Comparing Asset Pricing Models

• Author: JAY SHANKEN,FRANCISCO BARILLAS
• Date: 01 April 2018
• Published date: 01 April 2018
• DOI: http://doi.org/10.1111/jofi.12607

THE JOURNAL OF FINANCE •VOL. LXXIII, NO. 2 •APRIL 2018

Comparing Asset Pricing Models

FRANCISCO BARILLAS and JAY SHANKEN∗

ABSTRACT

A Bayesian asset pricing test is derived that is easily computed in closed form from

the standard F-statistic. Given a set of candidate traded factors, we develop a related

test procedure that permits the computation of model probabilities for the collection

of all possible pricing models that are based on subsets of the given factors. We find

that the recent models of Hou, Xue, and Zhang (2015a, 2015b) and Fama and French

(2015, 2016) are dominated by a variety of models that include a momentum factor,

along with value and profitability factors that are updated monthly.

GIVEN THE VARIETY OF PORTFOLIO-BASED factors that have been examined by re-

searchers, it is important to understand how best to combine them in a parsi-

monious asset pricing model for expected returns, one that excludes redundant

factors. Although there are standard econometric techniques for evaluating the

adequacy of a single model, a satisfactory statistical methodology for identi-

fying the best factor pricing model(s) is conspicuously lacking in investment

research applications. In this paper,we develop an easy-to-implement Bayesian

procedure that allows us to compute model probabilities for the collection of all

possible pricing models that can be formed from a given set of factors.

Beginning with the capital asset pricing model (CAPM) of Sharpe (1964)and

Lintner (1965), the asset pricing literature has attempted to understand the

determination of risk premia on financial securities. The central theme of this

literature is that a security’s risk premium should depend on the security’s

market beta or other measure(s) of systematic risk. In a classic test of the

CAPM that builds on Jensen (1968), Black, Jensen, and Scholes (1972) examine

the intercepts in time-series regressions of excess test portfolio returns on

market excess returns. Given the CAPM implication that the market portfolio

is efficient, these intercepts or “alphas” should be zero. A joint F-test of this

hypothesis is later developed by Gibbons, Ross, and Shanken (1989), henceforth

∗Barillas and Shanken are from the Goizueta Business School at Emory University.Shanken is

also from the National Bureau of Economic Research. Thanks to two anonymous referees, Rohit

Allena, Doron Avramov, Mark Fisher, Amit Goyal, Lubos Pastor, Seth Pruitt, Tim Simin, Ken

Singleton (the Editor), Rex Thompson, Pietro Veronesi, and seminar participants at the Northern

Finance Association meeting, Southern Methodist University,the Universities of Arizona, Geneva,

Hong Kong, Lausanne, Luxembourg, Michigan, Pennsylvania, and Singapore, Hong Kong Univer-

sity of Science and Technology, Imperial College, National University of Singapore, Ohio State

University,and Florida International University. We have read the Journal of Finance’s disclosure

policy and have no conflicts of interest to disclose.

DOI: 10.1111/jofi.12607

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716 The Journal of Finance R

GRS, who also explore the relation between the test statistic and standard

portfolio geometry.1

In recent years, a variety of multifactor asset pricing models have been

explored. Although tests of the individual models are routinely reported, these

tests often suggest “rejection” of the implied restrictions, especially when the

data sets are large (e.g., Fama and French (2016)). However, a relatively large

p-value may say more about imprecision in estimating a particular model’s

alphas than the adequacy of that model.2Simple statistical tools with which to

analyze the various models jointly in a model-comparison framework are thus

sorely needed. The information that our methodology provides about relative

model likelihoods complements that obtained from classical asset pricing tests

and is more in the spirit of the adage “it takes a model to beat a model.” 3

Like other asset pricing analyses based on alphas, we require that the bench-

mark factors are traded portfolio excess returns or return spreads. For exam-

ple, in addition to the market excess return, Mkt, the influential three-factor

model of Fama and French (1993), hereafter FF3, includes a book-to-market

or “value” factor, HML (high-low), and a size factor, SMB (small-big), based

on stock market capitalization. Although consumption growth and intertem-

poral hedge factors are not traded, one can always substitute (maximally cor-

related) mimicking portfolios for the nontraded factors.4Although this intro-

duces additional estimation issues, simple spread-portfolio factors are often

viewed as proxies for the relevant mimicking portfolios (e.g., Fama and French

(1996)).

We begin by analyzing the joint alpha restriction for a set of test assets in

a Bayesian setting.5Prior beliefs about the extent of model mispricing are

economically motivated and accommodate traditional risk-based views as well

as more behavioral perspectives. The posterior probability that the zero-alpha

restriction holds is then shown to be an easy-to-calculate function of the GRS F-

statistic. Our related model-comparison methodology is likewise computation-

ally straightforward. This procedure builds on results in Barillas and Shanken

(2017), who highlight the fact that, for several widely accepted criteria, model

comparison with traded factors only requires examination of each model’s abil-

ity to price the factors in the other models.

1See related work by Treynor and Black (1973) and Jobson and Korkie (1982).

2De Moor, Dhaene, and Sercu (2015) suggest a calculation that highlights the extent to which

differences in p-values may be influenced by differences in estimation precision across models, but

they do not provide a formal hypothesis test.

3Avramovand Chao (2006) also explore Bayesian model comparison for asset pricing models. As

we explain in the next two sections, their methodology is quite different from that developed here.

A recent paper by Kan, Robotti, and Shanken (2013) provides asymptotic results for comparing

model R2s in a cross-sectional regression framework. Chen, Roll, and Ross (1986) nest the CAPM

in a multifactor model with betas on macro-related factors included in cross-sectional regressions.

In other Bayesian applications, Malatesta and Thompson (1993) apply methods for comparing

multiple hypotheses in a corporate finance event-study context.

4See Merton (1973) and Breeden (1979), especially footnote 8.

5See earlier work by Shanken (1987b), Harvey and Zhou (1990), and McCulloch and Rossi

(1991).

Comparing Asset Pricing Models 717

The observation that all models are necessarily simplifications of reality and

hence must be false in a literal sense motivates an evaluation of whether a

model holds approximately, rather than as a sharp null hypothesis. Additional

motivation comes from recognizing that the factors used in asset pricing tests

are generally proxies for the relevant theoretical factors.6With these consid-

erations in mind, we extend our results to obtain simple formulas for testing

the more plausible approximate models. As a warm up, we consider all models

that can be obtained using subsets of the FF3 factors Mkt, HML, and SMB.A

nice aspect of the Bayesian approach is that it permits comparison of nested

models like CAPM and FF3, as well as the nonnested models {Mkt HML}and

{Mkt SMB}. Moreover, we are able to simultaneously compare all of the models,

as opposed to standard classical approaches that involve pairwise model com-

parison (e.g., Vuong (1989)). Over the period 1972 to 2015, the alphas for HML

when regressed on Mkt or Mkt and SMB are highly “significant,” whereas the

alphas for SMB when regressed on Mkt or Mkt and HML are modest. Our pro-

cedure aggregates all of this evidence, arriving at posterior probabilities with

our benchmark prior of 60% for the two-factor model {Mkt HML} and 39% for

FF3, with the remaining 1% split between CAPM and {Mkt SMB}.

In our main empirical application, we compare models that combine many

prominent factors from the literature. In addition to the FF3 factors, we con-

sider the momentum factor, UMD (up minus down), introduced by Carhart

(1997) and motivated by Jegadeesh and Titman (1993). We also include fac-

tors from the recently proposed five-factor model of Fama and French (2015),

hereafter FF5.7These are RMW (robust minus weak), which is based on

the profitability of firms, and CMA (conservative minus aggressive), which

is related to firms’ net investments. Hou, Xue, and Zhang (2015a,2015b),

henceforth HXZ, propose their own versions of the size (ME), investment

(IA), and profitability (ROE) factors, which we also examine. In particu-

lar, ROE incorporates the most recent earnings information from quarterly

data. Finally, we consider the value factor HMLmfrom Asness and Frazzini

(2013), which is based on book-to-market rankings that use the most recent

monthly stock price in the denominator. In total, we have 10 factors in our

analysis.

Rather than mechanically apply our methodology with all nine of the non-

market factors treated symmetrically, we structure the prior so as to recognize

that several of the factors are different versions of the same underlying con-

struct. Therefore, to avoid overfitting, we only consider models that contain

at most one of the factors in each of the following categories: size (SMB or

ME), profitability (RMW or ROE), value (HML or HMLm), and investment

(CMA or IA). The extension of our procedure to accommodate such “categorical

6Kandel and Stambaugh (1987) and Shanken (1987a) analyze pricing restrictions based on

proxies for the market portfolio or other equilibrium benchmark.

7We use the SMB factor from the FF5 model in this paper. Our findings are not sensitive to

whetherweusetheoriginalSMB factor, as the correlation between the two size measures is over

0.99.