Multi-period Mean-Expectile Portfolio Optimization under Wasserstein Ambiguity: Reformulation, Degeneracy and the Role of the Ground Metric

Paper: arXiv 2610.11917

Authors: Rupendra Yadav, Aparna Mehra

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.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 6.0/10 (capped from 8.0: transaction costs not modelled; see penalty flags)
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a highly complex mathematical framework for multi-period portfolio optimization using expectiles and Wasserstein ambiguity, which is a novel approach. It is supported by extensive numerical experiments on real-world data, demonstrating strong empirical rigor. The clarity is generally good, though the mathematical depth makes it challenging for non-specialists.

Research Flowchart

  flowchart TD
    A[Research Goal: Multi-period Mean-Expectile Portfolio Optimization under Wasserstein Ambiguity] --> B{Key Methodology: Envelope Theorem for Worst-case Expectiles};
    B --> C[Reformulation: Multi-period Tri-level Problem as 4 Parametric LPs];
    C --> D{Data/Inputs: 90 FTSE Constituents, 3341 Trading Days};
    D --> E[Computational Process: Numerical Experiments, Compare to CVaR];
    E --> F[Key Findings/Outcomes: 4 Structural Properties, Outperformance vs. CVaR, Degeneracy under Ground Metric];

Cited References (47)

1. Coherent measures of risk (1999)
2. Asset Management: A Systematic Approach to Factor Investing (2014) — link
3. On elicitable risk measures (2015)
4. Risk management with expectiles (2017)
5. Generalized quantiles as risk measures (2014)
6. Backtesting VaR and expectiles with realized scores (2019)
7. Quantitative concentration inequalities for empirical measures on non-compact spaces (2007)
8. Optimal investment policy in the time consistent mean–variance formulation (2013)

0 of 47 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.