Phase 16 - Lesson 16.3

Market Impact and Transaction Costs

Why trading moves the price against you, how that cost scales, and how to account for it honestly.

⏱ 50 min● Advanced🔗 Prereqs: 16.1–16.2

Builds on: 16.2’s Kyle’s-lambda and adverse selection are the roots of permanent impact.
Leads to: 16.4 optimizes a trade schedule against exactly these impact and risk costs.

Learning Objectives

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Distinguish temporary from permanent market impact and give the mechanism of each.
State the empirical square-root law of impact and contrast it with linear models.
Enumerate the components of implementation shortfall (transaction costs).
Implement a linear temporary/permanent impact cost model and compute total cost.
Explain why impact makes execution a trade-off rather than a ‘trade now’ decision.

Key Vocabulary

Market impact: The adverse price move caused by your own trading; the dominant cost for large orders.

Temporary impact: A transient price concession for demanding immediacy that decays after you stop trading.

Permanent impact: A lasting shift in the efficient price because trading reveals information (Kyle’s lambda).

Square-root law: Empirical rule: impact scales roughly as $\propto\sigma\sqrt{Q/V}$, sublinear in size.

Implementation shortfall: Total cost = execution price minus the decision (arrival) price, across the whole order.

Participation rate: Your trading as a fraction of market volume; a key driver of temporary impact.

Slippage: Realized execution price minus a benchmark (arrival mid or VWAP).

Intuition & Motivation

Intuition

Trading is not free even ignoring fees: the act of buying pushes the price up and selling pushes it down, so you are always fighting your own footprint. Two effects combine. Temporary impact is the concession you pay for immediacy - eat the book fast and prices back up as soon as you stop, so it depends on how aggressively you trade (your rate). Permanent impact is the market inferring information from your flow and repricing for good - it depends on how much you trade in total. Empirically impact is sublinear: doubling size does not double cost, it grows like a square root. Because trading fast costs impact but trading slow exposes you to price risk, execution becomes an optimization - the subject of 16.4.

Two kinds of impact

Model the price you pay when trading at rate $v_t$ (shares per unit time) as an efficient price plus a temporary concession:

\[\tilde P_t=\underbrace{P_0+\gamma\,X_t}_{\text{permanent}}+\underbrace{\eta\,v_t}_{\text{temporary}},\qquad X_t=\int_0^t v_s\,ds,\] (16.5)

where $X_t$ is cumulative quantity traded, $\gamma$ is the permanent-impact coefficient (a Kyle’s-lambda-like information term), and $\eta$ the temporary-impact coefficient. Permanent impact moves the reference price for everyone and persists; temporary impact affects only your own fills and decays when you stop.

Impact	Depends on	Persists?	Mechanism
Temporary	Trading rate $v_t$	No (decays)	Demanding immediacy; consuming resting depth
Permanent	Total quantity $X$	Yes	Information leakage; the market reprices

How impact scales: the square-root law

A robust empirical regularity across markets is that the cost of executing a metaorder of size $Q$ over a day scales as

\[\text{impact}\ \approx\ Y\,\sigma\,\sqrt{\frac{Q}{V}},\] (16.6)

where $\sigma$ is daily volatility, $V$ daily volume, and $Y$ an $O(1)$ constant. The sublinear square-root shape (not linear) is one of the most reproduced facts in microstructure: trading twice the size costs only about $\sqrt2\approx1.41$ times as much per the metaorder, so cost per share falls with size - but total cost still rises.

Key Idea

Linear impact (as in 16.5) is a tractable modeling choice that underlies Almgren–Chriss; the empirical square-root law (16.6) is the calibration reality. Use linear models to derive optimal schedules, but size expectations with the square-root law.

Transaction costs: implementation shortfall

The honest accounting benchmark is implementation shortfall (Perold): compare the final booked P&L to a paper portfolio that transacted instantly at the decision price $P_0$. It decomposes into:

Spread / immediacy cost - crossing the bid–ask (16.1–16.2).
Market impact - temporary + permanent (16.5).
Timing / opportunity cost - price drift while you wait, plus the cost of any unfilled residual.
Fees and taxes - commissions, exchange fees, stamp duties, minus rebates.

Worked Example - Total cost of a linear-impact execution

Buy $Q=100{,}000$ shares at constant rate over the day. Permanent $\gamma=2\times10^{-6}$ \$/share per share; temporary $\eta=1\times10^{-5}$ \$/share per (share/day).

Permanent cost: the price ends $\gamma Q=2\times10^{-6}\times10^5=0.20$ \$ higher; averaged over the fill (you pay half on the way up) it costs $\tfrac12\gamma Q^2=\tfrac12(2\times10^{-6})(10^5)^2=\$10{,}000$.

Temporary cost at rate $v=Q/T$ over horizon $T=1$ day: $\eta\,v\,Q=\eta Q^2/T=(10^{-5})(10^{10})/1=\$100{,}000$.

Total impact cost $\approx\$110{,}000$ on a $\$10{,}000{,}000$ notional (at \$100) - about $110$ bps. Trading slower shrinks the temporary term but lengthens exposure to price risk.

Interactive: measure slippage as you split an order

Slice the same parent order more finely and watch temporary impact fall while execution stretches out.

Interactive: a linear impact-cost model

Common Mistakes to Avoid

Confusing temporary and permanent impact: only temporary decays when you stop, and only it depends on your rate.
Assuming impact is linear in size for calibration: empirically it is sublinear (square-root), so linear extrapolation overstates big-order cost per share.
Benchmarking against the final price instead of the arrival price: implementation shortfall uses the decision price $P_0$.
Ignoring opportunity cost of unfilled shares: a ‘cheap’ passive strategy that under-fills can be the most expensive one.

Quant Practitioner Tips

Quote costs in basis points of notional so they compare across names and sizes.
Calibrate the square-root constant $Y$ from your own fills; it is stable but venue/name-dependent.
Separate alpha decay from impact: if your signal decays fast, paying more temporary impact to trade quickly can be optimal.
Always report implementation shortfall with all four components, including fees and unfilled residual.

Knowledge Check

Q1 Medium

Temporary market impact differs from permanent impact in that temporary impact:

Persists forever and depends on total size

Decays after you stop trading and depends on your trading rate

Is caused only by exchange fees

Is always larger than permanent impact

Q2 Medium

The empirical square-root law of market impact states that the cost of a metaorder scales approximately as:

Linearly in $Q$

$\propto\sigma\sqrt{Q/V}$, sublinearly in size

Independently of size

$\propto Q^2$

Q3 Easy

Implementation shortfall is measured relative to:

The closing price

The volume-weighted average price only

The decision (arrival) price when the order was generated

The next day’s open

Practical Exercise

You must buy $200{,}000$ shares (arrival price $\$50$). Using the linear model with $\gamma=1.5\times10^{-6}$ and $\eta=8\times10^{-6}$ (\$ units), (a) compute the impact cost of executing over $T=1$ day vs $T=4$ days. (b) Which term changes and why? (c) Name one reason you would not simply pick the slowest schedule.

▶ Show full solution

(a) Permanent $=\tfrac12\gamma Q^2=\tfrac12(1.5\times10^{-6})(2\times10^5)^2=\tfrac12(1.5\times10^{-6})(4\times10^{10})=\$30{,}000$ (independent of $T$). Temporary $=\eta Q^2/T=(8\times10^{-6})(4\times10^{10})/T=320{,}000/T$. For $T=1$: $\$320{,}000$, total $\$350{,}000$. For $T=4$: $\$80{,}000$, total $\$110{,}000$.

(b) Only the temporary term changes - it scales as $1/T$ because a longer horizon means a lower trading rate $v=Q/T$ and temporary impact is proportional to rate. Permanent impact depends on total quantity, which is fixed.

(c) Stretching to 4 days quadruples your exposure to price/timing risk: the stock can drift away from $\$50$ while you wait, and any alpha in the order decays. Minimizing impact alone ignores this variance - the reason optimal execution (16.4) balances impact against timing risk rather than just trading as slowly as possible.

After the reveal, answer for yourself: If the order carried a fast-decaying alpha signal, would you shift toward faster or slower execution?

Lesson Summary

Trading moves prices against you through temporary impact (a rate-dependent, decaying immediacy concession) and permanent impact (a quantity-dependent, lasting information effect, i.e. Kyle’s lambda). Empirically impact is sublinear - the square-root law $\propto\sigma\sqrt{Q/V}$ - while tractable linear models drive optimal-execution theory. Honest cost accounting uses implementation shortfall against the arrival price, and because fast trading costs impact while slow trading costs price risk, execution is inherently a trade-off.

Formula Sheet Additions

Linear impact model

\[\tilde P_t=P_0+\gamma X_t+\eta v_t\]

Permanent $\gamma X_t$ (persists, size-driven) plus temporary $\eta v_t$ (decays, rate-driven).

Square-root impact law

\[\text{impact}\approx Y\,\sigma\sqrt{Q/V}\]

Empirical, sublinear in size; used to calibrate expected cost.

Error Log Checklist

Did I separate temporary (rate) from permanent (quantity) impact?
Did I use the square-root law for calibration, not linear extrapolation?
Did I benchmark against the arrival price (implementation shortfall)?
Did I include opportunity cost of unfilled shares and fees?

Retrieval Practice

Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.

▶ Show retrieval prompts & answers

Q: Contrast temporary and permanent market impact.
A: Temporary impact is a rate-dependent immediacy concession that decays after you stop; permanent impact is a quantity-dependent, lasting repricing from information leakage (Kyle’s lambda).

Q: State the square-root law of impact and what makes it notable.
A: Impact $\approx Y\sigma\sqrt{Q/V}$: it is sublinear in order size, so cost per share falls with size - a highly reproduced empirical regularity.

Q: What is implementation shortfall and against what benchmark is it measured?
A: Total execution cost measured against the decision/arrival price $P_0$; it sums spread, market impact, timing/opportunity cost, and fees.

Flashcards

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Temporary vs permanent impact

Temporary: rate-driven, decays. Permanent: quantity-driven, persists (Kyle’s lambda).

Square-root law

impact $\approx Y\sigma\sqrt{Q/V}$; sublinear in size.

Implementation shortfall

Execution cost vs the arrival price: spread + impact + timing + fees.

Completion Checklist

I can explain the core ideas in my own words
I worked the derivations/examples by hand
I completed the interactive workbench(es)
I passed the knowledge check

Confidence / mastery rating

Personal notes

← 16.2 Next: 16.4 →

Impact	Depends on	Persists?	Mechanism
Temporary	Trading rate \(v_t\)	No (decays)	Demanding immediacy; consuming resting depth
Permanent	Total quantity \(X\)	Yes	Information leakage; the market reprices