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Sharpe Ratio in High Dimensions Cases of Maximum Out of Sample Constrained Maximum and Optimal Portfolio Choice

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Document pages: 46 pages

Abstract: In this paper, we analyze maximum Sharpe ratio when the number of assets in aportfolio is larger than its time span. One obstacle in this large dimensionalsetup is the singularity of the sample covariance matrix of the excess assetreturns. To solve this issue, we benefit from a technique called nodewiseregression, which was developed by Meinshausen and Buhlmann (2006). It providesa sparse weakly sparse and consistent estimate of the precision matrix, usingthe Lasso method. We analyze three issues. One of the key results in our paperis that mean-variance efficiency for the portfolios in large dimensions isestablished. Then tied to that result, we also show that the maximumout-of-sample Sharpe ratio can be consistently estimated in this largeportfolio of assets. Furthermore, we provide convergence rates and see that thenumber of assets slow down the convergence up to a logarithmic factor. Then, weprovide consistency of maximum Sharpe Ratio when the portfolio weights add upto one, and also provide a new formula and an estimate for constrained maximumSharpe ratio. Finally, we provide consistent estimates of the Sharpe ratios ofglobal minimum variance portfolio and Markowitz s (1952) mean varianceportfolio. In terms of assumptions, we allow for time series data. Simulationand out-of-sample forecasting exercise shows that our new method performs wellcompared to factor and shrinkage based techniques.

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