A Relaxed Approach to Estimating Large Portfolios and Gross Exposure

In this paper, using a novel proof technique, we show that the estimation error of the optimal portfolio variance decreases when the number of assets increases, in contrast to existing literature, notably results derived from factor based models. This proof is based on nodewise regression concept. Nodewise regression is a technique that helps us develop an approximate-relaxed inverse for the non-invertible sample covariance matrix of asset returns. The results are valid even when the number of assets, $p$ is larger than the sample size, $n$. We also estimate gross exposure of a large portfolio consistently with both growing and constant exposure cases, when $p>n$. Consistently estimating the growing gross exposure case is new, and shows the feasibility of such a portfolio. The main results use sub-Gaussian returns, but they are extended to weaker assumption of asset returns with bounded moments. Simulations verify and illustrate the theoretical results. In an out of sample forecasting application we show that portfolios based on our estimator outperform portfolios estimated with existing approaches in the finance literature in terms of out-of-sample variance and Sharpe ratio