factor momentum and the momentum factor - wild apricot

Factor Momentum and the Momentum Factor

Sina EhsaniNorthern Illinois University

Juhani LinnainmaaDartmouth College, NBER, and Research Affiliates

Q Group Fall 2020 Seminar

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Idea

Asset returns have a factor structure

Size, value, investment, profitability,. . . and momentum?Current asset pricing models: Hou, Xue, and Zhang (2015), Fama andFrench (2015), Stambaugh and Yuan (2016), . . .

What do we know about momentum?1 Momentum is truly everywhere: cross sections of stocks, corporate

bonds, corporate loans, mutual funds, hedge funds, currencies,commodities, credit default swaps,. . .

2 Other factors struggle to explain momentum

Fama and French three- and five-factor models do nothing to it

. . . unless we create a factor that directly or implicitly targetsmomentum

3 Momentum correlates negatively with, e.g., value

Potential for large diversification benefits

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This paper

This paper is about the connection between the “momentum factor” andall other factors

Rather than being unrelated to the other factors, momentum in factrelates to all of them.

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Key results

1 Factor returns are positively autocorrelated

2 Factor Momentum: momentum strategies that trade factors are veryprofitable

3 Factor momentum prices portfolios sorted by r12,2 better than themomentum factor

4 Factor momentum spans other forms of momentum as well:

Industry-adjusted momentum, industry momentum, intermediatemomentum, and Sharpe momentum

5 Factor momentum concentrates in more systematic (high-eigenvalue)factors

Consistent with factors reflecting mispricing in an economy with theabsence of near-arbitrage opportunity

6 More momentum in momentum-neutral factors

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Data: July 1963 through December 2019

22 off-the-shelf U.S. and Global ex. U.S. factors provided by AQR, KenFrench, and Rob Stambaugh:

Start Annual returnFactor Original study date Mean SD t-value

U.S. factors

Size Banz (1981) Jul 1963 2.7% 10.4% 1.97Value Rosenberg et al. (1985) Jul 1963 3.7% 9.7% 2.82Profitability Novy-Marx (2013) Jul 1963 3.1% 7.5% 3.13Investment Titman et al. (2004) Jul 1963 3.3% 6.9% 3.59Momentum Jegadeesh and Titman (1993) Jul 1963 7.8% 14.5% 4.02Accruals Sloan (1996) Jul 1963 2.8% 6.6% 3.19Betting against beta Frazzini and Pedersen (2014) Jul 1963 9.8% 11.2% 6.55Cash-flow to price Rosenberg et al. (1985) Jul 1963 3.4% 8.6% 2.94Earnings to price Basu (1983) Jul 1963 3.5% 8.9% 2.95Liquidity Pastor and Stambaugh (2001) Jan 1968 4.4% 11.6% 2.77Long-term reversals DeBondt and Thaler (1985) Jul 1963 2.5% 8.7% 2.16Net share issues Loughran and Ritter (1995) Jul 1963 2.8% 8.2% 2.52Quality minus junk Asness et al. (2014) Jul 1963 4.6% 7.7% 4.47Residual variance Ang et al. (2006) Jul 1963 1.6% 17.3% 0.68Short-term reversals Jegadeesh (1990) Jul 1963 6.0% 10.6% 4.21

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Data

Start Annual returnFactor Original study date Mean SD t-value

Global factors

Size Banz (1981) Jul 1990 1.1% 7.1% 0.83Value Rosenberg et al. (1985) Jul 1990 4.0% 7.4% 2.92Profitability Novy-Marx (2013) Jul 1990 4.3% 4.7% 4.91Investment Titman et al. (2004) Jul 1990 1.9% 6.1% 1.74Momentum Jegadeesh and Titman (1993) Nov 1990 7.9% 12.1% 3.54Betting against beta Frazzini and Pedersen (2014) Jul 1990 9.6% 9.7% 5.70Quality minus junk Asness et al. (2014) Jul 1990 6.3% 6.8% 5.06

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Autocorrelation in factor returns

Are factor premiums constant?

Condition on each factor’s performance over the prior year

Estimate a time-series regression of returns against an indicator variable

Indicator variable =

{1 if positive return,0 otherwise

Interpretation:

Intercept = average return next month conditional on a losing yearSlope = difference in returns conditional on gains vs. losses

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Intercept Slope

Anomaly α̂ t(α̂) β̂ t(β̂)

U.S. factors

Size −0.10 −0.62 0.58 2.51Value 0.04 0.20 0.41 1.78Profitability 0.04 0.22 0.34 1.67Investment 0.12 0.97 0.24 1.55Momentum 0.72 2.70 −0.09 −0.29Accruals 0.15 1.18 0.10 0.65Betting against beta −0.22 −0.63 1.32 3.53Cash-flow to price 0.13 0.78 0.24 1.16Earnings to price 0.10 0.62 0.30 1.46Liquidity 0.16 0.74 0.36 1.29Long-term reversals −0.25 −1.66 0.76 3.85Net share issues 0.17 1.32 0.09 0.49Quality minus junk 0.09 0.65 0.43 2.51Residual variance −0.46 −1.64 1.06 2.74Short-term reversals 0.49 1.43 0.01 0.04

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Intercept Slope

Anomaly α̂ t(α̂) β̂ t(β̂)

Global factors

Size −0.06 −0.39 0.28 1.33Value 0.04 0.15 0.47 1.77Profitability 0.14 1.03 0.26 1.62Investment −0.06 −0.41 0.38 1.94Momentum 0.67 1.77 0.02 0.04Betting against beta 0.19 0.58 0.84 2.30Quality minus junk 0.39 1.76 0.12 0.49

Pooled

All 0.06 0.72 0.45 4.22

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Time-series factor momentum

1964 1967 1969 1972 1975 1978 1980 1983 1986 1989 1991 1994 1997 2000 2002 2005 2008 2010 20130

2

4

6

8

10

12

14

Time

Total(com

pounded)return

Equal weighted

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1964 1967 1969 1972 1975 1978 1980 1983 1986 1989 1991 1994 1997 2000 2002 2005 2008 2010 20130

2

4

6

8

10

12

14

Time

Total(com

pounded)return

Equal weightedTime series winners

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1964 1967 1969 1972 1975 1978 1980 1983 1986 1989 1991 1994 1997 2000 2002 2005 2008 2010 20130

2

4

6

8

10

12

14

Time

Total(com

pounded)return

Equal weightedTime series winnersTime-series losers

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Relation to the cross-sectional stock momentum

Define a cross-sectional momentum strategy with N stocks

πmoms,t = (Rs,−τ − R̄−τ )(Rs,t − R̄t)

Suppose that stock returns obey a F -factor structure:

Rs,t =F∑

f =1

βfs rft + εs,t ,

The expected profit on the momentum strategy is then

E[πmomt ] =

F∑f =1

[cov(r f−t , r

ft )σ2

βf

]︸︷︷︸

factorautocovariance

+F∑

f =1

F∑g=1

f 6=g

[cov(r f−t , r

gt ) cov(βg

, βf )]

︸︷︷︸factor

cross-serial covariance

+1

N

N∑s=1

[cov(εs−t , ε

st )]

︸︷︷︸autocorrelation

in residuals

+ σ2η,

︸︷︷︸variation inmean returns

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Intuition

If asset returns follow a factor structure, cross-sectional momentum profitsemanate from a combination of

1 Factor autocovariance

2 Factor cross-serial covariance

3 Autocorrelation in asset-specific shocks

4 Persistent differences in mean returns (Conrad and Kaul 1998)

In this presentation we focus on the first channel by testing

UMD ∼F∑

f =1

[cov(r f−t , r

ft )σ2

βf

]︸︷︷︸

factorautocovariance

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Factor momentum versus stock momentum

Form momentum portfolios by sorting stocks into deciles by rt−12,t−2

Compare three asset pricing models:

1 Fama-French five-factor model

2 Fama-French five-factor model + UMD

3 Fama-French five-factor model + factor momentum

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Factor momentum versus stock momentum

Asset pricing modelFF5 FF5 + UMD FF5 + TSMOM

Decile α̂ α̂ b̂umd α̂ b̂tsmom

Losers −0.75 −0.10 −0.93 −0.04 −2.46(−4.05) (−0.94) (−36.59) (−0.28) (−20.06)

2 −0.35 0.13 −0.70 0.16 −1.78(−2.74) (2.08) (−46.76) (1.54) (−21.26)

9 0.08 −0.14 0.33 −0.11 0.66(1.08) (−2.46) (23.85) (−1.45) (11.04)

Winners 0.57 0.17 0.57 0.16 1.42(4.82) (2.32) (32.93) (1.60) (17.21)

Winners 1.33 0.27 1.51 0.20 3.88− Losers (4.91) (2.43) (56.81) (0.99) (23.13)

Avg. |α̂| 0.26 0.12 0.11GRS F -value 4.24 3.10 2.33GRS p-value 0.00% 0.04% 1.06%

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The many forms of stock momentum

Monthly FF5returns model

Momentum definition Reference r̄ SD t(r̄) α̂ t(α̂)

Individual stock momentum

Standard momentum Jegadeesh and Titman (1993) 0.64 4.22 3.93 0.70 4.28Ind.-adjusted momentum Cohen and Polk (1998) 0.41 2.64 3.96 0.50 4.93Industry momentum Moskowitz and Grinblatt (1999) 0.63 4.60 3.54 0.69 3.77Intermediate momentum Novy-Marx (2012) 0.48 3.02 4.12 0.56 4.81Sharpe ratio momentum Rachev et al. (2007) 0.55 3.59 3.94 0.63 4.51

Factor momentum

Factor momentum 0.33 1.20 7.01 0.29 6.21

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Spanning tests

Dependent variableIndividual Individualstock stock momentum Factor momentummomentum, UMD∗ α̂ FF5 FMOM α̂ FF5 UMD∗

Standard 0.00 Y 2.43 0.15 Y 0.20momentum (−0.04) (24.72) (4.44) (24.72)

Industry-adjusted 0.14 Y 1.23 0.16 Y 0.26momentum (1.67) (17.63) (4.07) (17.63)

Industry 0.02 Y 2.32 0.19 Y 0.15momentum (0.12) (18.83) (4.88) (18.83)

Intermediate 0.15 Y 1.41 0.16 Y 0.23momentum (1.51) (17.72) (4.15) (17.72)

Sharpe ratio 0.02 Y 2.12 0.14 Y 0.23momentum (0.19) (25.45) (4.20) (25.45)

All of above 0.14 Y .†

(4.30)†Note: This regression includes all six individual stock momentum factors as explanatory variables in addition to the five factors

of the Fama-French five-factor model: standard momentum, industry-adjusted momentum, industry momentum, intermediate

momentum, and Sharpe ratio momentum.

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Spanning UMD with momentum found in different sets offactors

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Economic mechanism: Kozak, Nagel, and Santosh (2018)model

Basic model

Some investors have a random sentiment-driven component to theirdemand

Arbitrageurs compensated for tilting their portfolios away from themarket portfolio to accommodate sentiment-investor demand

The mispricings that remain must align with factor risk

Arbitragers are reluctant to trade against such mispricings because itwould expose them to factor risk

Extended model

KNS consider an extended model in which sentiment is positivelyautocorrelated

We study autocorrelations within this extended model

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Economic mechanism

Extract principal components from the covariance matrix of returns

The autocorrelation of the kth principal component equals

cov(PC kt ,PC

kt+1) = λ2

kβ2kc0

[(1 + R2

f )φ− Rf − Rf φ2].

Factors positively correlate when sentiment is sufficiently persistent,φ ∈ ( 1

Rf, 1]

Why? Although arbitrageurs are aware that factors exhibit eitherreversals (when φ < 1

Rf) or momentum (when φ > 1

Rf), they are

reluctant to trade so aggressively that they would neutralize thispattern because, by doing so, they would assume factor risk.

What factors have more momentum in this model?

High-eigenvalue factors that line up with mispricings have moremomentumThis result parallels the distortion result in KNS: sentiment-drivendemand component δ has a large impact on SDF variance only when δlines up “primarily with the high-eigenvalue (volatile) PCs of assetreturns” (p. 1203).

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Factor momentum in high- and low-eigenvalue factor PCs

Use the data from Haddad, Kozak, and Santosh (2020) to examinethe extent to which factor momentum concentrates tohigh-eigenvalue principal components

54 return predictors—exclude the seven predictors that relate tomomentum or that combine momentum with other characteristics

Extract out-of-sample principal components:1 Compute eigenvectors using daily returns on the 47 factors up to

month t2 Compute monthly returns for the factor PCs up to month t + 1 using

these eigenvectors3 Demean and lever the factor PCs so that their variances up to month t

are equal to the variance of the average individual factor and that theiraverage returns up to month t are zero.

4 Construct a factor momentum strategy that is long factors withpositive average returns from month t − 11 to t and short factors withnegative average returns.

5 Compute the return on the resulting factor momentum strategy inmonth t + 1.

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Factor momentum in high- and low-eigenvalue factor PCs

Factor momentum in factor PCsIndependent Subsets of PCs ordered by eigenvaluevariable All 1–10 11–20 21–30 31–40 41–47

Alpha 0.12 0.18 0.13 0.12 0.07 0.07(6.63) (6.08) (5.11) (5.52) (2.72) (2.15)

FF5 factors Y Y Y Y Y Y

N 558 558 558 558 558 558R2 4.0% 2.0% 1.5% 4.2% 3.1% 3.3%

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Is factor momentum just individual stock momentum?

Isn’t factor momentum just individual stock momentum?

If the size factor, for example, has performed well over the prior year,then the stocks in this factor’s long leg have, by definition, higherpast returns than those in its short leg

Individual stock momentum alone would then predict that the sizefactor should continue to perform well after a year of a good returns.This incidental momentum effect could give rise to what looks likefactor momentum, even if momentum does not reside in factors.

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Constructing momentum-neutral factors

Examine the origins of momentum by measuring factor momentum inmomentum-neutral factors

How?

If a factor’s weights are wi ,t , distort these weights as little as possibleto find new weights xi such that

minxi

∑i

(wi − xi )2 s.t.

N∑i=1

xi = 0 and

N∑i=1

xi ri ,t−12,t−2 = 0

Weights xi equivalent to the residuals from a cross-sectionalregression of the original factor weights on past returns:

wi ,t = a + b ri ,t−12,t−2 + xi ,t .

Momentum-neutral factors typically earn similar premiums as theoriginal factors but with lower volatility ⇒ higher Sharpe andinformation ratios

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Factor momentum in momentum-neutral factors

Factor momentum strategyIndependent Momentum

variable Original neutralα̂ 0.12 0.00 0.10 0.04

(6.63) (0.12) (8.19) (4.62)

Original 0.56factor momentum (31.38)

Momentum-neutral 1.15factor momentum (31.38)

FF5 factors Y Y Y Y

N 558 558 558 558R2 4.0% 65.5% 5.7% 66.1%

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Conclusions

1 Almost all factors—both in the U.S. and globally—are positivelyautocorrelated

“Autocorrelated?” Risk premiums vary substantially over time.

2 Individual stock momentum strategies implicitly bet on these sameautocorrelations

3 Equity momentum is not a distinct factor—it is an aggregation of thetime-varying factor autocorrelations

Whether you make or lose money on momentum depends on whetherthe factors remain positively autocorrelated

4 More factor momentum in high-eigenvalue factors

Consistent with a mispricing explanation for factors, coupled with theabsence of near-arbitrage opportunities

5 Factor momentum is not incidental to individual stock momentum

More momentum in momentum-neutral factors

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factor momentum and the momentum factor - wild apricot

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