component against the other, and obtain thenew variance decomposition, but this approach has been shown toyield unstable results, as it depends on the ordering of the news termsin the Cholesky factorization. As discount rate and cash-flow shockcovariance is a property of the stock-level data, we leave the interpre-tation of our results open to the reader.


Table 4.4: Variance decomposition with constant transition matrix

This table shows the VAR implied decomposition of the beta-adjusted singlestock-level returns. The sample consists of all common stocks traded onNYSE/AMEX and NASDAQ exchanges from January 1964 to December2014, except those with prices below $1 and market cap below the 20thNYSE percentile.

DR CF DRCF -2COV CoR

Low-Vol 0.11 0.62 0.00 0.26 -0.49

High-Vol 0.14 0.54 0.02 0.30 -0.56

Low-High Vol 0.11 0.62 0.00 0.26 -0.49

4.6.2 Baseline results

Our baseline results refer to the VAR model with the portfolio indi-cator variables that are interacted with the stock-level state variablesthat define the present value identity, i.e. return, book-to-market, andprofitability. Table 4.5 presents the transition matrix for the volatility-sorted portfolios in our sample. We define three portfolios: low-volconsisting of 20% of stocks with the lowest three-year return volatility,high-vol consisting of 20% of stocks with the highest return volatil-ity, and mid-vol that contains all other stocks that are in between.The purpose of this asymmetry is to limit the number of parametersthat need to be estimated in the VAR, which is our specification isK× K− (K− 1). For instance, with only one aggregate state variableand three portfolios, the number of parameters we have to estimate is91. We note that our conclusions do not change materially if we usesymmetric portfolios.

Due to high persistence in volatility ranks, on a year-to-year basis,stocks are most likely to stay in the same portfolio (80.7% probabilityin the case of low-vol and 76.3% in the case of high-vol), althoughthere is a considerable amount of switching between adjacent port-folios (19.1% probability low-vol becomes mid-vol, and 26.6% thathigh-vol becomes mid-vol). Since we are interested in the long-termdynamics of the system, the transition matrix implies that over a 10-year period, a low-vol stock can end up in the high vol portfolio witha probability of 10.4%, and conversely, a high-vol stock can end up inthe low-vol portfolio with a probability of 18.7%.

Not reported in the table is that a new stock11 is 5.4 times morelikely to be added to the high-vol than to the low-vol portfolio, and

11 Note that we require at least three years of return data to sort stocks into portfolios.


Table 4.5: Transition matrix

This table shows the transition matrix for stocks transitioning from and tolow, mid, and high-volatility portfolios. The low and high-vol portfolios con-sist of 20% of stocks with the lowest and highest past return volatility, re-spectively, and the mid-vol portfolio contains the remaining 60%. The sam-ple consists of all common stocks traded on NYSE/AMEX and NASDAQexchanges from January 1964 to December 2014, except those with pricesbelow $1 and market cap below the 20th NYSE percentile. Volatility is de-fined as the standard deviation of previous 36-monthly return observations.Portfolios are value-weighted and rebalanced monthly.

LowVol (t-1) MidVol (t-1) HighVol (t-1)

LowVol (t) 80.70 7.26 0.08

MidVol (t) 19.15 87.84 23.65

HighVol (t) 0.15 4.90 76.26

Sum 100.00 100.00 100.00

a stock is 4.6 times more likely to disappear (e.g. due to default oracquisition) if it was in the high-vol than if it was in the low-volportfolio. This is consistent with the empirical and theoretical linksbetween equity volatility and default risk.

The VAR models are estimated using pooled least square with stan-dard errors clustered by time and firm. We add one aggregate statevariable at a time to the system. The reason for this choice is two-fold: (i) interactions between macro variables are outside of the scopeof this paper and are a subject of a very rich macro-finance literature.We acknowledge that some of our state variables are highly correlatedand may be capturing the same information set; (ii) each additionalstate variable requires at least eleven new parameters to be estimated,but depending on the restrictions we impose on the system, this canbe much bigger. Our aim is to preserve the parsimony of the model.

Table 4.6 shows the estimated transition matrices of the VAR modelwith the Cochrane and Piazzesi 2005 factor as the aggregate variable.Each one of the stock-level state variables is interacted with an asym-metric portfolio indicator, thus making the interpretation of param-eters conditional; i.e. probability-weighted. Since the extreme portfo-lios are equally populated with comparable transition probabilities,we are interested in comparing the coefficients that are interactedwith those two portfolio indicators, and in particular, how the macrovariable of interest, in this case CP, affects future beta-adjusted returns(i.e. alphas).


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Rtp1

,tbm

tp1

,tetp1

,tRtp2

,tbm

tp2

,tetp2

,tRtp3

,tbm

tp3

,tetp3

,tCPt

Rt+1p1

,t+1

0.0

8-0

.01

0.1

3-0

.01

0.0

1-0

.01

0.0

00.0

1-0

.01

-0.0

1

(1.5

7)

(-0.5

2)

(2.5

1)

(-1.6

2)

(4.7

0)

(-3

.71)

(-2

.01)

(5.6

4)

(-2

.56)

(-5

.39)

bm

t+1p1

,t+1

-0.2

70.7

50.0

8-0

.01

0.0

60.0

10.0

00.0

00.0

00.0

0

(-4.8

5)

(44.1

7)

(1.5

9)

(-1.0

0)

(10.1

5)

(1.3

0)

(0.3

0)

(-0

.13)

(0.6

8)

(3.5

5)

et+1p1

,t+1

0.1

0-0

.04

0.1

00.0

00.0

0-0

.01

0.0

00.0

10.0

00.0

0

(4.6

3)

(-4.8

3)

(2.7

5)

(-1.2

1)

(2.3

9)

(-2

.61)

(-1

.95)

(8.0

4)

(-2

.78)

(-8

.16)

Rt+1p2

,t+1

-0.0

80.0

4-0

.12

-0.0

10.0

30.0

1-0

.01

0.0

20.0

0-0

.02

(-2.4

1)

(5.1

7)

(-4.0

4)

(-0.1

5)

(2.1

3)

(0.4

8)

(-0

.62)

(3.4

5)

(-0

.95)

(-4

.64)

bm

t+1p2

,t+1

-0.1

10.1

60.1

0-0

.32

0.7

40.1

0-0

.02

0.1

80.0

10.0

9

(-3.6

7)

(9.7

0)

(2.0

8)

(-6.9

7)

(52.8

3)

(4.1

1)

(-1

.03)

(8.5

9)

(0.4

5)

(10.3

6)

et+1p2

,t+1

0.0

10.0

0-0

.04

0.1

1-0

.07

0.0

40.0

1-0

.01

0.0

2-0

.02

(1.3

5)

(0.2

1)

(-2

.57)

(10

.73)

(-13

.70)

(2.6

3)

(2.8

2)

(-3

.82)

(2.8

9)

(-16.0

7)

Rt+1p3

,t+1

-0.0

20.0

1-0

.04

-0.0

10.0

2-0

.03

-0.0

30.0

3-0

.04

0.0

2

(-1.8

9)

(4.5

3)

(-3

.83)

(-1

.24)

(7.7

9)

(-4

.88)

(-0

.78)

(1.8

4)

(-1

.35)

(2.4

5)

bm

t+1p3

,t+1

-0.0

10.0

1-0

.01

-0.0

70.0

40.0

0-0

.33

0.5

90.1

1-0

.10

(-1.4

2)

(1.4

6)

(-0

.96)

(-5

.95)

(8.1

9)

(0.2

7)

(-8

.48)

(30.5

9)

(3.1

9)

(-8

.86)

et+1p3

,t+1

0.0

20.0

00.0

20.0

3-0

.01

0.0

10.1

5-0

.09

0.0

70.0

1

(2.2

2)

(-2.9

0)

(3.2

6)

(5.9

7)

(-5

.90)

(3.0

4)

(13.9

4)

(-12.9

5)

(3.6

0)

(3.3

2)

CPt+1

0.4

1

(3.1

0)


We observe that an increase in CP is associated with a decrease infuture beta-adjusted returns for low-vol stocks and an increase in re-turn for high-vol stocks. In order to interpret the unconditional mag-nitude of our results, we need to multiply the estimated coefficientby the inverse of the probability that a stock is in the correspond-ing portfolio, which in the case of low and high-vol portfolios is 20%.This means that our results suggest big economic impacts: a marginalincrease (NB: this is a unit, not standard deviation) in CP leads to a2.65% drop in the low-vol beta-adjusted returns in the next year andover a 10% increase in the case of high-vol stocks. Both coefficientsare highly statistically significant with t-stats of -5.39 and 2.45, re-spectively.

Another way to visualize the long-term VAR implied effect of a CPshock on low and high-vol beta-adjusted returns is to plot the impulseresponses, where the shock to the aggregate variable is assumed tobe exogenous. This assumption is not unreasonable in our settings,as it is unlikely that there is economically significant feedback fromsingle firms to the macroeconomy. Figure 4.4 shows results for a onestandard deviation increase in CP.

Figure 4.4: Conditional return response to CP shock

This figure plots VAR implied conditional return response of stocks in the low andhigh-risk portfolios up to 10 years following an exogenous CP shock. The sampleconsists of all common stocks traded on NYSE/AMEX and NASDAQ exchangesfrom January 1964 to December 2014. Volatility is defined as the standard deviationof previous 36-monthly return observations. The significance bounds are obtainedusing the jackknife method.

The long-term economic impacts of a CP shock are pervasive: low-vol stocks severely underperform and high-vol stocks outperformeven 10 years following the initial shocks. As before, we report results


for conditional beta-adjusted returns. For instance, five years follow-ing a one standard deviation CP shock, low-vol stock alphas are, onaverage, over 5% down, ceteris paribus, and high-vol stock alphas areover 15% up. This dichotomy could not have been achieved with aconstant Γ .

4.6.3 Other state variables

We next consider the slope of the term structure as an alternative tothe CP factor. Consistent with the results above, we find that slopepredicts low and high-vol portfolio returns with coefficients of an op-posite sign. Cochrane and Piazzesi 2005 already showed that the infor-mation that is contained in the slope of the term structure about thefuture bond excess returns is largely subsumed by their tent-shapedfactor, and Koijen, Lusting, and van Nieuwerburgh 2017 show thatshocks to the CP factor and shocks to the slope contain related pric-ing information. For this reason, our prior exception was that CP andslope predict low-vol returns with the same sign. Table 4.7 showsthese results and Figure 4.5 plots the impulse responses.

Figure 4.5: Conditional return response to slope shock

This figure plots VAR implied conditional return response of stocks in thelow and high-risk portfolios up to 10 years following an exogenous shockto the slope of the term structure of interest rates. The sample consists ofall common stocks traded on NYSE/AMEX and NASDAQ exchanges fromJanuary 1964 to December 2014. Volatility is defined as the standard devi-ation of previous 36-monthly return observations. The significance boundsare obtained using the jackknife method.

In the appendix, Table 4.15 present results for the short rate, andTable 4.16 for the level of the term structure, both of which affect


Table4.

7:VAR

estimates

with

slope

Thistable

shows

theVA

Rtransition

Gamma

matrix

estimated

usingordinary

leastsquares

with

slopeof

theterm

structureof

interestrates

asthe

aggregatestate

variable.The

stock-levelstatevariables

areinteracted

with

theindicators

thatdeterm

inew

hethera

stockis

alow

,mid,or

high-riskat

anypoint

intim

e.Standarderrors

areclustered

bytim

eand

firm.The

sample

consistsof

allcomm

onstocks

tradedon

NY

SE/AM

EXand

NA

SDA

Qexchanges

fromJanuary

19

64

toD

ecember

20

14,exceptthose

with

pricesbelow

$1

andm

arketcapbelow

the2

0thN

YSE

percentile.Slopeis

thesecond

principalcomponent

estimated

from3

0to

1year

constant-maturity

government

bondportfolio

yields.

Rt p

1,t

bm

t p1

,tet p

1,t

Rt p

2,t

bm

t p2

,tet p

2,t

Rt p

3,t

bm

t p3

,tet p

3,t

SLt

Rt+1p1

,t+1

0.08

-0.

01

0.13

-0.

01

0.01

-0.

01

0.00

0.01

0.00

-0.

01

(1.

57)

(-0.

52)

(2.

51)

(-1.

62)

(4.

72)

(-3.

72)

(-2.

04)

(5.

67)

(-2.

45)

(-5.

55)

bm

t+1p1

,t+1

-0.

27

0.75

0.08

-0.

01

0.06

0.01

0.00

0.00

0.00

0.01

(-4.

85)

(44.

16)

(1.

59)

(-0.

99)

(10.

13)

(1.

30)

(0.

26)

(-0.

04)

(0.

47)

(3.

52)

et+1p1

,t+1

0.10

-0.

04

0.10

0.00

0.00

-0.

01

0.00

0.01

0.00

0.00

(4.

63)

(-4.

83)

(2.

75)

(-1.

22)

(2.

42)

(-2.

62)

(-1.

99)

(8.

02)

(-2.

64)

(-8.

38)

Rt+1p2

,t+1

-0.

08

0.04

-0.

12

-0.

01

0.03

0.01

0.00

0.02

0.00

-0.

03

(-2.

43)

(5.

17)

(-4.

04)

(-0.

16)

(2.

15)

(0.

47)

(-0.

60)

(3.

33)

(-0.

66)

(-4.

70)

bm

t+1p2

,t+1

-0.

11

0.16

0.10

-0.

32

0.74

0.10

-0.

02

0.19

0.00

0.12

(-3.

55)

(9.

59)

(2.

15)

(-6.

98)

(52.

70)

(4.

16)

(-1.

08)

(8.

90)

(0.

13)

(10.

38)

et+1p2

,t+1

0.01

0.00

-0.

04

0.11

-0.

07

0.04

0.01

-0.

01

0.02

-0.

02

(1.

29)

(0.

28)

(-2.

60)

(10.

70)

(-13.

68)

(2.

62)

(2.

94)

(-4.

18)

(3.

04)

(-16.

34)

Rt+1p3

,t+1

-0.

02

0.01

-0.

04

-0.

01

0.02

-0.

03

-0.

03

0.04

-0.

04

0.03

(-1.

78)

(4.

52)

(-3.

86)

(-1.

20)

(7.

74)

(-4.

87)

(-0.

79)

(1.

99)

(-1.

41)

(2.

64)

bm

t+1p3

,t+1

-0.

02

0.01

-0.

02

-0.

07

0.04

0.00

-0.

33

0.58

0.12

-0.

15

(-1.

76)

(1.

87)

(-1.

41)

(-6.

27)

(8.

69)

(0.

13)

(-8.

68)

(30.

04)

(3.

31)

(-9.

35)

et+1p3

,t+1

0.02

-0.

01

0.02

0.03

-0.

01

0.01

0.15

-0.

09

0.07

0.01

(2.

28)

(-3.

03)

(3.

40)

(6.

05)

(-6.

06)

(3.

09)

(13.

95)

(-12.

78)

(3.

59)

(2.

99)

SLt+1

0.60

(5.

12)


returns of the low and high-vol stocks in the opposite directions withhighly significant coefficients.

Inflation is another important determinant of the term structure,so we naturally ask if shocks to inflation have a differential impacton returns of low and high-vol stocks. Table 4.8 and Figure 4.6 showthese results. High-vol stocks, corrected for their market betas, pro-vide a good hedge against increases in inflation, and low-vol stocksunderperform during these periods.

Figure 4.6: Conditional return response to inflation shock

This figure plots VAR implied conditional return response of stocks in thelow and high-risk portfolios up to 10 years following an exogenous inflationshock. The sample consists of all common stocks traded on NYSE/AMEXand NASDAQ exchanges from January 1964 to December 2014. Volatility isdefined as the standard deviation of previous 36-monthly return observa-tions. The significance bounds are obtained using the jackknife method.

Lastly, Table 4.9 and Figure 4.7 show results when using marketvariance as the aggregate state variable. Consistent with results re-ported in previously discussed studies, we find that high-vol stockshedge against periods of unexpected increases in market variance,and low-vol stocks underperform in these periods. These results areperhaps counter-intuitive as one would expect low-vol stocks to pro-vide a hedge in periods when the market variance spikes, which aretypically associated with bad states of the world (recessions). Whilethis is certainly the case, the statements that we make are in termsof beta-adjusted returns: low-vol stocks do not outperform in thesestates, after adjusting for their low market betas. Barinov 2013 findsqualitatively similar results using the portfolio-level analysis of theidiosyncratic volatility anomaly: controlling for the market effects,


Table4.

8:VAR

estimates

with

inflation

Thistable

shows

theVA

Rtransition

Gamma

matrix

estimated

usingordinary

leastsquares

with

inflationas

theaggregate

statevariable.The

stock-levelstate

variablesare

interactedw

iththe

indicatorsthat

determine

whether

astock

isa

low,m

id,orhigh-risk

atany

pointin

time.Standard

errorsare

clusteredby

time

andfirm

.Thesam

pleconsists

ofall

comm

onstocks

tradedon

NY

SE/AM

EXand

NA

SDA

Qexchanges

fromJanuary

19

64

toD

ecember

20

14,except

thosew

ithprices

below$

1and

market

capbelow

the2

0thN

YSE

percentile.Inflationis

thelog

year-on-yeardifference

inthe

consumer

priceindex

multiplied

by1

00

forexposition.

Rt p

1,t

bm

t p1

,tet p

1,t

Rt p

2,t

bm

t p2

,tet p

2,t

Rt p

3,t

bm

t p3

,tet p

3,t

INFt

Rt+1p1

,t+1

0.09

-0.

01

0.13

-0.

01

0.01

-0.

01

0.00

0.01

-0.

01

-0.

06

(1.

58)

(-0.

55)

(2.

52)

(-1.

54)

(4.

67)

(-3.

67)

(-2.

00)

(5.

57)

(-2.

55)

(-5.

49)

bm

t+1p1

,t+1

-0.

27

0.75

0.08

-0.

01

0.06

0.01

0.00

0.00

0.00

0.04

(-4.

87)

(44.

16)

(1.

57)

(-1.

03)

(10.

21)

(1.

26)

(0.

69)

(-0.

53)

(1.

03)

(3.

15)

et+1p1

,t+1

0.11

-0.

04

0.10

0.00

0.00

-0.

01

0.00

0.01

0.00

-0.

04

(4.

64)

(-4.

85)

(2.

77)

(-1.

12)

(2.

26)

(-2.

55)

(-1.

95)

(8.

11)

(-2.

76)

(-7.

97)

Rt+1p2

,t+1

-0.

08

0.04

-0.

11

-0.

01

0.03

0.01

-0.

01

0.03

-0.

01

-0.

22

(-2.

34)

(5.

17)

(-4.

04)

(-0.

12)

(2.

05)

(0.

54)

(-0.

82)

(3.

80)

(-1.

39)

(-4.

64)

bm

t+1p2

,t+1

-0.

12

0.17

0.08

-0.

32

0.75

0.09

-0.

01

0.16

0.02

0.88

(-4.

23)

(10.

34)

(1.

73)

(-6.

95)

(53.

19)

(3.

83)

(-0.

64)

(7.

52)

(1.

16)

(9.

49)

et+1p2

,t+1

0.02

0.00

-0.

04

0.11

-0.

07

0.04

0.01

-0.

01

0.02

-0.

17

(1.

62)

(-0.

17)

(-2.

41)

(10.

68)

(-13.

74)

(2.

68)

(2.

27)

(-2.

46)

(2.

60)

(-15.

04)

Rt+1p3

,t+1

-0.

02

0.01

-0.

05

-0.

02

0.02

-0.

03

-0.

03

0.03

-0.

03

0.20

(-2.

02)

(4.

38)

(-3.

69)

(-1.

36)

(7.

97)

(-4.

94)

(-0.

71)

(1.

44)

(-1.

19)

(1.

76)

bm

t+1p3

,t+1

0.00

0.00

0.01

-0.

06

0.04

0.01

-0.

34

0.62

0.10

-1.

05

(0.

12)

(-0.

23)

(0.

89)

(-5.

16)

(6.

59)

(0.

92)

(-8.

43)

(30.

90)

(2.

65)

(-6.

84)

et+1p3

,t+1

0.02

0.00

0.01

0.03

-0.

01

0.01

0.15

-0.

09

0.07

0.11

(1.

98)

(-2.

42)

(2.

84)

(5.

75)

(-5.

35)

(2.

87)

(14.

00)

(-12.

89)

(3.

68)

(2.

44)

INFt+1

0.90

(13.

34)


high-ivol stocks outperform low-ivol stocks when aggregate variancespikes.

Figure 4.7: Conditional return response to market variance shock

This figure plots VAR implied conditional return response of stocks in thelow and high-risk portfolios up to 10 years following an exogenous shockto market variance. The sample consists of all common stocks traded on NY-SE/AMEX and NASDAQ exchanges from January 1964 to December 2014.Volatility is defined as the standard deviation of previous 36-monthly re-turn observations.The significance bounds are obtained using the jackknifemethod.

We consider a set of six macro state variables and find them to besignificant drivers of low-volatility anomaly returns. High-vol stockshedge against periods of increasing yields and shocks to market vari-ance, which could explain why their returns appear anomalouslylow from the standpoint of the CAPM. In the spirit of Merton 1974

ICAPM, if these periods correspond to those of poor investment op-portunities for the representative investor, a multi-factor model thatincorporates these shocks should reduce the returns that are left un-explained by the market factor alone.

4.6.4 Return variance decomposition

Table 4.10 shows the decomposition of the low-vol, high-vol, and theanomaly (long-short) portfolio beta-adjusted return variance for allthe specifications that we examine.

Similar to the constant Γ approach, on an anomaly-level, most ofthe return variance is driven by cash-flow news (CF) - around 46%,and only about 14% is driven by news about expected future discountrates (DR). These results, are once again hard to interpret at face value


Table4.

9:VAR

estimates

with

aggregatevariance

Thistable

shows

theVA

Rtransition

Gamma

matrix

estimated

usingordinary

leastsquaresw

ithstock

marketvariance

asthe

aggregatestate

variable.T

hestock-level

statevariables

areinteracted

with

theindicators

thatdeterm

inew

hethera

stockis

alow

,m

id,or

high-riskat

anypoint

intim

e.Standard

errorsare

clusteredby

time

andfirm

.Thesam

pleconsists

ofall

comm

onstocks

tradedon

NY

SE/AM

EXand

NA

SDA

Qexchanges

fromJanuary

19

64

toD

ecember

20

14,except

thosew

ithprices

below$

1and

market

capbelow

the2

0thN

YSE

percentile.Inflationis

thelog

year-on-yeardifference

inthe

consumer

priceindex.

Rt p

1,t

bm

t p1

,tet p

1,t

Rt p

2,t

bm

t p2

,tet p

2,t

Rt p

3,t

bm

t p3

,tet p

3,t

Svart

Rt+1p1

,t+1

0.08

-0.

01

0.12

-0.

01

0.01

-0.

01

0.00

0.01

0.00

-0.

09

(1.

55)

(-0.

49)

(2.

49)

(-1.

72)

(4.

75)

(-3.

75)

(-1.

78)

(5.

67)

(-2.

51)

(-5.

55)

bm

t+1p1

,t+1

-0.

27

0.75

0.08

-0.

01

0.06

0.01

0.00

0.00

0.00

0.06

(-4.

83)

(44.

16)

(1.

62)

(-0.

94)

(10.

02)

(1.

36)

(-1.

03)

(0.

87)

(0.

00)

(3.

54)

et+1p1

,t+1

0.10

-0.

04

0.10

0.00

0.00

-0.

01

0.00

0.01

0.00

-0.

06

(4.

62)

(-4.

81)

(2.

73)

(-1.

34)

(2.

59)

(-2.

70)

(-1.

71)

(7.

57)

(-2.

75)

(-8.

50)

Rt+1p2

,t+1

-0.

09

0.05

-0.

13

-0.

01

0.03

0.01

0.00

0.01

0.00

-0.

35

(-2.

52)

(5.

18)

(-4.

03)

(-0.

20)

(2.

29)

(0.

37)

(0.

01)

(2.

27)

(-0.

10)

(-4.

73)

bm

t+1p2

,t+1

-0.

09

0.15

0.13

-0.

31

0.74

0.11

-0.

04

0.23

-0.

01

1.45

(-2.

67)

(8.

64)

(2.

61)

(-6.

99)

(52.

26)

(4.

58)

(-1.

91)

(10.

18)

(-0.

49)

(10.

12)

et+1p2

,t+1

0.01

0.00

-0.

05

0.11

-0.

07

0.04

0.02

-0.

02

0.02

-0.

28

(0.

88)

(0.

87)

(-2.

81)

(10.

83)

(-13.

61)

(2.

52)

(4.

14)

(-6.

29)

(3.

23)

(-15.

59)

Rt+1p3

,t+1

-0.

02

0.01

-0.

04

-0.

01

0.02

-0.

03

-0.

04

0.04

-0.

04

0.28

(-1.

75)

(5.

13)

(-4.

18)

(-1.

14)

(7.

14)

(-4.

56)

(-0.

86)

(2.

49)

(-1.

48)

(1.

89)

bm

t+1p3

,t+1

-0.

04

0.02

-0.

05

-0.

08

0.05

-0.

01

-0.

30

0.54

0.13

-1.

68

(-3.

37)

(4.

42)

(-4.

03)

(-7.

15)

(11.

55)

(-0.

95)

(-8.

24)

(26.

89)

(3.

77)

(-9.

24)

et+1p3

,t+1

0.02

-0.

01

0.02

0.03

-0.

01

0.01

0.15

-0.

08

0.06

0.14

(2.

53)

(-3.

66)

(3.

83)

(6.

18)

(-6.

79)

(3.

30)

(13.

88)

(-12.

17)

(3.

53)

(3.

37)

Svart+1

0.42

(3.

16)


Table 4.10: Beta-adjusted return variance decomposition

This table shows the VAR implied decomposition of the beta-adjusted singlestock-level returns estimated using our new methodology. We consider aset of six aggregate state variables that are added to the VAR one at a time.These are: CP (Cochrane and Piazzesi (2005) tent-shaped return forecastingfactor), level and slope of the term structure, the short rate (Rf), inflation,and aggregate variance. The sample consists of all common stocks tradedon NYSE/AMEX and NASDAQ exchanges from January 1964 to December2014, except those with prices below $1 and market cap below the 20thNYSE percentile.

State Var Portfolio DR CF DRCF -2COV CoR

CPLow-Vol 0.12 0.57 0.00 0.27 -0.52

High-Vol 0.13 0.49 0.02 0.28 -0.57

Low-High 0.13 0.46 0.02 0.33 -0.67

SlopeLow-Vol 0.12 0.57 0.00 0.27 -0.52

High-Vol 0.12 0.49 0.02 0.28 -0.58

Low-High 0.13 0.46 0.02 0.33 -0.67

RfLow-Vol 0.13 0.54 0.00 0.28 -0.53

High-Vol 0.13 0.49 0.02 0.28 -0.57

Low-High 0.14 0.46 0.02 0.33 -0.65

LevelLow-Vol 0.12 0.56 0.00 0.28 -0.53

High-Vol 0.13 0.49 0.02 0.28 -0.57

Low-High 0.14 0.46 0.02 0.33 -0.65

InfLow-Vol 0.13 0.55 0.00 0.28 -0.52

High-Vol 0.12 0.49 0.02 0.28 -0.57

Low-High 0.13 0.46 0.02 0.33 -0.67

SvarLow-Vol 0.13 0.55 0.00 0.28 -0.52

High-Vol 0.12 0.49 0.02 0.29 -0.60

Low-High 0.13 0.45 0.02 0.34 -0.69


due to a very high, negative correlation between the two news com-ponents. On an anomaly-level, DR and CF news have a −2Cov termof around 30%, implying a correlation of -67%.

The use of aggregate state variables other than CP does not mate-rially change the conclusion of the analysis, but we do acknowledgethat a more richly specified VAR could lead to different conclusions.These variables would be quite hard to theoretically motivate and em-pirically find, as predicting single stock discount rates and cash-flowsis very difficult.

4.7 concluding remarks

Table 4.11 show the results of the spanning regressions, where weproject the excess returns on the volatility-sorted portfolios on themarket (CAPM), Fama and French 1993 three factors, and Fama andFrench 2015 five factors, respectively. As the new Fama-French factorsare available from the July of 1963 onward, our sample runs from July1963 till December 2015.

While the CAPM and the three factor model cannot explain thealpha of the low-minus-high risk portfolio, the addition of the twonew Fama-French factors, profitability and investment, is enough torender the alpha insignificant. Blitz and Vidojevic 2017 argue that theconclusion that the low-volatility anomaly is explained by the five-factor model is premature, given the lack of conclusive evidence fora positive relationship between risk and returns in the cross-sectionof single stocks, but a strong relationship between low-volatility andFama-French factors is indisputable. In this paper, we show that lowand high-vol stocks react in different manners to changes in a numberof macroeconomic indicators, which begs the question of whether theFama and French factors could be linked to macroeconomic risks, thatis, whether the motivation behind these factors could be grounded inICAPM, with these macro state variables describing a part of the in-vestors’ information set about expected future investment opportuni-ties. In fact, Koijen, Lusting, and van Nieuwerburgh 2017 argue thatthe business cycle risk can explain the value premium, and Petkova2005 also finds strong links between Fama-French factors and variousmacroeconomic indicators, giving support to the ICAPM interpreta-tion of these factors. Our results support these ideas and open a newavenue for research into the established asset pricing factor models.

4.7 concluding remarks 121

Table 4.11: Spanning regressions

This tables contains the estimated coefficients, t-statistics, and adjusted r-squares from spanning regressions of volatility-sorted portfolios excess re-turns on the (CAPM) one-, (Fama-French) three-, and (Fama-French) five-factor models . The sample consists of all common stocks traded on NY-SE/AMEX and NASDAQ exchanges from July 1963 to December 2015, ex-cept those with prices below $1. Volatility is defined as the standard de-viation of previous 36-monthly return observations. Portfolios are value-weighted and rebalanced monthly.

LowVol Q2 Q3 Q4 HighVol Low-High

CAPM

α 0.12 0.06 -0.06 -0.07 -0.43 0.55

(2.04) (1.19) (-0.84) (-0.57) (-2.24) (2.31)βmkt 0.76 1.06 1.28 1.49 1.69 -0.93

(59.24) (98.77) (76.38) (52.21) (39.26) (-17.45)r.sq 0.85 0.94 0.90 0.81 0.71 0.33

3-FM

α 0.08 0.01 -0.06 -0.05 -0.41 0.49

(1.67) (0.11) (-0.84) (-0.56) (-2.94) (2.88)βmkt 0.84 1.09 1.21 1.32 1.42 -0.58

(78.54) (99.05) (75.07) (56.92) (42.97) (-14.50)βsmb -0.25 -0.03 0.26 0.65 1.06 -1.31

(-16.57) (-1.91) (11.38) (19.87) (22.72) (-23.27)βhml 0.19 0.13 -0.11 -0.29 -0.45 0.64

(11.70) (7.80) (-4.46) (-8.13) (-8.90) (10.52)r.sq 0.91 0.94 0.92 0.89 0.85 0.67

5-FM

α -0.07 -0.08 0.03 0.18 -0.04 -0.03

(-1.74) (-1.85) (0.40) (1.92) (-0.30) (-0.20)βmkt 0.88 1.11 1.18 1.25 1.31 -0.42

(94.34) (103.92) (70.14) (55.12) (42.86) (-12.10)βsmb -0.18 0.02 0.23 0.55 0.87 -1.05

(-13.89) (1.57) (10.04) (17.47) (20.44) (-21.53)βhml 0.09 0.11 0.00 -0.09 -0.22 0.31

(5.06) (5.24) (0.13) (-2.02) (-3.64) (4.53)βrmw 0.30 0.23 -0.11 -0.45 -0.83 1.14

(16.89) (11.31) (-3.41) (-10.32) (-14.19) (16.89)βsma 0.21 0.04 -0.25 -0.43 -0.50 0.71

(7.85) (1.43) (-5.15) (-6.61) (-5.64) (7.02)r.sq 0.94 0.95 0.92 0.91 0.89 0.78


In summary, we identify a set a macroeconomic (aggregate) statevariables with significant predictive power over single stock returns.Using a novel methodology that builds on the present value iden-tity of Campbell and Shiller 1988 to decompose returns of dynamicportfolios into the discount rate and cash-flow news shocks, we in-corporate these aggregate state variables and quantitatively describetheir long-term impact on beta-adjusted returns of low and high-volstocks. The nuance of our approach is that we explicitly control forstock transitions from one portfolio to another as their characteristicschange over time. We apply this methodology to a set of volatility-sorted portfolio and find that most of the beta-adjusted return vari-ance on a long-short level is driven by the revision in expectationsabout future cash-flows. Furthermore, we find that various aggre-gate state variables significantly affect the low-high volatility port-folio beta-adjusted return spreads. In particular, positive shocks tothe Cochrane and Piazzesi 2005 CP factor, slope and level of the termstructure, the short rate, inflation, and aggregate variance are asso-ciated with high beta-adjusted returns for high-vol, and low returnsfor low-vol stocks. Thus, high-vol stocks hedge against these periods,which are in eyes of investors associated with bad states of the world.Our findings give support to other studies in this literature that docu-ment similar patterns using portfolio-level analysis and holds impor-tant practical implications.

4.8 appendix

4.8.1 Relationship between dividend yield, return volatility, and bond sen-sitivity

It is well known that dividends, as a share of cash-flow redistribu-tion, have been declining over time in favor of share repurchases, ren-dering the stability of the dividend yield process questionable. Thisraises the question whether these results are robust if we take netshare repurchases into account. We examine the dynamics of the netpayout yield, defined following Boudoukh et al. 2007 as the sum ofnet issuance yield and dividend yield12. The results are consistentwith the ones for dividend yield: low-vol stocks have significantlyhigher net payouts than high-vol stocks.

12 Net equality issuance is the monthly change in shares outstanding times the averageshare price. (See Boudoukh et al. 2007, p.885 footnote 10.)

4.8 appendix 123

Do stocks within the low-vol universe that have relatively higherdividend-price ratio co-vary more with bonds than the low yield-ing ones? To address this question, each month, we sort stocks intofive value-weighted portfolios on their 12-month dividend-price ratio(dividend yield) within the low-vol segment of the market, definedas the 20% of stocks with the lowest 36-month past volatility, andanalyze whether there is a relationship between their ex-post returnsand the bond market factor, dlvl13. Table 4.12 shows that as one movesfrom the low to the high yielding portfolio, the higher is the slope co-efficient on the bond factor. In fact, the low dividend-price, low-volportfolio does not even have a significant coefficient on the bond fac-tor. These results suggest that there is a strong association betweenthe dividend yield and bonds returns.

Table 4.12: Bond betas within low-risk segment

This table shows the performance characteristics and regression output forfive portfolios sorted on 12-month dividend-price ratio within the low-risksegment of the market, defined as the 20% of stocks with the lowest 36-month past volatility. The sample consists of all dividend paying commonstocks traded on NYSE/AMEX and NASDAQ exchanges from February1952 to December 2015, except those with prices below $1. The market fac-tor is the CRSP value-weighted market portfolio, and the bond factor, ‘dlvl’,is the negative of the AR1 innovations to the level of the term structure ofyields. Dividend yield is the ratio of 12-month gross and net stock-level re-turns. Volatility is defined as the standard deviation of previous 36-monthlyreturn observations. Portfolios are value-weighted and rebalanced monthly.

Low DY Q2 Q3 Q4 High DY

ex.ret 0.63 0.61 0.68 0.73 0.64

vol 4.21 3.89 3.74 3.61 3.96

sharpe 0.15 0.16 0.18 0.20 0.16

ex-ante DY 2.31 3.72 4.70 5.81 7.46

intercept 0.15 0.20 0.31 0.41 0.36

(1.86) (2.37) (3.47) (4.43) (3.11)βmkt 0.83 0.72 0.65 0.59 0.50

(43.52) (36.71) (31.96) (27.72) (18.59)βdlvl 0.03 0.08 0.09 0.11 0.16

(1.80) (5.17) (5.79) (6.75) (7.75)r.sq 0.71 0.64 0.58 0.51 0.34

13 Defined as the negative of the AR1 innovations to the level of the term structure


Do low-vol stocks co-vary with bonds only because they pay highdividends? Table 4.13 shows results for the five value-weighted port-folios sorted on their past 36-month volatility within the high-dividendyield segment of the market. The ex-ante dividend yield of these port-folio, measured as the cap-weighted average of dividend yields ofthe underlying stocks14, shows that all these portfolios have similarlevels of ex-ante mean DY, but we still see that the bond sensitivitydecreases with portfolio volatility. This indicates that dividend yieldsand volatility both seem to be associated with the equity bond sensi-tivity.

Table 4.13: Bond betas within high-dividend yield segment

This table shows the performance characteristics and regression output forfive portfolios sorted on 36-month past return volatility within the highdividend yield segment of the market, defined as the 20% of stocks withthe highest 12-month dividend yield. The sample consists of all dividendpaying common stocks traded on NYSE/AMEX and NASDAQ exchangesfrom February 1952 to December 2015, except those with prices below $1.The market factor is the CRSP value-weighted market portfolio, and thebond factor, ‘dlvl’, is the negative of the AR1 innovations to the level ofthe term structure of yields. Dividend yield is the ratio of 12-month grossand net stock-level returns. Volatility is defined as the standard deviation ofprevious 36-monthly return observations. Portfolios are value-weighted andrebalanced monthly.

Low Vol Q2 Q3 Q4 High Vol

ex.ret 0.76 0.73 0.72 0.69 0.96

vol 3.72 4.12 4.70 5.63 6.58

sharpe 0.20 0.18 0.15 0.12 0.15

ex-ante DY 6.64 6.49 6.35 6.34 6.75

intercept 0.49 0.35 0.24 0.13 0.29

(4.55) (3.29) (2.13) (0.92) (1.81)βmkt 0.49 0.67 0.82 0.97 1.14

(19.77) (27.17) (31.37) (30.21) (30.80)βdlvl 0.14 0.09 0.03 0.00 -0.04

(7.64) (5.09) (1.73) (-0.07) (-1.48)r.sq 0.37 0.50 0.56 0.54 0.55

14 Note that for the portfolio-level analysis of dividend-price, we calculated d-p as thedifference between the log gross and net portfolio returns, as detailed in the datasection

4.8 appendix 125

4.8.2 Other tables

Table 4.14: Summary statistics of aggregate state variables state variables

This table shows the full sample mean, median, and standard deviationof the aggregate state variables at annual frequency. These include the CPfactor, slope and level of the term structure, risk free rate, inflation, andaggregate stocks market variance. The sample starts in 1964 and ends in2014.

CP Slope Level RF Inflation Svar

mean 1.111 0.929 1.861 0.049 0.040 0.156

median 1.036 0.804 1.668 0.049 0.031 0.137

st dev 1.512 1.121 7.387 0.030 0.028 0.106


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4.8 appendix 127

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5B E H AV I O R A L H E T E R O G E N E I T Y I N R E T U R NE X P E C TAT I O N S A C R O S S E Q U I T Y S T Y L EP O RT F O L I O S

This chapter is jointwork with RemcoC.J. Zwinkels and

Philip A. Stork.

5.1 introduction

The notion that market prices exhibit dynamics consistent with the ra-tional expectations hypothesis is repeatedly challenged in the financeliterature. The theory of rational expectations, originally proposed byMuth 1961, postulates that the representative agent in the market isfully rational and sets the clearing prices and quantities of assets. Thelast couple of decades have seen a proliferation in theoretical andempirical work that produced a considerable amount of evidence forthe existence of agents with boundedly rational and heterogeneousbeliefs about future asset prices, who trade on these expectations,thus causing prices to deviate from their rational values. Depend-ing on which group of agents prevails in the market, different pricedynamics are generated. Structural asset pricing models that featureboundedly rational agents have shown promise in explaining empir-ically observed patterns, such as volatility clustering, skewed returndistributions, and persistent deviations from the fundamentals subse-quently followed by corrections, that remain a puzzle for the rationalmodels (see Lux 1998).

Brock and Hommes 1998 propose a discrete time1, structural as-set pricing model that features two types of agents in the market,who use different rules to form expectations about future prices. Onegroup of agents believe that the prices of assets revert toward theirfundamental values and, as such, their actions have stabilizing effectson market prices. This group of agents is referred to as the fundamen-talists. On the other hand, there is a group of investors who believethat patterns in past prices have predictive power over future asset

1 For treatment in continuous time, we refer the reader to He and Le 2012.

128

5.1 introduction 129

returns, and thus form expectations using heuristics based on pastprices. These traders are often referred to as the technical traders,noise traders, or simply chartists. In this framework, fundamentalistsresemble the ‘rational’ investors, however, they are only boundedlyrational as they do not take the existence of chartists into accountwhen forming price expectations. A key question is whether chartistscan persist in the market equilibrium, or their actions on average can-cel out, and the market ends up in a steady state where the pricesare set by the rational marginal investor (see Friedman 1953). If bothgroups of agents are identified in the market, the asset price dynam-ics are then determined by their interactions, that is, which group ispresent in a greater quantity determines whether prices converge to-ward or diverge away from their fundamentals. Agents in this typeof models are assumed to choose which expectation formation rulethey follow based on the past performance of the rule they had usedrelative to the one used by the other group. In addition, the mod-els feature an endogenous parameter - commonly referred to as theintensity of choice - that determines how sensitive the agent groupsare to the past performance when choosing which strategy to follow.A high value of this parameter indicates that agents are sensitive topast performance, and therefore there is a lot of switching betweengroups.

Heterogeneous agent models have been developed in many differ-ent specifications, and extensively tested in different contexts. Mostearly studies in this area used simulations in order to validate predic-tions of their respective models, however, empirical estimation usingreal-world data has been more frequently conducted over the last cou-ple of decades. Boswijk, Hommes, and Manzan 2007 are the first toempirically estimate a heterogeneous agent model (hereafter: HAM)featuring agents that switch between groups using the equity marketportfolio2 as the test asset, and find evidence for significant behav-ioral heterogeneity in beliefs about the future index levels. Chiarella,He, and Zwinkels 2014 estimate a somewhat different specificationof a HAM using quasi maximum likelihood on the same asset andfind confirming evidence. ter Ellen, Hommes, and Zwinkels 2017 ex-amine whether there is behavioral heterogeneity within various assetclasses3 using a uniform, generic heterogeneous agent model that al-

2 They proxy for the market portfolio with the S&P 500 index.3 Specifically, the authors consider various macro-economic time-series, such as CPI

and house prices, and more dynamic financial assets, such as foreign exchange, com-modities, and equities. They find that agents switch between groups more in the


lows them to make a direct comparison across estimated coefficients.They find evidence for the existence of heterogeneous agents withinall asset classes, except for equities.

To the best of our knowledge, no study has examined whether thereis behavioral heterogeneity within the equity market. In this paper,we estimate a HAM on a level of equity style portfolios, which al-lows us to uncover dynamics within the equity asset class that arepossibly obscure on the overall index level. This research questionis of high economic importance, given the increase in popularity ofthe style investing strategies over the past decade, facilitated by thefinancial innovation that enabled investors to access these strategiesat a low cost4. According to the 2018 annual survey of global assetowners5, conducted by FTSE Russell6, the evaluation and adoptionrate of the style investing strategies, that they refer to as smart beta,has been increasing steadily from 26% of respondents in 2015 to 48%in 2018. Another 17% of respondents in 2018 indicated that they arecurrently evaluating an allocation to the smart beta strategies. 41%of those with existing allocations have $10 billion or more investedin such strategies, 39% have between $1 and $10 billion, and the re-maining 20% have under $1 billion. Style investing, however, is not anew concept, as investors have been allocating to styles such as value,growth, small-cap, high dividend for decades.

Similar to Brock and Hommes 1998 and ter Ellen, Hommes, andZwinkels 2017, we assume there are two types of agents in the mar-ket, fundamentalists and chartists. We study the interaction betweenthese agents within a stylized heterogeneous agent model7, and testwhether the generated dynamics correspond to the empirically ob-served ones.

case of financial assets, but the heterogeneity in beliefs is more pronounced in caseof macro-economic series.

4 Over the past decade, there has been a proliferation of index funds that system-atically target style premia, as well as an increase in the number of quantitativelymanaged mutual funds that are often priced significantly cheaper than their funda-mentally managed counterparts.

5 In 2018, 185 asset owners from different regions responded to the survey. They haveestimated total assets under management of $3.5 trillion, and therefore represent asignificant part of financial markets.

6 "Smart beta: 2018 global survey findings from asset owners" can be found on thewebsite www.ftserussell.com.

7 Our HAM is specified in terms of return dynamics, as opposed to those of prices,which does not fundamentally alter the nature of the model, but does provide somemodeling convenience. ter Ellen and Zwinkels 2010 also estimate a version of amodel in returns for oil, and ter Ellen, Hommes, and Zwinkels 2017 show that amodel in return and model in prices lead to qualitatively similar results for varioustest assets that they consider.

www.ftserussell.com


The estimation of the heterogeneous agent models requires as aninput an exogenously determined fundamental value of the asset.While theoretical HAMs have no issues assuming exogenous funda-mental values that are used by the fundamentalists, for the empiricalvalidation of such models this input becomes of first-order impor-tance. Therefore, one has to make a choice of a specific model thatfundamentalists use to determine the value of an asset. The HAMdoes not require this estimate to be equal to the actual fundamentalvalue; it only requires that the expectation is consistent with that of arational agent8.

In order to empirically estimate the fundamentalists’ expected re-turns on stocks, our paper departs from the prior papers that validateHAM on an equity index level, where agents estimate fundamentalvalues based on a variant of the Gordon growth model or a simplemoving average of past prices. Instead, we build on the findings froma rich literature on empirical equity pricing that enable us to buildan expected return model on a single-stock level, and aggregate ex-pectations to a style-portfolio level. In particular, we assume that sixwell-documented stock-level characteristics - market beta, the ratioof the book value of equity to its market value, market capitaliza-tion, operating profitability, investment (change in total assets), andprice momentum drive the cross-sectional variation in stock returns.We estimate the monthly return premia associated with these char-acteristics using the classic Fama and MacBeth 1973 procedure on arolling basis, and each month, make a one period ahead expectedreturn forecast for each stock in the universe. This approach is simi-lar to that used in Lewellen 2015

9. Since our test assets are all hedge(long-short) portfolios, the cross-sectional differences in stock-levelcharacteristics drive the expected return variance of these portfolios,as the impact of the equity risk premium is largely removed throughhedging10. We find that the fit of our model matches well the first mo-ment of the empirical return distributions of these investment styles -i.e. expectations are, on average, realized.

8 The fundamentalists are only boundedly rational, as they do not take into accountthe existence of chartists into account, however, their trading strategy is based onthe reversion in prices toward the intrinsic (fundamental) value of the asset, whichresembles the strategy of the rational investor in the classical, rational framework.

9 We use a more parsimonious set of only six characteristics, and abstain from apply-ing parameter shrinkages that could raise over-fitting concerns.

10 The reason we use hedge portfolios is that the Fama and MacBeth 1973 procedure isused to predict cross-sectional differences in stock returns. We make no attempt toforecast the equity risk premium, which is the main driver of the return variance ofthe long-only portfolios.


Since our expected return model implies one month ahead expectedreturns, we estimate the HAM at a monthly frequency. We assumethat chartists form return expectation following a moving averagerule, in particular, an exponentially weighted moving average (EWMA),for which we consider various decay parameters for robustness. Theuse of a moving average rule is commonly employed to model chartists’expectations, and finds its support in the empirical evidence on amicro (individual)-level, coming from surveys and laboratory exper-iments11. For instance, using a laboratory experiment, Hommes etal. 2005 show that agents with the knowledge of the past returns, div-idend yields, and interest rates utilize technical rules to make returnforecasts. A number of papers provide evidence based on surveys forthe use of technical rules in the foreign exchange market (see, for in-stance, ter Ellen, Verschoor, and Zwinkels 2013). We refer the readerto ter Ellen and Verschoor 2018 and Lux and Zwinkels 2018 for areview of the recent literature on heterogeneous beliefs formation.

In our model, agents are also allowed to switch between groups.Each period agents evaluate the relative profitability of their tradingstrategy and decide whether to continue using their current expec-tation formation rule or switch to the one of the other group. Fur-thermore, we estimate models where investors choose their tradingstrategy based on short, medium, and long-term profits that each ofthe two expectation rules has generated.

Our results uncover evidence for behavioral heterogeneity in ex-pected return formation within the equity market. In our base caseHAM specification, where we assume that chartists form expectationsbased on an EWMA of past returns with a decay parameter of 0.5, weare able to identify both groups of agents in the market for all invest-ment styles except for profitability. The fundamentalists’ coefficientsare economically and statistically significant with the expected (pos-itive) sign for all styles, and as for chartists, for value, profitability,

11 As the cross-sectional price momentum is one of the most pervasive asset pricingfactors (see Fama and French 2008), we include it in the set of relevant pricing char-acteristics that the fundamentalists use to forecast stock and style portfolio returns.In our model, chartists form style return expectations based on a moving average ofthe realized strategy returns. This is a variant of the time-series momentum strategy.While there is some overlap between them, the two momentum strategies are distinctphenomena (see Moskowitz, Ooi, and Pedersen 2012, for detailed treatment of thistopic). Cross-sectional momentum strategies are based on the relative performanceof the single stocks, while time-series momentum strategies are based on the trendin the price of an asset, with no regard to the prices of other assets. In addition, withthe EWMA rule that we use to model chartists’ expectation, most weight is placedon the last month’s strategy return, while the cross-sectional momentum strategiesare typically based on stock return excluding the most recent month, due to theshort-term reversal effects, originally documented in Jegadeesh 1990.


and investment styles we find that chartists extrapolate past returns,and for momentum and size, chartists expect a reversion in returns -i.e. if winners/small-caps outperformed losers/large-caps this month,they expect them to underperform the following month. The chartists’coefficient in case of the profitability style is the only one that is statis-tically insignificant (a t-statistic of 1.48), albeit it is economically large.We find that these results are fairly robust to the choice of the EWMAdecay parameter that the chartists use to form expected return fore-casts, and some values of the decay parameter provide a better fit forsome styles than others. For instance, we find that in the case of valueand profitability, faster-moving averages (decay parameters of 0.8 or1) provide a better fit than the slower moving averages (decay parame-ters of 0.2 or 0.5), and the converse is true for the other three styles. Infact, with a decay parameter of 1, which implies that chartists formexpectations based on last month’s strategy return, both the funda-mentalists’, as well as the chartists’ coefficients for the profitabilitystyle are highly significant, with t-statistics of 4.45 and 11.52, respec-tively. In order to impose some structure on the model that guardsus against data-mining, we set the EWMA decay parameter to themid-range point (0.5) for all investment styles, as the base case speci-fication that we consider in various robustness tests.

Our results also indicate that there is more switching when agentsevaluate the profitability of their trading strategies over a short (onemonth) than over a longer term (twelve or twenty-four month) look-back period, consistent with the myopic behavior of investors thathas been extensively described in the behavioral finance literature(see, for instance Benartzi and Thaler 1995).

We also compare the performance of a full heterogeneous agentmodel with switching against two restricted models where it is eitherassumed that fundamentalist and chartists exist in fixed and equalproportions (i.e. there is no switching between the groups, and themarket is populated 50% by the fundamentalists and 50% by thechartists), or a model with no chartists in the market. We show that, inmost cases, the model without chartists has a worse fit than our fullmodel, and the model with a constant proportion of the two agentgroups is significantly worse than the full model with switching foronly investment and momentum styles. For the other three styles, theadded value of switching is more limited, according to the likelihoodratio test, although switching is necessary to identify the presence ofthe chartists in the case of all styles, except for value. We conclude


that the evidence for the presence of heterogeneous agents acrossthese equity styles is robust, however, allowing agents to switch be-tween the groups does not always lead to a better model fit.

These findings have implications for our understanding of whatdrives returns of these equity style portfolios. A rich asset pricingliterature subscribes to the risk-based explanations for the anoma-lously high returns12 generated by these style portfolios (see Famaand French 1993, 2015), consistent with the efficient markets hypoth-esis, and the notion of the rational marginal agent in financial mar-kets. Another strand of literature has been able to provide evidenceconsistent with the behavioral explanations that imply that the ob-served return patterns (i.e. positive style returns) are anomalies thatresult from various market inefficiencies. Our analysis casts doubt onthe rational, and gives support to the behavioral explanations behindthese anomalies, albeit we cannot dismiss the possibility that other at-tempts with different modeling choices may lead to other conclusions.Our hope is that future research in this area will be able to provideadditional insights into this debate.

The remainder of the paper is organized as follows: Section 2 presentsthe heterogeneous agent model and Section 3 presents the expectedreturn model that we use to estimate expected returns; Section 4 dis-cusses the data and Section 5 discusses the main results of the paperand a battery of robustness tests. Section 6 concludes the paper.

5.2 the model

The HAM that we estimate is a modification of the model used inBrock and Hommes (1997, 1998) and ter Ellen, Hommes, and Zwinkels2017. We estimate the same generic model across the five equity stylesseparately, assuming there is one risky asset13 in the economy at atime. Models that feature multiple risky assets have been shown tobe very difficult to estimate due to a high number of unidentifiedcoefficients14 (see Chiarella, Dieci, and Hung 2007, for a theoreticaltreatment). In this section, we derive the model.

12 Returns of the examined style portfolio cannot be explained by the standard equilib-rium asset pricing models, such as the Capital Asset Pricing Model, as proposed bySharpe 1964. From the perspective of this model, these portfolios have anomalouslyhigh returns.

13 Without a loss of generality, we assume that the only risky asset is the hedge styleportfolio, that it, investors are not able to invest in the long and the short leg of theportfolio separately.

14 A rich specification of a HAM would allow agents to also switch between riskyassets. In our context, that would mean that in certain periods, agents favor some

5.2 the model 135

Let Rt be the excess return15 on the risky asset, Rf,t be the risk-freerate of (total) return, and Wt be the value of the wealth portfolio, allat time t. The wealth portfolio evolves with the following dynamics:

Wt+1 =WtRf,t+1 + Rt+1zt. (5.1)

where zt is the demand for the risky asset, which investors solvefor using mean-variance optimization, that is, by maximizing the ex-pected return on the wealth portfolio for a given level of risk (i.e.expected variance of wealth) and a risk aversion parameter. Investorsare further assumed to have homogeneous preferences, and the vari-ance term is assumed to be constant over time and agents. We assumethere are H types of investors, with a fraction nht of investors of typeh, at time t. The analytic solution of this maximization is:

zht = Eht(Rt+1)/aσ2. (5.2)

where a is the coefficient of risk aversion and σ is the variance ofwealth. The investor specific demand, zht, is thus a linear function ofagents’ respective return expectations. As our heterogeneous agentmodel is specified in terms of return expectations, as opposed to ex-pected deviations from the fundamental price, we assume a presenceof a market maker16 who adjusts prices consistent with the excessdemand. Therefore, the change in the price (i.e. return) on the riskyasset is a function of the demand and supply:

Rt+1 =

H∑h=1

nhtzht+1 − st+1 (5.3)

where nht is the proportion of agents of type h at time t, and st+1

is the excess supply of the risky asset at time t+ 1. Without a loss ofgenerality, we set the excess supply to zero and obtain:

Rt+1 =

H∑h=1

nht{Eht(Rt+1)}. (5.4)

investment styles over others. Due to the complexities associated with such a model,we opt to leave those questions outside of the scope of this paper.

15 The return is gross of dividend, in excess of the risk-free rate.16 The heterogeneous agent models formulated in terms of deviations assume that the

market clears with a Walrasian auctioneer. Hommes 2006 provides a discussion onthe differences between these two approaches.


Let EFt(Rt+1) be the return that the fundamentalists expect onthe risky asset in the next period, and ECt(Rt+1) be the return thatchartists expect on the same asset, both calculated at time t.

The pricing equation becomes:

Rt+1 = nFtEFt(Rt+1) +nCtECt(Rt+1). (5.5)

The investors are also allowed to switch between the two groupsconditional on the relative performance (forecast error) of the groups.At the end of each period, investors evaluate the profitability of theirtrading strategy and decide whether to continue using the same ruleor switch to the other one. This process is repeated each month. Thefaction of each investor type, nht, is endogenous and given by:

nht = exp

(β

πht

πFt + πCt

)/Zt (5.6)

Zt =∑

h=F,C

exp

(β

πht

πFt + πCt

)(5.7)

which can further be rewritten as:

nFt =

(1+ exp

(βπFt − πCt

πFt + πCt

))−1

(5.8)

nCt =

(1+ exp

(βπCt − πFtπFt + πCt

))−1

(5.9)

where πht is the performance of group h at time t, and β deter-mines how sensitive the agents are to the relative performance ofthe two groups. High values of β imply that there is a lot of switch-ing between investor types, and a value of zero indicates that thereis no switching, and therefore a constant and equal proportion ofthe two types. In addition, we are interested in the extent to whichthe weighted returns that are expected by the fundamentalist andchartists converge to their future realizations. These quantities are de-scribed by the φF and φC coefficients, for the fundamentalists’ andthe chartists’, respectively. These parameters jointly determine the sta-bility of the system.

Lastly, the performance measure π, is defined in terms of the rela-tive ability of each investor group to forecast returns over the past Lperiods:

πFt =

L∑i=1

|EFt(Rt+i) − Rt+i|/L (5.10)

5.3 expected returns 137

πCt =

L∑i=1

|ECt(Rt+i − Rt+i)|/L. (5.11)

For positive values of the β parameter, the higher is the relativeprofitability of the strategy, the higher will be the proportion of agentsusing that strategy, as specified in equations 5.8 and 5.9.

5.3 expected returns

The estimation of the HAM requires an exogenously determined fun-damentalists’ expected return of the asset, EFt(Rt+1). In our model,this value is assumed to be used only by the fundamentalists, whosedemand for the risk asset is a direct function of its fundamental ex-pected return. If the one period ahead expected return is high, fun-damentalist demand the risky asset, thus increasing its price. Con-versely, if the expected return is low, they expect a reversion of theprice towards the fundamental value and sell the asset. In order toempirically estimate the expected returns on the long-short style port-folios, we apply the Fama and MacBeth 1973 technique on a single-stock level and aggregate our expected return forecasts into portfolioslevel estimates.

Leveraging on the evidence from the vast empirical asset pricingliterature17, we assume that cross-sectional differences in expected re-turns are driven by differences in stock-level characteristics. We useFama and MacBeth 1973 regressions to estimate return premia associ-ated with these characteristics, and subsequently make a one-periodahead expected return forecast for each stock in the universe, at eachpoint in time, by multiplying its current characteristics with the esti-mated premia. A similar approach is applied in Lewellen 2015, whostudies the cross-section of expected returns and concludes that Famaand MacBeth 1973 expected return forecasts are strongly related to fu-ture return realizations. Once we have estimated the expected stockreturns, in the second step we aggregate them into long-short styleportfolio return forecasts.

Therefore, each month, we run a cross-sectional regression of ex-cess returns of stocks at time t on their characteristics at time t− 1, toestimate premia associated with these characteristics:

Ri,t − Rf,t = αt + characteristici,t−1λt + εi,t (5.12)

17 See, for instance, Carhart 1997, Fama and French 1993, 2015, 2016, Lewellen 2015.


where λ is a vector of factor premia, α is the intercept, and ε is theerror term, all at time t.

In order to obtain rolling estimates of factor premia in month t, weaverage the estimated premia on each characteristic over the past 10

years (120 months).

λt =1′λt−N

N(5.13)

where N is the number of months in the look-back window (equalto 120), and 1 is a vector of ones of length N. The choice of the look-back window was guided by Lewellen 2015, and we confirm that ourresults are robust to this choice. As the first moment of the returndistribution is notoriously hard to estimate, and therefore requiresa long time-series, we also challenge the robustness of our resultsby using full-sample estimates of factor premia, and find the resultsqualitatively unchanged18.

This approach allows the estimated factor premia, λ, to vary overtime t, consistent with the evidence in the empirical asset pricingliterature that rejects the notion of static factor premia.

The expected return of stock i for time t+ 1, at the end of month tis equal to the sum of the product of the characteristics and associatedpremia:

Et(Ri,t+1) = characteristicTi,tλt. (5.14)

Once we have estimated the expected stock returns on each stock,at each point in time, we aggregate them into style portfolios in orderto obtain expected returns on the investment style that serve as testassets for the HAM estimation:

EFt(Rportfolio,t+1) =∑i

wi,tEt(Ri,t+1) (5.15)

where wi,t is the weight of stock i in the portfolio at the end of montht. The weight of a stock in the portfolio is proportional to its marketcapitalization, i.e. portfolios are value-weighted.

We assume that six characteristics that are well-documented inthe empirical asset pricing literature drive the cross-sectional differ-ences in expected stock returns. This assumption is fairly conserva-tive, given the mere number of characteristics that have been identi-fied as predictors of stock returns. Lewellen 2015, for instance, uses 15

characteristics. We choose to use a parsimonious set of six characteris-

18 These results can be furnished upon request.

5.3 expected returns 139

tics that have received the most attention in the academic community,but we acknowledge that a richer set of factors could enhance thefit of the model, particularly if tested ex-post19. The characteristicsthat we use are the ratio of book-to-market value of equity, marketcapitalization20, operating profitability, investment, and momentum.We also include the market beta due to its strong theoretical support,however, consistent with the literature on the low-risk premium (seeBlitz and Vidojevic 2017 for a comprehensive discussion), we find themarket beta not to be a significant predictor of cross-sectional differ-ences in stock returns, in the long-run.

In our expected return model, stock, and consequently portfolioexpected returns vary over time, as both the stock-level characteris-tics, as well as the factor premia vary over time. For instance, a stockcan start off as a small cap, but grow over time to become a largecap. Thus, in the early part of its half-life, this stock will have a posi-tive expected return contribution from its market capitalization (size)characteristics, but this will turn into a negative contribution in thelatter part of its existence. On the other hand, in certain periods, theestimated premium on the market capitalization characteristic can behigh, and in those instances the contribution to the stock’s expectedreturn of this factor will be high, provided a stock is exposed to it,and at other times this premium can be low, and thus depress theexpected returns.

The other type of agents in our model, the chartists, form expectedreturn forecasts by extrapolating past return patterns. In particular,we assume that the chartists use an exponentially weighted movingaverage rule and, for robustness, we test various levels of the decayparameter ranging from 0.2 to 1. The chartists’ expected return for-mation rule is specified as:

ECt(Rportfolio,t+1) = decayRi,t+

(1− decay)ECt−1(Rportfolio,t).(5.16)

19 Because of this choice, our results are on a conservative side. An in-sample fittedmodel is expected to lead to a better model fit but potentially lacks economic ratio-nale.

20 As it is commonly done in the literature, we log transform the book-to-market andmarket capitalization characteristics.


5.4 data

Our sample consists of common stocks (share codes 10 and 11) inthe CRSP database traded on the NYSE, AMEX, and NASDAQ ex-changes from June of 1963 till December of 2017

21. We exclude stockswith beginning-of-month prices below $1 and stocks with market cap-italization below the 20th percentile market capitalization of NYSE-listed stocks. These stocks are labeled as micro-caps, and accordingto Fama and French 2008 represent around 60% of stocks in the uni-verse but account for less than 3% of the total market capitalizationof the market portfolio. Therefore micro-caps have a big impact onFama and MacBeth 1973 estimation, but their importance in the over-all market is modest. The resulting universe consists of 1,509 stocks,on average, over the full sample period. Furthermore, the returns areadjusted for delistings.

We estimate stock-level market betas using univariate least squaresregressions of excess stock returns on the market factor over a sixty-month window (minimum twenty-four monthly return observations)22.The market capitalization (ME) of a stock is its price times the num-ber of shares outstanding. Book value is the sum of book value ofstockholders’ equity, balance sheet deferred taxes and investment taxcredit (if available), minus the book value of preferred stock. If avail-able, we use the redemption, liquidation, or par value to calculate thebook value of preferred stock. Stockholders’ equity is obtained eitherfrom Moody’s industrial manuals or Compustat. If it is not available,we measure stockholders’ equity preferably as the sum of book valueof common equity and the par value of preferred stock, or the bookvalue of assets minus total liabilities, if the first one is not available.For the fiscal years ending in 1993 or later, we do not add deferredtaxes to book equity due to changes in their treatment (FASB 109).The book value of equity is then divided by the market capitaliza-tion calculated at the end of the previous calendar year to obtain thebook-to-market ratio. Operating profitability is defined as annual rev-enues minus cost of goods sold, interest expense, and selling, general,and administrative expenses divided by book equity for the last fiscalyear end in t-1, and investment is the percentage change in firms’ to-tal assets from year t-2 to t-1. Accounting data for a given fiscal yearare updated once a year at the end of June of the following calendar

21 The start of the sample period is driven by the availability of the Compustat data.22 Results are robust to other measures of beta estimated using monthly or daily data.

5.5 results 141

year. The 12-2 month total return momentum is the total return frommonth t-12 to t-2.

The one-month U.S. Treasury bill rate, our proxy for the risk-freerate or return, is obtained from the website of Professor KennethFrench23.

All characteristics are winsorized at 1% and 99% levels to alleviateany concerns that our results are driven by outliers. For ease of in-terpretation, all right hand side variables in Fama and MacBeth 1973

regressions are standardized24, and reported t-statistics are calculatedusing Newey and West 1987 standard errors25.

5.5 results

5.5.1 Expected returns

Table 5.1 shows the estimated factor premia and their respective t-statistics, obtained using the Fama and MacBeth 1973 procedure overthe full-sample.

Table 5.1: Fama-MacBeth estimated premia

This table shows the output of the Fama and MacBeth 1973 procedure. Eachmonth, we run cross-sectional regressions of one period ahead stock returnson their current characteristics, and obtain a time-series of the estimated re-turn premia per unit of each characteristic. We average the estimates over thefull sample and calculate the corresponding standard errors and t-statistics,using the Newey and West 1987 correction with three lags. The sample runsfrom June of 1963 till December of 2017 and consists of all common stockstraded on the NYSE, AMEX, and NASDAQ exchanges, excluding pennystocks and micro-caps. The control characteristics are market beta, natu-ral logarithms of market capitalization and book-to-market ratio, operatingprofitability, investment (change in assets), and price momentum - all stan-dardized. * indicates significance at a 10% level, ** indicates significance ata 5% level, and *** indicates significance at a 1% level.

Constant Beta ln(Mcap) ln(BtM) OP INV MOM

coeff 0.73∗∗∗

0.00 -0.10∗∗∗

0.15∗∗∗

0.14∗∗∗ -0.14

∗∗∗0.26

∗∗∗

t-stat (3.39) (0.04) (-2.71) (3.01) (4.12) (-6.39) (4.55)

23 http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html24 This transformation does not significantly affect the outcome of estimation, but it

simplifies the interpretation of coefficients, as they represent returns rewarded perunit standard deviation change in the characteristic.

25 We use three lags, although this choice does not have a big impact on our results.




Consistent with the results found in prior papers26, we observethat all characteristics, except for the market beta, are priced in thecross-section of stock returns. Momentum has the highest estimatedpremium; one standard deviation higher momentum score than thatof the universe average leads to, ceteris paribus, an increase in ex-pected return of 0.26% a month. Similarly, one standard deviationhigher book-to-market and operating profitability characteristic in-crease stock’s expected return by 0.15% and 0.14%, respectively, andone standard deviation lower market capitalization and investmentincrease stock’s expected return by 0.10% and 0.14%, respectively. Allestimated premia, except for the market beta, are statistically signif-icant at the most conservative significance levels, and of high eco-nomic importance.

Figure 5.1 shows the estimated 10-year rolling estimates of thesepremia that we use to calculate the expected return forecasts. Whilethere is a significant amount of variation over time, the factor premiahave the correct sign in most 10-year periods. Two notable exceptionsare the momentum premium, that has turned even negative over thelast decade, and the size premium that has become significantly lessrobust after its publication in Banz 1981. The reason for the weakmomentum premium in the recent sample is the momentum crashthat occurred following the 2008 Financial Crisis. Due to the use ofa 10-year rolling window, the momentum crash of 2009 causes themomentum premium to be insignificant for years after the event.

We next test whether expected returns line up with the future real-ized returns by running Fama and MacBeth 1973 regressions of futureexcess stock returns on their expected values obtained using rollingand full-sample estimated factor premia. The sample starts in June of1973 as we lose 10 years of data for the rolling window estimation. Ifa model does a good job, we expect to see an estimated coefficient of1. The full-sample estimates of factor premia generate forecasts thatfit the real data quite well, with an estimated coefficient of 0.97 (t-statof 5.86) and not statistically different from 1 (t-stat of -0.20). The fitwith the 10-year rolling estimates of the premia is somewhat lower; inparticular, the expected returns slightly underestimate the future real-ized returns, however, the estimated coefficient of 0.85 is statisticallynot different from 1 (t-stat of -1.14). We conclude that both the fullsample, as well as the expected return estimates based on the 10-year

26 See Blitz and Vidojevic 2017 for a comprehensive discussion.

5.5 results 143

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rolling factor premia estimates of single stock returns generated byour parsimonious model fit the realized data quite well.

Panel A of Table 5.2 presents the average realized returns, standarddeviations, and t-statistics of a test of whether realized returns aredifferent from zero, for each long-short investment style over the June1973 till December 2017 sample period. Panel B of the same tableshows the correlations between styles. The momentum style had thehighest return of 0.55% a month, but it was also the most volatilewith a standard deviation of monthly returns of 5.88%. Profitability,on the other hand, has the lowest average return, and it is the onlyfactor for which raw returns are statistically insignificant (t-statisticof 1.47). If we inspect the correlation table, we note that profitabilityis strongly negatively correlated with size and value styles, and if wewere to correct for those negative exposures, the profitability emergeshighly significant - i.e. it’s alpha with respect to the other four factorsis 0.29% a month with a t-statistic of 2.39

27.

Table 5.2: Performance characteristics of styles

This table shows the performance characteristics and the return correlationbetween five equity investment styles. All numbers are shown in percentagepoints. The sample runs from June of 1973 till December of 2017 and con-sists of all common stocks traded on the NYSE, AMEX, and NASDAQ ex-changes, excluding penny stocks and micro-caps. All values but t-statisticsare presented in percentage points. * indicates significance at a 10% level,** indicates significance at a 5% level, and *** indicates significance at a 1%level.

Panel A: Performance

Size Value Profitability Investment MomentumReturn 0.39

∗∗0.34

∗∗0.20 0.38

∗∗∗0.55

∗∗

St. deviation 3.69 3.69 3.21 3.03 5.88

t-stat (2.44) (2.12) (1.47) (2.92) (2.17)

Panel B: Correlations

Size 100.00

Value 28.97 100.00

Profitability -47.46 -24.09 100.00

Investment -5.90 61.22 12.18 100.00

Momentum -8.82 -16.78 11.78 -4.18 100.00

Table 5.3 shows the average monthly expected and realized re-turns of each style portfolio, and the difference between the two.

27 This result is based on a full-sample regression of profitability on the other four (size,value, investment, and momentum) styles.

5.5 results 145

Table 5.3: Realized and expected returns

This table shows the expected and realized returns of the size, value, prof-itability, investment, and momentum equity styles, and the difference be-tween the two. Stock-level expected returns are obtained using rolling Famaand MacBeth 1973 regressions and are further aggregated to a style-portfoliolevel. Also shown is the correlation between the expected, and future real-ized returns, on a monthly level. The sample runs from June of 1973 tillDecember of 2017 and consists of all common stocks traded on the NYSE,AMEX, and NASDAQ exchanges, excluding penny stocks and micro-caps.All values (except t-statistics) are presented in percentage points. * indicatessignificance at a 10% level, ** indicates significance at a 5% level, and ***indicates significance at a 1% level.

Size Value Profitability Investment Momentum

Expected 0.32 0.37 0.01 0.44 0.65

Realized 0.39 0.34 0.20 0.38 0.55

Difference -0.07 0.03 -0.20 0.06 0.10

t-stat (-0.46) (0.19) (-1.41) (0.46) (0.38)

Corr (Exp,Real) 12.55 16.84 11.14 11.48 9.53

Also shown are the correlation coefficients between the two seriesestimated at a monthly frequency.

The value style has an expected monthly return of 0.37% a month,and a realized return of 0.34%, and the difference between the twoof 0.03% is statistically indistinguishable from zero (t-statistic of 0.19).Small stocks are expected to outperform big stocks by 0.32% a month,and the average realized outperformance was 0.39% a month, a dif-ference of -0.07%, which is also statistically insignificant (t-statistic of-0.52). The low investment stocks outperform high investment stocksby 0.44% in expectation, and 0.38% in reality and winners outper-formed losers by 0.55% a month in reality, and 0.65% in expectations,with differences for both investment, as well as momentum statisti-cally insignificant. We do observe a somewhat worse fit in case of theprofitability style, where the predicted return is almost 0.20% belowthe expected. Despite this mismatch in average values, the standarderror of the difference in means test is high enough to make this dif-ference insignificant. The reason why profitability style has such lowexpected returns is because of its strong, negative exposures to valuethat offsets most of its positive profitability exposure. This is a well-known property of profitable stocks; see Novy-Marx 2013.

We observe high correlations between monthly expected and sub-sequently realized returns. In the case of the value style, our model


achieves the best fit, with a correlation of 16.84%. Interestingly, in caseof the profitability style, for which the average expected and realizedreturns did not align as well as for other factors, we still observea high correlation between expected and realized monthly returns of11.14%. We conclude that the fit of our expected return model is goodenough to be used for estimation of a HAM.

Table 5.4 presents the average z-scores on each of the six character-istics that we assume predict stock returns for each of the five styleportfolios. For each style, the dominant return driver is precisely thecharacteristic that was used for sorting, but we also observe that othercharacteristics have an impact. For instance, the value style portfoliotends to go against profitability and investment. Size seems to also gomildly against profitability28, and profitability goes strongly againstbook-to-market, which contributes negatively to its average returns.Investment is positively related to book-to-market, which is not a sur-prising finding, as Fama and French 2016 show that the HML (value)factor is redundant in their five-factor model, to a large extent due toits loading on the investment factor.

Table 5.4: Characteristics

This table shows the characteristics of the size, value, profitability, invest-ment, and momentum equity styles. The sample runs from June of 1973 tillDecember of 2017 and consists of all common stocks traded on the NYSE,AMEX, and NASDAQ exchanges, excluding penny stocks and micro-caps.The characteristics are market beta, natural logarithms of market capitaliza-tion and book-to-market ratio, operating profitability, investment (change inassets), and price momentum - all standardized.

Beta Mcap BtM OP INV MOM

Size 0.48 -3.36 0.53 -0.54 0.14 0.02

Value -0.11 -0.71 2.72 -1.24 -0.35 0.02

Profitability -0.38 0.94 -1.44 2.18 -0.07 -0.03

Investment -0.33 -0.04 0.82 -0.19 -2.03 0.00

Momentum 0.03 0.04 0.04 0.02 -0.02 2.31

While Table 5.1 shows the full sample average characteristics, overtime hedge portfolio exhibit considerable variation. For instance, aftera good performance of value stocks, the value portfolio tends to havea higher momentum characteristic. Another example is that whensmall stocks rally, the can become too expensive, and consequently avalue-growth portfolio may be more populated by large stocks which

28 Note that for size, the smaller values are associated with higher expected returns.

5.5 results 147

become relatively cheaper. These dynamics are fully captured by ourexpected return model.

5.5.2 Heterogeneous agent model estimation results

We estimate a heterogeneous agent model of the following form:

Rt = c+nF,tφFRFt−1 +nC,tφCR

ewmat−1 + εt (5.17)

where c is the intercept, εt is the residual term, and the nF,t and nC,t

are defined in equations 5.8 and 5.9, respectively. In the base case spec-ification, we assume the value of the decay parameter of the movingaverage process (EWMA) of 0.5. As there is no theory to guide thischoice, for robustness we consider other values in subsection 5.3. Themodel is estimated using quasi maximum likelihood, a method com-monly employed in the estimation of heterogeneous agent models,and the starting values for φF, φC, and β are all set to 1

29.Table 5.5 presents the estimation output for each style portfolio. For

four of the five styles, size, value, investment, and momentum, wefind that both groups of heterogeneous agents are identified, as theφF and φC are both statistically significant. In the case of profitability,only the coefficient for the fundamentalists is statistically significant,however, the chartists’ coefficient is still economically significant andin line with that for the ‘investment’ style. In the case of size and mo-mentum, the chartists’ coefficient is negative, implying an expectedreversion in returns. When it comes to the economic magnitude ofthe coefficients, we note that all coefficients should be halved to ad-just for the average weights of the agents in the market. For all styles,fundamentalists tend to underestimate future style returns, as theiradjusted coefficients are above one, consistent with the finding thatthe Fama and MacBeth 1973 implied single-stock expected returnsare somewhat lower than the future realizations that we reported inthe subsection 5.1. The φF coefficients are fairly equal across styles,implying that fundamentalist behavior is consistent across styles. Onthe other hand, chartists tend to overestimate future style returns, astheir adjusted coefficient is below one. Unlike the fundamentalists’coefficients that are comparable across styles, chartists’ coefficientsdiffer significantly in terms of the economic magnitude, indicating

29 In some cases, researchers restrict the value of the β parameter to be non-negative.In our case, this restriction is never binding (β estimates are always positive) so wedo not impose it explicitly.


that there is more heterogeneity in their behavior across differentstyles. The estimate of the switching intensity (β) is positive in allcases, albeit it is statistically insignificant in all cases but one. This isnot uncommon for this type of models. Boswijk, Hommes, and Man-zan 2007 and Chiarella, He, and Zwinkels 2014 also find evidencefor the existence of both types of agents in the equity market, but apositive and insignificant intensity of switching parameter. Teräsvirta1994 shows that this outcome is not an issue so long as both groupsare identified and the model fit improves.

Table 5.5: Output of HAM estimation

This table shows the output of the estimation of a stylized heterogeneousagent model, where the test assets are five prominent equity style portfolios:size, value, profitability, investment, and momentum. φF, and φC are theestimated coefficients for the fundamentalists’ and chartists’ terms in themodel, respectively, and β is the intensity of switching parameter. The sam-ple runs from June of 1973 till December of 2017. * indicates significance ata 10% level, ** indicates significance at a 5% level, and *** indicates signifi-cance at a 1% level.


φF 2.78∗∗∗

2.53∗∗

2.61∗∗

2.83∗∗∗

2.60∗∗∗

(4.90) (2.07) (2.32) (6.67) (4.58)φC -0.13

∗∗∗0.35

∗∗0.22 0.18

∗∗ -0.44∗∗∗

(-2.78) (2.48) (1.48) (2.29) (-3.98)β 0.71 1.19 3.38 5.94 5.32

∗∗

(0.64) (0.48) (0.84) (1.54) (2.01)intercept 0.00 0.00 0.00 0.00 0.00

(-0.19) (-0.60) (1.22) (-0.89) (-1.09)

AIC -2009.77 -2015.36 -2152.46 -2230.20 -1517.99

Figure 5.2 shows the implied weights of chartists for the four stylesfor which they are identified with a significant coefficient. By con-struction of our model, the weight of fundamentalists is equal toone minus the weight of chartists. There is a considerable amountof switching between agent groups within the styles, with weightsoscillating around the average value of 0.5. Also plotted on the samegraphs are the smoothed conditional means30 that move around theaverage value. The switching is particularly pronounced in the caseof the investment and momentum styles, which corresponds to the

30 The conditional means are estimated using the loess method, based on smooth localregressions.

5.5 results 149

higher values of the estimated intensity of switching parameter thatwe observe for these two styles.

Table 5.6 shows that for each style portfolio, the average weightof chartists is around 50%, although there is a considerable amountof volatility over time, especially pronounced for the investment andmomentum portfolios. The lower panel of the table shows the corre-lations between the weights. We observe that the weights are lowlycorrelated, consistent with the fact that these style portfolios provideindependent sources of return. In the case of value and investment,we find this highest correlation of 13% on a monthly level, which isnot surprising given that these two styles are more closely related toeach other than any other style pair.

Table 5.6: Weight of chartists

This table shows the average and standard deviation of monthly weights ofthe chartists in the market over the full sample period for each of the fiveinvestment styles that we use as test assets. By construction, the weight ofthe fundamentalists is equal to one minus the weight of the chartists. Alsoshown are the correlations between the monthly weights of chartists acrossthe strategy. The sample runs from June of 1973 till December of 2017. Allvalues are presented in percentage points.


Average 50.52 50.28 49.41 54.93 49.34

St.Deviation 6.93 10.46 20.60 34.34 33.83

Correlations

Size 100.00

Value 6.24 100.00

Profitability 3.12 1.26 100.00

Investment -3.51 13.13 1.70 100.00

Momentum 2.09 3.76 -7.02 -4.39 100.00


Figure 5.2: Weight of chartists

This figure shows the estimated weights of the chartists in the market overtime, for the four investment styles for which both types of boundedly ratio-nal agents are identified in the base case HAM specification. These are thesize, value, investment, and momentum styles. By construction, the weightof the fundamentalists is equal to one minus the weight of the chartists. Thesample runs from June of 1973 till December of 2017.

5.5 results 151

5.5.3 Chartists’ expectation formation rules

Our base case model assumes that chartists form expected return fore-casts based on an exponentially weighted moving average (EWMA)of realized strategy returns with a decay parameter of 0.5. In thissubsection, we challenge the robustness of our results by consideringother values of this parameter.

We estimate the model specified in equation 5.17 with EWMA de-cay values of 0.2, 0.8, and 1. The model with the decay parameterequal to 1 effectively implies that chartists only consider the lastmonth’s return when forming expectations of future returns. Thelower/higher values of the decay parameter imply a longer/shortereffective moving average window. Table 5.7 shows the output of theestimation.

We note that the results are fairly consistent with those found inthe base case. Regardless of the value of the decay parameter, the fun-damentalists coefficient is statistically significant in all cases exceptfor one, with the decay parameter of 0.8 for the profitability strat-egy. This parameter is economically significant and comparable inmagnitude to the coefficients estimated for other values of the decayparameter, however, the standard error is also higher, resulting in at-statistic of 1.16. As for the chartists’ forecasts, the longer effectivemoving average windows yield more significant estimates in the caseof size and momentum, but the shorter windows are better for theother three styles. Most importantly, for each investment style, thereexists a value of the decay parameter for which both agent groups areidentified in the model. Since a priori we do not have a good way todecide which value to take as the base case, we opt for the mid-rangepoint of 0.5.

Tables 5.5 and 5.7 also report the AIC statistics of these HAMsthat can be directly compared, as all models have the same numberof parameters and observations. In the case of momentum, we seeclear improvements in model fit from using longer effective EWMAlook-back windows for chartists’ forecasts, with the AIC parameterdecreasing (improving) monotonically as the value of the decay pa-rameter decreases. We observe the same for size, although to a lesserextent, and for profitability we observe the opposite, with shorter win-dows working somewhat better. Taken together, these results implythat the model is robust to the choice of the effective look-back win-


Table 5.7: Output of HAMs with different EWMA decay parameters

This table shows the output of the estimation of the stylized heterogeneousagent models where chartists form expected return forecasts using an expo-nentially weighted moving average with various decay parameters. The testassets are the five prominent equity style portfolios: size, value, profitability,investment, and momentum. φF, and φC are the estimated coefficients forthe fundamentalists’ and chartists’ terms in the model, respectively, and βis the intensity of switching parameter. The sample runs from June of 1973

till December of 2017. * indicates significance at a 10% level, ** indicatessignificance at a 5% level, and *** indicates significance at a 1% level.

decay parameter = 0.2

Size Value Profitability Investment MomentumφF 4.08

∗∗∗2.61

∗∗∗2.80

∗∗∗2.90

∗∗∗2.84

∗∗∗

(6.50) (2.84) (6.84) (6.93) (5.89)φC -0.82

∗∗∗0.31 0.10

∗∗0.19 -1.81

∗∗∗

(-4.34) (1.12) (2.16) (1.30) (-4.05)β 0.78 0.83 1.13 7.34 4.71

∗∗∗

(1.42) (0.40) (0.55) (1.61) (2.88)intercept 0.00 0.00 0.00 0.00 0.00

(-0.68) (-0.82) (1.30) (-1.13) (-0.93)

AIC -2012.72 -2017.82 -2150.20 -2229.51 -1534.59

decay parameter = 0.8


∗∗∗2.77

∗∗3.05 2.83

∗∗∗1.96

∗∗∗

(7.41) (2.34) (1.16) (7.06) (3.12)φC -0.06

∗∗∗0.26

∗∗∗0.21

∗∗∗0.15

∗∗ -0.10∗∗∗

(-4.22) (2.54) (14.34) (2.19) (-2.91)β 1.05 0.80 1.26 5.32

∗7.26

(0.83) (0.47) (0.68) (1.66) (0.81)intercept 0.00 0.00 0.00 0.00 0.00

(-0.18) (-0.71) (1.21) (-0.81) (-0.38)

AIC -2008.77 -2016.08 -2153.92 -2230.57 -1510.97

decay parameter = 1


∗∗∗2.61

∗∗∗3.22

∗∗∗4.05

∗∗∗1.13

∗∗∗

(2.55) (2.89) (4.45) (2.60) (16.53)φC -0.01 0.21

∗∗∗0.17

∗∗∗0.13

∗∗∗0.04

(-0.26) (3.00) (11.52) (16.00) (0.80)β 0.48 1.18 1.25 1.57

∗0.96

(0.37) (0.52) (0.69) (1.84) (0.33)intercept 0.00 0.00 0.00 0.00 0.00

-0.21 -0.62 (1.22) (-1.25) (0.76)

AIC -2008.87 -2015.55 -2154.13 -2228.72 -1507.90

5.5 results 153

dow for chartists’ moving average rules, however, some values of thedecay parameter may favor more some strategies over others.

5.5.4 Robustness to look-back for past performance evaluation

In our model, at the end of each period, agents evaluate the profitabil-ity of their trading strategy and decide whether or not they want toswitch to the other group. The profitability of the strategy is definedin equations 5.10 for the fundamentalists and 5.11 for the chartistsas the average absolute error of the agent groups’ respective returnforecast. In the base case specification, we assume that agents makethe choice of whether to switch by only considering the relative prof-itability over the last month. Table 5.8 shows the output of the HAMswhere agents switching is based on longer-term profitability of theirrespective strategy.

We consider two look-back windows: medium-term 12-month, andlong-term 24-month look-back, to complement the base case with theshort-term look-back of 1 month. In both cases, we estimate our basecase HAM specification. We observe that the fundamentalists’ coeffi-cient is significant in all cases at both the medium, as well as the long-term strategies’ profitability look-back, except for size in the mediumterm, where it is economically significant, however, it falls under thesignificance threshold with a t-statistic of 1.32. A notable differenceis the φC coefficient, which is insignificant for all investment styles,except for value in the case of a long-term look-back window, indicat-ing that the agents in our model may be focusing on the more recentperformance of their strategies, more so than a long term one. Thisfinding is consistent with the results of Frankel and Froot 1990 whoshow that investors use speculative strategies for shorter, one monthhorizons, and more fundamental strategies for longer, one year hori-zons.

5.5.5 Restricted models

In this subsection, we compare the performance of a full heteroge-neous agent model with switching against two restricted versions.The first restricted model assumes that fundamentalist and chartistsexist in fixed and equal proportions, that is, there is no switching be-tween the groups. The second model assumes that only fundamentalinvestors exist. We evaluate to what extent the model fit improves as


Table 5.8: Output of HAM with longer profit look-back

This table shows the output of the estimation of the stylized heterogeneousagent models where agents switch between agent groups based on thelonger term profitability of their respective trading strategy. The test assetsare the five prominent equity style portfolios: size, value, profitability, in-vestment, and momentum. φF, and φC are the estimated coefficients for thefundamentalists’ and chartists’ terms in the model, respectively, and β isthe intensity of switching parameter. The upper panel shows results for a12-month look-back and the bottom panel for a 24-month look-back period.The sample runs from June of 1973 till December of 2017. * indicates signifi-cance at a 10% level, ** indicates significance at a 5% level, and *** indicatessignificance at a 1% level.

L=12

Size Value Profitability Investment MomentumφF 3.33 3.15

∗∗∗3.59

∗∗∗5.56

∗∗∗2.47

∗∗

(1.32) (3.51) (2.61) (3.73) (2.02)φC -0.11 0.27

∗0.10 0.09 0.00

(-0.57) (1.88) (0.65) (0.73) (0.00)β 3.98 0.98 1.00

∗∗∗3.27

∗∗0.98

(0.89) (0.17) (29.32) (2.36) (0.15)intercept 0.00 0.00 0.00 -0.01 0.00

AIC -1992.63 -1982.59 -2119.93 -2173.70 -1482.15

L=24


∗∗∗2.69

∗∗∗3.05

∗∗7.40

∗∗∗2.55

∗∗

(5.13) (2.79) (2.15) (4.04) (2.20)φC -0.01 0.40

∗∗0.13 0.07 0.00

(-0.10) (2.37) (0.86) (0.66) (-0.02)β 6.90 0.98 1.02

∗∗∗4.79

∗∗1.49

∗∗∗

(1.23) (0.09) (47.55) (2.21) (10.71)constant 0.00 0.00 0.00 0.00

∗0.00

(-0.66) (-1.13) (1.48) (-1.85) (-0.50)

AIC -1970.26 -1971.41 -2076.84 2138.25 -1442.19

5.5 results 155

the restrictions are loosened and switching between the groups is in-troduced using the likelihood ratio test. Table 5.9 presents the results.

In the base case specification of the HAM (with EWMA decay of0.5), for value, investment, and momentum styles, the model with-out chartists (mdl 2) has a significantly worse fit than the base casemodel with both agent groups and switching. In the case of two styles,size, and profitability, we find that the model with only fundamental-ists provides a similar fit to the full model, as the likelihood ratiostatistic is not statistically different from zero, although it is positive.When it comes to the model with both agent groups represented inthe constant and equal proportions (mdl 1), we note in all five casesthe likelihood ratio statistic is positive, but it is only significant inthe case of investment and momentum. This finding implies that theadded value of switching for the other three styles is more limited.For size, profitability, investment, and momentum, the chartists’ coef-ficient, φC, becomes insignificant, and in the model with switching,this coefficient is significant for all styles except for profitability.

As we showed in subsection 5.5.3, different values of the decayparameter for the EWMA model that chartists use to form returnexpectations work better for some styles than others. In Table 5.10,we show the likelihood ratio statistics that compare the fit of ourfull model with switching against the two restricted versions, whendifferent values of the decay parameters are used.

In the case of investment and momentum styles, slower movingaverage rules, with decay parameters of 0.5 or 0.2, provide a bettermodel fit. The switching model provides a better fit than either of thetwo restricted models. In the case of value and profitability, the fasterrules, with decay parameters of 0.8 and 1 provide a better fit. Whilethe model without chartists tends to perform worse for these styles,the added value of switching is limited here, and constant weights of0.5 of the agent groups in the market provide an equally good fit. Inthe case of size, the slower moving windows tend to improve the fitof the switching model, and with a decay of 0.2, the base case modelfares better than the restricted model without chartists, however, theadded value from switching is limited in this case as well.

Overall, while we find that the model that features heterogeneousagents in the market provides a better fit than a model with only fun-damentalist investors, the added value of switching is more limited.Only in the case of momentum and investment styles, the model with


Table5.

9:Restricted

models

This

tablecom

paresthe

performance

ofa

fullheterogeneousagent

modelw

ithsw

itchingagainst

two

restrictedversions.M

dl1

assumes

thatfunda-

mentalistand

chartistsexistin

fixedand

equalproportions-there

isno

switching

between

thegroups.M

dl2

assumes

thatonlyfundam

entalinvestorsexist.T

hetestassets

arethe

fiveprom

inentequitystyle

portfolios:size,value,profitability,investment,and

mom

entum.φF ,and

φC

arethe

estimated

coefficientsfor

thefundam

entalists’and

chartists’term

sin

them

odel,respectively,andβ

isthe

intensityof

switching

parameter.LR

-statisticis

thelikelihood

ratiostatistic.*

indicatessignificance

ata

10%

level,**indicates

significanceat

a5%

level,and***

indicatessignificance

ata

1%level.The

sample

runsfrom

Juneof

19

73

tillDecem

berof

20

17.

SizeV

alueProfitability

Investment

Mom

entum

mdl

1m

dl2

mdl

1m

dl2

mdl

1m

dl2

mdl

1m

dl2

mdl

1m

dl2

φF

3.07∗∗∗

1.44∗∗∗

2.86∗∗∗

1.76∗∗∗

2.89∗∗

1.63∗∗∗

2.42∗∗

1.41∗∗∗

2.49∗∗

1.24∗∗

(3.

19)

(3.

09)

(3.

03)

(3.

94)

(2.

25)

(2.

62)

(2.

16)

(2.

61)

(2.

31)

(2.

31)

φC

-0.

12

0.31∗∗

0.17

0.20

-0.

02

(-0.

81)

(2.

15)

(1.

13)

(1.

39)

(-0.

13)

constant0.

00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

(-0.

39)

(-0.

37)

(-1.

04)

(-1.

35)

(1.

26)

(1.

37)

(-0.

64)

(-0.

84)

(-0.

53)

(-0.

53)

LR-stat

0.39

1.05

1.37

5.99∗∗

1.42

2.70

12.

66∗∗∗

14.

59∗∗∗

9.46∗∗∗

9.47∗∗∗

5.5 results 157

Tabl

e5

.10

:LR

-sta

tist

ics

ofm

odel

sw

ith

diff

eren

tEW

MA

deca

ypa

ram

eter

s

Thi

sta

ble

com

pare

sth

epe

rfor

man

ceof

afu

llhe

tero

gene

ous

agen

tm

odel

wit

hsw

itch

ing

agai

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switching provides a better fit than both versions of the restrictedmodel.

5.5.6 Trading strategies

Another way to compare the performance of the various models is toconstruct trading strategies based on the HAM implied expected re-turns. We calculate expected returns on each style implied by thethree models that we estimate: the full HAM with switching, therestricted HAM with static weights, and the restricted model withonly fundamentalists31. For each model, we construct simple tradingstrategies where, each month, we overweight the styles with high,and underweight the styles with low expected one-month ahead re-turns, where the expected returns are compared across the styles. Thesimple weighing scheme that we apply assigns the weight of 6.67% tothe style with the lowest expected return, that progressively increasesby 6.67% for each higher ranked style32. We also construct a bench-mark 1/N strategy, that is invested equally in the five investmentstyles (all styles get a weight of 20%), and is rebalanced monthly.

Figure 5.3 shows the full sample performance of these strategies.A dollar invested in the 1/N portfolio in June of 1973 would haveyielded $6.90 at the end of 2017. The same amount invested in thestrategy based on expected returns implied by the full HAM withswitching would have yielded $16.31, that is 236% more. The strat-egy based on the restricted model with static weights slightly under-performs the full model, and the strategy based on the model withonly fundamentalists lies somewhere in between the full model-basedstrategy and the 1/N. These results provide additional evidence forthe added value of using models with heterogeneous agents to ex-plain asset return dynamics. Consistent with the results reported inthe previous subsection, we find that the added value of switching issomewhat more limited than the value of having both types of agentsin the model.

31 On caveat is that in order to construct the HAM based strategies, we use the full sam-ple estimates of HAM parameters, which means that our trading strategies are nottruly out-of-sample. Our trading strategy is used to validate the results reported inthe previous subsection. In order to construct a truly out of sample trading strategy,it would require that we estimate HAMs on a rolling basis, and given the relativelyshort length of our sample, this would pose challenges for that parameter stability.

32 The vector of weights is given by w = {6.67, 13.33, 20.00, 26.67, 33.33}, all in percent-age points.

5.5 results 159

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5.6 conclusion

The accumulating evidence that market prices, and consequently re-turns, exhibit dynamics that are inconsistent with the theory of ratio-nal expectations has prompted researchers to consider models thatdepart from the stringent assumptions of agent rationality. This gaverise to a relatively new and growing stream of finance literature thatstudies how boundedly rational agents with heterogeneous beliefsabout future asset values interact within markets and cause marketvalues to deviate from their fundamentals

We contribute to this literature by documenting evidence for the ex-istence of heterogeneous agents within equity markets, on a level offive prominent style investing portfolios. The contribution of our pa-per is two-fold. First, to the best of our knowledge, we are the first toestimate a heterogeneous agent model within the equity market, on alevel more granular than that of the equity market index. Our test as-sets are the characteristics-sorted portfolios, whose anomalously highreturns present one of the biggest puzzles in the empirical asset pric-ing literature. Second, we use the insights from the vast literature onthe role of stock-level characteristics in explaining the cross-section ofstock returns and link it with segregated literature on heterogeneousagent models.

We find evidence against the theory of rational expectations and infavor of theories that feature boundedly rational agents in financialmarkets.

6C O N C L U S I O N

Understanding what drives the prices of financial assets is the mainquestion of the research in asset pricing. Given the real world com-plexities of financial markets, this question is not easy to answer.Cochrane 2011 coins the term ’factor zoo’, referring to the fact thatthere are numerous documented asset pricing anomalies, that is, pat-terns in the data that appear to pose challenges to the establishedasset pricing theories. Harvey, Liu, and Zhu 2016 list hundreds ofanomalies in the cross-section of stock returns, and call for a useof more conservative methods of statistical inference when testingpotential factors. Many of these anomalies are a result of pure datasnooping (i.e. statistical patterns in data that are a result of chance),but others are pervasive factors that are yet to be linked to the eco-nomic fundamentals. One of the main challenges of the asset pricingtheories remains to explain the returns of these anomalies. This dis-sertation takes a step towards this goal. It consists of four chaptersthat address some of the prominent issues in this literature.

Chapter 2 challenges the conclusions of Novy-Marx 2014 and Famaand French 2016 that the low-risk anomaly, that is, the anomalouslyhigh/low returns of stocks with low/high past market betas is ex-plained by the Fama and French 1993 three-factor model augmentedwith a profitability factor. Both studies use time-series spanning teststo come to these conclusions and interpret the lack of significant al-phas for the beta-sorted portfolios as evidence against an indepen-dent low-risk anomaly. However, the time-series regressions are justone methodology that is utilized in the asset pricing literature to testif a model explains stock or portfolio returns, and when we evaluatethese claims using other testing methods, we find that the low-riskanomaly remains robust in the cross-section of stock returns. We con-sistently find that average stock returns do not increase with the mar-ket betas, regardless of which profitability factor is used as a controlvariable, or even if we apply various shrinkage methods to reduce

161

162 conclusion

the estimation errors in stock-level market betas. This chapter con-cludes that the low-risk anomaly continues to pose a challenge to theprominent asset pricing models.

Chapter 3 seeks to uncover the sources of the idiosyncratic momen-tum anomaly. The idiosyncratic momentum is closely related to oneof the most pervasive asset pricing anomalies - the total return (alsoknown as price) momentum, documented by Jegadeesh and Titman1993. The momentum anomaly refers to the empirical finding thatstocks that outperformed their peers over the recent past (winners)continue to outperform in the near to intermediate future, and stocksthat underperformed (losers) continue to generate poor returns. Thetotal return momentum anomaly presents one of the biggest chal-lenges for the established asset pricing theories, particularly thosethat are built on the foundations of perfectly rational agents in finan-cial markets.

Gutierrez and Pirinsky 2007 propose a momentum strategy in whichstocks are sorted on their idiosyncratic returns, that is, the stock-specific, residual returns that follow from regressions of total stockreturns on the three Fama and French 1993 factors. However, neitherthis nor the subsequent Blitz, Huij, and Martens 2011 study addressthe question of whether the idiosyncratic momentum is an indepen-dent factor that is not spanned by the other established factors, andin particular if the conventional momentum is included as a controlvariable.

Chapter 3 examines the economic rationale behind this anomaly.We show that the idiosyncratic momentum is a distinct phenomenonthat exists next to the conventional momentum and is not explainedby it. Furthermore, none of the established asset pricing factors, in-cluding the more recent factors of Hou, Xue, and Zhang 2015 andStambaugh and Yuan 2017, that can effectively explain the returnsof the conventional momentum factor, can explain the idiosyncraticmomentum. We also show that some of the prominent explanationsfor the conventional momentum, such as the non-linear crash risk orthe overconfidence and overreaction liked to the market states (bul-l/bear) or market dynamics (trending/reversing) can explain the id-iosyncratic momentum anomaly. We show that the long-term returndynamics of the idiosyncratic momentum support the underreactionhypothesis for its existence and that the factor generates robust re-turns across a range of developed and emerging markets. We con-

conclusion 163

clude that the two momentum phenomena stem from different mar-ket mechanisms.

Chapter 4 attempts to link the sources of the low-volatility anomalyto the macroeconomic state variables. We propose a novel approachto decompose returns of rebalanced portfolios into the discount rateand cash-flow news on a single-stock level that explicitly takes intoaccount stock’s dynamic exposures to aggregate shocks. Using a hostof macroeconomic indicators, we show that low-volatility stocks un-derperform following periods of increasing bond yields, inflation,and aggregate market variance and that high-volatility stocks hedgeagainst these macroeconomic states. Consistent with the ICAPM ofMerton 1974, provided that these macro state variables are proxies forinvestors’ long-term investment opportunities, our results imply in-vestors’ incentives for hedging against these risks as potential driversof the low-volatility anomaly.

The last chapter of the thesis links the literature on heterogeneousagents with the literature on the cross-section of expected stock re-turns and tests whether there is evidence for heterogeneity in ex-pected return formation within the equity market, on a level of equitystyle portfolios. The heterogeneous agent models have been utilizedin other areas of finance to model the behavior of, for instance, ex-change rates, macroeconomic variables, or various asset classes onthe aggregate, index level. We estimate a heterogeneous agent modelthat features two groups of boundedly rational agents on five promi-nent equity investment styles - value, size, profitability, investment,and momentum - and find evidence for behavioral heterogeneity inexpected return formation.

We show that a model that features two groups of agents, fun-damentalists and chartists, where the agents are further allowed toswitch between groups is able to forecast the expected style portfolioreturns better than a model that assumes that only the rational tradersexist in the market, or a model where the two agent groups exist infixed and equal proportions. However, there appears to be less incre-mental value from allowing the agents to switch, than from includingboth groups in the model. We also show that a simple trading strat-egy that overweights and underweights style portfolios relative to afixed-weight 1/N allocation based on their model implied one-monthahead expected returns significantly outperforms the static allocation.Our results thus cast doubt on the theories that assume perfect ra-

164 conclusion

tionality of the agents in financial markets, and give support to thebehavioral theories with heterogeneous agents.

Reflecting on the related work has been done over the past coupleof decades, a big part of it has been centered around finding a parsi-monious set of factors that explain what drives expected returns onstocks. Recently, many new asset pricing factor models have been pro-posed as alternatives to the highly influential Fama and French mod-els; for instance, the q-factor model of Hou, Xue, and Zhang 2015, thefour and five-factor models of Novy-Marx 2014, the model with mis-pricing factors of Stambaugh and Yuan 2017, the model with short-and long-horizon behavioral factors of Daniel, Hirshleifer, and Sun2018, the q5 factor model of Hou et al. 2018, the model of Barillasand Shanken 2018 that is constructed by taking the strongest factors,according to a Bayesian test, from other models. Yet to date no modelhas emerged as a clear winner.

Financial markets process information from the actions of numer-ous market participants and not all data points are equally useful. Inorder to cut through the noise, we use models that are abstract repre-sentations of the underlying processes. However, often times the teststhat we apply do not have enough power to reject one model in favorof another, or even if they do, this is only observed in some time pe-riods or geographies. To add another level of complexity, models canbe changing over time if agents in financial markets constantly learnand adapt. An entire stream of finance literature studies the learn-ing behavior of market participants. A highly influential McLean andPontiff 2016 paper estimates that after the publications of 97 assetpricing anomalies, there was a 32% decline in returns resulting inpost-publication trading. Their findings suggest that investors learnfrom academic studies and trade to eliminate any mispricing in themarket.

Calibrating models solely based on the past data can be prone toover-fitting. Ultimately sound economic theories are needed to safe-guard against this practice and help us to truly understand how finan-cial markets function. For this reason, I see a lot of promise for futureresearch that tries to understand the zoo of the already documentedanomalies through the lens of structural finance models.

Given the importance of these questions, I doubt that the proposedasset pricing models will ever cease to be challenged and altered. Andsuch is the nature of research - when confronted with new evidencewe learn and improve.

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S U M M A RY

Understanding what drives the prices of financial assets is the mainquestion of the research in asset pricing. This dissertation consists offour chapters that address some of the prominent issues in this field.

In Chapter 2, we take a critical look at the findings that the low-risk anomaly is explained by the newly proposed asset pricing mod-els which include a profitability factor. We argue that this conclusionis premature given the lack of empirical evidence for a positive rela-tionship between risk and return. We find that exposure to marketbeta in the cross-section is not rewarded with a positive premium, re-gardless of whether we control for the new factors in the five-factormodel. We also observe stronger mispricing for volatility than forbeta, which suggests that the low-volatility anomaly is the dominantphenomenon. We conclude that the low-risk anomaly is not explainedby the five-factor model.

In Chapter 3, we seek to uncover the drivers of the idiosyncraticmomentum anomaly. We show that: (i) idiosyncratic momentum is adistinct phenomenon that exists next to conventional momentum andis not explained by it; (ii) idiosyncratic momentum is priced in thecross-section of stock returns after controlling for established and re-cently proposed asset pricing factors, including the ones that explaina host of momentum-related anomalies; (iii) some of the prominentexplanations for the momentum premium, such as crash risk, and in-vestor overconfidence and overreaction linked to market states anddynamics cannot explain idiosyncratic momentum profits; (iv) long-term return dynamics of idiosyncratic momentum support the un-derreaction hypothesis for its existence; (v) idiosyncratic momentumgenerates robust returns across a range of developed and emergingmarkets.

In Chapter 4, we show that stocks with low past return volatility, of-ten labeled as low-vol stocks, generate high long-term risk-adjustedreturns (alphas), but underperform following periods of increasingbond yields, inflation, and aggregate market variance. High-vol stockshedge against these macroeconomic states. We propose a novel method-ology to decompose returns of rebalanced portfolios into the discountrate and cash-flow news on a single stock level that explicitly takes

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into account stock’s dynamic exposures to aggregate shocks. Pro-vided that these macro state variables are proxies for investors’ long-term investment opportunities, our results imply investors’ incentivesfor hedging against these risks as potential drivers of the low-riskanomaly.

In Chapter 5, we estimate a heterogeneous agent model on fiveprominent equity investment styles - value, size, profitability, invest-ment, and momentum - and find evidence for behavioral heterogene-ity in expected return formation. Our model features two groups ofboundedly rational investors, fundamentalists and chartists, whosedemand functions for the investment styles depend on their respec-tive expected style return forecasts. The fundamentalists form returnexpectations using a model based on time-varying stock-level charac-teristics and dynamic factor premia, and the chartists do so based onheuristics commonly employed by technical analysts, such as movingaverage rules. Our results cast doubt on the theories that assume per-fect rationality of the representative agent in financial markets, andgive support to the behavioral theories with heterogeneous agents.

on drivers of asset pricing factors dissertation.pdf · milan vidojevic´ vrije universiteit ......

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