# Reply to â€œ(Im)Possible Frontiers: A Commentâ€

*Thomas J. Brennan* and
*Andrew W. Lo*

*Critical Finance Review*, 2015, vol. 4, issue 1, 157-171

**Abstract:**
In Brennan and Lo (2010), a mean-variance efficient frontier is defined as â€œimpossibleâ€ 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 â€œpossibleâ€ 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.

**Keywords:** CAPM; Mean-Variance Analysis; Portfolio Optimization; Roll Critique; Shortselling; Long/Short (search for similar items in EconPapers)

**JEL-codes:** G11 G12 (search for similar items in EconPapers)

**Date:** 2015

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**Persistent link:** https://EconPapers.repec.org/RePEc:now:jnlcfr:104.00000026

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