Impossible Frontiers

Abstract

A key result of the Capital Asset Pricing Model (CAPM) is that the market portfoliothe 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 show that impossible frontiers are difficult to avoid. In particular, 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.

Multifactor Market Indexes

“…4 Numerous papers have found that various long-only market indexes are not mean-variance efficient portfolios (e.g., see Gibbons et al (1989); Gibbons (1982); Jobson and Korkie (1982); Kandel (1984); Shanken (1985Shanken ( , 1986; Kandel andStambaugh (1987a, 1987b); Gibbons et al (1989); Haugen and Baker (1991); MacKinlay and Richardson (1991); Zhou (1993); Brière et al (2013), and others). According to Brennan and Lo (2010), it is virtually impossible for long-only market indexes to be efficient portfolios. Supporting this proposition, many studies have found that short positions are needed to achieve efficiency (e.g., see Pulley (1981); Levy (1983); Kallberg and Ziemba (1983); Kroll et al (1984); Green and Hollifield (1992); Jagannathan and Ma (2003); Brennan and Lo (2010); Levy and Ritov (2010), and others).…”

“…According to Brennan and Lo (2010), it is virtually impossible for long-only market indexes to be efficient portfolios. Supporting this proposition, many studies have found that short positions are needed to achieve efficiency (e.g., see Pulley (1981); Levy (1983); Kallberg and Ziemba (1983); Kroll et al (1984); Green and Hollifield (1992); Jagannathan and Ma (2003); Brennan and Lo (2010); Levy and Ritov (2010), and others). More generally, Kothari et al (1995) have argued that the equity portfolio most highly correlated with the market portfolio is efficient.…”

The existence and persistence of financial anomalies: What have you done for me lately?

“…Factor contributions are 308 basis points, led by medium‐term momentum factor contributions of 148 basis points ( t = 8.06). The authors recently used the Boolean signal argument to address a footnote in Brennan and Lo (), which repeated several Wall Street “researchers” claiming that Markowitz MV analysis can produce impossible frontiers. We do not find such results with a reasonably specified upper stock bound and reasonable turnover constraints.…”