Folio Lab/Documentation

Distributionally Robust CVaR

A worst-case formulation of CVaR that hedges against ambiguity in the return distribution itself by optimising over a Wasserstein ball of plausible distributions.

Overview

Standard CVaR optimisation treats the empirical distribution of returns as if it were the truth. Distributionally Robust CVaR (DR-CVaR) recognises that the empirical distribution is itself an estimate, and instead minimises the worst-case CVaR over every distribution within a Wasserstein ball of radius around the empirical distribution.

The result is a portfolio that is provably immunised against small perturbations of the historical distribution — protecting it from the risk that the past is a noisy sample of the future.

Mathematical Formulation

Let be the empirical distribution from historical return observations. Define a Wasserstein ambiguity set:

where is the -Wasserstein distance. The DR-CVaR problem at confidence level is:

Esfahani & Kuhn (2018) show this admits a tractable convex reformulation. Folio Lab uses skfolio's DistributionallyRobustCVaR estimator with an automatically scaled radius derived from the median per-asset daily volatility, clamped to . If the solver is infeasible at the initial radius, the radius is progressively widened (1×, 2×, 5×, 10×) before raising.

Advantages & Limitations

Advantages

  • Out-of-sample guarantees: Bounds worst-case CVaR with high probability.
  • Hedges against estimation error: No pretence the empirical distribution is exact.
  • Convex: Tractable LP/SOCP formulation.
  • Theoretically principled: Replaces ad-hoc shrinkage with a tunable ball radius.

Limitations

  • Radius selection: Tuning matters; too large yields very defensive portfolios.
  • Computational cost: Heavier than plain CVaR.
  • Conservative: May give up upside in benign regimes.
  • Solver sensitivity: Requires careful conic solvers (CLARABEL, SCS).

References

  • Mohajerin Esfahani, P., & Kuhn, D. (2018). "Data-driven distributionally robust optimization using the Wasserstein metric." Mathematical Programming, 171(1-2), 115-166.
  • Rockafellar, R. T., & Uryasev, S. (2002). "Conditional value-at-risk for general loss distributions." Journal of Banking & Finance, 26(7), 1443-1471.
  • skfolio documentation — skfolio.optimization.DistributionallyRobustCVaR.