In this paper, we analyze maximum Sharpe ratio when the number of assets in a portfolio is larger than its time span. One obstacle in this large dimensional setup is the singularity of the sample covariance matrix of the excess asset returns. To solve this issue, we benefit from a technique called nodewise regression , which was developed by Meinshausen and Buhlmann (2006). It provides a sparse/weakly sparse and consistent estimate of the precision matrix , using the Lasso method. We analyze three issues. One of the key results in our paper is that mean-variance efficiency for the portfolios in large dimensions is established. Then tied to that result , we also show that the maximum out-of-sample Sharpe ratio can be consistently estimated in this large portfolio of assets. Furthermore, we provide convergence rates and see thatthe number of assets slow down the convergence up to a logarithmic factor . 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 .
We provide a new theory for nodewise regression when the residuals from a fitted factor model are used to apply our results to the analysis of the maximum Sharpe ratio when the number of assets in a portfolio is larger than its time span. We introduce a new hybrid model where factor models are combined with feasible nodewise regression. Returns are generated from an increasing number of factors plus idiosyncratic components (errors). The precision matrix of the idiosyncratic terms is assumed to be sparse, but the respective covariance matrix can be non-sparse. Since the nodewise regression is not feasible due to the unknown nature of errors, we provide a feasible-residual-based nodewise regression to estimate the precision matrix of errors as a new method. Next, we show that the residual-based nodewise regression provides a consistent estimate for the precision matrix of errors. In another new development , we also show that the precision matrix of returns can be estimated consistently, even with an increasing number of factors. Benefiting from the consistency of the precision matrix estimate of returns, we show that: (1) the portfolios in high dimensions are mean-variance efficient; (2) maximum out-of-sample Sharpe ratio estimator is consistent and the number of assets slows the convergence up to a logarithmic factor ; (3) the maximum Sharpe ratio estimator is consistent when the portfolio weights sum to one; and (4) the Sharpe ratio estimators are consistent in global minimum-variance and mean-variance portfolios .
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In this paper, we analyze
We provide a new theory for nodewise regression when the residuals from a fitted factor model are used to apply our results to the analysis of the
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One obstacle in this large dimensional setup is the singularity of the sample covariance
We introduce a new hybrid model where factor models are combined with feasible nodewise regression. Returns are generated from an increasing number of factors plus idiosyncratic components (errors). The precision
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excess asset returns. To solve this issue, we benefit from a technique called nodewise regression , which was developed by Meinshausen and Buhlmann (2006). It provides a sparse/weakly sparse and consistent estimate of
idiosyncratic terms is assumed to be sparse, but the respective covariance matrix can be non-sparse. Since the nodewise regression is not feasible due to the unknown nature of errors, we provide a feasible-residual-based nodewise regression to estimate
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, using the Lasso method. We analyze three issues. One of the key results in our paper is that mean-variance efficiency for the portfolios in large dimensions is established. Then tied to that result
of errors as a new method. Next, we show that the residual-based nodewise regression provides a consistent estimate for the precision matrix of errors. In another new development
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maximum
precision matrix of returns can be estimated consistently, even with an increasing number of factors. Benefiting from the consistency of the precision matrix estimate of returns, we show that: (1) the portfolios in high dimensions are mean-variance efficient; (2) maximum
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Sharpe ratio can be consistently estimated in this large portfolio of assets. Furthermore, we provide convergence rates and see thatthe
Sharpe ratio estimator is consistent and the
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slow down
slows
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. Then, we provide consistency of maximum Sharpe Ratio
; (3) the maximum Sharpe ratio estimator is consistent
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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
sum to one; and (4) the Sharpe ratio estimators are consistent in global minimum-variance and mean-variance portfolios