A promising alpha can decay before the backtest earns trust

A trading signal can be real and still be impossible to validate in time. Bonacorsi's paper formalizes that awkward case: evidence arrives while the signal's expected return decays, so research consumes part of the very opportunity it is trying to certify.

An alpha is expected return beyond the chosen economic baseline. Its half-life is the time required for that expected return to halve. The paper measures both learning and remaining trading value in Kullback–Leibler (KL) information, a measure of how distinguishable the signal law is from the no-alpha boundary.

Suppose expected signal return decays as μ₀e⁻λᵗ, its noise scale is σ, and S = μ₀/σ is instantaneous Sharpe. Even with an unlimited calendar horizon, squared signal strength integrates to a finite lifetime budget: I_life = S²h_1/2/(4 log 2), where h_1/2 is the half-life in the same time units used for S.

A level-α Gaussian test with target power 1−β requires I_crit = ½(z_1−α + z_1−β)². Reliable certification before decay therefore needs I_life ≥ I_crit.

def minimum_half_life(sharpe, z_false_deploy, z_power):
    separation = z_false_deploy + z_power
    required_information = 0.5 * separation * separation
    return 4.0 * log(2.0) * required_information / (sharpe * sharpe)

When M candidates share a familywise false-deployment budget α_F, Bonferroni gives each test α_F/M. That stricter tail threshold raises required information. The growth is only logarithmic in M asymptotically, but the signal's lifetime information is fixed.

The most important strict inequality in the paper is easy to miss. At I_life = I_crit, a test reaches the requested power only at terminal information time. Certification is feasible, but no post-certification opportunity remains. Positive deployable value requires strictly more information than the headline threshold.

For the illustrative Sharpe-2, half-life-5 signal, lifetime information is 7.213 nats. One declared candidate needs 4.282 nats, giving a capacity ratio of 1.68. Searching 100 candidates reduces that ratio to 0.69.

The paper also makes lifetime endogenous: arbitrage activity can shorten the remaining opportunity, so certification cost becomes an entry and crowding constraint. Its retrospective funding-rate illustration uses BTC-USDT and ETH-USDT perpetual futures from six exchanges, with 545 events and 543 finite lifetime scores, to ask whether a pre-outcome persistence estimate adds information beyond current funding magnitude.

Log the entire candidate library before selection, estimate decay out of sample, express Sharpe and half-life in compatible units, and report the capacity ratio beside conventional significance. Then stress-test correlation-aware multiplicity rules, non-exponential decay, parameter uncertainty, transaction costs, and value lost while waiting.

The Gaussian channel is a standardized mean-estimation problem. See the interactive linear-regression walkthrough for the from-scratch prediction machinery beneath many signal estimates.

The half-life table is paper-reported. All other plotted values are deterministic calculations from pure core functions; the Sharpe-2, half-life-5 example is illustrative. No ML library, external asset, or random draw is used.