6. Multifactor Pricing Models

the AI*T provides an approximate relation for expected asset returns with an unknown iiinnber ol unidentified factors. At this level rejection ol the theory is impossible (unless arbitrage opportunities exist) and as a consequence testability of the model depends on the introduction of additional assumptions.1

Ihe Arbitrage Ii icing Theory assumes that markets are competitive and Irieiionlcss and thai ihe return generating process for asset returns being considered is

Л\ = а. + Ъ+е, ((i.l.l)

1П = > (6.1.2)

I-lffl = a? < <t < oo, (6.1.3)

where H, is the return for asset /, <;, is the intercept of the factor model, b, is a (A.x 1) vector of factor sensitivities for asset i, f is a (Ax 1) vector of common factor realizations, and (, is the disturbance term. For the system of N assets,

R = a + Bf + c (6.1.4)

E[e I f = 0 (6.1.5)

l.[cr I f = E. (6.1.6)

In the system equation, R is an (Ax 1) vector with R = [/{) R> /\], a is an (Лх 1) vector with a = \/ц a-, ay], В is an (NxK) matrix with В = [b b . h.v Г, atul ( is an (IVx I) vector with e = [f ( > €,v]. We fuithcr assume that the factors account for the common variation in asset returns so that the disturbance term for large well-diversified portfolios vanishes. This requires that the disturbance terms be sufficiently uncorrelated across assets.

(liven this structure, Ross (1976) shows that the absence of arbitrage in large economies implies that

,i ~ i.Xu + RXK, (6.1.7)

where ;i is the (,V x I) expected return vector, A0 is the model zero beta parameter and is equal to the riskfree return if such an asset exists, and Ajc is a (K x I) vector of factor risk premia. I lere, and throughout the chapter,

Tliiie has been substantial debate mi lilt- testability nl the APT. Sh.inktii (l!)k >) anil Dybvij! .mil Kiiv. (IlMo iinviilr our inicirsiing cxi hange. i)hiyincs, friend, (iultrkin. and (uilickiu (1!1H 11 ,iim iiir\iiiui iIn- <*n11iiiit al iclcvanre ol the ntodcl.

~Л targe wvll-divnsilicd ] и tlolio is a jiot tlolio with a large nntuhci of stocks with weigh tings ol older -y

h.l. Theoretical Background

let i represent a conforming vector of ones. The relation in (6.1.7) is ap ; proximate as a finite number of assets can be arbitrarily mispriced. Becauseiffy ((>. 1.7) is only an approximation, it does not produce directly testable restric-. -л lions for asset returns. To obtain restrictions we need to impose additional7tx,w: structure so that the approximation becomes exact. .

Connor (1984) presents a competitive equilibrium version of the APT which has exact factor pricing as a feature. In Connors model the additional. jjjj requirements are that the market portfolio be well-diversified and that the factors be pervasive. The market portfolio will be well-diversified if no single * X\ asset in the economy accounts for a significant proportion of aggregate wealth. The requirement that the factors be pervasive permits investors to diversify away idiosyncratic risk without restricting their choice of factor risk exposure.

Dybvig (1985) and Grinblatt and Titman (1985) take a different approach. They investigate the potential magnitudes of the deviations from exact factor pricing given structure on the preferences of a representative agent. Both papers conclude that given a reasonable specification of the parameters of the economy, theoretical deviations from exact factor pricing are likely to be negligible. As a consequence empirical work based on the exact pricing relation is justified. j

Exact factor pricing can also be derived in an intertemporal asset pricing framework. The Intertemporal Capital Asset Pricing Model developed in Merton (1973a) combined with assumptions on the conditional distribution of returns delivers a multifactor model. In this model, the market portfolio serves as one factor and state variables serve as additional factors. The additional factors arise from investors demand to hedge uncertainty about future investment opportunities. Breeden (1979), Campbell (1993a, 1996), and Fama (1993) explore this model, and we discuss it in Chapter 8.

In this chapter, we will generally not differentiate the APT from the ICAPM. We will analyze models where we have exact factor pricing, tyat is,

ft = tAn+BA*. (J6.1.8)

There is some flexibility in the specification of the factors. Most empirical implementations choose a proxy for the market portfolio as one factor. I lowever, different techniques are available for handling the additional factors. We will consider several cases. In one case, the factors of the APi and the stale variables of die ICAPM need noi he traded portfolios. In other cases the factors are returns on portfolios. These factor portfolios are called mimicking portfolios because jointly they are maximally correlated with the factors. Exact factor pricing will hold with such portfolios. Huberman, Kandel, and Stambaugh (1987) and Breeden (1979) discuss this issue in the context of the APT and ICAPM, respectively.

6.2 Estimation and Testing

In this section we consider the estimation and testing of various forms of the ex tct factor pricing relation. The starling point for the econometric analysis of the model is an assumption about the time-scries behavior of returns. We will assume that returns conditional on the factor realizations are III) through time and jointly multivariate normal. This is a strong assumption, but it does allow for limited dependence in returns through the lime-series behavior of the factors. Furthermore, this assumption can be relaxed by casting the eslimation and testing problem in a Generalized Method of Moments framework as outlined in the Appendix. The GMM approach for multifactor models is just a generalization of the GMM approach to testing the CAPM presented in Chapter 5.

As previously mentioned, the multifactor models specify neither the number of factors nor the identification of the factors. Thus lo estimate and lest the model we need to determine the factors-an issue we will address in Section G.4. In this section we will proceed by taking the number of factors and their identification as given.

We consider four versions of the exact factor pricing model: (1) Factors arc portfolios of traded assets and a riskfree asset exists;1 (2) Factors are portfolios of traded assets and there is not a riskfree asset; (3) Factors are not portfolios of traded assets; and (4) Factors are portfolios of traded assets and the factor portfolios span the mean-variance frontier of risky assets. We vise maximum likelihood estimation to handle all four cases. Sec Shan ken (1992b) for a treatment of the same four cases using a cross-sectional regression approach.

Given the joint normality assumption for the returns conditional on the factors, we can construct a test of any of the four cases using the likelihood ratio. Since derivation of the test statistic parallels the derivation of the likelihood ratio test of the CAPM presented in Chapter 5, we will not repeat it here. The likelihood ratio test statistic for all cases takes the same general form. Defining J as the test statistic we have

J = - (т- ~ - К - lj [logE - log]£*], (6.2.1)

where £ and £ arc the maximum likelihood estimators of the residual covariance matrix for the unconstrained model and constrained model, respectively. T is the number of time-series observations, N is the number of ihcludcd portfolios, and К is the number of factors. As discussed in Chapter 5, the statistic has been scaled by (7* - - К - 1) rather than the usual 7 to improve the convergence of the finite-sample null distribution

to the large sample distribution. The large sample distribution of у under the null hypothesis will be chi-siuare with the degrees of freedom equal lo the number of restrictions imposed by the null hypothesis.

6.2. / Portfolios as factors with a Riskfree Asset

We first consider the case where the factors are traded portfolios and there exists a riskfree asset. The unconstrained model will be a A-factor model expressed in excess returns. Define Z, as an (Лх 1) vector of excess returns for N assets (or portfolios of assets). For excess returns, (he /C-faclor linear model is:

Z, = a + BZs, + e, (6.2.2)

E[e,l = 0 F.U,c,l = £

(6.2.3) (6.2.4)

K[ZA-,]

Ii[(Z,.i - цк) (ZKt - (tK)\ = ПА

Cov[Z,

= O.

(6.2.5) (6.2.6)

В is the (Л/ x K) matrix of factor sensitivities, ZA, is the {К x 1) vector of factor portfolio excess returns, and a and e, are (Лх I) vectors of asset return intercepts and disturbances, respectively. £ is the variance-covariance matrix of the disturbances, and ilf- is the variance-covariance matrix of the factor portfolio excess returns, while О is a (KxN) matrix of zeroes. Exact factor pricing implies that the elements of the vector a in (6.2.2) will be zero.

For the unconstrained model in (6.2.2) the maximum likelihood estimators are just the OLS estimators:

where

a = Д-ВДК (6.2.7)

]T)(Z, - fi)(zKI - £A)

= ]T(Z, - a - UZKI)(Z, - a - BZA-,), 1 i=i

(6.2.8)

(6.2.9)

See t(iiali(m (tlii.-h) and obs<m and KuiUic (I.WZ).

For tin с onsiraim-d model, will) a constrained to be /его, the maximum likelihood estimators arc

(6.2.10)

2* = .(Z,-B,ZK)(Z<-B,ZK). (fi.2.11)

The null hypothesis a equals /его can he tested using the likelihood ratio statistic J in (6.2.1). Under the null hypothesis the degrees of freedom ol the null distribution will be N since the null hypothesis imposes N restrictions.

In this case we can also construct an exact multivariate /-test of the null hypothesis. Defining f as the test statistic we have

,/t = U ~NN~k) 11 + /.ПкАкГаЕ a. (6.2.12)

where $i\ is the maximum likelihood estimator of ПА,

1

= ]C<Z -/ a-HZK~ Ак>- (6.2.13)

Under the null hypothesis, f\ is unconditionally distributed central / with N degrees of freedom in the numerator and (7 - N - Л.) degrees of freedom in the denominator. This test can be very useful since it can eliminate the problems that can accompany the use of asymptotic distribution theory. Jobson and Korkie (1985) provide a derivation of J\.

B =

zki za

6.2.2 Portfolios as Factors without a Riskfree Asset

In the absence of a riskfree asset, there is a zero-beta model that is a multi-factor equivalent of the Black version of the CAPM. In a multifactor context, the zero-beta porllolio is a portfolio with no sensitivity to any of the factors, and expected returns in excess of the zero-beta return are linearly related lo die columns of the matrix of factor sensitivities. The factors are assumed lo he portfolio returns in excess of the zero-beta return.

Define R, as an (Л/ x 1) vet tor of real returns for N assets (or portfolios of assets). For the unconstrained model, we have a A-faclor linear model:

R, = a + BRK( + £, (6.2.1-1)

E\c,\ = 0 (6.2. lf>)

E[f.,(.t\ = S (6.2.16)

E[KKl] = p./с, £[(R*i-МкЯ = (6..17)

Cov[RA- e,] = O. (6..18)

В is the (NxK) matrix of factor sensitivities, KK, is the (Kxl) vector of factor portfolio real returns, and a and 6, are (Л/х 1) vectors of asset return intercepts and disturbances, respectively. О is a (KxN) matrix of zeroes.

For the unconstrained model in (6.2.14) the maximum likelihood estimators are

(6.2.19)

£(Rjr, - Ajc) (Rk, - £*)

(6.2,20) (6421)

]T(R, - £)(RA-, - jj.K) (=1

1 2-

s = T- E(R-5-BR(R-s-BR)-71(

1 T 1 т

A = y-5ZR and Aa- = yY1Kki-

In the constrained model real returns enter in excess of the expected zero-beta portfolio return y0. For the constrained model, we have

where

Ri = iyo+B (Rki - tyo) + e, = (t-Bi)yo + BRK, + e,.

The constrained model estimators are:

(6.2.22)

B* =

£(R, - lyo) (Rjc - iM 1

. =1

(6.2.23)

= 7.£iR-byo-B(RK(-i.yo)]

x [R< ~ t-Yo - B*(Ra-, - Lyo)] y0 = [(i-BiJrd-Bi)]-1

x [(i-Bt)S* (A-B*/ix)].

(6.2.24)

(6.2.25)