03132cam a22003137 4500001000700000003000500007005001700012008004100029100002300070245008400093260006600177490004200243500001900285520140000304530006101704538007201765538003601837690007401873690009401947690011202041690012702153690014102280690018502421700001802606710004202624830007702666856003802743856003702781w14525NBER20160213172313.0160213s2008 mau||||fs|||| 000 0 eng d1 aBrennan, Thomas J.10aImpossible Frontiersh[electronic resource] /cThomas J. Brennan, Andrew W. Lo. aCambridge, Mass.bNational Bureau of Economic Researchc2008.1 aNBER working paper seriesvno. w14525 aDecember 2008.3 aA key result of the Capital Asset Pricing Model (CAPM) is that the market portfolio---the portfolio of all assets in which each asset's weight is proportional to its total market capitalization---lies on the mean-variance efficient frontier, the set of portfolios having mean-variance characteristics that cannot be improved upon. Therefore, the CAPM cannot be consistent with efficient frontiers for which every frontier portfolio has at least one negative weight or short position. We call such efficient frontiers "impossible", and derive conditions on asset-return means, variances, and covariances that yield impossible frontiers. With the exception of the two-asset case, we show that impossible frontiers are difficult to avoid. Moreover, as the number of assets n grows, we prove that the probability that a generically chosen frontier is impossible tends to one at a geometric rate. In fact, for one natural class of distributions, nearly one-eighth of all assets on a frontier is expected to have negative weights for *every* portfolio on the frontier. We also show that the expected minimum amount of shortselling across frontier portfolios grows linearly with n, and even when shortsales are constrained to some finite level, an impossible frontier remains impossible. Using daily and monthly U.S. stock returns, we document the impossibility of efficient frontiers in the data. aHardcopy version available to institutional subscribers. aSystem requirements: Adobe [Acrobat] Reader required for PDF files. aMode of access: World Wide Web. 7aG1 - General Financial Markets2Journal of Economic Literature class. 7aG11 - Portfolio Choice • Investment Decisions2Journal of Economic Literature class. 7aG12 - Asset Pricing • Trading Volume • Bond Interest Rates2Journal of Economic Literature class. 7aG14 - Information and Market Efficiency • Event Studies • Insider Trading2Journal of Economic Literature class. 7aG23 - Non-bank Financial Institutions • Financial Instruments • Institutional Investors2Journal of Economic Literature class. 7aG32 - Financing Policy • Financial Risk and Risk Management • Capital and Ownership Structure • Value of Firms • Goodwill2Journal of Economic Literature class.1 aLo, Andrew W.2 aNational Bureau of Economic Research. 0aWorking Paper Series (National Bureau of Economic Research)vno. w14525.4 uhttp://www.nber.org/papers/w1452541uhttp://dx.doi.org/10.3386/w14525