Distributionally Robust Optimization

Forecasting the U.S. Treasury Yield Curve: A Distributionally Robust Machine Learning Approach ArXiv ID: 2601.04608 “View on arXiv” Authors: Jinjun Liu, Ming-Yen Cheng Abstract We study U.S. Treasury yield curve forecasting under distributional uncertainty and recast forecasting as an operations research and managerial decision problem. Rather than minimizing average forecast error, the forecaster selects a decision rule that minimizes worst case expected loss over an ambiguity set of forecast error distributions. To this end, we propose a distributionally robust ensemble forecasting framework that integrates parametric factor models with high dimensional nonparametric machine learning models through adaptive forecast combinations. The framework consists of three machine learning components. First, a rolling window Factor Augmented Dynamic Nelson Siegel model captures level, slope, and curvature dynamics using principal components extracted from economic indicators. Second, Random Forest models capture nonlinear interactions among macro financial drivers and lagged Treasury yields. Third, distributionally robust forecast combination schemes aggregate heterogeneous forecasts under moment uncertainty, penalizing downside tail risk via expected shortfall and stabilizing second moment estimation through ridge regularized covariance matrices. The severity of the worst case criterion is adjustable, allowing the forecaster to regulate the trade off between robustness and statistical efficiency. Using monthly data, we evaluate out of sample forecasts across maturities and horizons from one to twelve months ahead. Adaptive combinations deliver superior performance at short horizons, while Random Forest forecasts dominate at longer horizons. Extensions to global sovereign bond yields confirm the stability and generalizability of the proposed framework. ...

Wasserstein Distributionally Robust Rare-Event Simulation ArXiv ID: 2601.01642 “View on arXiv” Authors: Dohyun Ahn, Huiyi Chen, Lewen Zheng Abstract Standard rare-event simulation techniques require exact distributional specifications, which limits their effectiveness in the presence of distributional uncertainty. To address this, we develop a novel framework for estimating rare-event probabilities subject to such distributional model risk. Specifically, we focus on computing worst-case rare-event probabilities, defined as a distributionally robust bound against a Wasserstein ambiguity set centered at a specific nominal distribution. By exploiting a dual characterization of this bound, we propose Distributionally Robust Importance Sampling (DRIS), a computationally tractable methodology designed to substantially reduce the variance associated with estimating the dual components. The proposed method is simple to implement and requires low sampling costs. Most importantly, it achieves vanishing relative error, the strongest efficiency guarantee that is notoriously difficult to establish in rare-event simulation. Our numerical studies confirm the superior performance of DRIS against existing benchmarks. ...

Conditional Risk Minimization with Side Information: A Tractable, Universal Optimal Transport Framework ArXiv ID: 2509.23128 “View on arXiv” Authors: Xinqiao Xie, Jonathan Yu-Meng Li Abstract Conditional risk minimization arises in high-stakes decisions where risk must be assessed in light of side information, such as stressed economic conditions, specific customer profiles, or other contextual covariates. Constructing reliable conditional distributions from limited data is notoriously difficult, motivating a series of optimal-transport-based proposals that address this uncertainty in a distributionally robust manner. Yet these approaches remain fragmented, each constrained by its own limitations: some rely on point estimates or restrictive structural assumptions, others apply only to narrow classes of risk measures, and their structural connections are unclear. We introduce a universal framework for distributionally robust conditional risk minimization, built on a novel union-ball formulation in optimal transport. This framework offers three key advantages: interpretability, by subsuming existing methods as special cases and revealing their deep structural links; tractability, by yielding convex reformulations for virtually all major risk functionals studied in the literature; and scalability, by supporting cutting-plane algorithms for large-scale conditional risk problems. Applications to portfolio optimization with rank-dependent expected utility highlight the practical effectiveness of the framework, with conditional models converging to optimal solutions where unconditional ones clearly do not. ...

On Cost-Sensitive Distributionally Robust Log-Optimal Portfolio ArXiv ID: 2410.23536 “View on arXiv” Authors: Unknown Abstract This paper addresses a novel \emph{“cost-sensitive”} distributionally robust log-optimal portfolio problem, where the investor faces \emph{“ambiguous”} return distributions, and a general convex transaction cost model is incorporated. The uncertainty in the return distribution is quantified using the \emph{“Wasserstein”} metric, which captures distributional ambiguity. We establish conditions that ensure robustly survivable trades for all distributions in the Wasserstein ball under convex transaction costs. By leveraging duality theory, we approximate the infinite-dimensional distributionally robust optimization problem with a finite convex program, enabling computational tractability for mid-sized portfolios. Empirical studies using S&P 500 data validate our theoretical framework: without transaction costs, the optimal portfolio converges to an equal-weighted allocation, while with transaction costs, the portfolio shifts slightly towards the risk-free asset, reflecting the trade-off between cost considerations and optimal allocation. ...

On Accelerating Large-Scale Robust Portfolio Optimization ArXiv ID: 2408.07879 “View on arXiv” Authors: Unknown Abstract Solving large-scale robust portfolio optimization problems is challenging due to the high computational demands associated with an increasing number of assets, the amount of data considered, and market uncertainty. To address this issue, we propose an extended supporting hyperplane approximation approach for efficiently solving a class of distributionally robust portfolio problems for a general class of additively separable utility functions and polyhedral ambiguity distribution set, applied to a large-scale set of assets. Our technique is validated using a large-scale portfolio of the S&P 500 index constituents, demonstrating robust out-of-sample trading performance. More importantly, our empirical studies show that this approach significantly reduces computational time compared to traditional concave Expected Log-Growth (ELG) optimization, with running times decreasing from several thousand seconds to just a few. This method provides a scalable and practical solution to large-scale robust portfolio optimization, addressing both theoretical and practical challenges. ...