A simple approach to arbitrage pricing theory

JOURNAL

OF ECONOMIC

THEORY

28, 183-191

(1982)

A Simple Approach to Arbitrage Pricing Theory GUR HUBERMAN * Graduate

School of Business, University Chicago, Illinois 60637

Received

August

18, 1980; revised

of Chicago.

March

16, 1981

1. INTRODUCTION The arbitrage theory of capital assetpricing was developed by Ross [9, 10, 1l] as an alternative to the mean-variance capital asset pricing model (CAPM), whose main conclusion is that the market portfolio is meanvariance efficient. Its formal statement entails the following notation. A given asseti has mean return Ei and the market portfolio has mean return E, and variance ui. The covariance between the return on asseti and the return on the market portfolio is uim, and the riskless interest rate is r. The CAPM assertsthat E,=rfAb,,

(1.1)

where A=E,-r,

and bi = ai,,&;

(1.2)

is the “beta coefficient” of asset i. Normality of the returns of the capital assetsor quadratic preferences of their holders are the assumptions which lead to (1.1~(1.2). Theoretically and empirically it is difftcult to justify the assumptions of the CAPM. Moreover, the CAPM has been under strong criticism becauseof its dubious empirical content (cf. [7]). The market portfolio is practically not obser* This is a modified version of the third chapter in my Ph.D. dissertation which was written at Yale University. I am grateful to Gregory Connor. who inspired some of the ideas in the paper. Comments from my advisor, SteveRoss, as well as from Jon Ingersoll, Uriel Rothblum and Michael Rothschild were helpful in my attempts to bring this work to a lucid form. This research was supported by NSF Grants ENG-78-25 182 and SOC-77-22301.

183 0022.0531/82/050183-09%02.00/0 Copyright ‘$ 1982 by Academic Press. Inc. All rights of reproduction in any form reserved.

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GUR

HUBERMAN

vable, and a statement on the market portfolio (such as the CAPM) is difficult to test empirically. Yet the linear relation (1.1) is appealing in its simplicity and in its ready interpretations. The arbitrage pricing theory [ 10, 1I ] is an alternative theory to mean-variance theories, an alternative which implies an approximately linear relation like (1. I). In [ IOJ Ross elaborated on the economic interpretation of the arbitrage pricing theory and its relation to other models, whereas in [ 1I] he provided a rigorous treatment of the theory. Recent interest in the APT is evident from papers elaborating on the theory (e.g., Chamberlin and Rothschild [I], Connor [4] and Kwon [5, 61) as well as on its empirical aspects (e.g., Chen [2, 31 and Roll and Ross 181). The main advantage of Ross’ arbitrage pricing theory is that its empirical testability does not hinge upon knowledge of the market’s portfolio. Unfortunately, Ross’ analysis is difftcult to follow. He does not provide an explicit definition of arbitrage and his proof-unlike the intuitively appealing introductory remarks in ] 11 ]-involves assumptions on agents’ preferences as well as “no arbitrage” assumptions. Here arbitrage is defined and the intuition is formalized to obtain a simple proof that no arbitrage implies Ross’ linear-like relation among mean returns and covariances. The main lines of the proof are illustrated in the following paragraphs. Consider an economy with n risky assets whose returns are denoted by .Xi (i = l,..., n) and they are generated by a factor model

2i=Ei+pi6+Ci

(i = I)...%n),

(1.3)

where the expectations E $= ECi = 0 (i = l,..., n), the ti are uncorrelated and their variances are bounded. Relying on results from linear algebra, express the vector E (whose ith component is Ei) as a linear combination of the vector e (whose ith component is l), the vector /3 (whose ith component is pi) and a third vector c which is orthogonal both to e and to /I. i In other words, one can always find a vector c such that

E=pe+@+c,

(1.4)

where p and y are scalars, ec 3

6

zl

ci =

0,

(1.5)

and (1.6) ’ Gregory

Connor

used this idea in an earlier

work

of his 141.

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PRICING

Next, consider a portfolio which is proportional to c, namely ac (a is a scalar). Note that it costs nothing to acquire such a portfolio because its components (the dollar amount put into each asset) sum to zero by (1.5). We shall call such a portfolio an arbitrage portfolio. Also, by (1.6) this is a zero-beta portfolio. The return on this portfolio is n

a.+==

\‘ Zici=a iE1

n

n

1 cf+a

1 ci.Fi,

i=l

i-l

(I.71

by virtue of the decomposition (1.4) and the orthogonality relations (1.5) and (1.6). It is important to notice that the expected return on the portfolio ac is proportional to a (and Cy=, cf), whereas an upper bound on the variance of its return is proportional to a* (and CL, cf). Suppose now that the number of assets n increases to infinity. Think of arbitrage in this environment as the oppurtunity to create a sequence of arbitrage portfolios whose expected returns increase to infinity while the variances of their returns decrease to zero. If the sum C;= i c: increased to infinity as n did, then one could find such arbitrage opportunities as follows. Set a = l/(x;= i c;)~‘~ and use the portfolio ac. The reason why such a choice of a will create the arbitrage is that the expected return on the portfolio is proportional to a (and with a = l/(Cy=i ~f)~‘~ it equals a Cy=i cf = (Cy=i c:)“~), while its variance is proportional to a* (and with a = l/(Cr=, ~f)“~ it equals a2 x1= 1cf = l/(Cr=i cf)“2). Therefore, if there are no arbitrage opportunities (as described above) the sum Cr= i CT cannot increase to infinity as n does. In particular, when the number of assets n is large, most of the cts are small and approximately zero. Going back to the original decomposition (1.4) we conclude that Ei x p + ypi for most of the assets. When motivating his proof, Ross [ 11, p. 3421 emphasized the role of “well-diversified” arbitrage portolios. He indicated that the law of large numbers was the driving force behind the diminishing contribution of the idiosyncratic risks Ei to the overall risks of the arbitrage portfolios. The portfolios presented above, ac, need not be well diversified, but they satisfy the orthogonality conditions (1.5) and (1.6). It is the judicious choice of the scalar a that enables us to apply an idea, which is in the spirit of the proof of the law of large numbers. Section 2 of this paper pesents the formal model, a precise statement of the result and a rigorous proof. In the closing section an attempt is made to interpret the linear-like pricing relation and to justify the no-arbitrage assumption in an equilibrated economy of von Neumann-Morgenstern expected utility maximizers.

186

GURHUBERMAN

2. ARBITRAGE

PRICING

The arbitrage pricing theory considers a sequence of economies with increasing sets of risky assets.In the lzth economy there are n risky assets whose returns are generated by a k-factor model (k is a fixed number). Loosely speaking, arbitrage is the possibility to have arbitrarily large returns as the number of available assetsgrows. We will show that in the absenceof arbitrage a relation like (1.1) holds, namely (2.9). Formally, in the nth economy, we consider an array of returns on risky assets (x”: : i = I,..., n). These returns are generated by a k-factor linear model of the form

~I=El+Pi;S”;+P~~~+...+Pi:,~+E”in

(i = 1, 2,..., n),

(2.1)

where

E@=O

(j = l,..., k),

EE;zJ’ = 0 and Var ,$’ < (T2

Et;=0 if

(i = I,..., n),

i#j, (i = l,..., n),

(2.2) (2.3) (2.4)

where 6’ is a fixed (positive) number. Using standard matrix notation we can rewrite (2.1) as

x’“=E”+/T’8”+P,

(2.5)

where p” is the n x k matrix whose elements are /I; (i = l,..., n; j = l,..., k). A portfolio c” E R” in the nth economy is an arbitrage portfolio if Fe” = 0, where e” = (1, l,..., 1) E R ‘. The return on a portfolio c is

z?(c)= CA?”= cE” + c/?” 6” + cc?‘.

(2.6)

Arbitrage is the existence of a subsequencen’ of arbitrage portfolios whose returns Zl(P’) satisfy iim Ef(c”‘) n’-rcc

= +oo,

(2.7)

and lim Var Z(c” ‘) = 0. n’+cc

(2.8)

In Section 3 we relate (2.7)-(2.8) to standard probabilistic convergence concepts, and discuss how von Neumann-Morgenstern expected utility maximizers view (2.7)-(2.8).

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In Theorem 1 we show that the absenceof arbitrage implies an approximation to a linear relation like (1.1). THEOREM 1. Suppose the returns on the risky investments satis(2.1 j-(2.4) and there is no arbitrage. Then for n = 1, 2,..., there exists p”, yy ,..., yz, and an A such that

2

E;-p”-

i=l

T’p;yj”

for

n = 1, 2,...

(2.9)

Jr,

Proof. Using the orthogonal projection of E” into the linear subspace spanned by e” and the columns of /I”, one obtains the representation E” = p”e + /?“y” + c”,

(2.10)

where y” E Rk, e”c” = 0,

(2.11)

p*cn= 0.

(2.12)

and

Note that 11 c” ]I* = x1= i (c~)’ = JJf=, (Ey - p” - Cj”=, yjn&)‘, and assumethat the result is false. Consequently, there is an increasing subsequence (n’) with lim I(c”‘I\ = sco n’+m Let p be fixed between -1 and -l/2, where

(2.13)

and consider the portfolio d”’ = a,a”‘,

a,. = IIc~‘I(~~.

(2.14)

By (2.1 l), d”’ is an arbitrage portfolio for each n’. Use (2.10)-(2.12) to see that its return z’(d”‘) = a,,, IIc”‘j12 + a,,,cn’?‘.

(2.15)

Ef(d”‘) = a,( JIc”‘l(’ = I(c”‘l\*+*p,

(2.16)

lim EZ(d”‘) = +a. n’-02

(2.17)

Note that

so (by (2.13)-(2.14)),

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GUR HUBERMAN

On the other hand (using (2.3), (2.4)) Var i?(d”‘) < 02ai, (]c”‘]J’ = s2 ]Jc”‘)(~+~~,

(2.18)

so (by (2.13)), lim Var z’(d”‘) = 0, n’-cc thus completing the proof.

1

Next, consider a stationary model, in which El = Ei and ,Bi =pij for all i, j and n. In other words, (2.5) is replaced by x”“=E+p~++P.

(2.5’)

The stationary model is the one considered originally by Ross ] 111. The nonstationary model is more general than the stationary model but its result (2.9) is not as elegantly presentable as the result in the stationary case (2.9’). THEOREM

(2.2t(2.4),

2. Suppose the returns on the risky investments satisfy (2.5’), and there is no arbitrage. Then there exist p, y1,..., yk such that ? Ei-p-tpijyj ,c, c j=l

)

2<*.

(2.9’)

ProoJ Consider the n x (k + 1) matrix B” whose (i, j) entry is 1 if j = 1 and pij-, if 1 3. Let n > fi be fixed. By permuting the columns of B” we may assume that its first r(s) columns can be expressed as linear combinations of the first r(6) columns. Define the set H” by 2

H” =

@, Y,,..., yk): f I


i=l

for

r(ii)
i

,

where A is the A whose existence was assertedin Theorem 1. Note that H” is nonempty (by Theorem l), compact for n > E and H” c H”“. Therefore, C-l:=, H” is nonempty. Since every k + 1 tuple @, y, ,..., yk) E OF=, H” satisfies (2.9’), the proof is complete. 1 Finally, we turn attention to the case where a risk free asset exists, i.e., where there is an additional assetin the nth economy, whose return, say, xi, satisfies (2.19) xg = r”0’

ARBITRAGE

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PRICING

Now look at excess returns of the risky assets(excessrelative to the riskless rate), i.e., at JyEX”y-r;, i = 1, 2)...) It. Note that any arbitrage portfolio (co, c, ,..., c,) E R”+ I of x,“, 2; ,..., 2:: (which of course satisfies CrZO ci = 0) is equivalent to a vector cc, ,..., c,) E R” indicating a wealth allocation among the risky assets.Using this idea one can go through the same analysis as in Theorem 1 with the excess returns vector y”, the decomposition (2.10) replaced by (2.100

En-ro”e=/3”y”+c”, and (2.11) deleted. Consequently, one has

Suppose the returns on the risky investments satisfy (2.1)-(2.4), there is a risk free asset satisfying (2.19) and there is no arbitrage. Then there exist y:, y:,..., yZ such that COROLLARY.

e El-r,“-G/jCyj”
i5) Remark.

for

n = 1, 2,... .

(2.20)

jY1

Analogously, a similar result holds for the stationary model.

3. DISCUSSION The interpretation of (2.9) or (2.9’) is straightforward: for most of the assets in a large economy, the mean return on an asset is approximately linearly related to the covariances of the asset’sreturns with economy-wide common factors. As the number of assetsbecomes large, the linear approximation improves and most of the assets’ mean returns are almost exact linear functions of the appropriate covariances. Next, consider the probabilistic implications of arbitrage returns satisfying (2.7)-(2.8). Given a sequence of random returns 4~“‘) which satisfy (2.7~(2.8), we can apply Chebychev’s inequality to see that along this sequence lim,,,, f(c”‘) = +co in probability (i.e., for all M > 0, limn,,, Pr(F(c”” > M} = 1). Furthermore, along a subsequencen^,a stronger convergence holds: lim,,, z”(c”^)= +a~ almost surely (i.e., for all M > 0, Pr{lim inf,+, z’(c”‘) > M} = 1). Are arbitrage portfolio which satisfy (2.7~(2.8) desirable for an expected utility maximizer? In other words, do (2.7~(2.8) suffice to assert that limn-roO EU(.F(c”)) = U(+co) for any monotone concave utility function (I? The negative answer is illustrated by the following examples.

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GURHUBERMAN

The first example considers a utility function which is -a~ for nonpositive wealth levels, whereas the second example is for an exponential utility which takes finite values for finite wealth levels. 1. The returns I?(P) are 0, n and 2n with probabilities l/n3, 1 - 2/n3, and l,/n3, respectively. The utility function V(X) =- I/X for x > 0 and U(X) = -co for x < 0. Then EC@(F)) = -co although Z(P) satisfies (2.7)-(2.8). 2. The returns .5(F) are -n, n, and 3n with probabilities l/n3, 1 - 2/n3, and l/n3, respectively. The utility function is U(x) = -exp(-x). Then EU(F(c”)) < -n3 exp(n), so lim,,, EU(z’(c”)) = -co, although (2.7)-(2.8) are met. General conditions which assert that (2.7 j(2.8) imply lim.+,EU(F(c”‘)) = U(+co) are not known. As shown in [ll, Appendix 21, utility functions which are bounded below or uniformly integrable utility functions will possessthis property. We conclude that one needs to make assumptions on agents’ preferences in order to relate existence of equilibria to absenceof arbitrage. This task is beyond the scope of this paper. However, it is straightforward to seethat if the economies satisfy the assumptionsmade by Ross (see [Ill, especially the first paragraph on p. 349 and Appendix 2), then no arbitrage can exist. In fact, a result of the type “no arbitrage implies a certain behavior of returns,” should involve no consideration of the preference structure of the agents involved. Our analysis is in this spirit, becauseit involves no assumptionson utilities. Other than the simple proof, this may be another contribution of this work.

1. G.

CHAMBERLAIN

asset markets,

AND M. ROTHCHILD, Arbitrage and mean-variance analysis of large mimeo. University of Wisconsin-Madison, 1980. Empirical evidence of the arbitrage pricing theory, mimeo, UCLA. 1980. The arbitrage pricing theory: Estimation and applications, mimeo. UCLA,

2. NAI-FU CHEN, 3. NAI-FU CHEN. 1980. 4. G. CONNOR, “Asset Prices in a Well-Diversified Economy,” Technical Report No. 47, Yale School of Organization and Management, July 1980. 5. Y. KWON. Counterexaples to Ross’ arbitrage asset pricing model, mimeo. University of Kansas, 1980. 6. Y. KWON, “On the negligibility of diversifiable risk components at the capital market equilibrium,” mimeo, University of Kansas, 1980. 7. R. ROLL, A critique of the asset pricing theory’s tests: Part I: On past and potential testability of the theory, Financial Econ. 4 (1977), 129-179. 8. R. ROLL AND S. ROSS, An empirical investigation of the arbitrage pricing theory. J. Fhance 35 (1980) 1073-l 103.

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9. S. Ross, “The General Validity of the Mean-Variance Approach in Large Markets,” Discussion Paper No. 12-72, Rodney L. White Center for Financial Research, University of Pennsylvania. 10. S. Ross, Return, risk and arbitrage, in “Risk and Return in Finance” (I. Friend and J. Bicksler, IS.), Ballinger, Cambridge, Mass., 1977. 11. S. ROSS, The arbitrage theory of capital asset pricing, J. Econ. Theory 13, No. 3 (1976) 341-360

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