Ross [1],[]) presents the aritrage pricing theory. The idea is that the structure of asset returns leads naturally to a odel of risk preia, for otherwise there would exist an opportunity for aritrage profit. Factor Model Assue that there exists a risk-free asset, and consider a factor odel for the excess return ξ on a set of assets: ξ = + B f + e. The ean excess return is the vector of risk preia. 1 Here and Ef)=0 Varf)=I, Ee)=0 Var e)=d. Here D is diagonal, and f and e are uncorrelated. One refers to f as factors. The eta coefficients B are also called factor loadings. 3 Systeatic Versus Non-Systeatic Risk Assue that ost of the coponents of B are not near zero. The diagonal eleents of D are not too large, and the nuer of assets n is large. Then the ter B f represents ost of the variation in the returns. Interpret B f as the systeatic risk, and e as the non-systeatic risk. One can argue that the non-systeatic risk can e eliinated y diversification, so the eta coefficients B should deterine the risk preiu. 4 Intuitive Arguent Ross gives the following intuitive arguent. Consider a portfolio x. Each coponent denotes the fraction of wealth invested in that asset, and 1 1 x is the fraction invested in the risk-free asset. The excess return on the portfolio is ξ x = x + f Bx + e x. Suppose that the portfolio is well-diversified: ost of the coponents of x are non-zero. By the law of large nuers, e x 0; diversification eliinates the non-systeatic risk. If the portfolio is chosen to eliinate the systeatic risk Bx = 0), then the resulting portfolio is nearly risk-free. Then the law of one price iplies x = 0. 5 6
Ross suarizes his arguent y the following: Bx = 0 x = 0. 1) Of course this arguent is not valid for an aritrary portfolio ut only for a well-diversified portfolio.) Exact Factor Model Consider first an exact factor odel, in which e = 0 so D = 0). Reark 1 In the exact factor odel, the law of one price is equivalent to the condition 1). 7 8 Law of One Price For the exact factor odel, the law of one price 1) says that is orthogonal to NB). By the fundaental theore of linear algera, ust lie in R B ). Thus we otain the following theore. Theore ) In the exact factor odel, the law of one price holds if only if the ean excess return is a linear coination of the eta coefficients, for soe. = B, ) 9 10 The Versus the Capital-Asset Pricing Model Like the capital-asset pricing odel, the systeatic risk eodied in the eta coefficients deterines the risk preia. However the reasoning is different. The capital-asset pricing odel is derived fro arket equiliriu, the equality of asset deand and supply. This equality iplies that the arket portfolio ust e efficient, and a typical investor holds the arket portfolio. In contrast, the aritrage pricing theory is derived fro an aritrage arguent, not a arket equiliriu arguent. The risk preia ) follow fro the factor structure of the asset returns. Asset supply is irrelevant to the arguent. If soe set of asset returns has the factor structure, then the conclusion follows for this set. 11 1
Weighted Least Squares We next suppose that the factor odel is not exact, that e 0. Then any value for is consistent with the law of one price the only portfolio with a constant excess return is x = 0). Nevertheless we put forward a duality arguent that ) isa good approxiation. We choose y weighted least squares. Prole 3 Prial) in [ 1 B ) D B ) ]. The prial is a weighted regression of the ean on the eta coefficients. The arguent is that the value is sall, for the optiu. 13 14 Dual An alternate axiization prole is dual to the prial. Prole 4 Dual) sup 1 D δ B =0 ). By definition, the indicator function δ is zero if the elongs to the set such that B = 0, and is otherwise. The prial and the dual are equivalent proles, in that either one can e calculated fro the other, and we explain their relationship. 15 The ojective function of the prial is jointly convex in and, and it follows that the value function V ) is convex. The choice variale is, and the perturation variale is. 16 Conjugate Definition 5 Conjugate) The conjugate of V ) is V ) :[, V )]. The conjugate is a convex function. Conjugate Duality Proposition 6 Conjugate Duality) Under general conditions, the conjugate of the conjugate is the original function, V )=V ). In atheatics, conjugate has any definitions, ut always the conjugate of the conjugate is the original; an exaple is the coplex conjugate. 17 18
Dual Definition 7 Dual) For a prial with value function V ), the dual is the axiization prole sup [, V )]. By conjugate duality, the optiu value in the dual is V ). Theore 8 No Duality Gap) The iniu value in the prial is the axiu value in the dual. 19 Calculation of the Dual fro the Prial Let us derive the dual prole 4) y calculating the conjugate: V ), [, V )] { [ 1, in { [, + sup 1 [, 1 B ) ) ]} D B B ) ) ]} D B B ) D B 0 ) ] Sustituting c := B separates the axiization into two parts: V ),c c,c + B 1 ) c D c,c 1 ) c D c = 1 D + δ B =0, + sup B, Prial Greater than or Equal to the Dual For this prole, let us verify directly the asic duality properties. Always the value of the prial is greater than or equal to the value of the dual. If B 0, then δ B =0 =, so the value of the dual is. Necessarily the value of the prial is greater than or equal to the value of the dual. otaining the dual prole 4). 1 If B = 0, then the prial less the dual is 1 B ) ) D B [, 1 ] D = 1 [ ) D B ] [ ) D D B ] 0 3) a su of squares. Here,B = B, = 0, = 0. Again, the value of the prial is greater than or equal to the value of the dual. 3 Prial Equal to the Dual In the prial, the first-order condition for a iniu is ) BD B = 0. Given the solution to the prial, solve the dual y setting ) = D B. By the first-order condition, B = 0, so the dual constraint is satisfied. Furtherore, the quadratic for 3) is zero. That the value of the prial equals the value of the dual proves that indeed we have the solution to oth proles. 4
Envelope Theore Applying the envelope theore to the prial yields ) V )/ = D B =, in which is the solution to the prial and is the solution to the dual. This relationship is a general duality result: the solution to the dual shows how the perturation variale affects the optiu value. The solution to the dual is a Lagrange ultiplier. First-Order Condition Osolete Even though this verification akes use of the first-order condition, a thee of duality theory is that the first-order condition is osolete. Because there is no duality gap, one can solve the prial and the dual siultaneously, y setting the prial equal to the dual. A systeatic procedure then finds the optiu values for the choice variales in the prial and the dual, to achieve this equality. 5 6 Econoic Interpretation of the Dual The dual has an econoic interpretation. The choice variale is a portfolio of investents in the risky assets, with 1 1 as the investent in the risk-free asset. The constraint B = 0 says that the portfolio is chosen to e uncorrelated with the factors; the variaility of the return arises solely fro the non-systeatic risk. Thus is the excess return on the portfolio, and D is the variance of the return. 7 The dual reseles the derivation of the separation theore with sall risks, in which the ojective function is a linear function of the ean excess return and the variance. Just as for the separation theore, the solution to the dual axiizes the ratio of the ean excess return to the standard deviation, D, 4) here suject to the constraint that the portfolio return is uncorrelated with the factors. Furtherore, the square of the axiu value of this ratio is the optiu value of the dual. 8 Efficient Frontier Define s as the axiu value of the ratio 4). It is the slope of an efficient frontier, suject to the constraint that the portfolio return is uncorrelated with the factors. The slope s of the efficient frontier is of course greater than or equal to s. Upper Bound to the Weighted Su of Squares As there is no duality gap, in [ 1 B ) D B ) ] = s s. 5) Thus the slope s of the efficient frontier provides an upper ound to the weighted su of squares. The slope provides an upper ound to how far the predicted ean B can deviate fro the actual ean. 9 30
Many Assets A key property is that this upper ound is independent of the nuer n of assets. Conclusion 9 ) If the nuer of assets is large, it follows that for ost assets. B 6) Otherwise the upper ound would e violated. The approxiation ay e poor for a few assets, ut for ost assets the approxiation ust e excellent. 31 Trivial Case Note that this conclusion holds even for the trivial case B = 0, for which B = 0. Then the duality relation 5) says that so 1 D = s s, 0 is a good approxiation. For ost assets, the ean excess return is near zero. 3 Irrelevance of Non-Systeatic Risk? Ross s point of view is that the error e is non-systeatic risk, and this risk should e eliinated y portfolio diversification. Hence the non-systeatic risk should have no effect on ean returns. If this point of view is true, then s should e sall, sall even if s is large. By the duality relation 5), it would then follow that the approxiation 6) would e extreely good. Diagonal Variance That the variance of the error e is diagonal is iportant, and allows one to see the error as non-systeatic risk. The duality relation 5) holds regardless of whether D is in fact diagonal. If the coponents of e were highly correlated, then the approxiation 6) ight e poor for any assets. For exaple, for the trivial case B = 0, there is no presuption that ost of the coponents of should e near zero. 33 34 References [1] S. Ross. Return, risk, and aritrage. In I. Friend and J. L. Bicksler, editors, Risk and Return in Finance, pages 189 18. Ballinger, Caridge, MA, 1977. HG4539R57. [] S. A. Ross. The aritrage theory of capital asset pricing. Journal of Econoic Theory, 133):341 360, Deceer 1976. HB1J645. 35