A Critique of the Arbitrage Pricing Theory Matthew Pollard 17th November 2007

Abstract This short note proves that the pricing equation of the Arbitrage Pricing Theory (Ross, 1976) is a mathematical identity. The result of APT is that expected returns on any portfolio are approximately linear in factor loadings of K risk factors. We show that the expected return on any portfolio can be expressed exactly as the sum of K risk factors where at least two factors are (in-sample) mean-variance efficient portfolios. Such portfolios always exist, hence empirical tests of the APT should always accept the model. It is also shown that if the factors do not exactly equal the return on an efficient portfolio, but are still correlated, the APT also holds approximately we prove a bound on the difference.

1

Introduction

The Arbitrage Pricing Theory (Ross, 1976) is an important result in the financial economics literature. The result of Arbitrage Pricing Theory is that expected returns on any portfolio are approximately linear in factor loadings of K risk factors, E[Ri ] ≈ α + β1 γ1 + β2 γ2 + ... + βK γK , where (β1 , ..., βK ) is a vector of factor loadings for asset i, and (γ1 , ..., γK ) is a vector of factor risk premia. Roll (1977) showed that mean-variance efficiency of a portfolio is a sufficient and necessary condition for the capital asset pricing model (CAPM) to hold. This has been termed the “Roll Critique.” For any mean-variance efficient portfolio with expected return E[Rp ], then for all assets i, E[Ri ] − E[Rzp ] = βi (E[Rp ] − E[Rzp ]) where E[Rzp ] the unique mean-variance efficient portfolio satisfying Cov(Rp , Rzp ) = 0. We generalize Roll’s result for the APT and prove that the pricing equation of the APT is a mathematical identity. The expected return on any portfolio can be expressed exactly as the sum of K risk factors where at least two factors are sample mean-variance efficient portfolios. Given any dataset of returns, there exists an uncountable number of in-sample mean-variance efficient portfolios. Consequently, empirical tests of the APT where the set of possible factors include portfolio returns should always accept the model. An example is the Fama-French SMB and HML factors, which are returns on size and value portfolios respectively. We also show that if two factors are correlated to meanvariance portfolios, the APT also holds approximately and we provide a bound on the difference.

1

2

The Identity

Suppose there are N risky assets, each earning return R = (Ri )N i=1 . Denote the expected N return vector µ = (ERi )i=1 and variance-covariance matrix Σ = (E(Ri −µi )(Rj −µj ))N i,j=1 . We do not assume a risk-free asset exists. Let x = (x1 , ..., xN ) denote a vector of portfolio P T weights, with N i=1 xi = 1. The return on portfolio p is Rp = xp R, the expected return is ERp = xTp µ, variance V ar(Rp ) = xTp ΣV xp . A mean-variance efficient portfolio, xe , solves xe = arg min(xT ΣV x) subject to xe = ERe and xTe 1 = 1.

Theorem 1 (Roll’s Spanning Result) For any asset i, and for any nth moment efficient portfolio xe , we we have the relation E[Ri ] = E[Rze ] + βi (E[Re ] − E[Rze ]), where βi =

Cov(Ri ,Re ) V ar(Re )

(1)

and xze is any portfolio satisfying Cov(Rze , Re ) = 0.

Proof See section A of the appendix. Immediately we see that (1) is a K = 1 version of the APT, where α = E[Rzp ] and γ = E[Rp ] − E[Rzp ]. We shall construct a K factor version of (1).

Theorem 2 (APT Identity) For any any integer K ≥ 1 and and set of efficient portfolios, xe,k , the following relation holds, E[Ri ] = α + β1 γ1 + β2 γ2 + ... + βK γK , where γk = E[Re,k − Rze,k ], and xze,k is any portfolio satisfying Cov(Re,k , Rze,k ) = 0. Proof See section B of the appendix.

Corollary 1 If a risk-free asset rf exists, then equation (4) reduces to ERi − rf =

1 [βi,1 (ERe,1 − rf ) + βi,2 (ERe,2 − rf ) + ... + βi,K (ERe,K − rf )]. K

This follows immediately since Cov(rf , ERe,k ) = 0 for all k (and similarly for higher moments). Beyond this condition, choice of xze,k was arbitrary.

Theorem 3 (Correlated Factor Identity) Let K = 1 and Re be the return of an nth moment efficient portfolio. Then for any factor F with return RF and with Corr(RK , Re ) = ρ, then for all assets i, 1 E[Ri ] = α + βi,F γF + O −1 . |ρ|2 

where βi,F =



Cov(Ri , RF ) for n = 2, and similarly for n = 3, n = 4. V ar(RF )

Proof See section A.3 of the appendix. 2

Corollary 2 (Approximate APT Identity) Let K ≥ 1 and the factors k be correlated to nth moment efficient portfolios with correlation ρk = Corr(Rk , Re,k ). Then 

E[Ri ] = α + βi,1 γ1 + βi,2 γ2 + ... + βi,K γK − O where βi,k =

1 −1 mink |ρk |2



Cov(Ri , Rk ) . V ar(Rk )

Corollary 3 Suppose a risk-free asset exists with return rf . Then: 

E[Ri ] − rf = βi,1 γ1 + βi,2 γ2 + ... + βi,K γK − O where βi,k =

3

Cov(Ri ,Rk ) V ar(Rk )

and γk = (ERk −

1 ρk

r

V ar(Rk ) V ar(Re,k ) (rf

1 −1 mink |ρk |2



− ζ)).

Conclusion

We prove that the linear pricing result of the APT can be derived, without assumptions, by Roll (1977) spanning result of any two mean-variance efficient portfolios. This identity holds in any set of asset returns, and therefore empirical tests of the APT should always accept the APT model when portfolio returns are considered for factors. We also derive the APT equation when the factors do no equal efficient portfolios, but are instead correlated. The pricing error depends on the absolute value of correlation and has order O( |ρ1k | − 1).

A A.1

Appendix Proof of Theorem 1

Let xe denote any mean-variance efficient portfolio other than the minimum variance portfolio. Let Σ = (σi,j )N ×N denote the matrix of asset covariances. xe solves the optimization problem 1 xe = arg min xT Σx subject to xe = ERe , xTe 1 = 1. 2 The necessary and sufficient first order conditions for a solution are: Σxe = λµ + κ1

(2)

0

ERe = xe µ 0

1 = xe 1. Solving these FOCs for x0e yields xe = (λΣ−1 µ + κΣ−1 1), x0e µ = E[Re ] where λ and κ are λ=

1 1 (CE[Rp ] − A), γ = (B − AE[Rp ]) . D D 3

(3)

and where the scalars (A, B, C, D) are defined as 0

0

A = µ Σ−1 1, B = µ Σ−1 µ, C = 1Σ−1 1, D = BC − A2 . By definition of covariance, Cov(Rp , Rq ) = x0p Σxq . for all portfolios xp and xq . Suppose xp is efficient. Using the first order (2) it follows that Cov(Rp , Rq ) = (λΣ−1 µ + γΣ−1 1)0 Σxq = λµ0 Σ−1 Σxq + γ10 Σ−1 xq 0

= λµ xq + γ10 xq = λE[Rq ] + γ. Substituting in values of λ and γ yields the expression C A E[Rp ] − D C 

Cov(Rp , Rq ) =



E[Rq ] −

A C



+

1 . C

Let xzp denote any portfolio satisfying Cov(Rp , Rzp ) = 0. Such a portfolio always exists and lies on the frontier. The expected return E[Rzp ] is given by rearranging the covariance expression: D A C2 E[Rzp ] = − . A C E[Rp ] − C The variance of Rp is Cov(Rp , Rp ). Dividing the above by V ar(Rp ) yields Cov(Rp , Rq ) V ar(Rp ) Isolating E[Rq ] and setting

C D

=



E[Rp ] − C D

Cov(Rp ,Rq ) V ar(Rp )



C D



C D



1 C  A C

E[Rp ] −

E[Rp ] −

A C

βq V ar(Rp )

=

C D



E[Rq ] − A C

2

+

A C



+

1 C

1 C

= βq ,

βq V ar(Rp )

=



E[Rp ] −

βq V ar(Rp ) −

E[Rq ] =

A C

E[Rp ] −

A C

+

+

A C D A C2 − C E[Rp ] −

A C

 + E[Rzp ].

Working on the first term yields: βq V ar(Rp ) C D



E[Rp ] −

A C



= βq

C D



E[Rp ] −

C D



A C

2

E[Rp ] −

A C

+

1 C



D A C2 = βq E[Rp ] − + C E[Rp ] − = βq (E[Rp ] − E[Rzp ])

Substituting this this back in to , we get E[Rq ] = βq (E[Rp ] − E[Rzp ]) + E[Rzp ]. 4

! A C

A.2

Proof of Theorem 2

Pick any K efficient portfolios; (1) holds true for each one, so summing over K yields KE[Ri ] =

K X

E[Rzpk ] +

K X

βi,k (E[Rpk ] − E[Rzpk ])

k=1

k=1

Divide by K: E[Ri ] = and setting α =

1 K

K K X 1 X E[Rpk ] − E[Rzpk ] βi,k E[Rzpk ] + K k=1 K k=1

PK

k=1 E[Rzpk ],

(E[Rpk ] − E[Rzpk ]) = Kγi gives us

E[Ri ] = α + β1 γ1 + β2 γ2 + ... + βK γK

(4)

where the factor loadings reflect the normalised covariance between Ri and market factor Rpk .

A.3

Proof of Theorem 3

By Theorem 1, we have E[Ri ] = α + β1 γ1 where βi γ1 =

Cov(Ri ,Re,1 ) V ar(Re,1 ) [ERe

− ERze ]. Since Corr(RK , Re ) = ρ, we have Re = ζ + ξRF + ε

where Eε = 0, V ar(ε) = 9ξ Plugging this in yields\ βi γ1 = = =



1 |ρ|2



− 1 , and Cov(RF , ε) = 0; see appendix for proof.

ξCov(Ri , RF ) + Cov(Ri , ε) [ζ + ξERF − ERze ] ξ 2 V ar(RF ) + V ar(ε) Cov(Ri , ε) Cov(Ri , RF ) 1 [ERe,1 − ERZe,1 ] + [ERF − (ERze − ζ)] V ar(Re,1 ) V ar(RF ) + V ar(ε)/ξ 2 ξ 1 Cov(Ri , ε) Cov(Ri , RF )   [ERF − (ERze − ζ)] [ERe,1 − ERZe,1 ] + V ar(Re,1 ) ξ V ar(RF ) + 9 1 2 − 1 |ρ|

Using the fact that

a b+c

=

a b

Cov(Ri , RF ) V ar(RF ) + 9



1 |ρ|2



−1

−

ac b(b+c) ,

=

Cov(Ri , RF )



1 |ρ|2

−1



Cov(Ri , RF )   − V ar(RF ) V ar(RF )2 + 9V ar(RF ) |ρ|1 2 − 1 

= βi,F − O

1 −1 |ρ|2



so Cov(Ri , ε) 1 1 [ERe,1 − ERZe,1 ] + βi,F [ERF − (ERze − ζ)] − O −1 V ar(Re,1 ) ξ |ρ|2 

β1,i γ1 =

If we assume Cov(Ri , ε) = 0, then we have 1 E[Ri ] = α + βi,F γF − O −1 . |ρ|2 

where βi,F =

Cov(Ri ,RF ) V ar(RF ) ,

γF = ERF − 1ξ (ERze − ζ). 5





References [1] Ross, S., “The Arbitrage Theory of Capital Asset Pricing,” Journal of Economic Theory, 13, 341-360, 1976. [2] Roll, R., “A critique of the asset pricing theory’s tests Part I: On past and potential testability of the theory,” Journal of Financial Economics, 4,2, 129-176, 1977.

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