4.2 Arbitrage Pricing Model, APM

Empirical evidence indicates that the CAPM

beta does not completely explain the cross

section of expected asset returns. This sug-

gests that additional factors may be required.

Ross (1976)† introduced the Arbitrage Pric-ing Theory (APT) as an alternative to the

CAPM.

The basic assumption is that there are a

number of, say K, common risk factors gen-

erating the returns so that

Ri = ai+ bif + i,

with

E[ i|f] = 0

E[ 2i ] = σ2i ≤ σ2 <∞,

†Ross, S. (1976). The arbitrage theory of capitalasset pricing. Journal of Economic Theory, 13, 341{360.

58

and

E[ i j] = 0, whenever i = j,

where i = 1, . . . , N, the number of assets, aiis the intercept term of the factor model, biis a K×1 coe±cient vector of factor sensitiv-ities (loadings) for asset i, f is a K×1 vectorof common factors, and i is the disturbance

term.

Without loss of generality we may assume

that the common factors have zero mean,

i.e., E[f ] = 0, which implies that ai = E[Ri] =

µi are the mean returns.

In the matrix form the return generating model

is

R= µ+Bf +(17)

E[ |f ] = 0

and

E[ |f] = §,

59

where R= (R1, R2, . . . , RN) , µ = (µ1, . . . , µN) ,

B = (b1, . . . ,bN) is an N × K factor load-

ing matrix, = ( 1, . . . , N) , and § is a N ×N matrix (assumed diagonal in the original

Ross model). Furthermore, it is assumed

that K N.

The derivation of the APM relies on the no

arbitrage assumption.

Let w = (w1, . . . , wN)' be an arbitrage strat-

egy. Then

w ι =N

i=1

wi = 0,(18)

and the implied portfolio should be riskfree

(or more precisely its starting value should

be zero, non-negative in the meantime with

probability one, and have a strictly positive

expected end value).

The return of the portfolio is

Rp = w R= w a+w Bf +w .

60

In order to make the portfolio riskfree, the

market risk w Bf and unsystematic risk w

must be eliminated. The unsystematic risk

can be eliminated by letting N be large, so

that Var(w ) = w §w = Ni=1w

2i σ2i . The

weights wi are of order 1/N, soNi=1w

2i σ2i →

0 as N →∞.

Next in order to eliminate the market risk the

weights must be selected such that

w B= 0(19)

In the language of linear algebra the columns

of the expanded matrix ~B = (ι,B) spans a

K +1-dimensional linear subspace, call it V ,

in IRN . Because N > K + 1 all vectors lying

in the orthogonal complement, V ⊥, of V are

valid candidates for w to satisfy conditions

(18) and (19).†

†More precisely V = {y ∈ IRN : y = ~Bx, x ∈ IRK+1}and V ⊥ = {z ∈ IRN : z y = 0 ∀ y ∈ V }. Note furtherthat IRN = V ∪V ⊥, actually IRN = V ⊕V ⊥, the directsum of V and V ⊥.

61

Given an arbitrage strategy w that satis¯es

(19), we get with large N approximately (be-

cause w ≈ 0)

Rp =N

i=1

wiµi = w µ(20)

which is a riskfree return.

The absence of arbitrage implies that any ar-

bitrage portfolio must have a zero return. In

other words

Rp = w µ = 0,(21)

which implies that the expected return vec-

tor µ is orthogonal to w. But then it is a

vector in the linear space V and hence of the

form

µ = ~Bλ,(22)

where λ= (λ0,λ1, . . . ,λK) ∈ IRK+1.

In other words the expected returns of the

single assets are of the form

µi = λ0 + bi1λ1 + . . .+ biKλK(23)

62

If there is a riskfree asset with return Rf =µ0, it has by de¯nition, zero exposure on thecommon market risk factors. That is b0j = 0,

j = 1, . . . ,K. Then from (23) with b0j = 0,

we get

Rf = λ0,(24)

and we can rewrite (23) as

µi = Rf + λ1bi1 + · · ·+ λKbiK.(25)

This is the APT equilibrium model of the

expected asset returns.

Because bij is a sensitivity to the jth commonrisk factors it is natural to interpret that λjrepresents the risk premium (the price of risk)

for factor j in the equilibrium.

Note. Generally if there is no riskfree return

λ0 can be interpreted the zero-beta return.

In the matrix form the APM for the expected

equilibrium returns is

E[Rt] = µ = ιλ0 +Bλ(26)

where λ0 = Rf is the riskfree return if it

exists, and λ= (λ1, . . . ,λK) .

63

Estimation and Testing of APT

Assumption: Returns are normally and tem-

porally independently distributed.

APT does neither specify the factors nor the

number of factors. We consider four ver-

sions:

Factors are

(1) portfolios of traded assets and a riskfree asset exists;

(2) portfolios of traded assets and a riskfree asset does not exist;

(3) not portfolios of traded assets;

(4) portfolios of traded assets and the factor portfolios span the

mean-variance frontier of risky assets.

The derivation of the test statistics is analo-gous to the CAPM case. Relying on normal-ity the LR test statistic is of the form

J = − T − N2−K − 1 log |§|− log |§∗| ,(27)

where |§| and |§∗| are the unconstrained

and constrained ML-estimators, respectively.

Again as before the asymptotic null distribu-

tion of J is chi-square with degrees of free-

dom equal to the number of restrictions im-

posed by the null hypothesis.

64

(1) Portfolios as Factors with a Riskfree Asset

Denote the unconstrained form of the factor

model (17) in this case as

Zt = a+BZKt+ t(28)

with

E[ t] = 0, E[ t t] = §,

E[ZKt] = µK, E[ZKt−µK)(ZKt−µK) ] = −Kand

Cov[ZKt, t] = 0,

where B is the N ×K matrix of factor sensi-

tivities, ZKt is the K×1 vector of factor port-folio excess returns, and a and t are N × 1vectors of intercepts and error terms, respec-

tively.

The APM implies that a = 0. In order to test

this with the LR test wee need to estimate

the unconstrained and constrained model.

65

Model (28) is a seemingly unrelated regres-

sion (SUR) case, but because each regression

equation has the same explanatory variables

the ML-estimators are just the OLS estima-

tors

a= µ− BµK,

B =T

t=1

(Zt − µ)(ZKt − µK)T

t=1

(ZKt − µK)(ZKt − µK)−1

,

§ =1

T

T

t=1

(Zt − a− BZKt)(Zt − a− BZKt) ,

µ =1

T

T

t=1

Zt, µK =1

T

T

t=1

ZKt.

The constrained estimators are

B∗ = T

t=1

ZtZKt

T

t=1

ZKtZKt

−1(29)

and

§∗ = 1

T

T

t=1

(Zt − B∗ZKt)(Zt − B∗ZKt) .

66

The exact multivariate F -test becomes then

J1 =T −N −K

N1+ µK−

−1K µK

−1a §−1a,(30)

where

−K =1

T

T

t=1

(ZK − µK)(ZK − µK) .(31)

Under the null hypothesis J1 ∼ F(N,T −N −K).

(2) Portfolios as Factors without a Riskfree Asset

Let RKt = (R1t, . . . , RKt) be portfolios that

are factors of the APT model, and denote

the related factor model (17) as

Rt = a+BRKt+ t.(32)

If there does not exist a riskfree asset, then

as in the CAPM there exist a portfolio which

is uncorrelated with the portfolios in RKt.

Let γ0 denote the expected return of this

portfolio.

67

Then in the APM λ0 = γ0 and the con-

strained factor model is

Rt = ιγ0 +B(RKt − ιγ0) + t

= (ι−Bι)γ0 +BRKt+ t,

so that

a = (ι−Bι)γ0.The constrained ML-estimators are

B∗ =T

t=1

(Rt − ιγ0)(RKt − ιγ0)T

t=1

(RKt − ιγ0)(RKt − ιγ0)−1

,

§∗ =1

T

T

t=1

Rt − ιγ0 − B∗(RKt − ιγ0) Rt − ιγ0 − B∗(RKt − ιγ0) ,

andγ0 = (ι− B∗ι) §∗−1(ι− B∗ι) −1 (ι− B∗ι) §∗−1(ι− B∗µ) .

Again a suitable iterative procedure must be

implemented in the estimation.

The null hypothesis

H0 : a = (ι−Bι)γ0can be tested again with the likelihood ratio

test J of the form (27), which is asymptot-

ically chi-squared distributed with N − 1 de-grees of freedom under the null hypothesis.

68

(3) Macroeconomic Variables as Factors

Let fK be macroeconomic factors of the APM,

with E[fKt] = µfK. Then (17) becomes

Rt = a+BfKt+ t(33)

with parameter structure and unconstrained

ML estimators similar as derived in the case

of (28).

To formulate the constrained parameters con-

sider the expected value of (33)

µ = a+BµfK(34)

If the APM holds then this should be equal

to (26). So equating these we get for a

a = ιλ0 +B(λ− µfK).Denoting γ0 = λ0 and γ1 = λ−µfK, an K×1vector, the restricted model is

Rt = ιλ0 +Bγ1 +BfKt+ t.(35)

69

Again estimating the parameters, the APM

implied null hypothesis

H0 : a = ιλ0 +Bγ1(36)

can be tested with the likelihood ratio test of

the form (27) which in this case is under the

null hypothesis asymptotically chi squared dis-

tributed with degrees of freedom N −K − 1.

(4) Factor Portfolios Spanning the Mean-Variance Fron-

tier

Consider again the factor portfolio model (32).

If the factor portfolios in RKt span† the mean-

variance frontier then in (26) λ0 is zero (c.f.

the zero-beta case considered earlier).

This form of APM imposes the restriction

H0 : a = 0 and Bι = ι(37)

on (32).

†We say that a set of vectors S = {x1, . . . , xn} spansthe linear space V , if each y ∈ V can be representedas a linear combination of elements of S, i.e., if y ∈ Vthen y = a1x1+· · ·+anxn for some a= (a1, . . . , am) ∈IRn.

70

Example. Zero-beta version of CAPM (a two-factor model). LetRmt denote the market portfolio (a MVP) and R0t the associatedzero-beta portfolio. Then K = 2, B = (β0m, βm): (N × 2), andRKt = (R0t, Rmt) : (2× 1), so that

Rt = a+ β0mRmt+ βmRmt+ t.

As found earlier a= 0, and β0m+ βm = ι.

Again estimating the constrained and uncon-

strained model, we can use the LR-statistic

(27) to test the null hypothesis (37).

The degrees of freedom are 2N. These come

from N restrictions in a = 0 and N restric-

tions in B to satisfy Bι = ι.

Again, assuming that the multivariate nor-

mality holds, there exist an exact test

J2 =T −N −K

N

|§∗||§| − 1 ,(38)

which is F [2N,2(T −N −K)]-distributed un-der the null hypothesis.

71

Estimation of Risk Premia and Expected

Returns

As given in (26) the expected return of the

assets are under the APT

µ = λ0ι+Bλ.

To make the model operational one needs

to estimate the riskfree or zero beta return

λ0, factor sensitivities B, and the factor risk

premia λ.

The appropriate estimation procedure varies

across the four cases considered above. The

principle is that we use the appropriate re-

stricted estimators in each case to estimate

the parameters of (26).

72

For example, in the excess return case of

(28), B is estimated by (29), and

λ= µK =1

T

T

t=1

ZKt,(39)

which is the average excess return of the

market factors.

An interesting question then might be if the

factors are jointly priced. That is to test the

null hypothesis

H0 : λ= 0(40)

An appropriate test procedure is the mul-

tivariate mean test (known as Hotelling T2

test)

J3 =T −KTK

λ Var[λ]−1

λ,(41)

where

Var[λ] =1

T−K =

1

T 2

T

t=1

(ZKt − µKt)(ZKt − µKt) .(42)

Asymptotically J3 ∼ F(K,T −K) under thenull hypothesis (40).

73

Signi¯cance of individual factors, i.e., null hy-

potheses

H0 : λj = 0(43)

can be tested with the t-test

J4 =λj√vjj,(44)

which is asymptotically N(0,1)-distributed

under the null hypothesis, and where vjj is

the jth diagonal element of Var[λ], j = 1, . . . ,K.

Note. Test of individual factors is sensible only if the factors aretheoretically speci¯ed. If they are empirically speci¯ed they donot have clear-cut economic interpretations.Note. Another way to estimate factor risk premia is to use atwo-pass cross-sectional regression approach. In the ¯rst passthe factor sensitivities (B) are estimated and in the second passthe premia parameters of the regression

Zt = λ0tι+ Bλ+ ηt(45)

can be estimated time-period-by-time-period. Here B is identical

to the unconstrained estimator of B.

74

Selection of Factors

The factors of APM must be speci¯ed. This

can be on statistical or theoretical basis.

Statistical Approaches

Linear factor model as given in (17), in gen-

eral form is

Rt = a+Bft+ t(46)

with

E[ t t|ft] = §.(47)

In order to ¯nd the factors there are two

primary statistical approaches factor analysis

and principal component analysis

75

Factor Analysis

Using statistical factor analysis the assump-

tion is that there are K common latent (not

directly observable) common factors that af-

fect the stock prices. Especially it is assumed

that the common factors capture the cross-

sectional covariances between the asset re-

turns.

The factor model is of the form (46), which

with the above additional assumption implies

the following structure to the covariance ma-

trix of the returns

−= B©B +D(48)

where © = Cov(ft) is a K×K covariance ma-

trix of the common factors, and D= Cov( )

is an N ×N diagonal matrix of the residuals

(unique factors).

76

All the parameter matrices on the right hand

side of the decomposition (48) (altogether

(NK+K(K+1)/2+N parameters) are un-

known. Furthermore the decomposition is

not unique, because (given that © is positive

de¯nite) we can always write

© = GG

so that rede¯ning B as BG

−= BB +D.(49)

Even in this case B is not generally unique,

because again de¯ning C = BT where T is

an arbitrary K×K matrix such that TT = I,

we get − = CC + D. Matrix T is called a

rotation matrix.

The APT does not give the number of com-

mon factors K. Thus the ¯rst task is to de-

termine the number of factors. Empirically

this can be done with various statistical cri-

teria using factor analysis packages.

77

A popular method is to select as many fac-

tors as there are eigenvectors larger than one

computed from the return correlation matrix.

Modern statistical packages provide also ex-

plicit statistical tests as well as various crite-

rion functions for the purpose.

An example of FA approach is in

Lehmann, B. and D. Modest (1988). The

empirical foundations of the Arbitrage Pric-

ing Theory, Journal of Financial Economics,

21, 213{254.

78

Principal Component

Principal component analysis is another tool

for deriving common factors. This however is

a more technical approach. Furthermore usu-

ally di®erent components are obtained from

correlation matrix than from covariance ma-

trix. Nevertheless, there is no clear-cut re-

search results which one, PCA or FA, should

be a better choice in APT analysis.

An example of PCA application is in

Connor, G. and R. Korajczyk (1988). Risk an

return in an equilibrium APT: Application of

a new test methodology.Journal of Financial

Economics, 21, 255{290.

79

Theoretical Approaches

One approach is to macroeconomic and ¯-

nancial market variables that are thought to

capture systematic risk of the economy.

Chen, Roll and Ross (1986) Journal of Busi-

ness, 31, 1067{1083, use ¯ve factors:

(1) long and short government bond yield

spread (maturity premium), expected in°a-

tion,

(2) unexpected in°ation,

(3) industrial production growth, yield spread

between high- and low-grade bonds (default

premium),

(4) aggregate consumption growth and

(5) oil prices.

Another approach is to specify di®erent char-

acteristics of ¯rms and form a portfolio of

these. For example: market value of equity,

PE-ratio and book value to market value.

80

Some Empirical Results

The evidence supporting the exact factor pric-

ing, i.e., model (26), are mixed.

Especially di±culties are to explain the "size"

e®ect and the "book to market" e®ect.

Nevertheless APM seem to provide an attrac-

tive alternative to the single-factor CAPM.

81