TY  - JOUR
AU  - Chamberlain,Gary
AU  - Rothschild,Michael
TI  - Arbitrage, Factor Structure, and Mean-Variance Analysis on Large Asset Markets
JF  - National Bureau of Economic Research Working Paper Series
VL  - No. 996
PY  - 1984
Y2  - 1984
UR  - http://www.nber.org/papers/w0996
L1  - http://www.nber.org/papers/w0996.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. These portfolios span the mean-variance efficient set. We resolve the question of when a market with many assets permits so much diversification that risk-free investment opportunities are available. Ross 112, 141 showed that if there is a factor structure, then the mean returns are approximately linear functions of factor loadings. We define an approximate factor structure and show that this weaker restriction is sufficient for Ross' result. If the covariance matrix of the asset returns has only K unbounded eigenvalues, then there is an approximate factor structure and it is unique. The corresponding K eigenvectors converge and play the role of factor loadings. Hence only a principal component analysis is needed in empirical work.
ER  -