For an AMZN parent order, the choice between 60 minutes and 120 may matter less than the choice of fill schedule. In Kulkarni and Nayar's example, every window from 16 to 270 minutes comes within 1% of maximum Sharpe. The paper's second half supplies the warning. A schedule cheaper than TWAP under the authors' cost model can surrender half the trade's Sharpe when measured against the P&L used to choose the window. The authors report that loss. Their abstract says they quantify the schedule's robustness cost in Sharpe ratio, and their conclusion acknowledges that a pessimistic envelope "can cost a substantial fraction of the Sharpe ratio." They regard the cost as the price of robustness. A desk can use the broad window band while remaining wary of the schedule.
This is a theoretical paper, with no empirical claim or reported trading performance. Its AMZN figures illustrate the model using inputs that are partly assumed. Our backtest below is our own construction and should be judged on those terms.
Why waiting can reduce risk
The trader has a forecast return μ_f over a horizon t_f, set to 19.2 bp over 1950 minutes in the AMZN example, and buys in the forecast's direction during a window T. Prices combine a drift m(t), representing the forecast as it realizes, with fractional Brownian motion of Hurst exponent H. Impact follows a concave power law whose exponent β lies in the cited range 0.4 to 0.7. The objective is Sharpe on the trade's own P&L: forecast-horizon price less average fill price and impact.
Slower execution reduces that P&L's variance. A fill late in the window shares more of its price path with the horizon price, leaving less noise between the two. At H = 1/2, the variance factor V(T) is exactly t_f - 2T/3 and declines as T increases. Treating noise during execution as separate risk, added in quadrature, reverses the sign. Once the covariance is included, urgency comes from alpha realized while the order remains open. As the paper says, "Urgency comes only from alpha that is realized while the order is being worked." For event alpha paid only at the horizon, the slowest admissible window wins if impact does not increase with duration.
The regime turns on two signs. The first belongs to a, the duration elasticity of impact: it measures how fixed-size impact changes as the order is stretched. When impact scales with volatility over the window, a equals β - H, the authors' impact-persistence gap. The second belongs to ν, net urgency after subtracting the exposure relief of later fills from alpha realization. If ν keeps one sign throughout the window, an interior optimum exists to leading order in T/t_f only when both signs are positive. Other sign combinations favor the fastest or slowest permitted pace. The authors flag exceptions: at a = 0 and H ≲ 0.35, ν starts negative and the optimum is interior (T* ≈ 13 min at β = H = 0.3). With both signs negative and |a| > min(1, 2H), Sharpe is single-peaked, though the structural prior excludes that case. At an interior optimum, impact divided by alpha equals ν/(a - ε_V); ε_V measures exposure-relief elasticity. Order size, volume and the impact coefficient drop out of that condition, making it checkable by a TCA team without a calibrated impact model.
AMZN's broad band
The worked example takes AMZN volume of 1.05 × 10^5 shares a minute and volatility of 10.6 bp per root-minute from one January 2026 session. Its 10^4-share order, forecast, β = 0.6, C = 1 and τ_α = t_f are assumptions. With H = 1/2 and therefore a = 0.1, the closed form gives 99 minutes. Its correction gives 86; exact maximization gives 87, with impact equal to 8.6% of alpha. Even a factor-of-two error in T costs only 0.2% to 0.3% of Sharpe. Treating AMZN as one name in a diversified book lowers the optimum from 99 to 37 minutes, a factor of 2.7. The 37-minute choice still lies inside the 1% band.
The band is the useful result. The authors recommend reporting it instead of a single optimum. Hold a at 0.1 and vary H from 0.3 to 0.7: T* runs through 114, 93, 87, 91 and 105 minutes. Those differences look small beside the 16 to 270 minute band. Its shape is asymmetric. In this example, choosing a window too short costs less Sharpe than choosing one too long.
The sign of a is less forgiving. Under the structural prior, T* takes values of 134, 117 and 87 minutes at H of 0.3, 0.4 and 0.5; at H = 0.6, it falls to the 0.5-minute floor. The authors note that meta-order studies find little dependence of impact on duration, placing a near zero. Classifying a trade also requires estimating a more precisely than ε_V, about 1.5% at 87 minutes. Getting a duration elasticity within 0.015 from execution data is a demanding condition. The authors identify another tension: their local impact law, with a fixed scale, implies a = β rather than the near-zero empirical value. They leave a propagator-model fix open. The regime map faces the harder encounter with data; the flat Sharpe band has more room for error.
Can the envelope savings hold up against uniform fills?
After choosing the window, the paper changes objectives. It minimizes impact plus a per-share delay cost w(t). Set w = θσ1 t^H, where θ counts standard deviations of adverse drift as a robustness level. Participation is proportional to (λ - w(t))^(1/β): the schedule starts fast and tapers smoothly to zero. Once normalized, its shape depends only on β and H, and the authors express it with the incomplete beta function. Against TWAP over the same active duration, envelope cost falls by 10.4% to 16.9%; at β = H = 0.5, the saving is 14.8%.
Envelope cost is the measure behind that saving. For β = H = 1/2, normalize TWAP impact to 1. The proposed schedule then incurs 1.47 in impact and 1.47 in delay, while TWAP incurs 2.45 in delay. Its arrival-price slippage exceeds TWAP's. The authors explicitly warn that savings under the price process "are not implied by this calculation and require simulation."
At T = 87 minutes, scoring the schedule on the original Sharpe P&L gives it 50% of uniform Sharpe at θ = 1. It stops trading after 7.7 minutes and incurs average impact of 10.5 bp, versus 1.7 bp for uniform fills. Sharpe retention rises to 79% at θ = 0.3 and 93% at θ = 0.1. At the critical θ_c ≈ 0.07, below which trading occupies the full window, it reaches 96%. Using the alpha reading w = m, which agrees with the P&L mean, produces an almost uniform schedule and recovers uniform Sharpe to within 10^-4.
The authors say the envelope "buys robustness against adverse paths at a cost in expected Sharpe, and θ sets the exchange rate." Yet the Sharpe calculation already incorporates path variance through the fBm covariance. The adverse behavior being insured against would have to sit outside that model, perhaps a misspecified H or a forecast decaying faster than linearly. The authors acknowledge that real decay is usually front-loaded. They do not test whether either possibility earns back the θ = 1 schedule's 50% haircut. The stopping time is sensitive too: make the impact scale depend on the active horizon instead of the nominal window, and t* at θ = 1 shifts from 7.7 minutes to about 1. The paper limits its robustness claim to normalized shape across θ. Trading stops at a movable point.
Our AMZN backtest, 2020 to mid-2024
The paper supplies no trading performance to compare with our figures. The figures printed above this article come from one automated backtest of our own using AMZN 1-minute bars from 2 January 2020 to 1 July 2024. This is one automated construction based on the paper's idea, rather than a test of its claims or a verdict on the authors' work. At 10:59 ET, a ridge regression of 20-minute returns fitted on the previous 60 sessions chooses the side of one parent order each day. Its size is 1% of projected shares across a 90-minute horizon. We select a window using the paper's uniform-entry Sharpe objective, then compare a capped front-loaded schedule with TWAP under a 10% minute-volume cap. We exit over the final 10 minutes and charge commissions of $0.004 a share, with a $1 minimum, on every fill.
The result was weak.
Total return came to -9.37%, Sharpe to -0.29 and maximum drawdown to -22.29%. Annualized volatility was 7.29%, Sortino -0.32 and Calmar -0.10. These figures bear more heavily on our forecast than on the fill schedule. They exclude market impact, the subject of the paper, leaving timing as the main difference between the front-loaded and TWAP legs. Our design differs from the paper in other ways as well. Exiting over the final 10 minutes gives up some forecast alpha, so our chosen window does not optimize round-trip Sharpe. The paper calculates its 10.4% to 16.9% envelope-cost savings at constant volume. Its volume-time mapping holds cleanly only at H = 1/2; our minute volumes change, so those savings do not describe our schedules. Weakness in this run belongs first to our build.
A 1% band spanning more than an order of magnitude in the AMZN example is worth taking to a desk. Before adopting the front-loaded schedule, set θ against the 96% Sharpe retained at θ_c ≈ 0.07 and measure each version on realized shortfall with impact included.
Our backtest stops at 2024-07-01, and everything after that date is deliberately left untouched so the same strategy can be checked out of sample later.