Two equal-variance forecasts can have sharply different trading value. Their frictionless Markowitz values match, yet Smerlak calculates what the gap costs. In one power-law impact market, his best Ornstein-Uhlenbeck (OU) signal has a net value of 0.1624. A perfectly predictable sinusoid at the cheapest frequency sets the ceiling at 0.2849. Speed the OU signal up tenfold and it retains only 0.17 of that ceiling.
This is an analytical paper, with no data, sample period or universe. Its numbers come from one-dimensional quadratures at illustrative parameters. The power-law examples set νm = 3/4 and β = γ = s² = 1, with no inventory penalty and no instantaneous cost. The backtest figures below belong to a strategy we built from the idea. The author supplies no backtest figures for comparison.
From forecast to position
One asset carries unimpacted price drift α, the predictor, alongside martingale noise. Trading at rate u changes the position π, which can also bleed passively at rate β. Smerlak uses an options book as the example: its vega falls as maturity approaches, and maintaining exposure takes fresh trades. For stock, β = 0. Costs include an inventory penalty λ and propagator impact, under which a kernel g maps past trading into price displacement. Together they give the friction symbol f̂(ω) = λ + (β² + ω²)ĝ(ω), the cost of holding a position oscillating at frequency ω.
Wiener-Hopf factorization separates f̂ into a causal factor and its anticausal mirror. The resulting policy first builds a friction-weighted forecast z from expected future drift E[α at t+s], weighting each forecast by market costs. A causal filter then turns z into a position. Net value equals half the mean square of z; gross signal gain equals exactly twice net value.
The familiar cases help locate the new result. With instantaneous cost only, the filter is a single exponential moving average (EMA). It moves partway toward the risk-adjusted target α/λ, reduced by b/(θs + b) to account for expected signal decay (here b = √(λ/η)). Exponential impact gives two EMAs with positive weights summing to one. Smerlak matches both cases to the Gârleanu-Pedersen solutions. His new stationary-investment case uses power-law impact; Forde and coauthors studied finite-horizon liquidation. For a kernel decaying as t^-(2νm-1), with 1/2 < νm < 1, the optimal trading rate becomes a fractional derivative of the signal, of order 1-νm. Its nonrational transfer has no exact finite-dimensional linear state-space representation. The implication can look odd on a blotter: "a sufficiently declining signal can induce selling even while it remains positive."
Why equal variance buys different value
The bound is v ≤ s²/(2 inf f̂). Variance placed at costly frequencies erodes value, as does future drift that past data cannot predict after friction weighting. The frequency loss alone can approach the whole value. Even a random-phase sinusoid, fully predictable from its past, "can be arbitrarily small despite perfect predictability and unchanged variance" when high-frequency friction grows.
Optimizing speed changes the comparison between smooth and rough signals. In the same power-law market (νm = 3/4), a Matérn predictor with νs = -2 reaches 0.2122 at a preferred rate of about 1.4064. OU reaches 0.1624 at rate 1/3. Across the Matérn family, the top value is near 0.2314, below 0.2849. Hold the rate instead at a slow 0.01 and the ranking reverses: the values are 0.0365, 0.0490 and 0.0730 for νs = -0.4, 0 and 0.3. Rougher signals put less variance at low frequencies, where replenishing exposure costs more. Under pure exponential impact (θm = 0.5, γ = 1.25, β = 1, bound 0.4), all smoothness levels approach the same rate-optimized supremum.
The speed result has a condition
The headline optimum is θs* = β(1-νm)/νm, or 1/3 in the illustrated market. Slow, preferred and fast OU signals retain 0.30, 0.57 and 0.17 of the bound. Their anticipative values are 0.37, 0.68 and 0.37. Exposure replenishment creates the γβ²|ω|^(2νm-2) friction term, which punishes one end of the speed range. Rapid adjustment creates the γ|ω|^(2νm) term, which punishes the other.
The qualification is explicit. Smerlak's abstract says "Under power-law impact with exposure bleeding"; discussing β > 0 with λ = 0, the body says "This choice matters for the conclusions." Ordinary holdings with inventory risk have β = 0 and λ > 0. There, f̂ = λ + ω²ĝ reaches its minimum at zero frequency, slower signals are favored, and every Matérn exponent approaches s²/(2λ) as θs → 0. Set β = λ = 0 and these markets have no finite bound. A cash equity book can use the filter without carrying over the medium-speed optimum.
A daily equity book
The OU policy is practical to express. Produce a forecast for each name and estimate its decay rate θs. Choose λ, η, γ and θm. Scale the forecast by (θs + θm)/(η(θs + b1)(θs + b2)), then distribute it between two EMAs at rates b1 < θm < b2. History can supply the forecast and its decay. Daily price bars cannot supply the three impact numbers; estimating them takes trade-level data, and they determine the filter's shape.
Our run, 2020 to mid-2024
We built a daily strategy using US stocks and ETFs. At each close it selected up to 20 of each from 200 screened candidates, subject to a $10 million trailing 60-session dollar-volume floor. Per-name rolling OLS forecasts used a 5-session return feature and 252 past pairs. We estimated decay from 126 forecast pairs, using 0.1 per session when estimation failed. We set λ = 0.04 and β = 0, the regime in which the paper favors slower signals. Our impact scenario, η = 0.04, γ = 0.012 and θm = 0.05 per session, was an assumption fixed before the run. We capped each name at 10% of capital and gross leverage at 4.0. Every fill incurred commissions of $0.0040 a share, minimum $1. Short borrow, margin financing and market impact are not modelled.
From 2020-01-03 to 2024-07-01, our construction returned 8.78% in total. Sharpe was 0.07 and Sortino 0.09, with 26.70% annualized volatility. Maximum drawdown reached -55.70%, producing a Calmar of 0.03. Our build performed weakly. The paper reports no performance of its own against which to compare it.
The result speaks to our construction, with substantial limits. We intended next-open execution, yet our fills were market-on-close. Our trade records do not say whether the two-EMA or single-EMA policy was active, so this run cannot distinguish them. Position caps break the paper's optimality identities when they clip holdings. The annual candidate-name list has no publication date, leaving room for hindsight. Rolling OLS forecasts may also depart from OU dynamics.
Impact needs a desk's trades
Smerlak cites Bouchaud and coauthors' inference of algebraic impact decay, while warning that the decay law "remains market- and estimation-dependent." Historical daily bars cannot reveal the impact of our own trades. A backtest must therefore assume γ, θm and νm, even though the fractional filter's order is precisely 1-νm. We encountered the same obstacle with a threshold rule whose calibration never reached its impact regime (/articles/dynamic-programming-for-value-at-risk-markov-decision-processes-in-continuous-sp). As the paper puts it, "Empirical use requires estimating predictive strength jointly with signal dynamics and trading costs."
Testing the speed result calls for a desk's own metaorder record to pin νm inside (1/2, 1), and a book that genuinely bleeds exposure, such as a vega book.
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.