The ground metric can decide whether changing the radius changes a long-only book at all. Yadav and Mehra's DR-CVaR benchmark chose the same FTSE portfolio under an ℓ∞ ground metric across six settings. The radius doubled from 0.005 to 0.010, and the trade-off weight doubled from 0.10 to 0.20. Every setting delivered 6.54% a year and a Sharpe of 0.390. The paper's text describes a fivefold increase in ρ; its table shows twofold. The authors anticipated the collapse in Proposition 4 and present the result as confirmation.

Their main proposal replaces CVaR with an expectile. The τ-expectile is the loss level where τ times expected excess above it equals (1−τ) times expected shortfall below it. For τ ≥ 0.5 it is coherent. It is also elicitable, allowing forecasts to be scored against realized losses. Rockafellar and Uryasev give CVaR an easy worst-case form as a minimum of an expectation. The expectile lacks that form, so the authors use an envelope theorem: its worst-case value is the root in q of the worst-case expectation of L + κ(L−q)⁺, minus q, with κ = (2τ−1)/(1−τ). At their τ = 0.95, κ is 18. The integrand is the maximum of two affine pieces, which brings standard type-1 Wasserstein duality into play.

The model rebalances every 21 trading days using the trailing 250 daily return vectors as scenarios. It minimizes (1−λ) times the negative worst-case mean plus λ times the worst-case expectile, subject to a mean at or above the equal-weight sample mean. The problem has four linear-program branches. Each becomes linear when a scalar dual multiplier is fixed. The authors search over that multiplier on a coarse grid, then refine locally: the value function need not be unimodal, and golden-section search could settle at a local minimum. They call these parametric linear programs and state that each is linear only at a fixed multiplier. In practice, on our reading, their "exact" reformulation relies on a one-dimensional search without a global optimality certificate.

How much does the ground metric matter?

The adversary can shift each of the 250 daily scenarios at a transport cost. With an ℓ1 cost, shifting the asset with the largest weight is the cheapest way to hurt portfolio return. The objective penalty scales with that maximum weight: c·ρ·max weight, where c = 1 + λκ, or 2.80 at λ = 0.10. With an ℓ∞ cost, all assets move together. Damage scales with the sum of the weights, which is always 1 on the simplex. The penalty is then constant across allocations.

The reported portfolios follow that distinction. In the six ℓ∞ cells with ρ ≥ 0.005, the dual multiplier landed exactly at its kink on all 160 rebalance dates. Case A is the regime in which λ affects weights; its share of dates was 0.00. Across all nine cells, the ℓ∞ model never beat 1/N, whose Sharpe was 0.614. At (ρ, λ) = (0.005, 0.10), changing the same CVaR model to ℓ1 raised effective names from 7.15 to 18.59 and lowered the mean largest weight from 22.4% to 6.2%. The authors call the ground metric "an economic modeling choice". These results give that phrase substance.

The expectile Sharpe lead

For the controlled comparison, the ambiguity set, ℓ1 metric, radius and λ stay fixed. Only the risk functional changes. The expectile model leads on Sharpe in all nine cells. Its mean gap is +0.243; the largest is +0.502 at (0.005, 0.15). Studentized block-bootstrap tests in the Ledoit-Wolf style find all six gaps at ρ ∈ {0.005, 0.010} significant at 5%. None of the three gaps at ρ = 0.001 is significant. At λ = 0.10, the radius coefficients differ by 3.5% (2.80 against 2.90), making the radius match close.

At (0.005, 0.10), the books have 18.64 and 18.59 effective names. The expectile book earns 13.94% a year against 7.19%. The paper concludes that "the proposed model selects a less risky portfolio." Yet the benchmark table gives the expectile book higher volatility (14.40% vs 12.60%), deeper maximum drawdown (33.82% vs 31.83%) and higher daily CVaR0.95 (0.0216 vs 0.0189).

The Sharpe edge comes from return.

The authors acknowledge a likely reason for the return difference: the two functionals at 0.95 have different scales. On the same proposed portfolio, e0.95 is 0.0105 and CVaR0.95 is 0.0216. The 0.95 expectile sets a tail target about half as large as the 0.95 CVaR, so more return accompanied by more risk is plausible. The similar-diversification comparison applies to that one cell. At ρ = 0.001, the expectile book is more concentrated at every λ: 4.98 effective names against 10.04 at λ = 0.10, 7.14 against 12.02 at 0.15, and 9.29 against 12.25 at 0.20.

When the trade-off drops out

Sharpe declines as ρ increases, from 1.005 at (0.001, 0.10) to 0.543 at (0.010, 0.20). The critical radius marks where the empirical tail term drops out. The ratio of ρ to that radius stays below 0.336, so the expectile remains active in every cell. The derivation leaves out the mean-return constraint, making this ratio approximate for the full model. The mean-risk trade-off is less persistent. Case A share reaches 0.21 at (0.010, 0.20). On the remaining 79% of dates, λ drops out and the model minimizes worst-case expectile at the return floor. The authors flag this and ask readers to report Case A share alongside performance. That practice is worth keeping.

With ρ = 0, the nominal expectile model has the table's best Sharpe, 1.209. It also has 2.07 effective names, a 57.6% mean top weight and a 50.6% drawdown. At ρ = 0.005 and λ = 0.10, the worst-case layer reduces drawdown to 33.8% while surrendering 0.241 of Sharpe (0.968 vs 1.209). On these figures, the radius buys a shallower drawdown at the cost of Sharpe; it adds no risk-adjusted return over the nominal model.

Ninety FTSE constituents, gross of costs

The sample contains 90 FTSE constituents from LSEG Refinitiv, covering 20 September 2012 to 26 June 2026. It runs for 3,591 days, including 3,341 out-of-sample days across 160 rebalances. Membership is fixed at the sample's end. The authors concede the survivorship and argue that, because it affects every model, the ranking remains intact. Main results are gross of costs; the supplement reports cost sensitivity. We did not find turnover in the main text. The introduction says a stationary-bootstrap radius calibration is assessed, while the main text reports only a fixed 3×3 grid and identifies no calibrated ρ.

We could not rerun the test. It requires daily price history for UK listed names at this breadth, whereas our coverage is US equities. A US universe would test a different book. Its results would not stand in for these.

The ℓ1-versus-ℓ∞ finding travels furthest: it applies to any long-only book with linear portfolio losses under a type-1 Wasserstein ball, regardless of market. For the scale issue, the authors report both functionals for every model rather than rescale one. I would find the expectile-versus-CVaR spread persuasive as a risk claim if it survived matching the models on realized CVaR instead of nominal confidence level.