Critical Finance Review > Vol 4 > Issue 1

Reply to “(Im)Possible Frontiers: A Comment”

Thomas J. Brennan, Northwestern University, USA, Andrew W. Lo, MIT Sloan School of Management, USA,
Suggested Citation
Thomas J. Brennan and Andrew W. Lo (2015), "Reply to “(Im)Possible Frontiers: A Comment”", Critical Finance Review: Vol. 4: No. 1, pp 157-171.

Published: 29 Jun 2015
© 2015 T. J. Brennan and A. W. Lo
Financial markets: Portfolio theory
CAPMMean-Variance AnalysisPortfolio OptimizationRoll CritiqueShortsellingLong/Short

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In this article:
1. The Set of Possible Portfolios is not Dense
2. Finding the Closest Frontier that is Possible
3. Limitations of LR’s Impossible-to-possible Transformation
4. Ingersoll’s Critique
5. Conclusion


In Brennan and Lo (2010), a mean-variance efficient frontier is defined as “impossible” if every portfolio on that frontier has negative weights, which is incompatible with the Capital Asset Pricing Model (CAPM) requirement that the market portfolio is mean-variance efficient. We prove that as the number of assets n grows, the probability that a randomly chosen frontier is impossible tends to one at a geometric rate, implying that the set of parameters for which the CAPM holds is extremely rare. Levy and Roll (2014) argue that while such “possible” frontiers are rare, they are ubiquitous. In this reply, we show that this is not the case; possible frontiers are not only rare, but they occupy an isolated region of mean-variance parameter space that becomes increasingly remote as n increases. Ingersoll (2014) observes that parameter restrictions can rule out impossible frontiers, but in many cases these restrictions contradict empirical fact and must be questioned rather than blindly imposed.