TY - JOUR
AU - Chamberlain, Gary
AU - Rothschild, Michael
TI - Arbitrage and Mean-Variance Analysis on Large Asset Markets
JF - National Bureau of Economic Research Technical Working Paper Series
VL - No. 15
PY - 1981
Y2 - July 1981
DO - 10.3386/t0015
UR - http://www.nber.org/papers/t0015
L1 - http://www.nber.org/papers/t0015.pdf
N1 - Author contact info:
Gary Chamberlain
Department of Economics
Littauer Center 123
Harvard University
Cambridge, MA 02138
Tel: 617/495-1869
Fax: 617/495-8570
E-Mail: gary_chamberlain@harvard.edu
Michael Rothschild
531 14th Street
Santa Monica, CA 90402
Tel: 310-394-6010
Fax: 310-593-4401
E-Mail: mrothsch@princeton.edu
AB - We examine the implications of arbitrage in a market with many assets. The absence of arbitrage opportunities implies that the linear functionals that give the mean and cost of a portfolio are continuous; hence there exist unique portfolios that represent these functionals. The mean variance efficient set is a cone generated by these portfolios. Ross [16, 18J showed that if there is a factor structure, then the distance between the vector or mean returns and the space spanned by the factor loadings is bounded as the number of assets increases. We show that if the covariance matrix of asset returns has only K unbounded eigenvalues, then the corresponding K eigenvectors converge and play the role of factor loadings in Ross' result. Hence only a principal components analysis is needed to test the arbitrage pricing theory. Our eigenvalue conditional can hold even though conventional measures of the approximation error in a K factor model are unbounded. We also resolve the question of when a market with many assets permits so much diversification that risk-free investment opportunities are available.
ER -