Price learning can hurt a short-horizon book when its prior drift is close to the truth. Pohlschmidt and Zhu give that claim a boundary: at v = 0.09, the drift error must be under about 21 points a year, against a prior standard deviation of 30 points.
This is a theory paper. It reports no backtest, no Sharpe and no market data; every number comes from one illustrative calibration solved numerically in R. The market run below therefore cannot confirm or refute an empirical claim by the authors. They make none.
Where the loss comes from
One stock has known volatility and an unknown constant drift. The investor starts with a Gaussian prior, N(m, v), CARA utility (constant absolute risk aversion) and a finite horizon. Three investors share the same initial wealth and belief. The naive investor retains her date-t belief and integrates over it, shrinking myopic demand by q = σ²/(σ²+(T−t)v). The price observer Kalman-filters the drift from prices. The attentive investor also receives an independent signal, whose precision depends on attention capacity κ. Capacity measures the permitted rate of entropy reduction under Shannon entropy or the Tsallis family indexed by ρ.
The strategies and value functions are explicit. Two coefficients governing the scenario results solve ODEs; only the price observer has a closed-form solution for them. Before conditioning on a true drift, the ranking is straightforward: price learning has strictly positive value, attention adds value, and value rises strictly in κ.
Now hold each optimal strategy fixed and evaluate it at one true drift μ, averaging over return shocks. The authors call the result scenario-conditional value. Against the naive investor, it is (e^{−r(T−t)}/γ)[B·Δ² − C]. The initial position gap Δ is a scaled distance between the naive position and the full-information position. B lies in (0, a(t,0)); multiplying Δ² makes it the benefit from correcting that gap. C lies in (0, 1/2), independently of the scenario. It charges the value of learning for the fluctuations caused by trading on noisy belief revisions.
At Δ = 0, learning is strictly worse at every finite capacity. Averaging across scenarios still gives a positive learning term in the value function, ½ln(1+2vB) − C. In the baseline calibration (m = 0.08, v = 0.09, σ = 0.3, r = 0.04, T = 5), the true drift is one point above the prior mean. Yet q = 1/6 produces Δ = −0.0433: the naive investor has shrunk her position heavily.
The authors keep the distinction between these evaluations clear. Their conclusion says information should be assessed "jointly with the trading strategy it induces, the investment horizon, and the information-processing technology." Their scenario map adds to the ex ante criterion without displacing it.
How close is close enough?
The sharp result is a τ² limit as the remaining horizon shrinks. The abstract states: "Over sufficiently short horizons, the scenario-conditional value of learning is negative when belief error is small relative to posterior uncertainty." For price observation, the condition is (m−μ)² < v/2. Any fixed positive capacity makes the scenario-conditional loss worse. At v = 0.09, this means a drift error of about 21 points a year, compared with a prior standard deviation of 30 points. Much of the investor's prior mass lies outside that band, which is narrower than one prior standard deviation.
The authors call the winning case for learning "confident but wrong": a tight prior paired with a large error. It is an apt warning for short-dated Bayesian tilts.
The limit does not specify how short "short" must be. Figure 3 maps maturity against belief error for the attentive investor at κ = 0.5. It shows a broad area where total value relative to the naive investor is negative. Its centre remains at the zero-gap point m−μ = (1−q)(m−r); as maturity grows, q falls and the point shifts right. Across a substantial area at longer maturities, attention beats the price observer yet still loses to the naive investor.
At the five-year baseline horizon, the investor's own prior assigns probability 0.247 to the price observer losing to the naive investor. For the attentive investor at κ = 5, that probability falls to 0.041. Plain Kalman learning loses in one in four scenarios, even with zero trading costs.
I expect costs to make learning look worse. C captures risk from position swings, rather than the cost of executing them. The authors raise the issue themselves: transaction costs "would make belief-driven portfolio adjustments directly costly, but they could also reduce the frequency and magnitude of these adjustments." Their net effect "remains an open question." Cutting adjustments in that way requires a no-trade band, absent from their optimal rule. My bet is that costs push 0.247 up for a daily rebalancer because the naive book barely trades. The authors leave the sign open.
The capacity dial
No trader can read κ from a tape. It is an exogenous measure of the auxiliary signal's capacity, always fully used in the model; the authors leave its endogenization for future work. They describe the assumption that price observation is costless as a normalization.
At v = 0.09, reaching a 10% underperformance target requires κ = 0.903. A 1% target requires 80.821. As uncertainty rises, the required capacity falls: for the 1% target, it drops from 229.891 at v = 0.01 to 52.889 at v = 0.16. At v = 0.16, price learning alone already meets a 20% target. The authors warn that lower relative odds at higher variance do not imply lower absolute risk.
The abstract gives the longer-run capacity result too. At a given horizon, enough capacity beats any fixed finite-capacity benchmark in every drift scenario. Eventual dominance, the authors caution, does not imply that value rises monotonically in κ at intermediate capacities.
To close 90% of the worst-case gap to perfect learning, Shannon requires κ = 2.816. Tsallis ρ = 4 requires 936.088, roughly 330 times as much. A weak signal initially worsens the worst case relative to price-only learning. Break-even comes at 0.107 under Shannon and 2.090 at ρ = 4. At high capacity, the shortfall decays like κ⁻¹ for 1 ≤ ρ < 3, like ln(1+κ)/κ at ρ = 3, and like κ^{−2/(ρ−1)} above.
K1 and K2, the constants in those rates, are unspecified. The rates therefore establish orders of magnitude rather than an exact capacity for a chosen reduction, a limit the authors acknowledge. Nor does a market observable identify which ρ an analyst feed follows. The useful result here is the ordering under different ways of accounting for entropy.
Our US stock run, 2020 to 2024
Our run reaches only a narrower part of the theory. We could not construct the attentive investor: market data provide no independent, capacity-constrained Gaussian signal about latent drift. Historical prices cannot reveal true fixed drifts either, leaving the scenario-conditional claims beyond empirical checking. Daily rebalancing discretizes the continuous-time rule. The price-observer book is our own construction from the authors' idea.
We used the highest-volume US stocks in each year from January 2020 to July 2024. For each name, a trailing window of daily log returns supplied the prior mean; a shorter trailing window supplied σ, with v = σ². The cash rate stayed at the last fed funds print before entry. We applied the paper's γ = 0.02 with wealth measured in millions of dollars, our choice of unit. A fixed long-only dollar sleeve per stock clipped positions. We charged a per-share commission subject to a per-order minimum. Our rules called for next-open fills, whereas our cost model priced fills at the close. We have not reconciled those two conventions.
The clip dominates.
Given our volatilities and horizon, the unclipped dollar target exceeds the sleeve except at the smallest positive posterior excess drifts. In practice, the book is close to a long-or-flat switch on the sign of m − r, with the year's most-traded names held near equal weight.
From January 2020 to July 2024, it returned 152.81% in total. Sharpe was 1.07, Sortino 1.41 and annualized volatility 31.77%. The worst peak-to-trough fall was 57.39%, with a Calmar of 0.40. Those risk figures belong to a near-equal-weight basket of the year's most-traded names. In particular, the 57.39% drawdown is the long basket's drawdown; it says nothing about the paper's interior-solution rule.
We have figures for the price-observer book alone and no results for a matching naive book. They therefore tell us nothing about whether learning beat a frozen prior. The paper reports no performance figures of its own for comparison. Whether this book looks weak or strong, its result belongs to a clipped, one-pass construction rather than serving as a verdict on the authors' work.
Exact theory, frictionless calibration
The split into B·Δ² and a fixed C is exact. So is the near-maturity condition (m−μ)² < v/2, which a desk can reason with. The 0.247 probability and every κ requirement come from one frictionless calibration with an unobservable capacity dial. A costed simulation near maturity, under assumed drifts, would move me if the naive book won inside the (m−μ)² < v/2 band and lost outside it.
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.