Interpreting Factor Models

• DOI: http://doi.org/10.1111/jofi.12612
• Author: SHRIHARI SANTOSH,STEFAN NAGEL,SERHIY KOZAK
• Published date: 01 June 2018
• Date: 01 June 2018

THE JOURNAL OF FINANCE •VOL. LXXIII, NO. 3 •JUNE 2018

Interpreting Factor Models

SERHIY KOZAK, STEFAN NAGEL, and SHRIHARI SANTOSH∗

ABSTRACT

Weargue that tests of reduced-form factor models and horse races between “character-

istics” and “covariances” cannot discriminate between alternative models of investor

beliefs. Since asset returns have substantial commonality, absence of near-arbitrage

opportunities implies that the stochastic discount factor can be represented as a func-

tion of a few dominant sources of return variation. As long as some arbitrageurs are

present, this conclusion applies even in an economy in which all cross-sectional vari-

ation in expected returns is caused by sentiment. Sentiment-investor demand results

in substantial mispricing only if arbitrageurs are exposed to factor risk when taking

the other side of these trades.

REDUCED-FORM FACTOR MODELS ARE ubiquitous in empirical asset pricing. In

these models, the stochastic discount factor (SDF) is represented as a func-

tion of a small number of portfolio returns. In equity market research, models

such as the three-factor SDF of Fama and French (1993) and various exten-

sions are popular with academics and practitioners alike. These models are

reduced-form because they are not derived from assumptions about investor

beliefs, preferences, and technology that prescribe which factors should appear

in the SDF. What interpretation should one give such a model if it works well

empirically?

That there exists a factor representation of the SDF is almost a tautology.1

The economic content of the factor-model evidence lies in the fact that not only

do covariances with the factors explain the cross section of expected returns, but

the factors also account for a substantial share of the time-series comovement

of stock returns. As a consequence, an investor who wants to benefit from the

∗Serhiy Kozak is with the Stephen M. Ross School of Business, University of Michigan. Stefan

Nagel is with the University of Chicago Booth School of Business, NBER, and CEPR. Shrihari

Santosh is with the Robert H. Smith School of Business, University of Maryland. Weare grateful for

comments from Kent Daniel, David Hirshleifer, Stijn van Nieuwerburgh, Ken Singleton, Annette

Vissing-Jorgensen, two anonymous referees, and participants at the American Finance Association

Meetings, Copenhagen FRIC conference, NBER Summer Institute, and seminars at the University

of Cincinnati, Florida, Luxembourg, Maryland, Michigan, MIT, Nova Lisbon, Penn State, and

Stanford. The authors read the Journal of Finance’s disclosure policy and have no conflicts of

interest to disclose.

1If the law of one price (LOP) holds, one can always construct a single-factor or multifactor

representation of the SDF in which the factors are linear combinations of asset payoffs (Hansen

and Jagannathan (1991)). Thus, the mere fact that a low-dimensional factor model “works” has no

economic content beyond the LOP.

DOI: 10.1111/jofi.12612

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expected return spread between, say, value and growth stocks or recent winner

and loser stocks must invariably take on substantial factor risk exposure.

Researchers often interpret the evidence that expected return spreads are

associated with exposures to volatile common factors as a distinct feature of “ra-

tional,” as opposed to “behavioral,” asset pricing models. For example, Cochrane

(2011, p. 1075) writes,

Behavioral ideas—narrow framing, salience of recent experience, and so

forth—are good at generating anomalous prices and mean returns in in-

dividual assets or small groups. They do not [ . . . ] naturally generate

covariance. For example, “extrapolation” generates the slight autocorre-

lation in returns that lies behind momentum. But why should all the

momentum stocks then rise and fall together the next month, just as if

they are exposed to a pervasive, systematic risk?

In a similar vein, Daniel and Titman (1997) and Brennan, Chordia, and

Subrahmanyam (1998) suggest that one can test for the relevance of behavioral

effects on asset prices by looking for a component of expected return variation

associated with stock characteristics (such as value/growth, momentum, etc.)

that is orthogonal to factor covariances. This view that behavioral effects on

asset prices are distinct from and orthogonal to common factor covariances is

pervasive in the literature.2

Contrary to this standard interpretation, we argue that there is no such clear

distinction between factor pricing and behavioral asset pricing. If sentiment—

which we use as a catch-all term for distorted beliefs, liquidity demands, or

other distortions—affects asset prices, the resulting expected return spreads

between assets should be explained by common factor covariances in a sim-

ilar way as in standard rational expectations asset pricing models. The rea-

son is that the existence of a relatively small number of arbitrageurs should

be sufficient to ensure that near-arbitrage opportunities—that is, trading

strategies that earn extremely high Sharpe ratios (SRs)—do not exist. To

take up Cochrane’s example, if stocks with momentum did not rise and fall

together next month to a considerable extent, the expected return spread

between winner and loser stocks would not exist in the first place because

arbitrageurs would have picked this low-hanging fruit. Arbitrageurs neutral-

ize components of sentiment-driven asset demand that are not aligned with

2For example, Brennan, Chordia, and Subrahmanyam (1998, p. 346) describe the reduced-

form factor model studies of Fama and French as follows: “ .. . Fama and French (FF) (1992a, b,

1993b, 1996) have provided evidence for the continuing validity of the rational pricing paradigm.”

The standard interpretation of factor pricing as distinct from models of mispricing also appears

in more recent work. To provide one example, Hou, Karolyi, and Kho (2011, p. 2528) write that

“Some believe that the premiums associated with these characteristics represent compensation

for pervasive extra-market risk factors, in the spirit of a multifactor version of Merton’s (1973)

Intertemporal Capital Asset Pricing Model (ICAPM) or Ross’s (1976) Arbitrage Pricing Theory

(APT) (Fama and French (1993,1996), Davis, Fama, and French (2000)), whereas others attribute

them to inefficiencies in the way markets incorporate information into prices (Lakonishok, Shleifer,

and Vishny (1994), Daniel and Titman (1997), Daniel, Titman, and Wei(2001)).”

Interpreting Factor Models 1185

common factor covariances, but are reluctant to trade aggressively against

components that would expose them to factor risk. Only in the latter case

can the sentiment-driven demand have a substantial impact on expected re-

turns. These conclusions apply not only to the equity factor models that we

focus on here, but also to no-arbitrage bond pricing models and currency factor

models.

We start by analyzing implications of the absence of near-arbitrage opportu-

nities for the reduced-form factor structure of the SDF.For typical sets of assets

and portfolios, the covariance matrix of returns is dominated by a small number

of factors. These empirical facts combined with the absence of near-arbitrage

opportunities imply that the SDF can be represented to a good approximation

as a function of these few dominant factors.3This conclusion also applies to

models with sentiment-driven investors, as long as arbitrageurs eliminate the

most extreme forms of mispricing.

If this reasoning is correct, then it should be possible to obtain a low-

dimensional factor representation of the SDF based purely on information

from the covariance matrix of returns. We show that a factor model with a

small number of principal component (PC) factors does about as well as popu-

lar reduced-form factor models in explaining the cross section of expected re-

turns on anomaly portfolios. Thus, there does not seem to be anything special

about the construction of the reduced-form factors proposed in the literature—

purely statistical factors do just as well. For typical test asset portfolios, their

return covariance structure essentially dictates that the first few PC factors

must explain the cross section of expected returns.4Otherwise, near-arbitrage

opportunities would exist.

Testsof characteristics versus covariances, like those pioneered in Daniel and

Titman ( 1997),look for variation in expected returns that is orthogonal to factor

covariances. Ex-post and in sample such orthogonal variation always exists,

perhaps even with statistical significance according to conventional criteria.

It is an open question, however, whether such near-arbitrage opportunities

are a robust and persistent feature of the cross section of stock returns. To

address this question, we perform a pseudo out-of-sample exercise. Splitting

the sample period into two subsamples, we extract the PCs from the covariance

matrix of returns in one subperiod and then use the portfolio weights implied

by the first subsample PCs to construct factors out of sample in the second

subsample. While factors beyond the first few PCs contribute substantially to

3This notion of the absence of near-arbitrage is closely related to the interpretation of the

arbitrage pricing theory (APT) in Ross (1976). When discussing the empirical implementation

of the APT in a finite-asset economy, Ross suggests bounding the maximum squared SR of any

arbitrage portfolio at twice the squared SR of the market portfolio. However, our interpretation of

APT-type models differs from some of the literature. For example, Fama and French (1996)regard

the APT as a rational pricing model. We disagree with this narrow interpretation, as the APT is

just a reduced-form factor model.

4The number of factors depends heavily on the underlying space of test assets. For instance, for

the Fama-French 5 ×5 size and book-to-market (B/M) sorted portfolios, there are three dominant

factors. For payoff spaces with weaker factor structure, the number of dominant factors is larger.