Wasserstein-Kelly Portfolios: A Robust Data-Driven Solution to Optimize Portfolio Growth

We introduce a robust variant of the Kelly portfolio optimization model, called the Wasserstein-Kelly portfolio optimization. Our model, taking a Wasserstein distributionally robust optimization (DRO) formulation, addresses the fundamental issue of estimation error in Kelly portfolio optimization by defining a ``ball" of distributions close to the empirical return distribution using the Wasserstein metric and seeking a robust log-optimal portfolio against the worst-case distribution from the Wasserstein ball. Enhancing the Kelly portfolio using Wasserstein DRO is a natural step to take, given many successful applications of the latter in areas such as machine learning for generating robust data-driven solutions. However, naive application of Wasserstein DRO to the growth-optimal portfolio problem can lead to several issues, which we resolve through careful modelling. Our proposed model is both practically motivated and efficiently solvable as a convex program. Using empirical financial data, our numerical study demonstrates that the Wasserstein-Kelly portfolio can outperform the Kelly portfolio in out-of-sample testing across multiple performance metrics and exhibits greater stability.