Multi-period Mean-Expectile Portfolio Optimization under Wasserstein Ambiguity: Reformulation, Degeneracy and the Role of the Ground Metric
Abstract
Expectiles are the only law-invariant risk measures that are both coherent and elicitable. Unlike Conditional Value-at-Risk (CVaR), however, they do not admit a Rockafellar--Uryasev representation that admits tractable Wasserstein reformulations. We address this difficulty by developing an envelope theorem for worst-case expectiles that characterizes the worst-case expectile over a Wasserstein ambiguity set as the unique root of a worst-case expectation with a two-piece affine integrand. This representation permits direct application of standard Wasserstein duality. Using this result, we reformulate a multi-period tri-level mean--expectile portfolio problem as four parametric linear programs with constraints. We establish four structural properties of the proposed model: an endogenously damped price of robustness, a decision-dependent critical radius beyond which the expectile tail component becomes inactive, exact recovery of the nominal model at zero ambiguity, and a characterization of how the Wasserstein ground metric determines whether the limiting portfolio becomes more concentrated or more diversified. Numerical experiments on 90 FTSE constituents over 3,341 out-of-sample trading days show that the expectile model outperforms a CVaR model matched on ambiguity set, radius, ground metric, and trade-off weight in all nine parameter cells---significantly so whenever the radius is non-trivial. The experiments further confirm the predicted degeneracy under the ground metric.
Related
- A true single-level reformulation for pessimistic bilevel optimization
- A splitting proximal method in portfolio optimization with entropic value-at-risk
- Susceptibility optimization for wisdom of crowds: A social power reformulation and realization framework
- Portfolio-Based Constrained Multi-Objective Bayesian Optimization for Materials Design
- Multi-period Value-at-Risk Constrained Portfolio Optimization via DC Programming
BibTeX
@misc{yadav2026multiperiod,
title = {{Multi-period Mean-Expectile Portfolio Optimization under Wasserstein Ambiguity: Reformulation, Degeneracy and the Role of the Ground Metric}},
author = {Rupendra Yadav and Aparna Mehra},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11917},
eprint = {2610.11917},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11917},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Multi-period Mean-Expectile Portfolio Optimization under Wasserstein Ambiguity: Reformulation, Degeneracy and the Role of the Ground Metric
AU - Rupendra Yadav
AU - Aparna Mehra
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11917
UR - https://arxiv.org/abs/2610.11917
AB - Expectiles are the only law-invariant risk measures that are both coherent and elicitable. Unlike Conditional Value-at-Risk (CVaR), however, they do not admit a Rockafellar--Uryasev representation that admits tractable Wasserstein reformulations. We address this difficulty by developing an envelope theorem for worst-case expectiles that characterizes the worst-case expectile over a Wasserstein ambiguity set as the unique root of a worst-case expectation with a two-piece affine integrand. This representation permits direct application of standard Wasserstein duality. Using this result, we reformulate a multi-period tri-level mean--expectile portfolio problem as four parametric linear programs with constraints. We establish four structural properties of the proposed model: an endogenously damped price of robustness, a decision-dependent critical radius beyond which the expectile tail component becomes inactive, exact recovery of the nominal model at zero ambiguity, and a characterization of how the Wasserstein ground metric determines whether the limiting portfolio becomes more concentrated or more diversified. Numerical experiments on 90 FTSE constituents over 3,341 out-of-sample trading days show that the expectile model outperforms a CVaR model matched on ambiguity set, radius, ground metric, and trade-off weight in all nine parameter cells---significantly so whenever the radius is non-trivial. The experiments further confirm the predicted degeneracy under the ground metric.
ER -