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: 8.0/10
  • 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];