Before revision:  Then, we provide consistency of maximum Sharpe Ratio when the portfolio weights add up to one, and also provide a new formula and an estimate for constrained maximum Sharpe ratio . Finally, we provide consistent estimates of the Sharpe ratios of global minimum variance portfolio and Markowitz's (1952) mean variance portfolio. In terms of assumptions, we allow for time series data. Simulation and out-of-sample forecasting exercise shows that our new method performs well compared to factor and shrinkage based techniques .
After revision:  Then, we provide consistency of maximum Sharpe Ratio when the portfolio weights sum to one; and (4) the Sharpe ratio estimators are consistent in global minimum-variance and mean-variance portfolios .