We propose to impose a weighted l1 and squared l2 norm penalty on the portfolio weights to improve out-of-sample (OOS) performances of portfolio optimization when the number of assets becomes large. We show that under certain conditions, the realized risk of the optimal minimum variance portfolio (MVP) obtained from the strategy can asymptotically be lower than those of benchmark portfolios with a high probability. Our theoretical results imply that penalty parameters for the weighted-norm penalty can be specified as a simple function of the number of assets and sample size. With the theoretical results, we also develop an automatic calibration procedure for choosing the penalty parameters. We demonstrate superior OOS performances of the weighted-norm MVP with two real data sets. Finally, we propose several alternative norm penalties and show that their OOS performances are comparable to the weighted-norm strategy.

Approximate factor models and their extensions are widely used in economic analysis and forecasting due to their ability to extracting useful information from a large number of relevant variables. In these models, candidate predictors are typically subject to some common components. In this paper we propose a new method for robustly estimating the approximate factor models and use it in risk assessments. We consider a class of approximate factor models in which the candidate predictors are additionally subject to idiosyncratic large uncommon components such as jumps or outliers. By assuming that occurrences of the uncommon components are rare, we develop an estimation procedure to simultaneously disentangle and estimate the common and uncommon components. We then use the proposed method to investigate whether risks from the latent factors are priced for expected returns of Fama and French 100 size and book-to-market ratio portfolios. We find that while the risk from the common factor is priced for the 100 portfolios, the risks from the idiosyncratic factors are not. However, we find that model uncertainty risks of the idiosyncratic factors are priced, suggesting that with effective diversifications, only the predictable idiosyncratic risks can be reduced, but the unpredictable ones may still exist. We also illustrate how the proposed method can be adopted on evaluating value at risk (VaR) and find it can delivery comparable results as the conventional methods on VaR evaluations.