Active portfolio management measures success relative to a benchmark. Beat the S&P 500 by 2% and you've earned your fees. Lose to it by 2% and you haven't. But the penalties are asymmetric: underperforming by 5% is far worse than outperforming by 5% is good. Clients flee underperformance; they merely expect outperformance.
Pesenti and Nguyen (arXiv: 2603.20580) build this asymmetry into portfolio construction using the alpha-Bregman-Wasserstein divergence — a distance measure between probability distributions that penalizes upside and downside deviations differently.
The standard Wasserstein distance (optimal transport) treats gains and losses symmetrically. The Bregman-Wasserstein divergence introduces asymmetry via a convex function. The alpha-Bregman-Wasserstein divergence parameterizes the asymmetry with alpha, allowing the investor to specify exactly how much worse underperformance is than outperformance is good.
The framework constrains the portfolio's return distribution to lie within a specified divergence from the benchmark's return distribution. The investor maximizes expected utility from the excess return subject to this constraint and a budget constraint. The result: an optimal quantile function that trades off excess return against the risk of deviation from the benchmark, with the asymmetry baked in.
The through-claim: the geometry of risk is not symmetric, and portfolio construction should reflect this. Traditional tracking error (variance of excess returns) treats over- and under-performance equally. But the lived experience of investing is asymmetric: losses compound, trigger redemptions, and destroy careers; equivalent gains merely meet expectations. The optimal portfolio under asymmetric divergence constraint doesn't just minimize tracking error — it reshapes the entire distribution of excess returns to be asymmetrically cautious on the downside.
Pesenti & Nguyen, 2603.20580. Quantitative finance / portfolio management / optimal transport / Wasserstein distance / benchmark tracking.