Slow Decay of Impact in Equity Markets¶

Authors: X. Brokmann, E. Sérié, J. Kockelkoren, J.-P. Bouchaud Institution: Capital Fund Management, Paris Year: 2014 (arXiv July 2014; published Market Microstructure and Liquidity 2015) arXiv: 1407.3390 Categories: q-fin.TR, cond-mat.stat-mech

Plain-language abstract¶

After a large meta-order finishes executing, does its price impact decay to zero (mechanical, purely technical) or plateau at some fraction of the peak (a signature that the trade was informed and the information has been permanently impounded)? Prior empirical work was split. This paper uses CFM's proprietary meta-order data + prediction signals and a deconvolution method to separate the impact of a single meta-order from the impact of all correlated subsequent trades. Finding: after proper deconvolution, impact decays essentially all the way to zero — possibly as a power-law — over ~10 trading days. The apparent plateau seen in the raw, un-deconvolved data is a statistical artefact of order-flow autocorrelation (order splitting across days), not genuine permanent impact. The square-root law of peak impact is confirmed with $\delta \approx 0.6$.

Key contributions¶

Decomposition of measured meta-order impact:

$$\mathbb{E}[p(t + \tau) - p(t) \mid \text{metaorder at } t] = \theta(Q) \cdot I(\tau) + \alpha \cdot H(\tau)$$

where: - $\theta(Q) = \epsilon(Q) Y_0 \sigma (Q/V)^\delta$ is the instantaneous square-root impact ($\delta \approx 0.6$ empirically; $Y_0$ order 1), - $I(\tau)$ is the mechanical impact propagator decaying from $I(0) = 1$ to $I(\infty) \approx 0$, - $\alpha$ is the predictor amplitude (zero for uninformed trades), - $H(\tau)$ is the way the prediction realises over time (e.g., $H(\tau) = 1 - e^{-\Gamma \tau}$ for an OU predictor).

Deconvolution method to separate $I(\tau)$ from the raw measured profile. Shows that autocorrelated order flow (the same trader executing a series of same-signed orders over days) artificially inflates the apparent long-horizon plateau when no deconvolution is applied.
Toy-model demonstration: an optimising investor with an OU signal and instantaneously-decaying true impact produces a raw measured impact that appears to decay slowly — purely because the investor's continued trading keeps reinforcing the original direction. Proves the artefact is quantitatively what's seen in raw data.
Empirical calibration on CFM meta-order data:
Volume fractions $Q/V$ in 0.1–5%.
Predictor horizons $\Gamma^{-1}$ from 10 to 100 days.
Signals market-neutral, unit-variance-normalised.
Execution near 100% (no selection bias).
Confirms square-root law: $\delta \approx 0.6$, within the empirical 0.4–0.7 band reported across other studies.
Finds impact decays to zero (or a very small residual) after deconvolution. Contradicts the $2/3$-of-peak plateau prediction from the no-arbitrage argument of [Farmer et al. 2013] when applied without deconvolution. Reconciles with that argument's spirit by noting that residual permanent impact should be attributable to true information content ($\alpha \cdot H(\infty)$), not to the mechanical impact $I(\tau)$.

Method summary¶

Signal & prices¶

Daily observation times $t$ when meta-orders start.
Execution prices $p_x(t)$ and daily mids $p_d(t)$.
Strike slippage $r_x(t) = (p_x(t) - p_d(t))/p_d(t)$.
Daily return $r_d(t) = (p_d(t+1) - p_d(t))/p_d(t)$.

Raw vs deconvolved impact¶

Raw impact $I_\text{raw}(\tau)$: average price change $\tau$ days after meta-order completion, over all meta-orders. Contaminated by the continued trading of the same investor on correlated signals.
Deconvolved impact $I(\tau)$: obtained by subtracting the contribution of all correlated subsequent meta-orders (using the investor's signal autocorrelation structure) from $I_\text{raw}$.

Deconvolution via toy model¶

In the OU-signal toy model, the optimal position is:

\[\pi(t) = \phi_0 \int_{-\infty}^t e^{-\omega(t-t')} s(t') \, dt'\]

with a specific $\omega$ depending on the predictor horizon $\Gamma$ and impact coefficient $\gamma$. This gives an analytic formula for $I_\text{raw}$ as a convolution of the "true" $I$ with the position-process autocorrelation — which can be inverted.

Functional form of $I(\tau)$¶

Consistent with a power-law decay $I(\tau) \sim \tau^{-\beta}$ at long times, or a stretched-exponential; the data can't distinguish between these cleanly but both imply $I(\infty) \approx 0$.

Main results¶

Peak impact matches square-root law with $\delta \approx 0.6$.
Deconvolved $I(\tau)$ decays to within statistical noise of zero over 10 trading days.
Raw $I_\text{raw}(\tau)$ shows the widely reported 1/3-2/3 plateau artefact — this is now explained quantitatively by the toy model.
Holds across US / Europe / Japan equity datasets (CFM trades across multiple markets); approximately universal.
The residual long-horizon impact that remains after deconvolution is attributed to the genuine information content of informed trades (i.e., $\alpha$), not to a permanent mechanical footprint.

Limitations¶

Proprietary data: CFM-only; impossible to reproduce without access. But the methodology generalises.
Quasi-linear assumption: meta-orders assumed to add linearly. Known to be an approximation — the paper acknowledges this explicitly. True short-time impact is strongly concave (square-root), not linear; the quasi-linear framework is a convenient long-time abstraction.
10-day horizon is the empirical ceiling due to noise; can't rule out tiny residual impact beyond.
Single functional-form fit: power-law vs stretched-exponential is not discriminated; both decay to zero.
Equities only: futures, FX, crypto may show different decay patterns — untested here.
Informed vs uninformed decomposition via $\alpha$ is model-structural; extracting $\alpha$ empirically depends on the predictor's known structure.

Connections¶

Direct empirical support for the propagator framework surveyed in bouchaud-farmer-lillo-propagator. That paper advocated a transient-impact view; this paper is the definitive empirical test on proprietary meta-orders.
Contrasts with permanent-impact view (fixed permanent impact per trade) — shows it can't survive proper deconvolution.
Relevant for optimal-execution practitioners: if mechanical impact decays fully, the "cost-of-trading" calculation used in Almgren-Chriss / MPC frameworks should use the propagator form, not a fixed permanent component. Impacts practical execution cost estimation.
Complementary to price-impact-order-book-events: CKS give the linear OFI→mid-price coefficient at sub-minute horizons; this paper gives the shape of meta-order impact decay over days. Together: short-horizon contemporaneous impact plus long-horizon relaxation.
Motivates the impact-feedback kernel in reality-gap-lob-simulation — the power-law decay of that paper's simulator kernel is calibrated to be consistent with findings like these.
Connects to adverse-selection: the residual $\alpha \cdot H(\infty)$ is the permanent component attributable to information content — the signature of informed trading in the Kyle / Glosten-Milgrom sense.