Table 5. Turnover, out-of-sample portfolio variance and Sharpe ratio for optimal mean-variance portfolios (Mean-var) and minimum-variance portfolios (Min-var), corresponding to different estimators of mean and covariance, in the case “short selling allowed”.

	Mean-var			Min-var
Estimators	Turnover	${({\widehat{\sigma }}^k)}^2$	${\widehat{SR}}^k$	Turnover	${({\widehat{\sigma }}^k)}^2$	${\widehat{SR}}^k$
MLE	1.5665	0.0105	0.0368	0.2507	0.0021	-0.0378
MPE, α = 0.1	1.4717	0.0161	0.0727	0.2109	0.0020	-0.0282
MPE, α = 0.2	1.9083	0.0292	0.0880	0.1838	0.0019	-0.0322
MPE, α = 0.25	2.4381	0.0400	0.0911	0.1753	0.0019	-0.0367
SE, ɛ* = 0.25	1.4119	0.0126	0.0631	0.2283	0.0020	-0.0291
SE, ɛ* = 0.5	1.5506	0.0218	0.0878	0.1966	0.0019	-0.0302