02106cam a22002417 4500001000600000003000500006005001700011008004100028100002300069245012800092260006600220490005100286500001500337520111700352530006101469538007201530538003601602700002501638710004201663830008601705856003701791856003601828t0015NBER20150523034947.0150523s1981 mau||||fs|||| 000 0 eng d1 aChamberlain, Gary.10aArbitrage and Mean-Variance Analysis on Large Asset Marketsh[electronic resource] /cGary Chamberlain, Michael Rothschild. aCambridge, Mass.bNational Bureau of Economic Researchc1981.1 aNBER technical working paper seriesvno. t0015 aJuly 1981.3 aWe 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. aHardcopy version available to institutional subscribers. aSystem requirements: Adobe [Acrobat] Reader required for PDF files. aMode of access: World Wide Web.1 aRothschild, Michael.2 aNational Bureau of Economic Research. 0aTechnical Working Paper Series (National Bureau of Economic Research)vno. t0015.4 uhttp://www.nber.org/papers/t001541uhttp://dx.doi.org/10.3386/t0015