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

Paper: arXiv 2302.13979

Authors: Jonathan Yu-Meng Li

Abstract

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.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 6.0/10 (capped from 7.0: transaction costs not modelled; see penalty flags)
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a strong theoretical extension to classical Kelly growth-optimal portfolio theory by applying Wasserstein Distributionally Robust Optimization (DRO). It successfully translates complex convex optimization and measure-theoretic distribution bounds into a practical quantitative framework with solid out-of-sample empirical validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Mitigate estimation error in Kelly Portfolio using Wasserstein DRO] --> B[Inputs: Historical Financial Return Data]
    B --> C[Methodology: Define Wasserstein Ball of distributions around empirical data]
    C --> D[Model Formulation: Formulate min-max robust log-optimal objective]
    D --> E[Computation: Solve robust growth-optimal portfolio as a Convex Program]
    E --> F[Outcomes: Outperforms classic Kelly in out-of-sample growth, stability, and robustness]