lectures on the investment capm - 1. the q-factor...

Lectures on the Investment CAPM

1. The q-Factor Model

Lu Zhang1

1The Ohio State Universityand NBER

Shanghai University of Finance and EconomicsDecember 2015

Prof. Lu Zhang (2015) Lecture 1: The q-Factor Model SUFE 1 / 115

Theme

Four lectures based on Zhang (2015, The Investment CAPM), withadditional details from the original articles referenced:

1 The q-factor model

2 The structural investment CAPM

3 Quantitative investment theories

4 The big picture: Past, present, and future


The Investment CAPMA two-period stochastic general equilibrium model

Three de�ning characteristics of neoclassical economics:

Rational expectations

Consumers maximize utility, and �rms maximize market value

Markets clear


The Investment CAPMThe consumption CAPM

A representative household maximizes:

U(Ct) + ρEt [U(Ct+1)]

subject to:

Ct +∑i

PitSit+1 =∑i

(Pit + Dit)Sit

Ct+1 =∑i

(Pit+1 + Dit+1)Sit+1

The �rst principle of consumption:

Et [Mt+1rSit+1] = 1 ⇒ Et [r

Sit+1]− rft = βMit λMt


The Investment CAPMHeterogeneous �rms

An individual �rm i maximizes:

Pit + Dit ≡ max{Iit}

[ΠitKit − Iit −

a

2

(IitKit

)2Kit + Et [Mt+1Πit+1Kit+1]

]

The �rst principle of investment:

1 = Et

[Mt+1

Πit+1

1 + a(Iit/Kit)

]Pit+1 + Dit+1

Pit≡ rSit+1 = r Iit+1 ≡

Πit+1

1 + a(Iit/Kit)

Capital budgeting implies cross-sectionally varying expected returns


The Investment CAPMImplications

The evidence that characteristics predicting returns is consistent with theinvestment CAPM, does not necessarily mean mispricing

The consumption CAPM and the investment CAPM deliver identicalexpected returns in general equilibrium:

rft + βMit λMt = Et [rSit+1] =

Et [Πit+1]

1 + a(Iit/Kit)

Consumption: Covariances are su�cient statistics of Et [rSit+1]

Investment: Characteristics are su�cient statistics of Et [rSit+1]


Outline

1 The q-Factor Model

2 Factors WarThe Empire Strikes BackConceptual ComparisonEmpirical ComparisonLagging the De�ator: The Irrelevance of Gross Pro�tabilityIndependent Comparison

3 Understanding the Low Risk Anomaly

4 Notes


The q-Factor Model

Outline




4 Notes


The q-Factor Model

The q-Factor ModelHou, Xue, and Zhang (2015a, Digesting anomalies: An investment approach, RFS)

rit − rft = αiq + βiMKTMKTt + βiME rME,t + βi

I/A rI/A,t + βiROE rROE,t + εi

MKTt , rME,t , rI/A,t , and rROE,t are the market, size, investment, andROE factors, respectively

βiMKT

, βiME, βi

I/A, and βiROE

are factor loadings


The q-Factor Model

The q-Factor ModelIntuition: The investment factor

q and high investment, and high discount rates give rise to low marginal q and low investment. This

discount rate intuition is probably most transparent in the capital budgeting language of Brealey,

Myers, and Allen (2006). In our setting capital is homogeneous, meaning that there is no difference

between project-level costs of capital and firm-level costs of capital. Given expected cash flows,

high costs of capital imply low net present values of new projects and in turn low investment, and

low costs of capital imply high net present values of new projects and in turn high investment.12

Figure 1. The Investment Mechanism

-X-axis: Investment-to-assets

6Y -axis: The discount rate

0

High investment-to-assets firms

SEO firms, IPO firms, convertible bond issuers

High net stock issues firms

Growth firms with low book-to-market

Low market leverage firms

Firms with high long-term prior returns

High accrual firms

High composite issuance firms

��

��

Low investment-to-assets firms

Matching nonissuers

Low net stock issues firms

Value firms with high book-to-market

High market leverage firms

Firms with low long-term prior returns

Low accrual firms

Low composite issuance firms

The negative investment-expected return relation is conditional on expected ROE. Investment

is not disconnected with ROE because more profitable firms tend to invest more than less prof-

itable firms. This conditional relation provides a natural portfolio interpretation of the investment

mechanism. Sorting on net stock issues, composite issuance, book-to-market, and other valuation

ratios is closer to sorting on investment than sorting on expected ROE. Equivalently, these sorts

12The negative investment-discount rate relation has a long tradition in economics. In a world without uncertainty,Fisher (1930) and Fama and Miller (1972, Figure 2.4) show that the interest rate and investment are negativelycorrelated. Intuitively, the investment demand curve is downward sloping. Extending this insight into a world withuncertainty, Cochrane (1991) and Liu, Whited, and Zhang (2009) demonstrate the negative investment-expectedreturn relation in a dynamic setting with constant returns to scale. Carlson, Fisher, and Giammarino (2004)also predict the negative investment-expected return relation. In their real options model expansion options areriskier than assets in place. Investment converts riskier expansion options into less risky assets in place. As such,high-investment firms are less risky and earn lower expected returns than low-investment firms.

23


The q-Factor Model

The q-Factor ModelIntuition: The ROE factor

High ROE relative to low investment means high discount rates:

Suppose the discount rates were low

Combined with high ROE, low discount rates would imply high netpresent values of new projects and high investment

So discount rates must be high to counteract high ROE to induce lowinvestment

Price and earnings momentum winners as well as low distress �rms tend tohave higher ROE and earn higher expected returns


The q-Factor Model

The q-Factor ModelIntuition: Beyond investment and ROE

Just like the consumption CAPM saying that betas are su�cient statisticsof expected stock returns, the investment CAPM says a few characteristics(investment and pro�tability) are su�cient

Confront the investment CAPM with an exhaustive array of anomalies


The q-Factor Model

The q-Factor ModelWhy a factor model?

The investment CAPM is a characteristics-based model, but the q-factormodel is a factor pricing model:

Stock returns are better measured than accounting variables, also withhigher frequency

Estimating the structural model involves speci�cation errors absentfrom the factor model

Relative to cross-sectional regressions: Robust to outliers (microcaps),value- and equal-weighting, nonparametric form


The q-Factor Model

The q-Factor ModelFactor construction

Construct rME,t , rI/A,t , and rROE,t with a triple two-by-three-by-three sorton size, investment, and ROE

Variable de�nitions:

Size: Stock price times shares outstanding from CRSP

Investment, I/A: Annual changes in total assets (item AT) divided bylagged total assets

ROE: Income before extraordinary items (item IBQ) divided byone-quarter-lagged book equity


The q-Factor Model

The q-Factor ModelFactor construction

NYSE breakpoints: 50-50 for size, 30-40-30 for I/A, and 30-40-30 for ROE

Annual sort in June on the market equity at the June end

Annual sort in June of year t on I/A for the �scal year ending incalendar year t − 1

Monthly sort at the beginning of each month on ROE with the mostrecently announced quarterly earnings

Timing consistent with the investment CAPM and empirical practice


The q-Factor Model

The q-Factor ModelProperties of the q-factors, 1/1972�12/2012

m α βMKT βSMB βHML βUMD

rME 0.31 0.23 0.17(2.12) 0.04 0.02 0.99 0.17

0.01 0.02 0.99 0.19 0.03

rI/A 0.45 0.52 −0.15(4.95) 0.33 −0.06 −0.02 0.39

0.28 −0.05 −0.02 0.41 0.05

rROE 0.58 0.63 −0.11(4.81) 0.77 −0.09 −0.33 −0.20

0.50 −0.03 −0.33 −0.10 0.28

rI/A rROE MKT SMB HML UMD

rME −0.11 −0.31 0.25 0.95 −0.07 0.01rI/A 0.06 −0.36 −0.22 0.69 0.05rROE −0.19 −0.38 −0.09 0.50


The q-Factor Model

The q-Factor ModelSharpe ratios

Sharpe ratios Maximum Sharpe ratios

MKT SMB HML UMD rME rI/A rROE CAPM FF3 Carhart q

0.10 0.06 0.13 0.16 0.10 0.24 0.22 0.10 0.21 0.30 0.43


The q-Factor Model

The q-Factor ModelTesting portfolios

Use an extensive array of anomaly portfolios (nearly 80), scope comparablewith the largest in the literature

Green, Hand, and Zhang (2013)

Harvey, Liu, and Zhu (2015)

McLean and Ponti� (2015)

NYSE breakpoints and value-weighted decile returns to alleviate the impactof microcaps

Fama and French (2008)


The q-Factor Model

The q-Factor ModelTesting portfolios across six categories, nearly 80

Panel A: Momentum (plus six momentum-reversal variables)

SUE-1, earnings surprise SUE-6, earnings surprise(1-month holding period), (6-month holding period),Foster, Olsen, and Shevlin (1984) Foster, Olsen, and Shevlin (1984)Abr-1, cumulative abnormal stock Abr-6, cumulative abnormal stockreturns around earnings announcements returns around earnings announcements(1-month holding period), Chan, (6-month holding period), Chan,Jegadeesh, and Lakonishok (1996) Jegadeesh, and Lakonishok (1996)RE-1, revisions in analysts' earnings RE-6, revisions in analysts' earningsforecasts (1-month holding period), forecasts (6-month holding period),Chan, Jegadeesh, and Lakonishok (1996) Chan, Jegadeesh, and Lakonishok (1996)R6-1, price momentum (6-month prior R6-6, price momentum (6-month priorreturns, 1-month holding period), returns, 6-month holding period),Jegadeesh and Titman (1993) Jegadeesh and Titman (1993)R11-1, price momentum, (11-month I-Mom, industry momentum,prior returns, 1-month holding period), Moskowitz and Grinblatt (1999)Fama and French (1996)


The q-Factor Model


Panel B: Value versus growth

B/M, book-to-market equity, A/ME, market leverage,Rosenberg, Reid, and Lanstein (1985) Bhandari (1988)Rev, reversal, De Bondt and Thaler (1985) E/P, earnings-to-price, Basu (1983)EF/P, analysts' earnings forecasts-to-price, CF/P, cash �ow-to-price,Elgers, Lo, and Pfei�er (2001) Lakonishok, Shleifer, and Vishny (1994)D/P, dividend yield, O/P, payout yield, Boudoukh, Michaely,Litzenberger and Ramaswamy (1979) Richardson, and Roberts (2007)NO/P, net payout yield, Boudoukh, SG, sales growth,Michaely, Richardson, and Roberts (2007) Lakonishok, Shleifer, and Vishny (1994)LTG, long-term growth forecasts Dur, equity duration,of analysts, La Porta (1996) Dechow, Sloan, and Soliman (2004)


The q-Factor Model


Panel C: Investment

ACI, abnormal corporate investment, I/A, investment-to-assets,Titman, Wei, and Xie (2004) Cooper, Gulen, and Schill (2008)NOA, net operating assets, Hirshleifer, 4PI/A, changes in PPEHou, Teoh, and Zhang (2004) plus changes in inventory scaled by assets,

Lyandres, Sun, and Zhang (2008)IG, investment growth, NSI, net stock issues,Xing (2008) Ponti� and Woodgate (2008)CEI, composite issuance, NXF, net external �nancing,Daniel and Titman (2006) Bradshaw, Richardson, and Sloan (2006)IvG, inventory growth, IvC, inventory changes,Belo and Lin (2011) Thomas and Zhang (2002)OA, operating accruals, Sloan (1996) TA, total accruals, Richardson, Sloan,

Soliman, and Tuna (2005)POA, percent operating accruals, Hafzalla, PTA, percent total accruals, Hafzalla,Lundholm, and Van Winkle (2011) Lundholm, and Van Winkle (2011)


The q-Factor Model


Panel D: Pro�tability

ROE, return on equity, ROA, return on assets,Haugen and Baker (1996) Balakrishnan, Bartov, and Faurel (2010)RNA, return on net operating assets, PM, pro�t margin, Soliman (2008)Soliman (2008)ATO, asset turnover, CTO, capital turnover,Soliman (2008) Haugen and Baker (1996)GP/A, gross pro�ts-to-assets, F , F -score,Novy-Marx (2013) Piotroski (2000)TES, tax expense surprise, TI/BI, taxable income-to-book income,Thomas and Zhang (2011) Green, Hand, and Zhang (2013)RS, revenue surprise, NEI, number of consecutive quartersJegadeesh and Livnat (2006) with earnings increases,

Barth, Elliott, and Finn (1999)FP, failure probability, O, O-score, Dichev (1998)Campbell, Hilscher, and Szilagyi (2008)


The q-Factor Model


Panel E: Intangibles and other characteristics

OC/A, organizational capital-to-assets, BC/A, brand capital-to-assets,Eisfeldt and Papanikolaou (2013) Belo, Lin, and Vitorino (2014)Ad/M, advertisement expense-to-market, RD/S, R&D-to-sales,Chan, Lakonishok, and Sougiannis (2001) Chan, Lakonishok, and Sougiannis (2001)RD/M, R&D-to-market, RC/A, R&D capital-to-assets, Li (2011)Chan, Lakonishok, and Sougiannis (2001)H/N, hiring rate, OL, operating leverage,Belo, Lin, and Bazdresch (2014) Novy-Marx (2011)G , corporate governance, AccQ, accrual quality, Francis, Lafond,Gompers, Ishii, and Metrick (2003) Olsson, and Schipper (2005)Ind, industries, Fama and French (1997)


The q-Factor Model


Panel F: Trading frictions

ME, the market equity, Ivol, idiosyncratic volatility,Banz (1981) Ang, Hodrick, Xing, and Zhang (2006)Tvol, total volatility, Svol, systematic volatility,Ang, Hodrick, Xing, and Zhang (2006) Ang, Hodrick, Xing, and Zhang (2006)MDR, maximum daily return, β, market beta,Bali, Cakici, and Whitelaw (2011) Frazzini and Pedersen (2014)D-β, Dimson's beta, Dimson (1979) S-Rev, short-term reversal, Jegadeesh (1990)Disp, dispersion of analysts' Turn, share turnover,earnings forecasts, Datar, Naik, and Radcli�e (1998)Diether, Malloy, and Scherbina (2002)1/P, 1/share price, Dvol, dollar trading volume,Miller and Scholes (1982) Brennan, Chordia, and Subrahmanyam (1998)Illiq, Absolute return-to-volume,Amihud (2002)


The q-Factor Model

The q-Factor ModelOverview of empirical results

About one half of nearly 80 anomalies are insigni�cant with NYSEbreakpoints and value-weighted decile returns

Across 35 signi�cant anomalies, the q-factor model outperforms theFama-French 3-factor and Carhart models:

The average magnitude of high-minus-low alphas: .2% per month inq, .33% in Carhart, and .55% in Fama-French

The number of signi�cant high-minus-low alphas: 5 in q, 19 inCarhart, and 27 in Fama-French

The number of rejections by the GRS test: 20 in q, 24 in Carhart, and28 in Fama-French


The q-Factor Model

The q-Factor ModelInsigni�cant anomalies in the broad cross section

R6-1 A/ME Rev EF/P D/P O/P SG LTG ACI NXF

m 0.48 0.43 −0.39 0.45 0.27 0.35 −0.27 0.01 −0.27 −0.30tm 1.43 1.82 −1.57 1.73 0.94 1.53 −1.34 0.02 −1.70 −1.55

TA RNA PM ATO CTO F TES TI/BI RS O

m −0.19 0.13 0.10 0.22 0.20 0.37 0.32 0.13 0.29 −0.08tm −1.31 0.61 0.40 1.11 1.11 1.28 1.92 0.86 1.82 −0.37

BC/A RD/S RC/A H/N G AccQ ME Ivol Tvol MDR

m 0.18 0.01 0.32 −0.25 0.03 −0.18 −0.24 −0.54 −0.37 −0.31tm 0.73 0.06 1.27 −1.47 0.09 −0.79 −0.90 −1.56 −0.95 −0.94

β D-β S-Rev Disp Turn 1/P Dvol Illiq

m −0.13 0.07 −0.31 −0.33 −0.12 −0.00 −0.26 0.27tm −0.36 0.30 −1.39 −1.24 −0.43 −0.01 −1.30 1.14


The q-Factor Model

The q-Factor ModelSigni�cant anomalies in the momentum category

SUE-1 SUE-6 Abr-1 Abr-6 RE-1 RE-6 R6-6 R11-1 I-Mom |ave|

m 0.45 0.24 0.73 0.30 0.89 0.60 0.85 1.18 0.51αFF 0.55 0.39 0.84 0.38 1.20 0.94 1.12 1.52 0.68 0.85αC 0.34 0.18 0.62 0.19 0.56 0.37 0.06 0.09 −0.18 0.29αq 0.16 0.02 0.64 0.26 0.12 0.03 0.24 0.24 0.00 0.19

tm 3.59 2.17 5.50 3.11 3.43 2.58 3.17 3.52 2.33tFF 4.50 3.62 5.93 3.89 4.81 4.52 4.47 4.99 3.25tC 2.62 1.69 4.37 2.06 2.56 2.15 0.51 0.67 −1.11tq 1.12 0.18 4.07 2.18 0.43 0.14 0.71 0.54 0.01

|αFF | 0.17 0.13 0.16 0.11 0.27 0.23 0.19 0.26 0.15 0.19

|αC | 0.11 0.09 0.12 0.08 0.11 0.09 0.10 0.13 0.06 0.10

|αq| 0.05 0.07 0.13 0.07 0.10 0.11 0.08 0.13 0.13 0.10

pFF 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.09pC 0.00 0.00 0.00 0.01 0.16 0.12 0.00 0.00 0.45pq 0.42 0.04 0.00 0.02 0.46 0.08 0.00 0.01 0.03


The q-Factor Model

The q-Factor Modelq-factor loadings and q-characteristics, anomalies in the momentum category

SUE-1 SUE-6 Abr-1 Abr-6 RE-1 RE-6 R6-6 R11-1 I-Mom

βMKT −0.08 −0.06 −0.06 −0.03 −0.05 −0.07 −0.09 −0.14 −0.11βME 0.10 0.09 0.07 0.09 −0.15 −0.19 0.27 0.40 0.31βI/A 0.02 −0.11 −0.13 −0.16 0.04 −0.12 −0.07 0.04 −0.03βROE 0.48 0.45 0.28 0.18 1.33 1.12 1.02 1.48 0.82tβMKT −1.82 −1.53 −1.31 −1.20 −0.76 −1.24 −1.17 −1.43 −1.72tβME 1.94 1.27 0.67 1.82 −1.42 −1.98 1.43 1.74 1.86tβI/A 0.18 −0.97 −1.25 −2.24 0.25 −0.82 −0.27 0.12 −0.13tβROE 5.75 5.95 3.26 2.94 10.09 9.96 5.31 5.67 4.90

ME 0.69 0.75 −0.01 0.03 0.77 0.87 0.40 0.52 0.62I/A −1.46 −0.96 −1.37 −1.13 −0.80 0.72 −4.07 −3.83 −1.18ROE 5.80 3.38 1.59 1.49 6.58 6.47 4.14 5.34 1.61tME 4.91 5.38 −0.29 1.31 8.75 9.65 4.92 4.95 3.67tI/A −3.30 −2.57 −2.36 −2.58 −1.22 1.13 −5.54 −4.66 −1.79tROE 16.46 19.07 13.38 15.47 29.77 27.86 16.00 17.06 10.24


The q-Factor Model

The q-Factor ModelSigni�cant anomalies in the value minus growth category

B/M E/P CF/P NO/P Dur |ave|

m 0.70 0.59 0.52 0.66 −0.54αFF 0.01 0.05 0.01 0.52 −0.06 0.13αC −0.01 0.01 −0.06 0.49 −0.08 0.13αq 0.21 0.17 0.22 0.36 −0.27 0.25

tm 2.88 2.63 2.44 3.23 −2.59tFF 0.04 0.34 0.08 3.51 −0.44tC −0.06 0.03 −0.40 3.33 −0.56tq 1.15 0.76 1.04 2.38 −1.32|αFF | 0.07 0.10 0.08 0.17 0.11 0.11

|αC | 0.06 0.09 0.07 0.15 0.08 0.09

|αq| 0.08 0.10 0.14 0.12 0.08 0.10

pFF 0.19 0.18 0.43 0.00 0.15pC 0.29 0.38 0.37 0.00 0.41pq 0.35 0.13 0.02 0.00 0.72


The q-Factor Model

The q-Factor Modelq-factor loadings and q-characteristics, the value versus growth anomalies

B/M E/P CF/P NO/P Dur

βMKT −0.03 −0.12 −0.15 −0.18 0.11βME 0.46 0.25 0.19 −0.32 −0.23βI/A 1.45 0.99 1.01 1.03 −0.85βROE −0.51 −0.09 −0.24 0.02 0.24tβMKT −0.59 −2.02 −2.41 −3.86 1.67tβME 5.37 1.90 1.66 −4.40 −1.61tβI/A 12.74 5.76 6.79 10.25 −5.69tβROE −5.98 −0.66 −1.78 0.19 1.87

ME −2.46 −0.73 −0.89 1.23 0.41I/A −9.70 −1.11 −5.63 −14.43 3.95ROE −5.68 0.21 −0.72 1.18 0.49tME −10.31 −4.09 −4.57 7.74 5.23tI/A −17.06 −1.20 −5.47 −13.81 2.73tROE −29.57 1.30 −4.83 7.27 2.22


The q-Factor Model

The q-Factor ModelSigni�cant anomalies in the investment category

I/A NOA 4PI/A IG NSI CEI IvG IvC OA POA PTA |ave|

m −0.42 −0.38 −0.51 −0.41 −0.68 −0.57 −0.41 −0.45 −0.30 −0.46 −0.40αFF −0.15 −0.52 −0.41 −0.26 −0.64 −0.50 −0.29 −0.38 −0.37 −0.32 −0.29 0.38αC −0.09 −0.41 −0.36 −0.20 −0.54 −0.40 −0.19 −0.30 −0.33 −0.25 −0.27 0.30αq 0.14 −0.38 −0.26 0.05 −0.26 −0.22 −0.03 −0.28 −0.56 −0.12 −0.10 0.22

tm −2.45 −2.55 −3.43 −2.93 −4.13 −2.96 −2.77 −3.05 −2.32 −3.02 −2.57tFF −1.09 −3.30 −2.93 −1.99 −4.28 −3.72 −2.10 −2.61 −2.84 −2.42 −2.06tC −0.61 −2.69 −2.48 −1.51 −3.58 −2.93 −1.34 −1.97 −2.32 −1.88 −1.82tq 1.08 −1.90 −1.85 0.39 −1.75 −1.50 −0.20 −1.84 −3.90 −0.87 −0.67|αFF | 0.12 0.17 0.13 0.13 0.18 0.15 0.11 0.12 0.13 0.11 0.11 0.13

|αC | 0.10 0.14 0.12 0.11 0.15 0.15 0.10 0.10 0.12 0.11 0.10 0.12

|αq| 0.09 0.12 0.14 0.09 0.11 0.12 0.11 0.08 0.15 0.12 0.08 0.11

pFF 0.01 0.00 0.00 0.00 0.00 0.00 0.03 0.01 0.00 0.00 0.01pC 0.02 0.00 0.01 0.00 0.00 0.00 0.11 0.04 0.00 0.01 0.02pq 0.01 0.00 0.00 0.01 0.02 0.01 0.08 0.56 0.00 0.00 0.11


The q-Factor Model

The q-Factor Modelq-factor loadings and q-characteristics, anomalies in the investment category

I/A NOA 4PI/A IG NSI CEI IvG IvC OA POA PTA

βMKT 0.02 −0.02 0.05 −0.02 0.04 0.24 −0.03 0.04 0.03 −0.01 0.06βME −0.11 0.06 −0.05 −0.11 0.17 0.26 0.12 0.00 0.28 0.15 0.21βI/A −1.37 −0.01 −0.77 −0.82 −0.68 −1.06 −0.96 −0.65 −0.02 −0.90 −0.90βROE 0.15 −0.01 0.16 −0.07 −0.32 −0.12 0.05 0.18 0.29 0.05 0.04tβMKT 0.62 −0.55 1.33 −0.71 0.99 6.27 −0.77 1.01 0.80 −0.19 1.50tβME −1.81 0.54 −0.94 −1.95 2.24 3.79 2.85 −0.07 4.41 3.20 3.28tβI/A −15.50 −0.04 −6.98 −10.91 −6.14 −13.11 −11.81 −5.49 −0.21 −9.61 −8.72tβROE 2.29 −0.12 1.93 −1.06 −4.07 −1.42 0.56 1.95 4.59 1.04 0.55

ME 0.88 0.07 0.63 0.22 −1.37 −1.79 0.26 0.19 −0.24 −0.36 −0.36I/A 83.89 55.72 61.16 34.03 27.04 14.80 37.85 44.80 10.15 11.12 16.14ROE 1.63 −1.24 0.84 0.50 −1.71 −1.41 0.42 1.07 0.88 1.02 0.36tME 7.75 1.71 7.56 6.16 −6.44 −7.77 4.27 4.86 −5.13 −9.39 −6.76tI/A 32.74 18.28 30.77 22.38 13.85 14.37 23.42 34.89 5.19 7.90 12.93tROE 10.00 −7.89 5.10 3.24 −11.87 −9.07 3.31 8.13 5.06 7.42 2.56


The q-Factor Model

The q-Factor ModelSigni�cant anomalies in the pro�tability category

ROE ROA GP/A NEI FP |ave|

m 0.80 0.62 0.34 0.39 −0.67αFF 1.17 1.00 0.50 0.63 −1.44 0.95αC 0.85 0.67 0.45 0.43 −0.67 0.61αq 0.05 0.09 0.11 0.18 −0.17 0.12

tm 3.11 2.70 2.18 3.31 −1.98tFF 5.43 5.40 3.25 6.03 −6.44tC 4.03 3.59 2.85 3.73 −3.79tq 0.37 0.72 0.71 1.68 −0.57|αFF | 0.24 0.23 0.14 0.23 0.23 0.21

|αC | 0.15 0.14 0.14 0.15 0.12 0.14

|αq| 0.09 0.07 0.11 0.09 0.13 0.10

pFF 0.00 0.00 0.01 0.00 0.00pC 0.00 0.04 0.01 0.00 0.00pq 0.05 0.75 0.38 0.05 0.00


The q-Factor Model

The q-Factor Modelq-factor loadings and q-characteristics, anomalies in the pro�tability category

ROE ROA GP/A NEI FP

βMKT −0.10 −0.14 0.05 0.02 0.44βME −0.41 −0.38 0.03 −0.10 0.43βI/A 0.10 −0.10 −0.24 −0.30 0.17βROE 1.50 1.31 0.52 0.63 −1.61tβMKT −2.57 −4.48 1.20 0.88 6.46tβME −6.56 −6.41 0.51 −2.53 2.45tβI/A 1.05 −1.23 −2.35 −3.78 0.63

tβROE 20.71 16.86 7.08 10.83 −8.79ME 2.81 2.66 0.39 2.34 −3.09I/A 3.56 5.32 −1.29 5.35 −3.91ROE 16.95 14.71 3.94 4.32 −8.74tME 10.56 10.50 10.84 11.58 −10.76tI/A 4.16 6.88 −2.05 11.45 −4.18tROE 29.02 27.97 23.88 27.36 −25.56


The q-Factor Model

The q-Factor ModelSigni�cant anomalies in the intangibles and trading frictions categories

OC/A Ad/M RD/M OL Svol |ave|

m 0.56 0.79 0.63 0.39 −0.60αFF 0.61 0.15 0.22 0.37 −0.66 0.40αC 0.40 0.32 0.31 0.33 −0.62 0.40αq 0.09 0.11 0.60 −0.05 −0.37 0.24

tm 4.07 2.96 2.31 2.06 −2.57tFF 4.52 0.79 0.93 1.91 −2.88tC 2.97 1.37 1.40 1.76 −2.59tq 0.66 0.39 2.40 −0.27 −1.42|αFF | 0.15 0.13 0.17 0.11 0.19 0.15

|αC | 0.13 0.18 0.21 0.12 0.16 0.16

|αq| 0.11 0.11 0.27 0.12 0.11 0.14

pFF 0.00 0.18 0.02 0.07 0.01pC 0.00 0.07 0.01 0.06 0.06pq 0.02 0.07 0.00 0.09 0.20


The q-Factor Model

The q-Factor Modelq-factor loadings and q-characteristics, intangibles and trading frictions anomalies

OC/A Ad/M RD/M OL Svol

βMKT −0.13 0.04 0.16 −0.06 0.04βME 0.25 0.50 0.66 0.26 0.31βI/A 0.35 1.42 0.21 0.16 −0.21βROE 0.51 −0.27 −0.58 0.54 −0.43tβMKT −3.74 0.50 2.51 −1.22 0.53tβME 5.69 2.85 6.75 2.63 2.30tβI/A 3.52 6.03 1.21 1.34 −1.30tβROE 7.12 −1.37 −4.10 4.85 −3.54ME −1.31 −1.34 −4.39 −1.31 −0.19I/A −13.77 −10.71 −3.22 −5.71 0.66ROE 1.52 −3.33 −2.80 1.86 −0.64tME −9.44 −9.83 −9.47 −8.43 −3.95tI/A −11.38 −12.17 −2.54 −4.70 1.37tROE 7.97 −12.60 −9.21 11.42 −4.03


The q-Factor Model

The q-Factor Model25 size and book-to-market portfolios

Low 2 3 4 High H−L Low 2 3 4 High H−L

m αFF (|αFF | = 0.10)

Small 0.08 0.72 0.84 0.95 1.11 1.02 −0.54 0.02 0.13 0.18 0.16 0.702 0.32 0.69 0.86 0.87 0.99 0.67 −0.21 0.00 0.09 0.07 0.04 0.253 0.38 0.71 0.77 0.77 1.02 0.65 −0.09 0.04 0.03 0.00 0.13 0.224 0.52 0.59 0.73 0.74 0.84 0.32 0.14 −0.02 0.03 0.00 0.02 −0.12Big 0.40 0.54 0.54 0.61 0.56 0.16 0.16 0.11 0.09 0.00 −0.16 −0.32

tm tFF (pFF = 0.00)

Small 0.20 2.02 2.57 3.07 3.30 4.59 −4.84 0.23 1.58 2.53 1.97 5.662 0.90 2.20 3.03 3.29 3.31 2.93 −2.59 0.01 1.36 0.92 0.45 2.193 1.15 2.49 3.03 3.05 3.93 2.76 −1.21 0.52 0.35 0.02 1.28 1.704 1.73 2.35 2.91 3.08 3.24 1.43 1.74 −0.23 0.27 0.03 0.17 −0.87Big 1.70 2.53 2.71 3.04 2.43 0.79 2.66 1.29 0.92 −0.02 −1.31 −2.32


The q-Factor Model

The q-Factor Model25 size and book-to-market portfolios

Low 2 3 4 High H−L Low 2 3 4 High H−L

αC (|αC | = 0.11) αq (|αq| = 0.11)

Small −0.48 0.03 0.12 0.18 0.22 0.70 −0.25 0.27 0.31 0.30 0.32 0.572 −0.18 0.03 0.09 0.10 0.04 0.22 −0.14 0.02 0.03 0.07 0.10 0.243 −0.04 0.04 0.09 0.03 0.16 0.20 −0.01 −0.03 −0.04 −0.01 0.14 0.154 0.15 −0.01 0.07 0.03 0.09 −0.06 0.18 −0.14 −0.01 0.02 0.06 −0.12Big 0.17 0.07 0.07 −0.03 −0.13 −0.31 0.10 −0.04 0.06 −0.01 −0.04 −0.13

tC (pC = 0.00) tq (pq = 0.00)

Small −4.00 0.36 1.58 2.59 2.53 5.72 −1.48 2.24 3.09 3.68 2.72 2.912 −2.28 0.37 1.34 1.40 0.44 1.88 −1.21 0.29 0.37 0.67 0.89 1.253 −0.50 0.53 0.93 0.28 1.40 1.43 −0.09 −0.30 −0.37 −0.05 1.16 0.924 1.87 −0.16 0.74 0.24 0.75 −0.42 1.50 −1.58 −0.06 0.21 0.44 −0.61Big 2.89 0.91 0.70 −0.36 −1.02 −2.12 1.32 −0.49 0.65 −0.06 −0.23 −0.70


The q-Factor Model

The q-Factor ModelThe key contribution of Hou, Xue, and Zhang (2015a)

The q-factor model largely summarizes the cross section of average stockreturns, capturing most (but not all) anomalies that plague theFama-French 3-factor model


Factors War

Outline




4 Notes


Factors War

Factors WarA quote from German philosopher Arthur Schopenhauer (1788�1860)

�All truth passes through threestages. First, it is ridiculed. Second,it is violently opposed. Third, it isaccepted as being self-evident.�


Factors War The Empire Strikes Back

Factors WarThe Fama-French (2015a, b) �ve-factor model

rit − rft = ai + bi MKTt + si SMBt + hi HMLt + ri RMWt + ci CMAt + eit

MKTt , SMBt ,HMLt ,RMWt , and CMAt are the market, size, value,pro�tability, and investment factors, respectively

bi , si , hi , ri , and ci are factor loadings



Factors WarThe q-factor model predates the Fama-French �ve-factor model by 3�6 years

Neoclassical factors July 2007

An equilibrium three-factor model January 2009Production-based factors April 2009A better three-factor model June 2009

that explains more anomaliesAn alternative three-factor model April 2010, April 2011

Digesting anomalies: An investment approach October 2012 , August 2014

Fama and French (2013): A four-factor model for June 2013the size, value, and pro�tabilitypatterns in stock returns

Fama and French (2014): November 2013 , September 2014

A �ve-factor asset pricing model



Factors WarExcerpts from Swedroe's �Battle of New Factor Models�

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Index Investor Corner

Swedroe: Battle Of New Factor ModelsByLarry SwedroeAugust 07, 2015Share:

In their groundbreaking paper, “Digesting Anomalies: An Investment Approach,” Kewei Hou, Chen Xue and Lu Zhang proposed a new four-factor asset pricing model that goes a long way toward explaining many of the anomalies neither the Fama-French three-factor nor subsequent four-factor models could explain.

The study, which was published in the March 2015 issue of The Review of Financial Studies, covered the period 1972 through 2012.

The authors called their model the q-factor model. Specifically, their four factors are:

• The market excess return (beta)• The difference between the return on a portfolio of small-cap stocks and the return on a

portfolio of large-cap stocks• The difference between the return on a portfolio of low-investment stocks and the

return on a portfolio of high-investment stocks

Page 1 of 7

12/5/2015http://www.etf.com/sections/index-investor-corner/swedroe-battle-new-factor-models?nop...

�Fama & French Examine The Q-Model�

�Through their research, �nancialeconomists continue to advance ourunderstanding of how �nancial marketswork and how prices are set. TheFama-French three-factor model was asigni�cant improvement on thesingle-factor capital asset pricing model.Mark Carhart moved the needle further byadding momentum as a fourth factor. Andthe authors of the q-theory have madefurther signi�cant advancements, which inturn motivated the development of thecompeting FF �ve-factor model (myemphasis).�



Factors WarA little dark humor



Factors WarFactor construction

SMB, HML, RMW, and CMA from double 2× 3 sorts from interacting sizewith B/M, OP, and Inv

Size: Stock price times shares outstanding from CRSP

B/M: Per Davis, Fama, and French (2000)

OP: Revenues minus costs of goods sold, SG&A, and interest expense,all divided by current book equity

Inv: Annual changes in total assets divided by lagged assets

All annual sorts



Factors WarMotivation from valuation theory

Motivation from the Miller-Modigliani (1961) valuation model:

Pit

Bit=

∑∞τ=1 E [Yit+τ −4Bit+τ ]/(1 + ri )

τ

Bit,

Fama and French (2006, 2015a) derive three predictions:

A lower Pit/Bit means a higher ri , all else equal

A higher E [Yit+τ ] means a higher ri , all else equal

A higher E [4Bit+τ ]/Bit means a lower ri , all else equal


Factors War Conceptual Comparison

Factors WarCritique I: IRR 6= the one-period-ahead expected return

Fama and French (2015a, p. 2): �Most asset pricing research focuses onshort-horizon returns�we use a one-month horizon in our tests. If eachstock's short-horizon expected return is positively related to its internalrate of return�if, for example, the expected return is the same for allhorizons�the valuation equation... (our emphasis).�

Assumption clearly contradicting price and earnings momentum



Factors WarIRR estimates per Gebhardt, Lee, and Swaminathan (2001)

2× 3 2× 2 2× 2× 2× 2

Return IRR Di� Return IRR Di� Return IRR Di�

SMB 0.26 0.06 0.20 0.27 0.07 0.20 0.25 0.05 0.20[t] 1.89 9.61 1.45 1.89 10.18 1.43 1.94 9.98 1.56HML 0.23 0.27 −0.04 0.17 0.19 −0.02 0.17 0.19 −0.02[t] 1.38 40.30 −0.23 1.42 35.93 −0.14 1.37 40.11 −0.12RMW 0.32 −0.08 0.40 0.21 −0.06 0.27 0.23 0.01 0.22

[t] 2.65 −15.74 3.34 2.63 −13.25 3.38 2.78 4.18 2.64

CMA 0.28 0.05 0.23 0.20 0.03 0.17 0.17 −0.01 0.18[t] 2.75 9.22 2.27 2.67 8.83 2.25 2.68 −3.46 2.84



Factors WarCritique II: HML separate in theory but redundant in the data

HML redundant in Fama and French (2015a), inconsistent with theirmotivation based on valuation theory

Consistent with the investment CAPM:

Et [rSit+1] =

Et [Πit+1/Ait+1] + 1

1 + a(Iit/Ait),

in which the denominator = Pit/Bit

Without the redundant HML, the �ve-factor model becomes (a noisyversion of) the q-factor model



Factors WarCritique III: The expected investment-return relation is likely positive

Reformulating valuation theory with Et [rit+1]:

Pit =Et [Yit+1 −4Bit+1] + Et [Pit+1]

1 + Et [rit+1],

Pit

Bit=

Et

[Yit+1

Bit

]− Et

[4Bit+1

Bit

]+ Et

[Pit+1

Bit+1

(1 + 4Bit+1

Bit

)]1 + Et [rit+1]

,

Pit

Bit=

Et

[Yit+1

Bit

]+ Et

[4Bit+1

Bit

(Pit+1

Bit+1− 1)]

+ Et

[Pit+1

Bit+1

]1 + Et [rit+1]

.

Recursive substitution: A positive Et [4Bit+τ/Bit ]-Et [rit+1] relation

Evidence: Lettau and Ludvigson (2002), Fama and French (2006)



Factors WarCritique IV: Past investment does not forecast future investment

Total assets ≥ $5 millions and book equity ≥ $2.5 millions

Bit+τ−Bit+τ−1Bit+τ−1

∣∣∣ 4TAitTAit−1

Bit+τ−Bit+τ−1Bit+τ−1

∣∣∣ 4BitBit−1

OPit+τ |OPit

τ γ0 γ1 R2 γ0 γ1 R2 γ0 γ1 R2

1 0.09 0.22 0.05 0.09 0.21 0.06 0.03 0.80 0.55

2 0.10 0.10 0.01 0.10 0.10 0.02 0.06 0.67 0.36

3 0.10 0.07 0.01 0.10 0.06 0.01 0.07 0.59 0.28

4 0.10 0.05 0.00 0.10 0.06 0.00 0.09 0.53 0.22

5 0.10 0.05 0.00 0.10 0.03 0.00 0.10 0.49 0.19

6 0.10 0.05 0.00 0.10 0.03 0.00 0.10 0.46 0.16

7 0.10 0.05 0.00 0.10 0.03 0.00 0.11 0.43 0.14

8 0.10 0.03 0.00 0.10 0.01 0.00 0.12 0.40 0.12

9 0.10 0.03 0.00 0.10 0.01 0.00 0.12 0.38 0.12

10 0.09 0.04 0.00 0.10 0.02 0.00 0.13 0.37 0.11


Factors War Empirical Comparison

Factors WarFactor spanning tests, q-factors, 1/1967�12/2014

m αC βMKT βSMB βHML βUMD

rME 0.32 0.01 0.01 0.97 0.17 0.03(2.42) (0.25) (1.08) (67.08) (7.21) (1.87)

rI/A 0.43 0.29 −0.06 −0.04 0.41 0.05(5.08) (4.57) (−4.51) (−1.88) (13.36) (1.93)

rROE 0.56 0.51 −0.04 −0.30 −0.12 0.27(5.24) (5.58) (−1.39) (−4.31) (−1.79) (6.19)

α5 b s h r c

rME 0.05 0.00 0.98 0.02 −0.01 0.04(1.39) (0.39) (68.34) (1.14) (−0.21) (1.19)

rI/A 0.12 0.01 −0.05 0.04 0.07 0.82(3.35) (0.73) (−2.86) (1.60) (2.77) (26.52)

rROE 0.45 −0.04 −0.11 −0.24 0.75 0.13(5.60) (−1.45) (−2.69) (−3.54) (13.46) (1.34)



Factors WarFactor spanning tests, new Fama-French factors, 1/1967�12/2014

m αC βMKT βSMB βHML βUMD

SMB 0.28 −0.02 0.00 1.00 0.13 0.00(1.92) (−1.24) (0.96) (89.87) (8.07) (0.11)

HML 0.36 −0.00 0.00 −0.00 1.00 −0.00(2.57) (−1.79) (1.79) (−1.69) (13282.85) (−0.87)

RMW 0.27 0.33 −0.04 −0.28 −0.00 0.04(2.58) (3.31) (−1.32) (−3.20) (−0.03) (0.81)

CMA 0.34 0.19 −0.09 0.03 0.46 0.04(3.63) (2.83) (−4.42) (0.86) (13.52) (1.51)



Factors WarFactor spanning tests, new Fama-French factors, 1/1967�12/2014

αq βMKT βME βI/A βROE

SMB 0.05 −0.00 0.94 −0.09 −0.10(1.48) (−0.17) (62.40) (−4.91) (−5.94)

HML 0.03 −0.05 0.00 1.03 −0.17(0.28) (−1.33) (0.03) (11.72) (−2.17)

RMW 0.04 −0.03 −0.12 −0.03 0.53(0.42) (−0.99) (−1.78) (−0.35) (8.59)

CMA 0.01 −0.05 0.04 0.94 −0.11(0.32) (−3.63) (1.68) (35.26) (−3.95)



Factors WarAcross 36 signi�cant anomalies with NYSE breakpoints and value-weighted returns,Hou, Xue, and Zhang (2015b, A comparison of new factor models), 1/1967�12/2013

The average magnitude of high-minus-low alphas:.20% in q, .36% in FF5, .33% in Carhart

The number of signi�cant high-minus-low alphas:7 in q, 19 in FF5, 21 in Carhart

The number of rejections by the GRS test:25 in q, FF5, and Carhart

The q-factor model outperforms the �ve-factor model the most in themomentum and pro�tability categories



Factors WarSigni�cant momentum anomalies with NYSE-VW, alphas

SUE-1 Abr-1 Abr-6 RE-1 RE-6 R6-6 R11-1 I-Mom |ave|m 0.41 0.73 0.30 0.78 0.52 0.83 1.20 0.58 0.67αC 0.35 0.62 0.18 0.49 0.31 0.07 0.18 −0.11 0.29

αq 0.15 0.64 0.26 0.06 −0.02 0.22 0.26 0.03 0.21

a 0.44 0.85 0.44 0.86 0.66 0.97 1.25 0.61 0.76tm 3.65 5.58 3.10 3.05 2.35 3.44 4.00 2.91tC 2.95 4.40 2.04 2.38 1.83 0.70 1.41 −0.72 4

tq 1.12 4.21 2.25 0.22 −0.07 0.68 0.65 0.11 2

ta 3.74 5.87 4.23 3.23 2.86 3.38 3.45 2.45 8

|αC | 0.10 0.12 0.08 0.10 0.09 0.09 0.13 0.05 0.10

|αq| 0.06 0.13 0.07 0.11 0.12 0.09 0.15 0.12 0.11

|a| 0.11 0.16 0.08 0.20 0.17 0.17 0.23 0.21 0.17pC 0.00 0.00 0.00 0.05 0.06 0.00 0.00 0.41 5pq 0.39 0.00 0.01 0.16 0.02 0.00 0.00 0.03 6pa 0.02 0.00 0.00 0.01 0.01 0.00 0.00 0.00 8



Factors WarSigni�cant momentum anomalies with NYSE-VW, betas

SUE-1 Abr-1 Abr-6 RE-1 RE-6 R6-6 R11-1 I-Mom

βME 0.10 0.07 0.08 −0.19 −0.18 0.22 0.33 0.25βI/A 0.03 −0.14 −0.17 0.09 −0.07 0.01 0.12 0.09

βROE 0.46 0.28 0.18 1.31 1.10 1.01 1.46 0.82tβME 1.88 0.70 1.81 −2.11 −1.98 1.25 1.53 1.55tβI/A 0.32 −1.31 −2.27 0.52 −0.45 0.06 0.39 0.39

tβROE 5.76 3.18 2.86 9.82 9.36 5.40 5.80 4.95s −0.03 −0.05 0.01 −0.42 −0.40 −0.08 −0.05 0.01h −0.17 −0.20 −0.12 −0.16 −0.27 −0.54 −0.71 −0.37r 0.14 −0.11 −0.12 0.55 0.41 0.11 0.35 0.12c 0.18 0.13 −0.07 −0.02 0.02 0.39 0.67 0.39ts −0.45 −0.63 0.16 −3.79 −4.23 −0.56 −0.27 0.06th −1.67 −1.80 −1.74 −0.94 −1.76 −2.44 −2.47 −1.79tr 1.75 −1.15 −1.73 3.28 2.80 0.44 1.13 0.51tc 1.25 0.84 −0.60 −0.05 0.07 1.25 1.62 1.24



Factors WarSigni�cant value anomalies with NYSE-VW, alphas

B/M Rev E/P CF/P NO/P Dur |ave|m 0.64 −0.46 0.55 0.46 0.65 −0.46 0.54αC −0.05 −0.07 −0.02 −0.10 0.51 0.01 0.13

αq 0.16 −0.16 0.11 0.15 0.36 −0.18 0.19

a −0.02 0.08 0.05 0.01 0.22 −0.05 0.07tm 2.88 −2.02 2.67 2.30 3.27 −2.39tC −0.42 −0.39 −0.13 −0.76 3.45 0.09 1

tq 0.96 −0.93 0.53 0.77 2.45 −0.92 1

ta −0.17 0.45 0.37 0.09 1.57 −0.34 0

|αC | 0.07 0.10 0.08 0.07 0.15 0.06 0.09

|αq| 0.09 0.08 0.12 0.15 0.12 0.08 0.11

|a| 0.06 0.05 0.09 0.13 0.11 0.05 0.08pC 0.08 0.26 0.15 0.11 0.00 0.43 1pq 0.13 0.25 0.04 0.00 0.00 0.36 3pa 0.44 0.42 0.13 0.03 0.00 0.72 2



Factors WarSigni�cant value anomalies with NYSE-VW, betas

B/M Rev E/P CF/P NO/P Dur

βME 0.48 −0.64 0.29 0.20 −0.32 −0.26βI/A 1.40 −1.17 1.01 1.00 1.03 −0.86βROE −0.53 0.72 −0.11 −0.26 0.02 0.27tβME 5.86 −7.83 2.34 1.91 −4.31 −1.93tβI/A 13.01 −10.49 6.27 7.31 10.33 −6.36tβROE −6.24 7.47 −0.78 −2.03 0.21 2.22s 0.52 −0.67 0.32 0.25 −0.25 −0.32h 1.16 −0.47 1.38 1.29 0.46 −1.18r −0.27 0.39 0.16 0.00 0.51 0.02c 0.33 −0.77 −0.38 −0.24 0.54 0.29ts 10.93 −6.65 5.96 4.92 −3.96 −5.04th 15.84 −3.70 14.16 14.05 5.57 −11.93tr −3.80 4.44 2.14 0.01 6.83 0.25tc 3.06 −4.57 −3.00 −1.93 4.33 2.28



Factors WarSigni�cant investment anomalies with NYSE-VW, alphas

ACI I/A NOA 4PI/A IG NSI CEI IvG IvC OA POA PTA |ave|m −0.31 −0.46 −0.39 −0.50 −0.42 −0.72 −0.57 −0.38 −0.46 −0.28 −0.43 −0.42 0.45αC −0.19 −0.17 −0.43 −0.34 −0.21 −0.61 −0.45 −0.19 −0.31 −0.33 −0.24 −0.31 0.39

αq −0.17 0.09 −0.39 −0.23 0.02 −0.31 −0.25 −0.01 −0.28 −0.54 −0.09 −0.15 0.21

a −0.30 0.04 −0.44 −0.31 −0.08 −0.32 −0.25 −0.12 −0.38 −0.52 −0.12 −0.11 0.25

tm −2.12 −2.86 −2.82 −3.67 −3.33 −4.56 −3.18 −2.72 −3.30 −2.24 −3.08 −3.03tC −1.23 −1.20 −3.08 −2.55 −1.84 −4.35 −3.71 −1.37 −2.23 −2.46 −2.01 −2.29 8

tq −0.96 0.72 −2.10 −1.81 0.18 −2.22 −1.85 −0.09 −1.95 −3.81 −0.68 −1.03 3

ta −1.91 0.34 −2.62 −2.61 −0.75 −2.43 −2.33 −0.99 −2.92 −4.10 −1.03 −0.87 6

|αC | 0.11 0.10 0.14 0.11 0.10 0.15 0.14 0.09 0.10 0.11 0.12 0.11 0.12

|αq | 0.13 0.10 0.10 0.13 0.09 0.11 0.12 0.10 0.07 0.14 0.12 0.08 0.11

|a| 0.15 0.11 0.10 0.11 0.06 0.10 0.10 0.10 0.08 0.12 0.12 0.08 0.10pC 0.00 0.01 0.00 0.01 0.00 0.00 0.00 0.07 0.03 0.00 0.00 0.00 11pq 0.00 0.00 0.00 0.00 0.02 0.01 0.01 0.08 0.45 0.00 0.00 0.01 10pa 0.00 0.00 0.02 0.01 0.28 0.01 0.01 0.04 0.20 0.00 0.00 0.02 10



Factors WarSigni�cant investment anomalies with NYSE-VW, betas

ACI I/A NOA 4PI/A IG NSI CEI IvG IvC OA POA PTAβME −0.29 −0.14 0.10 −0.08 −0.15 0.17 0.25 0.08 −0.01 0.29 0.14 0.15

βI/A 0.13 −1.37 −0.07 −0.79 −0.76 −0.72 −1.04 −0.94 −0.68 −0.05 −0.94 −0.86βROE −0.19 0.17 0.01 0.16 −0.07 −0.30 −0.11 0.05 0.18 0.27 0.06 0.05tβME

−4.99 −2.35 1.01 −1.63 −2.60 2.41 3.92 1.80 −0.13 4.85 3.32 2.44

tβI/A

1.01 −16.71 −0.48 −7.70 −10.41 −6.96 −14.03 −12.40 −6.09 −0.51 −10.91 −8.76tβROE

−2.10 2.58 0.08 2.02 −1.18 −3.85 −1.44 0.60 1.98 4.16 1.21 0.62

s −0.26 −0.10 0.16 −0.03 −0.13 0.12 0.24 0.11 0.07 0.31 0.18 0.11h 0.16 −0.18 0.46 0.01 −0.08 −0.12 −0.41 −0.08 0.03 0.01 −0.16 −0.23r −0.03 0.03 0.08 0.25 −0.14 −0.66 −0.41 0.07 0.35 0.40 −0.04 −0.23c −0.02 −1.14 −0.52 −0.75 −0.60 −0.57 −0.60 −0.74 −0.64 0.01 −0.75 −0.57ts −4.15 −1.52 2.25 −0.58 −2.53 2.58 4.82 2.12 1.29 5.90 4.41 1.76th 1.58 −2.56 5.11 0.16 −1.23 −1.89 −6.50 −0.91 0.39 0.14 −2.94 −2.36tr −0.33 0.34 0.64 3.62 −1.90 −9.56 −5.71 0.72 3.77 6.18 −0.63 −2.76tc −0.10 −10.88 −3.58 −6.88 −5.45 −5.36 −6.39 −7.32 −4.66 0.10 −8.47 −4.67



Factors WarSigni�cant pro�tability anomalies with NYSE-VW, alphas

ROE ROA GP/A RS NEI |ave|m 0.68 0.58 0.40 0.31 0.38 0.47αC 0.79 0.64 0.51 0.49 0.42 0.57

αq −0.03 0.06 0.20 0.21 0.18 0.14

a 0.51 0.50 0.21 0.53 0.46 0.44tm 2.95 2.54 2.75 2.15 3.34tC 4.15 3.46 3.51 3.41 3.92 5

tq −0.24 0.49 1.39 1.41 1.72 0

ta 3.57 3.43 1.58 3.73 4.57 4

|αC | 0.15 0.13 0.15 0.12 0.13 0.14

|αq| 0.10 0.07 0.12 0.08 0.09 0.09

|a| 0.11 0.15 0.10 0.15 0.15 0.13pC 0.00 0.05 0.00 0.00 0.00 4pq 0.01 0.79 0.19 0.04 0.03 3pa 0.01 0.06 0.08 0.00 0.00 3



Factors WarSigni�cant pro�tability anomalies with NYSE-VW, betas

ROE ROA GP/A RS NEI

βME −0.39 −0.38 0.04 −0.13 −0.09βI/A 0.08 −0.09 −0.31 −0.40 −0.32βROE 1.50 1.32 0.54 0.61 0.65tβME −6.44 −6.50 0.76 −2.41 −2.32tβI/A 0.88 −1.12 −3.21 −4.56 −4.36tβROE 21.14 17.12 7.58 7.99 11.41

s −0.48 −0.48 0.11 −0.25 −0.17h −0.27 −0.25 −0.45 −0.47 −0.35r 1.43 1.25 0.89 0.28 0.45c 0.20 0.03 0.19 −0.02 −0.08ts −6.22 −6.13 2.25 −4.12 −3.67th −2.57 −2.95 −4.46 −5.53 −5.45tr 12.18 10.52 9.56 3.33 6.49tc 1.27 0.18 1.46 −0.17 −0.77



Factors WarSigni�cant intangibles-trading frictions anomalies with NYSE-VW, alphas

OC/A Ad/M RD/M OL Svol |ave|m 0.58 0.78 0.64 0.46 −0.55 0.60αC 0.41 0.31 0.31 0.39 −0.59 0.40

αq 0.13 0.11 0.60 0.02 −0.34 0.24

a 0.33 −0.06 0.38 0.09 −0.34 0.24tm 4.59 2.99 2.40 2.65 −2.46tC 3.34 1.38 1.43 2.28 −2.51 3

tq 1.03 0.40 2.46 0.14 −1.37 1

ta 2.61 −0.32 1.57 0.56 −1.36 1

|αC | 0.12 0.19 0.21 0.12 0.16 0.16

|αq| 0.11 0.11 0.27 0.11 0.11 0.14

|a| 0.11 0.12 0.21 0.09 0.11 0.13pC 0.00 0.08 0.00 0.02 0.03 4pq 0.01 0.09 0.00 0.03 0.13 3pa 0.00 0.28 0.00 0.06 0.25 2



Factors WarSigni�cant intangibles-trading frictions anomalies with NYSE-VW, betas

OC/A Ad/M RD/M OL Svol

βME 0.24 0.51 0.66 0.30 0.31

βI/A 0.29 1.40 0.20 0.11 −0.21βROE 0.51 −0.26 −0.58 0.54 −0.42tβME 5.66 2.92 6.84 3.10 2.32

tβI/A 2.98 6.01 1.13 0.95 −1.32tβROE 6.92 −1.34 −4.08 4.88 −3.53s 0.21 0.65 0.60 0.37 0.25h −0.13 1.03 0.05 0.04 −0.06r 0.55 0.47 −0.52 0.89 −0.56c 0.40 0.14 0.41 0.06 −0.12ts 4.49 6.93 6.96 5.75 2.35th −1.70 7.17 0.33 0.44 −0.39tr 5.13 4.41 −2.86 10.50 −4.02tc 2.92 0.68 1.99 0.45 −0.57



Factors WarSummary, NYSE-VW

Except for R&D-to-market, the q-factor model performs well relative to theFama-French �ve-factor model:

The q-factor model outperforms the �ve-factor model in themomentum and pro�tability categories by a big margin

The q-factor model outperforms in the investment category

The models largely comparable in the value-versus-growth category,but the �ve-factor model has a slight edge



Factors WarAcross 50 signi�cant anomalies with all-but-micro breakpoints and equal-weighted returns

The average magnitude of high-minus-low alphas:.24% in q, .41% in FF5, .40% in Carhart

The number of signi�cant high-minus-low alphas:16 in q, 34 in FF5, 37 in Carhart

The number of rejections by the GRS test:37 for q, 35 for FF5, and 39 for Carhart

The q-factor model continues to outperform the �ve-factor model the mostin the momentum and pro�tability categories



Factors WarSigni�cant momentum anomalies with ABM-EW, alphas

SUE-1 SUE-6 Abr-1 Abr-6 RE-1 RE-6 R6-1 R6-6 R11-1 I-Mom |ave|m 0.72 0.30 0.97 0.46 0.79 0.44 1.08 0.92 1.24 0.68 0.76αC 0.58 0.21 0.87 0.31 0.47 0.20 0.16 0.03 0.23 −0.01 0.31

αq 0.31 −0.04 0.85 0.31 0.26 −0.06 0.34 0.04 0.37 0.13 0.27

a 0.70 0.31 1.02 0.52 0.86 0.48 1.12 0.90 1.35 0.62 0.79tm 6.39 3.36 8.74 5.61 4.08 2.65 3.86 3.82 4.27 3.47tC 5.40 2.60 8.54 3.47 2.78 1.41 0.84 0.19 1.77 −0.08 5

tq 3.06 −0.53 5.55 2.14 1.53 −0.37 0.84 0.11 0.88 0.48 3

ta 6.50 3.41 7.94 4.68 4.62 2.87 3.10 2.68 3.62 2.44 10

|αC | 0.16 0.13 0.19 0.14 0.15 0.12 0.13 0.10 0.09 0.04 0.13

|αq| 0.11 0.10 0.19 0.17 0.13 0.15 0.16 0.14 0.11 0.10 0.14

|a| 0.19 0.08 0.19 0.11 0.23 0.14 0.18 0.16 0.27 0.21 0.18pC 0.00 0.00 0.00 0.00 0.01 0.16 0.00 0.00 0.03 0.30 8pq 0.00 0.01 0.00 0.00 0.01 0.01 0.00 0.00 0.00 0.20 9pa 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 10



Factors WarSigni�cant momentum anomalies with ABM-EW, betas

SUE-1 SUE-6 Abr-1 Abr-6 RE-1 RE-6 R6-1 R6-6 R11-1 I-Mom

βME 0.01 −0.03 0.10 0.13 0.04 −0.01 0.53 0.47 0.52 0.33βI/A 0.13 0.07 0.00 −0.05 −0.15 −0.09 0.02 0.12 −0.06 0.16

βROE 0.62 0.56 0.22 0.25 0.94 0.87 1.16 1.21 1.36 0.70tβME 0.26 −0.91 1.35 2.07 0.72 −0.21 1.85 2.30 2.13 1.88tβI/A 2.17 1.35 0.01 −0.48 −1.35 −1.03 0.04 0.41 −0.17 0.69

tβROE 8.42 12.45 2.80 3.12 8.78 11.16 4.07 5.10 5.14 4.02s −0.13 −0.15 0.01 0.03 −0.12 −0.16 0.20 0.11 0.14 0.11h −0.21 −0.19 −0.19 −0.14 −0.08 −0.13 −0.67 −0.60 −0.87 −0.35r 0.25 0.23 −0.07 −0.02 0.30 0.32 0.22 0.26 0.19 0.10c 0.26 0.15 0.23 0.04 −0.23 −0.13 0.65 0.57 0.65 0.49ts −2.58 −3.43 0.19 0.66 −1.38 −2.34 0.91 0.74 0.76 0.74th −2.32 −2.66 −2.20 −1.93 −0.56 −1.10 −1.95 −2.23 −2.83 −1.60tr 4.00 4.25 −0.72 −0.19 2.89 3.58 0.47 0.76 0.50 0.36tc 2.46 1.62 1.88 0.26 −1.07 −0.70 1.30 1.38 1.47 1.55



Factors WarSigni�cant value anomalies with ABM-EW, alphas

B/M E/P CF/P NO/P Dur A/ME Rev |ave|m 0.77 0.65 0.73 0.64 −0.64 0.67 −0.64 0.68αC 0.17 0.29 0.25 0.42 −0.17 0.02 −0.22 0.22

αq 0.06 0.25 0.20 0.20 −0.01 −0.17 −0.30 0.17

a −0.01 0.18 0.13 0.24 −0.06 −0.26 −0.20 0.15tm 3.29 3.20 3.46 3.46 −2.93 2.56 −3.50tC 1.35 2.48 1.87 3.41 −1.15 0.10 −1.31 2

tq 0.27 1.33 0.96 1.46 −0.04 −0.69 −1.89 0

ta −0.12 1.46 0.95 1.94 −0.49 −1.85 −1.30 0

|αC | 0.13 0.15 0.15 0.19 0.14 0.13 0.13 0.15

|αq| 0.14 0.06 0.06 0.11 0.11 0.14 0.08 0.10

|a| 0.04 0.06 0.05 0.06 0.04 0.08 0.04 0.05pC 0.09 0.11 0.10 0.00 0.08 0.18 0.047 2pq 0.21 0.42 0.23 0.048 0.06 0.10 0.12 1pa 0.75 0.49 0.63 0.03 0.48 0.25 0.39 1



Factors WarSigni�cant value anomalies with ABM-EW, betas

B/M E/P CF/P NO/P Dur A/ME Rev

βME 0.11 0.00 0.01 −0.25 −0.07 0.07 −0.36βI/A 1.82 1.15 1.33 1.08 −1.18 1.97 −1.13βROE −0.10 −0.01 0.04 0.27 −0.34 −0.08 0.44tβME 0.99 0.00 0.04 −5.48 −0.61 0.48 −5.74tβI/A 8.32 7.92 7.82 10.01 −7.43 8.40 −9.19tβROE −0.59 −0.11 0.26 2.66 −1.88 −0.40 4.13s 0.15 0.03 0.06 −0.25 −0.09 0.13 −0.36h 1.29 1.12 1.26 0.44 −1.11 1.49 −0.45r 0.39 0.38 0.53 0.57 −0.71 0.52 0.21c 0.51 −0.01 0.04 0.54 −0.05 0.42 −0.65ts 3.66 0.72 1.21 −4.66 −1.55 2.78 −4.44th 23.11 16.35 18.64 6.55 −12.35 20.90 −4.02tr 3.93 4.75 6.28 6.51 −6.44 5.35 1.64tc 4.32 −0.06 0.40 4.98 −0.30 3.45 −3.74



Factors WarSigni�cant investment anomalies with ABM-EW, alphas

ACI I/A NOA 4PI/A IG NSI CEI NXF

m −0.32 −0.72 −0.56 −0.64 −0.41 −0.82 −0.65 −0.67αC −0.19 −0.47 −0.63 −0.50 −0.22 −0.62 −0.54 −0.54αq −0.13 −0.30 −0.71 −0.34 −0.03 −0.26 −0.28 −0.23a −0.26 −0.34 −0.79 −0.45 −0.12 −0.34 −0.43 −0.33tm −3.74 −4.84 −3.40 −5.07 −4.30 −5.49 −3.90 −3.80tC −2.10 −3.84 −3.94 −4.28 −2.43 −4.73 −4.69 −4.34tq −1.32 −2.52 −3.26 −2.72 −0.34 −2.02 −2.12 −1.70ta −2.85 −3.21 −4.89 −4.27 −1.45 −3.04 −3.99 −2.55|αC | 0.17 0.19 0.21 0.21 0.15 0.20 0.20 0.22

|αq| 0.10 0.14 0.17 0.17 0.12 0.14 0.12 0.13

|a| 0.07 0.08 0.15 0.12 0.04 0.10 0.07 0.09pC 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00pq 0.15 0.01 0.00 0.00 0.41 0.00 0.00 0.00pa 0.09 0.01 0.00 0.00 0.89 0.00 0.00 0.00



Factors WarSigni�cant investment anomalies with ABM-EW, alphas

IvG IvC OA POA TA PTA |ave|m −0.50 −0.50 −0.30 −0.43 −0.46 −0.51 0.53αC −0.34 −0.43 −0.33 −0.24 −0.38 −0.43 0.42

αq −0.29 −0.42 −0.53 −0.14 −0.45 −0.35 0.32

a −0.38 −0.50 −0.51 −0.23 −0.38 −0.33 0.39tm −4.32 −4.30 −2.50 −3.88 −3.99 −5.06tC −3.24 −3.80 −2.35 −2.63 −2.89 −4.82 14

tq −2.35 −3.37 −4.05 −1.51 −4.13 −3.88 10

ta −3.55 −5.12 −4.79 −2.74 −3.68 −3.62 13

|αC | 0.18 0.17 0.18 0.18 0.17 0.19 0.19

|αq| 0.10 0.11 0.16 0.16 0.13 0.14 0.14

|a| 0.08 0.10 0.12 0.13 0.07 0.09 0.09pC 0.00 0.00 0.00 0.00 0.00 0.00 14pq 0.00 0.00 0.00 0.00 0.00 0.00 12pa 0.00 0.00 0.00 0.00 0.04 0.00 12



Factors WarSigni�cant investment anomalies with ABM-EW, betas

ACI I/A NOA 4PI/A IG NSI CEI NXF

βME −0.14 0.02 −0.03 0.01 0.01 0.07 0.29 0.23

βI/A −0.15 −1.25 0.00 −0.82 −0.77 −0.90 −0.94 −0.89βROE −0.15 0.15 0.18 0.02 −0.08 −0.43 −0.31 −0.38tβME −3.78 0.46 −0.19 0.11 0.29 1.19 6.47 5.09

tβI/A −2.09 −15.01 −0.01 −8.50 −12.00 −8.86 −10.62 −8.09tβROE −2.19 2.09 1.47 0.25 −1.30 −5.10 −3.64 −4.26s −0.10 0.08 0.11 0.08 0.05 0.04 0.33 0.27h 0.11 −0.20 0.61 0.06 −0.09 −0.21 −0.41 −0.14r −0.03 0.02 0.51 0.06 −0.13 −0.80 −0.45 −0.59c −0.24 −0.97 −0.65 −0.82 −0.63 −0.65 −0.40 −0.68ts −2.67 1.75 1.68 1.83 1.29 0.92 5.62 5.17th 1.76 −2.97 5.91 0.81 −1.96 −3.98 −5.75 −2.02tr −0.41 0.19 4.32 0.89 −2.01 −9.90 −5.67 −5.96tc −2.60 −8.90 −5.41 −8.64 −6.75 −7.02 −4.28 −6.19



Factors WarSigni�cant investment anomalies with ABM-EW, betas

IvG IvC OA POA TA PTA

βME 0.08 0.10 0.22 0.18 0.05 0.09

βI/A −0.68 −0.55 −0.19 −0.77 −0.65 −0.58βROE 0.05 0.17 0.43 −0.01 0.42 0.04tβME 1.94 2.38 4.07 5.60 1.16 2.14

tβI/A −7.46 −6.07 −1.69 −12.18 −7.89 −9.28tβROE 0.79 2.85 4.84 −0.22 4.90 0.72s 0.13 0.17 0.25 0.23 0.05 0.06h −0.03 0.00 0.01 −0.21 −0.12 −0.18r 0.10 0.32 0.62 −0.02 0.38 −0.14c −0.58 −0.47 −0.13 −0.47 −0.48 −0.35ts 3.03 5.27 6.22 6.71 0.93 1.50th −0.49 0.07 0.15 −4.71 −1.32 −2.99tr 1.49 5.21 8.35 −0.33 4.71 −2.46tc −6.31 −5.68 −1.08 −6.42 −3.26 −3.53



Factors WarSigni�cant pro�tability anomalies with ABM-EW, alphas

ROE ROA GP/A RS NEI CTO F TES O |ave|m 1.00 0.90 0.65 0.57 0.47 0.36 0.58 0.32 −0.28 0.57αC 0.91 0.82 0.53 0.67 0.44 0.23 0.49 0.27 −0.42 0.53

αq 0.10 0.12 −0.06 0.26 0.05 −0.15 0.25 0.03 −0.23 0.14

a 0.56 0.51 0.00 0.59 0.36 −0.18 0.48 0.32 −0.31 0.37tm 4.60 4.03 3.67 4.49 4.28 1.98 2.57 2.52 −1.98tC 4.42 3.87 3.15 5.61 4.25 1.29 2.72 2.25 −3.23 8

tq 0.71 0.89 −0.41 2.39 0.70 −0.79 1.28 0.28 −1.51 1

ta 4.14 3.52 0.01 5.07 4.02 −1.32 2.67 2.65 −2.21 7

|αC | 0.19 0.18 0.16 0.16 0.21 0.15 0.18 0.14 0.17 0.17

|αq| 0.12 0.14 0.14 0.09 0.11 0.12 0.12 0.10 0.13 0.12

|a| 0.12 0.14 0.08 0.12 0.19 0.09 0.13 0.09 0.08 0.12pC 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.09 0.00 8pq 0.00 0.00 0.01 0.01 0.00 0.01 0.08 0.54 0.02 7pa 0.00 0.00 0.03 0.00 0.00 0.02 0.01 0.19 0.08 7



Factors WarSigni�cant pro�tabilities anomalies with ABM-EW, betas

ROE ROA GP/A RS NEI CTO F TES O

βME −0.12 −0.12 0.25 −0.11 −0.03 0.44 −0.23 0.10 0.16βI/A 0.24 0.12 0.23 −0.17 −0.08 −0.18 0.38 −0.32 0.29

βROE 1.50 1.40 0.84 0.76 0.80 0.64 0.68 0.51 −0.54tβME −1.11 −1.28 2.98 −2.61 −1.13 4.19 −3.03 2.01 2.89tβI/A 2.24 1.08 1.75 −2.69 −2.20 −1.47 2.46 −2.93 2.37

tβROE 20.45 18.27 8.00 13.52 26.12 6.84 5.60 6.97 −5.74s −0.15 −0.17 0.35 −0.20 −0.12 0.59 −0.28 0.02 0.17h −0.03 −0.07 −0.19 −0.35 −0.25 −0.08 0.27 −0.27 0.27

r 1.59 1.48 1.44 0.56 0.65 1.16 0.68 0.22 −0.60c 0.20 0.14 0.45 0.11 0.10 0.01 0.00 −0.13 −0.07ts −1.98 −2.25 7.73 −4.24 −2.72 13.27 −2.93 0.34 3.45th −0.22 −0.69 −2.80 −4.90 −4.00 −1.31 2.22 −3.67 3.17

tr 16.11 14.48 18.02 6.98 10.97 16.02 5.20 2.32 −4.95tc 1.32 0.90 3.84 1.09 1.01 0.06 0.01 −1.01 −0.60



Factors WarSigni�cant intangibles-trading frictions anomalies with ABM-EW, alphas

OC/A Ad/M RD/M H/N OL Svol MDR S-Rev Disp Dvol |ave|m 0.35 0.64 0.94 −0.50 0.41 −0.45 −0.66 −0.57 −0.44 −0.37 0.53αC 0.38 0.20 0.65 −0.28 0.38 −0.47 −0.73 −0.67 −0.63 −0.15 0.45

αq 0.39 −0.25 0.78 −0.05 −0.04 −0.15 −0.16 −0.62 −0.04 0.03 0.25

a 0.45 −0.32 0.69 −0.11 −0.01 −0.18 −0.28 −0.43 −0.33 0.08 0.29tm 3.72 2.15 3.69 −3.62 2.18 −2.18 −2.14 −2.61 −1.98 −2.64tC 3.67 0.84 2.70 −2.68 2.14 −2.23 −3.62 −2.94 −3.47 −1.32 8

tq 3.12 −0.84 2.66 −0.44 −0.22 −0.76 −0.76 −1.74 −0.25 0.20 2

ta 4.38 −1.45 2.70 −1.19 −0.08 −0.92 −2.02 −1.45 −2.32 0.73 4

|αC | 0.15 0.23 0.27 0.16 0.15 0.17 0.19 0.20 0.16 0.08 0.18

|αq| 0.17 0.23 0.36 0.12 0.13 0.16 0.12 0.19 0.10 0.12 0.17

|a| 0.10 0.14 0.22 0.08 0.07 0.08 0.09 0.15 0.07 0.06 0.11pC 0.00 0.10 0.01 0.00 0.05 0.06 0.00 0.02 0.01 0.02 7pq 0.00 0.047 0.00 0.00 0.11 0.04 0.03 0.00 0.29 0.00 7pa 0.00 0.23 0.04 0.01 0.18 0.31 0.04 0.03 0.07 0.15 5



Factors WarSigni�cant intangibles-trading frictions anomalies with ABM-EW, betas

OC/A Ad/M RD/M H/N OL Svol MDR S-Rev Disp Dvol

βME 0.00 0.14 0.63 0.09 0.26 0.18 0.67 0.05 0.28 −0.64βI/A 0.04 1.67 0.28 −1.14 0.29 −0.16 −1.19 0.16 −0.14 −0.74βROE 0.03 0.27 −0.37 −0.02 0.47 −0.52 −0.78 0.19 −1.04 0.02

tβME 0.07 1.02 4.61 2.39 3.82 1.19 4.65 0.20 4.61 −6.85tβI/A 0.34 8.52 1.05 −15.65 2.29 −0.87 −6.23 0.58 −2.08 −8.45tβROE 0.35 1.38 −2.02 −0.32 4.51 −4.00 −4.44 0.98 −14.58 0.20

s −0.02 0.25 0.50 0.13 0.30 0.15 0.65 −0.06 0.28 −0.69h −0.07 0.98 −0.28 −0.25 −0.07 0.09 −0.64 −0.31 0.02 −0.43r −0.02 1.01 −0.39 −0.20 0.86 −0.60 −1.24 −0.13 −1.06 −0.33c 0.07 0.55 0.76 −0.82 0.41 −0.23 −0.53 0.50 −0.02 −0.32ts −0.56 3.16 5.07 2.89 5.45 1.23 10.53 −0.37 5.32 −12.70th −1.06 10.27 −2.08 −4.82 −0.88 0.62 −4.86 −1.31 0.25 −6.24tr −0.26 9.44 −1.67 −2.63 8.81 −3.78 −12.43 −0.42 −16.58 −5.39tc 0.59 3.95 2.73 −7.73 2.98 −1.23 −3.54 1.86 −0.17 −3.68



Factors WarSummary, ABM-EW

The q-factor model outperforms the Fama-French �ve-factor model:

The q-factor model outperforms the �ve-factor model in themomentum and pro�tability categories by a big margin

The two models are largely comparable in the value-versus-growth,investment, intangibles, and trading frictions categories



Factors WarRobustness to alternative factor constructions

The q-factor model outperforms the �ve-factor model with alternativefactor constructions:

2× 2× 2, 2× 3, and 2× 2 for the q-factors

2× 2× 2× 2 and 2× 2 for the new Fama-French factors



Factors WarProperties of alternative q-factors, 1/1967�12/2013

2× 2× 2 m αC βMKT βSMB βHML βUMD

rME 0.33 −0.03 0.05 1.03 0.19 0.02(2.27) (−0.79) (4.82) (74.66) (6.66) (1.24)

rI/A 0.26 0.18 −0.07 −0.02 0.28 0.03(3.90) (3.63) (−5.38) (−0.56) (10.14) (1.31)

rROE 0.40 0.36 −0.01 −0.20 −0.08 0.18(5.00) (4.99) (−0.63) (−3.53) (−1.52) (5.50)

a b s h r c

rME −0.01 0.05 1.03 0.02 0.01 0.08(−0.39) (5.32) (76.52) (0.88) (0.24) (2.19)

rI/A 0.07 −0.02 −0.04 0.01 0.00 0.59(2.20) (−2.47) (−2.94) (0.61) (0.21) (22.79)

rROE 0.29 −0.01 −0.05 −0.17 0.59 0.10(4.80) (−0.51) (−1.67) (−3.23) (12.90) (1.26)




2× 3 m αC βMKT βSMB βHML βUMD

rME 0.30 −0.05 0.04 1.03 0.19 0.01(2.07) (−1.49) (4.54) (87.41) (6.85) (0.53)

rI/A 0.29 0.13 −0.08 0.01 0.48 0.02(3.07) (1.88) (−4.57) (0.45) (15.75) (0.76)

rROE 0.57 0.52 −0.03 −0.31 −0.10 0.27(4.44) (4.40) (−0.89) (−3.51) (−1.06) (5.04)

a b s h r c

rME −0.02 0.04 1.02 0.05 −0.03 0.02(−0.79) (5.29) (100.22) (3.31) (−1.22) (0.78)

rI/A −0.04 0.00 −0.03 0.07 −0.04 0.90(−1.34) (0.52) (−1.89) (3.07) (−2.03) (31.81)

rROE 0.37 −0.02 −0.06 −0.24 0.97 0.16(4.02) (−0.63) (−1.22) (−3.09) (12.90) (1.41)





rME 0.29 −0.07 0.06 1.05 0.19 0.00(1.96) (−1.91) (6.34) (88.91) (6.57) (0.15)

rI/A 0.19 0.10 −0.08 0.01 0.32 0.01(2.68) (1.99) (−5.47) (0.32) (13.36) (0.30)

rROE 0.40 0.35 0.00 −0.21 −0.07 0.19(4.36) (4.04) (0.17) (−3.07) (−1.06) (4.61)

a b s h r c

rME −0.04 0.05 1.05 0.04 −0.03 0.03(−1.61) (8.20) (103.46) (2.83) (−1.36) (1.20)

rI/A 0.00 −0.03 −0.03 0.05 −0.08 0.59(0.05) (−3.09) (−2.95) (3.00) (−4.48) (24.00)

rROE 0.23 0.01 −0.02 −0.16 0.72 0.09(3.59) (0.55) (−0.64) (−2.97) (12.82) (1.04)



Factors WarProperties of the new Fama-French factors, alternative constructions, 1/1967�12/2013

2× 2× 2× 2 m αC βMKT βSMB βHML βUMD

SMB 0.29 0.03 −0.02 0.94 0.08 0.00(2.23) (1.26) (−3.18) (66.68) (5.26) (0.52)

HML 0.29 0.06 −0.02 −0.06 0.69 0.00(2.74) (1.66) (−1.82) (−3.73) (47.39) (0.10)

RMW 0.27 0.25 −0.02 −0.16 0.20 0.00(3.78) (3.97) (−1.24) (−3.35) (6.09) (0.00)

CMA 0.16 0.14 −0.08 −0.03 0.15 0.02(2.84) (2.88) (−5.51) (−1.55) (6.23) (0.97)




2× 2× 2× 2 αq βMKT βME βI/A βROE

SMB 0.07 −0.03 0.89 −0.11 −0.05(1.88) (−1.89) (48.08) (−3.89) (−2.38)

HML 0.09 −0.06 −0.05 0.65 −0.07(0.89) (−1.90) (−0.81) (8.97) (−1.05)

RMW 0.13 −0.03 −0.10 0.16 0.21(1.68) (−1.19) (−1.82) (2.84) (4.33)

CMA −0.00 −0.04 −0.02 0.44 −0.02(−0.01) (−3.79) (−1.09) (19.61) (−0.68)





SMB 0.28 −0.03 0.02 1.01 0.14 0.00(2.03) (−1.45) (3.20) (92.98) (8.18) (0.05)

HML 0.27 0.03 −0.02 −0.03 0.71 0.00(2.58) (1.17) (−3.37) (−3.99) (68.49) (0.22)

RMW 0.19 0.23 −0.01 −0.19 −0.02 0.01(2.56) (3.27) (−0.37) (−2.94) (−0.32) (0.44)

CMA 0.24 0.14 −0.08 0.02 0.32 0.02(3.33) (2.90) (−5.38) (0.69) (12.31) (1.08)




2× 2 αq βMKT βME βI/A βROE

SMB 0.05 0.01 0.95 −0.08 −0.10(1.40) (0.71) (59.89) (−4.16) (−5.71)

HML 0.04 −0.06 −0.03 0.76 −0.12(0.51) (−2.09) (−0.58) (11.78) (−2.12)

RMW 0.03 0.00 −0.09 −0.05 0.36(0.50) (0.07) (−1.78) (−0.89) (7.87)

CMA 0.03 −0.06 0.03 0.63 −0.09(0.93) (−5.34) (1.28) (30.82) (−3.76)



Factors WarPerformance across 36 signi�cant anomalies with NYSE-VW

The average magnitude of high-minus-low alphas:

q: .20% in 2× 3× 3, .26% in 2× 2× 2, .26% in 2× 3, .29% in2× 2

FF5: .36% in 2× 3, .40% in 2× 2× 2× 2, .37% in 2× 2

The number of signi�cant high-minus-low alphas:

q: 7 in 2× 3× 3, 10 in 2× 2× 2, 11 in 2× 3, 11 in 2× 2

FF5: 19 in 2× 3, 19 in 2× 2× 2× 2, 20 in 2× 2

The number of rejections by the GRS test:

q: 25 in 2× 3× 3, 20 in 2× 2× 2, 25 in 2× 3, 23 in 2× 2

FF5: 25 in 2× 3, 23 in 2× 2× 2× 2, 25 in 2× 2



Factors WarPerformance across 50 signi�cant anomalies with ABM-EW

The average magnitude of high-minus-low alphas:

q: .24% in 2× 3× 3, .29% in 2× 2× 2, .29% in 2× 3, .31% in2× 2

FF5: .41% in 2× 3, .46% in 2× 2× 2× 2, .43% in 2× 2

The number of signi�cant high-minus-low alphas:

q: 16 in 2× 3× 3, 20 in 2× 2× 2, 21 in 2× 3, 22 in 2× 2

FF5: 34 in 2× 3, 36 in 2× 2× 2× 2, 35 in 2× 2

The number of rejections by the GRS test:

q: 37 in 2× 3× 3, 37 in 2× 2× 2, 36 in 2× 3, 36 in 2× 2

FF5: 35 in 2× 3, 34 in 2× 2× 2× 2, 35 in 2× 2


Factors War Lagging the De�ator: The Irrelevance of Gross Pro�tability

Factors WarWhich ROE factor?

Fama and French (2006): The expected pro�tability e�ect insigni�cantlypositive in second-stage cross-sectional regressions

Fama and French (2008): Pro�tability sorts �produce the weakest averagehedge portfolio returns�

Novy-Marx (2013): The gross pro�tability e�ect, arguing that gross pro�tsare a cleaner measure of true economic pro�ts than earnings, as expensedinvestments reduce earnings, but do not increase book equity; adopted byFama and French (2015a)

Hou, Xue, and Zhang (2015a, b): Monthly sorts on ROE, quarterlyearnings-to-lagged book equity



Factors WarLagging the de�ator: The irrelevance of gross pro�tability, 1/1967�12/2014

Low 2 3 4 5 6 7 8 9 High H−L m.a.e.

Gross pro�ts-to-current assets, annual sorts

m 0.35 0.40 0.46 0.49 0.62 0.57 0.51 0.49 0.61 0.73 0.38tm 1.66 2.04 2.21 2.32 3.05 2.76 2.28 2.23 2.91 3.41 2.62αC −0.15 −0.19 −0.12 −0.08 0.06 0.03 0.14 0.18 0.24 0.35 0.49 0.15tC −1.51 −2.53 −1.42 −0.86 0.74 0.38 1.52 1.98 3.05 3.63 3.39 [0.00]α5 0.07 −0.16 −0.10 −0.14 0.01 0.01 0.03 0.07 0.16 0.27 0.19 0.10t5 0.85 −1.98 −1.13 −1.57 0.17 0.16 0.36 0.77 2.25 2.58 1.46 [0.04]αq 0.02 −0.16 −0.11 −0.10 0.04 0.08 0.16 0.19 0.15 0.20 0.18 0.12tq 0.21 −1.76 −1.21 −1.13 0.50 1.02 1.65 1.65 1.82 1.90 1.24 [0.11]

Gross pro�ts-to-lagged assets, annual sorts

m 0.46 0.45 0.49 0.54 0.63 0.61 0.50 0.49 0.60 0.61 0.16tm 2.30 2.36 2.37 2.55 3.28 2.91 2.42 2.21 2.89 2.69 1.04

Scaling by lagged assets makes more sense, as current assets areaccumulated by the end of current period



Factors WarThe importance of monthly sorts, 1/1976�12/2014

Low 2 3 4 5 6 7 8 9 High H−L m.a.e.

Gross pro�ts-to-lagged assets, monthly sorts

m 0.39 0.59 0.46 0.51 0.62 0.71 0.65 0.58 0.72 0.91 0.51tm 1.56 2.94 1.98 2.22 2.81 3.21 2.75 2.58 3.22 3.87 3.40αC −0.19 −0.03 −0.24 −0.20 −0.06 0.11 0.12 0.12 0.23 0.37 0.56 0.17tC −1.66 −0.38 −2.57 −2.14 −0.71 1.35 1.24 1.53 2.63 3.76 3.81 [0.00]α5 0.03 0.05 −0.19 −0.24 −0.17 0.00 0.01 −0.03 0.14 0.33 0.30 0.12t5 0.26 0.70 −2.00 −2.51 −1.94 0.00 0.08 −0.31 1.55 2.96 2.14 [0.02]αq 0.05 0.04 −0.16 −0.19 −0.08 0.07 0.16 0.00 0.14 0.25 0.20 0.11tq 0.40 0.47 −1.54 −2.03 −0.84 0.78 1.27 −0.04 1.47 2.12 1.41 [0.12]




No need for gross pro�ts in monthly sorts

Low 2 3 4 5 6 7 8 9 High H−L m.a.e.

ROA (earnings-to-lagged assets), monthly sorts

m 0.20 0.44 0.57 0.68 0.57 0.75 0.72 0.63 0.70 0.78 0.58tm 0.56 1.58 2.25 3.30 2.87 3.61 3.58 3.05 3.25 3.41 2.53αC −0.42 −0.15 −0.03 0.07 −0.04 0.12 0.11 0.08 0.13 0.27 0.68 0.14tC −2.70 −1.38 −0.35 0.80 −0.57 1.48 1.59 0.94 1.82 3.61 3.61 [0.00]α5 −0.14 −0.16 −0.13 0.06 −0.06 0.03 0.02 −0.11 0.04 0.24 0.39 0.10t5 −1.30 −1.33 −1.38 0.75 −0.73 0.35 0.27 −1.56 0.60 3.07 2.85 [0.10]αq 0.16 0.10 0.10 0.17 −0.04 0.06 0.03 −0.07 −0.01 0.13 −0.02 0.09tq 1.39 1.01 0.98 1.74 −0.49 0.61 0.43 −0.85 −0.10 1.60 −0.21 [0.20]




Using gross pro�ts in monthly sorts does more harm than good

Low 2 3 4 5 6 7 8 9 High H−L m.a.e.

ROE (earnings-to-lagged book equity), monthly sorts

m 0.16 0.43 0.50 0.56 0.68 0.58 0.68 0.71 0.68 0.88 0.72tm 0.45 1.57 2.02 2.69 3.36 2.68 3.34 3.37 3.17 3.93 2.79αC −0.49 −0.15 −0.04 −0.03 0.08 −0.03 0.09 0.12 0.10 0.31 0.80 0.14tC −3.09 −1.40 −0.38 −0.30 0.99 −0.32 1.30 1.60 1.46 3.42 3.72 [0.00]α5 −0.24 −0.15 0.00 −0.02 0.05 −0.03 0.02 0.04 0.10 0.15 0.39 0.08t5 −2.02 −1.29 0.01 −0.17 0.55 −0.32 0.23 0.48 1.45 1.71 2.68 [0.33]αq 0.09 0.13 0.28 0.10 0.11 0.00 0.05 −0.05 0.03 0.03 −0.05 0.09tq 0.75 1.41 2.99 1.05 1.24 0.01 0.63 −0.59 0.37 0.39 −0.40 [0.06]


Factors War Independent Comparison

Factors WarIndependent comparison by impartial observers

Barillas and Shanken (2015): A Bayesian procedure for model comparison

Shanken (2015): Model probabilities overwhelmingly in favor of theq-factor model (97% versus 3%)

Cooper and Maio (2015): The q-factor model shows the best convergencewith the intertemporal CAPM

Stambaugh and Yu (2015): Con�rm Hou, Xue, and Zhang's (2015b) factorspanning tests and empirical comparison tests


Understanding the Low Risk Anomaly

Outline




4 Notes



The Low Risk AnomalyCan the q-factor model explain the idiosyncratic volatility anomaly? 1/1967�12/2014

Low 2 3 4 5 6 7 8 9 High H−L m.a.e.

NYSE breakpoints and value-weights

m 0.50 0.68 0.65 0.66 0.61 0.60 0.69 0.65 0.63 −0.01 −0.51tm 3.17 3.91 3.49 3.20 2.73 2.54 2.66 2.26 1.90 −0.02 −1.62α 0.15 0.27 0.19 0.16 0.07 0.03 0.07 −0.02 −0.10 −0.81 −0.96 0.19tα 1.84 3.66 2.73 2.46 0.84 0.33 0.85 −0.17 −0.77 −4.41 −4.02 [0.00]α3 0.11 0.26 0.18 0.14 0.06 −0.01 0.10 −0.03 −0.09 −0.84 −0.94 0.18t3 1.52 3.83 2.74 2.05 0.75 −0.07 1.17 −0.37 −0.86 −6.24 −5.43 [0.00]αC 0.05 0.21 0.14 0.12 0.09 0.01 0.18 0.05 0.03 −0.60 −0.65 0.15tC 0.77 2.96 1.87 1.70 1.18 0.08 2.09 0.52 0.24 −3.89 −3.39 [0.00]α5 −0.07 0.12 0.03 0.02 −0.03 −0.06 0.13 0.02 0.11 −0.41 −0.34 0.10t5 −0.99 1.54 0.37 0.35 −0.39 −0.73 1.51 0.16 1.12 −3.45 −2.25 [0.00]αq −0.12 0.06 0.00 0.01 0.01 −0.02 0.23 0.13 0.23 −0.21 −0.08 0.10tq −1.43 0.75 −0.04 0.12 0.10 −0.22 2.12 1.29 1.98 −1.48 −0.46 [0.00]




Low 2 3 4 5 6 7 8 9 High H−L m.a.e.

All-but-micro breakpoints and equal-weights

m 0.59 0.73 0.81 0.87 0.86 0.83 0.83 0.75 0.57 −0.04 −0.63tm 3.70 3.85 3.88 3.95 3.55 3.21 2.94 2.36 1.61 −0.10 −1.95α 0.25 0.30 0.32 0.35 0.29 0.22 0.17 0.02 −0.22 −0.89 −1.14 0.30tα 2.96 3.79 4.01 4.21 3.49 2.37 1.67 0.18 −1.42 −4.24 −4.47 [0.00]α3 0.08 0.14 0.18 0.20 0.17 0.13 0.12 0.03 −0.20 −0.81 −0.90 0.21t3 1.20 2.20 2.79 3.54 3.03 2.31 2.10 0.41 −2.16 −5.85 −4.92 [0.00]αC 0.08 0.17 0.22 0.25 0.21 0.15 0.16 0.09 −0.09 −0.60 −0.68 0.20tC 1.12 2.50 3.37 4.05 3.33 2.54 2.70 1.21 −0.97 −3.90 −3.44 [0.00]α5 −0.06 −0.01 0.06 0.11 0.14 0.13 0.23 0.21 0.10 −0.34 −0.28 0.14t5 −0.89 −0.13 1.08 2.03 2.49 2.24 3.51 3.03 1.19 −2.95 −1.96 [0.00]αq −0.06 0.00 0.09 0.13 0.17 0.16 0.29 0.32 0.22 −0.14 −0.08 0.16tq −0.79 −0.03 1.21 1.95 2.43 2.58 3.82 3.68 2.15 −0.91 −0.38 [0.00]




Low 2 3 4 5 6 7 8 9 High H−L m.a.e.

All breakpoints and value-weights

m 0.56 0.56 0.59 0.60 0.67 0.53 0.49 0.16 −0.15 −0.72 −1.28tm 3.50 3.10 2.78 2.51 2.48 1.68 1.42 0.44 −0.35 −1.66 −3.48α 0.19 0.11 0.06 0.01 0.03 −0.18 −0.26 −0.64 −0.97 −1.50 −1.69 0.39tα 2.67 2.01 1.05 0.15 0.27 −1.34 −1.63 −3.64 −4.37 −5.56 −5.42 [0.00]α3 0.17 0.11 0.08 0.06 0.08 −0.12 −0.18 −0.58 −0.98 −1.57 −1.74 0.39t3 2.87 2.24 1.31 0.75 1.04 −1.15 −1.48 −4.34 −6.06 −7.18 −7.16 [0.00]αC 0.13 0.10 0.12 0.09 0.11 −0.02 −0.06 −0.40 −0.67 −1.26 −1.39 0.30tC 2.09 2.07 1.89 1.27 1.36 −0.21 −0.45 −3.18 −3.67 −5.09 −5.03 [0.00]α5 0.04 0.03 0.06 0.09 0.21 0.08 0.14 −0.19 −0.44 −0.96 −1.01 0.22t5 0.67 0.64 0.77 1.08 2.71 0.90 1.31 −1.66 −2.88 −4.50 −4.16 [0.00]αq 0.02 0.04 0.10 0.12 0.24 0.21 0.25 −0.03 −0.17 −0.66 −0.68 0.19tq 0.29 0.72 1.30 1.38 2.82 1.91 1.93 −0.24 −0.97 −2.67 −2.39 [0.00]




Low 2 3 4 5 6 7 8 9 High H−L m.a.e.

All breakpoints and equal-weights

m 0.65 0.79 0.93 0.97 0.98 0.93 0.93 0.71 0.64 0.38 −0.27tm 3.71 3.72 3.85 3.61 3.35 2.89 2.61 1.87 1.52 0.82 −0.75α 0.33 0.33 0.41 0.40 0.37 0.27 0.23 −0.01 −0.09 −0.35 −0.68 0.28tα 3.30 3.61 4.07 3.42 2.75 1.81 1.28 −0.03 −0.38 −1.12 −2.29 [0.00]α3 0.14 0.15 0.21 0.22 0.21 0.12 0.08 −0.17 −0.28 −0.57 −0.71 0.22t3 1.75 2.40 3.77 3.58 3.33 1.81 0.93 −1.60 −1.79 −2.49 −2.91 [0.00]αC 0.15 0.19 0.28 0.30 0.33 0.27 0.28 0.09 0.06 −0.19 −0.34 0.21tC 1.86 3.25 4.76 4.81 4.88 3.50 2.70 0.64 0.30 −0.67 −1.15 [0.00]α5 0.01 0.04 0.14 0.19 0.26 0.24 0.30 0.11 0.13 −0.07 −0.08 0.15t5 0.09 0.65 2.57 3.01 3.55 2.96 2.91 0.81 0.68 −0.28 −0.30 [0.00]αq 0.05 0.09 0.21 0.28 0.39 0.40 0.53 0.41 0.55 0.39 0.33 0.33tq 0.59 1.20 2.77 3.36 4.01 3.44 3.59 2.05 1.92 1.05 0.88 [0.00]



The Low Risk AnomalyCan the q-factor model explain the beta anomaly? 1/1967�12/2014

Low 2 3 4 5 6 7 8 9 High H−L m.a.e.


m 0.52 0.61 0.64 0.71 0.54 0.57 0.55 0.49 0.51 0.30 −0.22tm 3.41 4.09 3.85 3.98 2.80 2.65 2.43 1.84 1.80 0.81 −0.65α 0.27 0.31 0.27 0.30 0.09 0.07 0.03 −0.12 −0.15 −0.52 −0.79 0.21β 0.49 0.59 0.72 0.81 0.89 0.98 1.03 1.19 1.29 1.62 1.13 [0.01]tα 2.35 3.11 3.26 3.15 1.08 0.76 0.28 −1.09 −1.20 −3.09 −3.25α3 0.08 0.15 0.17 0.21 −0.01 0.00 −0.04 −0.19 −0.20 −0.49 −0.57 0.15t3 0.72 1.81 2.05 2.49 −0.16 −0.01 −0.42 −1.78 −1.69 −3.13 −2.51 [0.01]αC 0.04 0.12 0.13 0.16 −0.04 −0.01 0.01 −0.14 −0.10 −0.27 −0.31 0.10tC 0.34 1.38 1.46 1.88 −0.45 −0.07 0.10 −1.30 −0.89 −1.62 −1.27 [0.22]α5 −0.04 0.06 0.03 −0.02 −0.22 −0.20 −0.24 −0.39 −0.31 −0.32 −0.27 0.18t5 −0.38 0.68 0.35 −0.27 −2.85 −2.60 −2.59 −3.76 −2.60 −1.94 −1.18 [0.00]



The Low Risk AnomalyCan the q-factor model explain the beta anomaly? 1/1967�12/2014

Low 2 3 4 5 6 7 8 9 High H−L m.a.e.


αq −0.11 0.00 0.00 −0.07 −0.22 −0.19 −0.22 −0.40 −0.19 −0.10 0.01 0.15βMKT 0.58 0.67 0.79 0.92 0.98 1.06 1.10 1.24 1.28 1.43 0.85 [0.02]βME 0.08 0.01 0.00 −0.05 −0.05 −0.04 0.00 0.08 0.11 0.32 0.24βI/A 0.54 0.46 0.37 0.44 0.42 0.33 0.28 0.26 −0.02 −0.57 −1.11βROE 0.14 0.12 0.14 0.25 0.16 0.16 0.17 0.22 0.05 −0.33 −0.47tq −0.84 −0.05 −0.05 −0.71 −2.38 −2.06 −2.07 −3.38 −1.49 −0.55 0.05tMKT 17.91 24.70 31.59 35.31 43.28 34.45 40.15 37.91 32.40 28.41 11.70tME 1.41 0.32 0.09 −0.92 −1.00 −0.98 0.00 1.25 1.57 4.67 2.62tI/A 5.27 6.75 5.82 4.98 5.14 3.81 3.06 2.35 −0.20 −5.23 −6.69tROE 1.79 1.87 2.30 4.23 2.69 2.20 2.26 2.71 0.68 −2.83 −2.81



The Low Risk AnomalyCan the q-factor model explain the betting-against-beta factor? 1/1967�12/2014

Panel A: The Carhart model

αC βMKT βSMB βHML βUMD R2

rBAB 0.54 0.07 0.01 0.53 0.20 0.22(3.31) (1.33) (0.22) (5.96) (3.54)

Panel B: The Fama-French (2015a) �ve-factor model

α5 b s h r c R2

rBAB 0.47 0.09 0.13 0.29 0.48 0.37 0.24(3.05) (1.61) (2.20) (2.70) (5.36) (2.43)

Panel C: The Hou-Xue-Zhang (2015a, b) q-factor model

αq βMKT βME βI/A βROE R2

rBAB 0.31 0.07 0.14 0.69 0.38 0.20(1.78) (1.32) (2.26) (6.07) (4.02)


Notes

Outline




4 Notes


Notes

NotesThe q-factor model is built on a rich empirical literature in �nance and accounting

�[As] the q-factor model uses the Fama-French three- and �ve-factormodels as a straw man, it is only natural to acknowledge our enormousintellectual debt to Fama and French (p. 30).�

�It is evident that the intellectual designs of the q-factor model, includingits econometric tests, factor construction, formation of testing portfolios,and above all, the taste of the economic question, are all deeply in�uencedby Fama and French (p. 31).�


Notes

NotesDi�erent forms of the investment premium

Titman, Wei, and Xie (2004)

Anderson and Garcia-Feijòo (2006)

Cooper, Gulen, and Schill (2008)

Lyandres, Sun, and Zhang (2008)

Butler, Cornaggia, Grullon, and Weston (2011)

Cooper and Priestley (2011)


Notes

NotesThe cross-sectional variation of the investment premium

Li and Zhang (2010)

Lam and Wei (2011)

Lipson, Mortal, and Schill (2011)

Titman, Wei, and Xie (2013)

Watanabe, Xu, Yao, and Yu (2013)


Notes

NotesApplications in the accrual anomaly

Sloan (1996); Richardson, Sloan, Soliman, and Tuna (2005)

Fair�eld, Whisenant, and Yohn (2003)

Wu, Zhang, and Zhang (2010)


Notes

NotesThe balance sheet operating accruals deciles

Low 2 3 4 5 6 7 8 9 High H−L m.a.e.

Panel A: Average returns, 1/1967�12/2014

m 0.61 0.55 0.56 0.63 0.62 0.60 0.62 0.42 0.44 0.32 −0.29tm 2.40 2.61 3.06 3.42 3.09 3.19 3.19 2.00 1.93 1.14 −2.13

Panel B: Two-factor regressions, 1/1967�12/2014

α 0.17 0.06 0.05 0.11 0.10 0.06 0.22 −0.03 0.05 0.07 −0.10 0.09βMKT 1.09 0.99 0.93 0.93 0.93 0.91 0.92 0.96 1.05 1.15 0.07 [0.01]βI/A −0.26 −0.04 0.09 0.10 0.10 0.16 −0.13 −0.09 −0.31 −0.79 −0.53tα 1.48 0.67 0.78 1.41 1.35 0.88 3.54 −0.35 0.62 0.78 −0.66tMKT 27.84 37.17 56.71 35.95 42.16 52.43 43.58 47.76 41.83 43.73 1.46tI/A −2.45 −0.29 1.77 1.79 1.71 1.93 −2.66 −1.28 −4.32 −12.70 −4.03


Notes


Low 2 3 4 5 6 7 8 9 High H−L m.a.e.

Panel C: The q-factor model, 1/1967�12/2014

αq 0.33 0.16 0.13 0.09 0.11 −0.04 0.14 −0.12 −0.03 −0.09 −0.42 0.12βMKT 1.07 1.00 0.94 0.94 0.95 0.94 0.94 0.97 1.05 1.10 0.03 [0.00]βME −0.06 −0.12 −0.12 −0.03 −0.08 −0.03 −0.04 0.05 0.05 0.37 0.43βI/A −0.28 −0.05 0.08 0.10 0.10 0.17 −0.13 −0.07 −0.30 −0.75 −0.47βROE −0.22 −0.10 −0.07 0.04 0.02 0.17 0.13 0.13 0.09 0.10 0.31tq 2.65 1.39 1.81 1.01 1.36 −0.48 2.10 −1.57 −0.35 −1.00 −2.96tMKT 29.98 38.57 52.20 36.98 45.00 57.17 43.81 45.38 37.99 46.56 0.86tME −1.21 −2.39 −3.57 −0.94 −2.14 −0.86 −1.21 1.72 1.42 11.77 7.60tI/A −3.11 −0.45 1.62 1.76 1.78 2.41 −3.01 −1.25 −4.48 −15.72 −5.19tROE −2.55 −1.27 −2.09 0.80 0.46 3.28 2.87 3.10 1.78 2.03 4.08


Notes


Low 2 3 4 5 6 7 8 9 High H−L m.a.e.

Panel D: The q-factor model, the cash-based ROE factor, 1/1972�12/2014

α̃q 0.14 0.12 0.12 0.10 0.15 −0.02 0.13 −0.07 0.02 −0.03 −0.17 0.09

β̃MKT 1.11 1.02 0.94 0.94 0.92 0.91 0.93 0.95 1.04 1.11 0.00 [0.07]

β̃ME 0.03 −0.09 −0.09 −0.03 −0.10 −0.06 −0.07 0.00 0.01 0.29 0.26

β̃I/A −0.16 −0.05 0.03 0.10 0.06 0.12 −0.10 −0.03 −0.26 −0.64 −0.48β̃ROE −0.09 −0.03 0.00 0.05 −0.03 0.21 0.11 0.02 −0.04 −0.14 −0.05t̃q 1.07 1.16 1.44 1.15 1.73 −0.22 1.87 −0.74 0.18 −0.37 −1.09t̃MKT 26.54 39.37 51.24 34.02 46.76 49.24 38.93 42.14 33.86 41.07 0.07t̃ME 0.69 −1.95 −3.22 −1.20 −2.57 −1.89 −2.09 −0.13 0.40 8.13 4.50t̃I/A −1.51 −0.44 0.66 1.97 1.14 2.12 −2.45 −0.59 −4.20 −11.79 −4.26t̃ROE −0.92 −0.24 0.02 0.75 −0.43 2.17 1.93 0.34 −0.46 −2.52 −0.48


Notes

NotesPost-earnings-announcement drift

Ball and Brown (1968)

Bernard and Thomas (1989, 1990)


Notes

NotesFundamental analysis

Graham and Dodd (1934)

Ou and Penman (1989)

Lev and Thiagarajan (1993)

Abarbanell and Bushee (1997, 1998)

Piotroski (2000)

Penman and Zhu (2015)


lectures on the investment capm - 1. the q-factor...

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