We consider a continuous time investment problem in a multi-asset Black-Scholes market with the following features: The assets' drifts are not known and constitute a source of model ambiguity. However, there is a prior distribution (knowledge) on the possible drifts. Our investor is ambiguity averse and wants to maximize a mean-variance criterion for the terminal wealth where ambiguity aversion is incorporated in a smooth way. We consider here the criterion introduced in Maccheroni et al. 2013 where the variance is decomposed and each part is weighted differently to account for different levels of market risk and model ambiguity aversion. We use a novel approach to find the optimal dynamic investment strategy within the class of all adapted strategies which allow for learning. We also present a number of numerical results which help to understand how the model parameters affect the optimal investment strategy. In general it turns out that ambiguity averse investors invest less in the risky assets.

We consider the problem of estimating the true Sharpe ratio of an asset selected for having the highest observed in-sample Sharpe ratio among many assets. We discuss estimators based on the polyhedral lemma, James Stein shrinkage, debiasing the expected maximum Sharpe ratio, thresholding and empirical Bayes. We test these estimators in simulations, computing bias and root mean square error across different values of sample size, number of assets, and spread and shape of population Sharpe ratios. We also compute rank correlation of the estimators against the underlying quantity, simulating how these estimators might be used to compare or rank the output of different teams which perform this selection process. We find that the James Stein estimator provides the best performance across many different realistic values of the relevant parameters, followed by the GMLEB estimator of Jiang and Zhang. These results are fairly robust to correlation of asset returns, with some caveats.

We describe a post hoc test for the Sharpe ratio, analogous to Tukey's test for pairwise equality of means. The test can be applied after rejection of the hypothesis that all population Signal-Noise ratios are equal. The test is applicable under a simple correlation structure among asset returns. Simulations indicate the test maintains nominal type I rate under a wide range of conditions and is moderately powerful under reasonable alternatives.

This paper studies a type of periodic utility maximization problem for portfolio management in incomplete stochastic factor models with convex trading constraints. The portfolio performance is periodically evaluated on the relative ratio of two adjacent wealth levels over an infinite horizon, featuring the dynamic adjustments in portfolio decision according to past achievements. Under power utility, we transform the original infinite horizon optimal control problem into an auxiliary terminal wealth optimization problem under a modified utility function. To cope with the convex trading constraints, we further introduce an auxiliary unconstrained optimization problem in a modified market model and develop the martingale duality approach to establish the existence of the dual minimizer such that the optimal unconstrained wealth process can be obtained using the dual representation. With the help of the duality results in the auxiliary problems, the relationship between the constrained and unconstrained models as well as some fixed point arguments, we derive and verify the optimal constrained portfolio process for the original problem over an infinite horizon.