Arbitrage and Mean-Variance Analysis on Large Asset Markets

58 Pages Posted: 28 Dec 2006 Last revised: 19 Jul 2010

See all articles by Gary Chamberlain

Gary Chamberlain

Harvard University - Department of Economics; National Bureau of Economic Research (NBER)

Michael Rothschild

Princeton University; National Bureau of Economic Research (NBER)

Date Written: July 1981

Abstract

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.

Suggested Citation

Chamberlain, Gary and Rothschild, Michael, Arbitrage and Mean-Variance Analysis on Large Asset Markets (July 1981). NBER Working Paper No. t0015. Available at SSRN: https://ssrn.com/abstract=579727

Gary Chamberlain (Contact Author)

Harvard University - Department of Economics ( email )

Littauer Center
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Michael Rothschild

Princeton University ( email )

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United States
5102509635 (Phone)

National Bureau of Economic Research (NBER) ( email )

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United States

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