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

01 ABSTRACT

Expectiles are the only law-invariant risk measures that are both coherent and elicitable, but unlike CVaR they lack a tractable Wasserstein reformulation. The authors use an envelope theorem to express the worst-case expectile as the unique root of a worst-case expectation with a two-piece affine integrand, and reformulate a multi-period tri-level mean-expectile portfolio problem as four parametric linear programs. Numerical experiments on 90 FTSE constituents over 3,341 out-of-sample trading days show 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.

02 KEY FINDINGS

1. An envelope theorem for worst-case expectiles characterizes them as the unique root of a worst-case expectation with a two-piece affine integrand, enabling direct application of standard Wasserstein duality.
2. A multi-period tri-level mean-expectile portfolio problem is reformulated as four parametric linear programs with constraints, and four structural properties are established: an endogenously damped price of robustness, a decision-dependent critical radius, exact recovery of the nominal model at zero ambiguity, and characterization of how the Wasserstein ground metric determines whether the limiting portfolio becomes more concentrated or diversified.
3. Experiments on 90 FTSE constituents over 3,341 out-of-sample trading days show 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, and confirm the predicted degeneracy under the ground metric.