“…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).…”