Can stochastic discount factor models explain the cross‐section of equity returns?

• Author: Pongrapeeporn Abhakorn,Peter N. Smith,Michael R. Wickens
• Date: 01 January 2016
• Published date: 01 January 2016
• DOI: http://doi.org/10.1016/j.rfe.2016.01.001

Can stochastic discount factor models explain the cross-section of

equity returns?☆

Pongrapeeporn Abhakorn

a,

⁎, Peter N. Smith

b

, Michael R. Wickens

c

a

FiscalPolicy Office, Ministry of Financeof Thailand, PhayataiRoad, 10400, Thailand

b

Departmentof Economics and RelatedStudies, Universityof York,YO10 5DD, UK

c

CardiffBusiness School and Departmentof Economics and RelatedStudies, University of York,YO10 5DD, UK

abstractarticle info

Articlehistory:

Received11 November 2014

Receivedin revised form 21 December2015

Accepted1 January 2016

Availableonline 7 January 2016

We propose a multivariate test of the capital asset pricing model (C-CAPM) of the cross-sectional variation

in equity returns in which we compare cross-sectional variation in equit y returns to the cross-sec tional var-

iation in their conditional covariance with stochastic discountfactors. We use a multivariate generalized

heteroskedasticity in mean model to estimate 25 portfolios that are formed on size and the book-to-

market ratio. Each portfolio is allowed to have its ownno-arbitrage condition. We find that although the

conditional covariances of returns with consumption exhibit negative variation across size, they do not

vary across the book-to-market ratio. Thus, C-CAPM can capture the size effect, but not the value effect.

The fit is, however, improved by allowing the coefficients on the consumption covariances to be different.

The value effect appears to be associated with the book-to-market ratio as well as size. On its own the

book-to-market ratio does not generate additional information about average returns to C-CAPM. A

possible explanation for these findings is that both small and low book-to-market ratio firms are expectedto

have higher ratesof growth.

© 2016 ElsevierInc. All rights reserved.

Keywords:

Risk premium

Equityreturn

Stochasticdiscount factor

No-arbitragecondition

1. Introduction

Size and value effects have long been recognized as “anomalies”

both in the capital asset pricing model (CAPM) literature (summa-

rized in Fama & French, 2006 and 2008), and in the consumption-

based CAPM (C-CAPM) framework with power utility (see

Cochrane, 2008). This paper tests whether s tochastic discount factor

(SDF) models that satisfy no-arbitrage restrictions can explain the

behavior of a cross-section of returns on 25 portfolios sorted by

firm size and their book-to-market ratio (the 25 Fama–French

portfolios). We e xamine whether portfolios of sto cks have different

returns due to differe nt conditional covarianc es between the returns

and the relevant discount factors, or because the coefficients of the

discount factors v ary by portfolio characteristi c. This provides a test

of no-arbitrage as finding either effect would imply that

no-arbitrage does not hol d.

Insteadof modeling separate no-arbitrage conditionsfor the returns

on the 25 Fama–French portfolios, we mo del them simultaneously

employing an SDF framework. We use a multi variate generalized

autoregressive conditional heteroskasticity in mean model (MGM) as

in Smith and Wickens(2002). This methodology is in contrast to most

of the time-series econometric models of equity returns in the litera-

ture,which are univariatemodels and donot include conditionalcovari-

ances(see for example Ludvigson,2012). Smith, Sorensen, and Wickens

(2008)used the approach adoptedin this paper, examiningvarious SDF

models, includingthe standard C-CAPM, to generate models involving

macroeconomic variables.Abhakorn, Smith, and Wickens (2013) esti-

matethe MGM for the standardC-CAPM for eachof the 25 Fama–French

portfolios, and find that the fit of the modelis significantly improvedby

the inclusionof the firm book-to-market valueratio (HML) factor. This

paper extendstheir analysis by estimatingall 25 Fama–Frenchportfolio

returns simultaneously and tes ting for each asset-pricing model

whether theconditional covariances ofthese returns with the relevant

discount factors can adequate ly explain the excess return s of these

portfolios.

We find that C-CAPMis rejected by the no-arbitrage test. Themodel

can explainthe size effect,as the conditional covarianceof consumption

with firm sizeis negative, but not thevalue effect, as the conditional co-

varianceof consumptionwith the book-to-marketratio does not vary as

requiredacross the book-to-marketquintiles. We find thatthe value ef-

fect tends to be slightly lower for port folios in the highest book-to-

market quintile –indicatinga lower risk premium –than for portfolios

with the lowestbook-to-market quintiles.Allowing the coefficients on

Reviewof Financial Economics 28 (2016)56–68

☆The views expressed in the paper are those of the authors and d o not necessarily

representthose of the Fiscal PolicyOffice.

⁎Correspondingauthor.

E-mailaddresses: pongrapeeporn.a@mof.go.th(P. Abhakorn), peter.smith@york.ac.uk

(P.N.Smith), mike.wickens@york.ac.uk (M.R. Wickens).

http://dx.doi.org/10.1016/j.rfe.2016.01.001

1058-3300/©2016 Elsevier Inc. All rightsreserved.

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