6.7.3 Temporary vs Permanent Impact

Basic Prerequisites: 02 · The Kyle Model and Spread Decomposition & the Roll Model.

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1. Intuition & Practical Objective

Every impact you observe is the sum of two things that behave nothing alike:

• Permanent impact - the part of the price move that stays. The market has genuinely re-priced the asset (because your flow was informative, or because a large imbalance permanently cleared). Sell a stock and watch the price settle 20bp lower for good: that is permanent impact. It is schedule-independent: it depends on how much you trade, not on how you slice it.
• Temporary impact - the part that decays. You are consuming liquidity faster than it replenishes, so the price you pay is temporarily worse; once you stop, the book refills (“resilience”) and the price snaps back. It is schedule-sensitive: trade more slowly and it shrinks.

Why this is the most important distinction in execution. If you mistake temporary impact for permanent, you will conclude that trading slowly is pointless (it isn’t - it removes the temporary part). If you mistake permanent for temporary, you will schedule an order assuming the price will revert when it never will - and you will systematically lose money on every large trade. The realized cost of your execution (the VWAP impact) is a mixture: it is worse than the permanent impact but better than the instantaneous peak, and where it lands between them is precisely what your execution algorithm controls.

The one-sentence essence. “Permanent impact is the price you truly pay for the liquidity of the asset and is fixed by size; temporary impact is the price you pay for trading fast and is a dial your execution schedule turns.”

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2. Mathematical Ground Truth & Derivations

2.1 The Almgren–Chriss decomposition

Sell $X$ shares in $N$ discrete slices, $n_k$ shares at time $t_k$, so ${\sum }_k{n}_k=X$. The price has a permanent drift and a temporary concession:

$$S_k=S_0+\gamma \sum _{j\le k}{n}_j(\text{permanent}),{\tilde{S}}_k=S_k+ϵ\mathrm{sgn}(n_k)+\tilde{\eta }\frac{n_k}{\tau }(\text{temporary execution price}).$$

Here $g(v)=\gamma v$ is the permanent impact function and $h(v)=ϵ\mathrm{sgn}(v)+\tilde{\eta }v$ the temporary one; $ϵ$ is the fixed (spread-plus-fee) cost and $\tilde{\eta }$ the variable concession per unit trade rate. The realized cost of the program is ${\sum }_k{n}_k{\tilde{S}}_k-X{S}_0$, whose expectation is

$$\boxed{\mathbb{E}[x]=\frac{1}{2}\gamma X^2+ϵ\sum _k|n_k|+\tilde{\eta }\sum _k{n}_k^2},\tilde{\eta }=\eta -\frac{1}{2}\gamma ,$$

and whose variance is $\mathrm{Var}[x]=\frac{1}{2}{\sigma }^2{\sum }_k{\tau }_k{x}_k^2$ (the uncertain future price of the shares you still hold). Two immediate consequences:

• The permanent term $\frac{1}{2}\gamma X^2$ is independent of the schedule - every child order size drops out of it (the sum telescopes). You cannot schedule permanent impact away.
• The temporary term $\tilde{\eta }{\sum }_k{n}_k^2$ is minimised by spreading the order out (making the $n_k$ small). Trading very slowly drives it toward zero - the “min-impact” / TWAP limit.

Trading off expected cost against timing risk yields the efficient frontier, and for linear impact the optimal trajectory is the closed form

$$x_j=\frac{\sinh (\kappa (T-t_j))}{\sinh (\kappa T)}X,\kappa \approx \sqrt{\frac{\tilde{\lambda }{\sigma }^2}{\tilde{\eta }}}(\text{Gatheral’s }\kappa ),$$

which interpolates between constant-rate (risk-neutral, $\kappa \to 0$) and front-loaded (risk-averse, $\kappa \to \infty$) execution. (The scheduling problem itself is Pillar 2; here the point is the arithmetic of the two components.)

2.2 The temporary component is exactly what the price does back

Define the transient pressure $T_n$ as an AR(1)-like decay: $T_n=\rho T_{n-1}+\eta n_n$, with recovery factor $\rho \in [0,1)$ (resilience). Then the observed mid is $m_n+T_n$ where $m_n$ carries the permanent drift. After execution stops ($n_n=0$), $T_n\to {\rho }^{n}T\to 0$ and the price settles at $m_{\infty }=S_0+\gamma X$ - the permanent level. The gap between the peak and the permanent level is the temporary impact that “comes back”.

2.3 The statistical version - generalized Roll

Hasbrouck’s generalized Roll model makes the same split in transaction data:

$$\Delta p_t=c(q_t-q_{t-1})+\lambda q_t+u_t,$$

with $q_t\in \{+1,-1\}$ the trade sign, $c$ the transitory (order-processing / inventory / bid-ask bounce) component and $\lambda$ the permanent (adverse-selection) component. The spread is $2(c+\lambda )$, and the identified random-walk variance is ${\sigma }_w^2={\lambda }^2+{\sigma }_u^2={\gamma }_0+2{\gamma }_1$. The transitory component is exactly the $q_t-q_{t-1}$ “bounce”; the permanent component is the $\lambda q_t$ that shifts the efficient price. (Glosten–Harris 1988 estimate the same decomposition with size-dependent $\lambda$; Huang–Stoll 1997 is the three-way ordering/inventory/adverse-selection version.)

2.4 Resilience and the finite-book correction

If the book refills immediately, all impact is temporary and total cost is zero for infinitesimal trade rates - absurd, so permanent impact must exist. If the book never refills, all impact is permanent - equally wrong. The truth is a decay: Obizhaeva & Wang (2013) show that with a resilient book the optimal strategy is a block at the start, a block at the end, and a linear ramp between - precisely because the temporary impact you create will partly come back, and you can re-use that resilience.

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3. Computational Implementation - the same order, four speeds

One sell of $X=-10^6$ shares, executed at four speeds (10, 25, 100, 500 child orders). Permanent coefficient $\gamma =+2\times 10^{-6}$/share (so a sell has negative signed flow and pushes the price down), transient coefficient $\eta =+8\times 10^{-6}$/share, per-slice decay $\rho =0.90$. Stdlib only.

Read the table:

• Permanent impact is constant at $-2.0000$ across a fifty-fold speed range - the per-share permanent impact $\gamma X=2\times 10^{-6}\times (-10^6)=-$$2 (in total dollars, $\frac{1}{2}\gamma X^2=-$$1M), schedule-independent.
• Peak (adverse) impact falls from $-7.21$ to $-2.16$ as execution slows: the temporary component $\to 0$, and the peak converges on the permanent level.
• Realized VWAP impact is always between the two ($-4.41$ down to $-1.16$): it is what you actually pay, and it is the quantity your execution schedule controls.

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4. Failure Modes & First-Principles Breakdowns

1. Double-charging the permanent part. A common bug: charging the full observed impact to every slice and also counting the permanent drift - effectively paying $\gamma X$ twice. The Almgren–Chriss bookkeeping charges $\frac{1}{2}\gamma X^2$ once (the $\frac{1}{2}$ comes from the fact that the shares you sell early escape the full permanent drop).
2. Assuming the price reverts when it doesn’t. If the flow is informed, “temporary” impact can become permanent the moment the market infers the information - the Kyle half-information result (page 02). Distinguishing temporary from permanent in real data requires waiting and observing the post-trade decay (the Almgren et al. 2005 methodology: $S_{\text{post}}-S_0$ gives permanent, $\bar{S}-S_0$ gives realized).
3. Ignoring resilience. Treating the book as statically thin over-charges repeated trading; treating it as instantly-resilient under-charges the first block. Obizhaeva–Wang’s block-ramp-block solution is the correct minimal model when resilience matters.
4. Confusing the transitory bounce with information. In the generalized Roll model, a negative first autocovariance (${\gamma }_1=-c(c+\lambda )$) is the bid-ask bounce; a persistent positive response to signed flow is the permanent $\lambda$. Mislabelling the bounce as information (or vice-versa) corrupts every downstream estimate.
5. The linear/quadratic trap. The clean $\frac{1}{2}\gamma X^2+\tilde{\eta }\sum n_k^2$ form relies on linear impact; real temporary impact is concave (page 04), so the quadratic penalty is a local approximation that mis-sizes large orders.

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5. References

• Almgren, R. & Chriss, N. (2000), Optimal execution of portfolio transactions, Journal of Risk 3(2), 5–39. Permanent/temporary split, the linear cost function, the efficient frontier, the $\sinh$ trajectory. Primary PDF in corpus (Almgren_2000_...).
• Almgren, R., Thum, C., Hauptmann, H. & Li, H. (2005), Direct estimation of equity market impact, Risk 18(7), 57–62. The empirical operationalisation: pre-trade, post-trade, realized prices; $\gamma ,\eta$ fitted. Primary PDF in corpus.
• Obizhaeva, A. & Wang, J. (2013), Optimal trading strategy and supply/demand dynamics, J. Financial Markets 16(1), 1–32. Resilience and the block-ramp-block solution.
• Hasbrouck, J. (2007), Empirical Market Microstructure
• Glosten, L. R. & Harris, L. E. (1988), Estimating the components of the bid/ask spread, JFE 21(1), 123–142. The size-dependent transitory/permanent decomposition. Primary PDF in corpus.

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6. Connected Graph Bridges

• Back: 02 · The Kyle Model · Spread Decomposition & the Roll Model
• Forward: 04 · The Square-Root Law · Index Hub
• Pillar 2: Almgren–Chriss Optimal Execution (the scheduling problem built on this decomposition) · Limit Order Book Mechanics (resilience measured in the book)