The Schedule Problem: Impact Against Timing Risk

The previous lesson ended with a conflict. Making it precise turns it into a solvable problem.

Let the order be worked over a horizon T. Trading faster means a larger quantity per unit time, and impact grows with the rate at which liquidity is consumed. So expected cost falls as T rises.

Meanwhile the unexecuted portion of the order remains exposed to whatever the price does. Over a longer horizon the price has more time to wander, and the variance of the final outcome grows with T. So risk rises as T rises.

One is an expectation and the other a variance, which is what makes the problem tractable: it has the same shape as portfolio selection. Choose the schedule that minimises expected cost plus a penalty times variance, and the penalty is where preference enters.

Robert Almgren and Neil Chriss set this out in "Optimal Execution of Portfolio Transactions", Journal of Risk, volume 3, 2000, pages 5 to 39, framing liquidation over a fixed horizon as a mean-variance trade-off between impact costs from trading too quickly and timing risk from trading too slowly.

The model separates impact into the two components the microstructure path established, because they enter the optimisation differently.

Temporary impact depends on the rate of trading right now. Push shares through faster and you pay more per share at this instant, but the effect does not persist: stop trading and the book replenishes. This is the term the schedule can control.

Permanent impact depends on cumulative quantity. The market infers information from the total order flow, and that inference does not revert. Crucially, it depends on how much you trade rather than how fast, so it is the same for every schedule that completes the order.

That asymmetry is why the optimisation is possible at all. A cost that is identical under every schedule cannot influence the choice among schedules. It sets the floor on what the order costs and then drops out of the decision.

So the entire scheduling question reduces to trading temporary impact against timing risk, with permanent impact as a fixed toll.

Because the problem is mean-variance, its solution is a frontier rather than a single schedule. For every level of risk there is a minimum achievable expected cost, and vice versa.

The risk-aversion parameter selects a point on it. At one extreme, indifference to risk gives the slowest schedule, trading at a constant tiny rate to minimise expected cost while accepting maximum exposure to drift. At the other, extreme risk aversion gives immediate execution, paying whatever impact is required to eliminate uncertainty at once.

Between them the optimal trajectory front-loads: more of the order is executed early, with the rate decaying over time. The intuition is that shares not yet traded are the ones carrying risk, so reducing the outstanding quantity early buys risk reduction, and the shape balances that against the impact of trading fast.

That is a genuine result rather than an obvious one. The naive schedule, trading a constant quantity per unit time, is optimal only for a trader indifferent to risk. Anyone with positive risk aversion should trade a declining schedule.

A simplified version makes the shape visible without the full derivation.

import math

def schedule_cost(T, qty=1_000_000, adv=20_000_000, sigma_daily=0.02,
                  temp_bps=100.0, risk_aversion=0.1):
    """T in days. Returns (expected impact bps, timing risk bps, objective).
    temp_bps is the impact at 100% participation; cost is concave in rate."""
    rate   = (qty / T) / adv                        # participation rate
    impact = temp_bps * math.sqrt(rate)
    risk   = 1e4 * sigma_daily * math.sqrt(T) / 2   # average outstanding
    return impact, risk, impact + risk_aversion * risk

for T in (0.25, 0.5, 1, 2, 5, 10):
    i, r, obj = schedule_cost(T)
    print(f"T={T:>5}d  impact={i:5.1f}  risk={r:6.1f}  objective={obj:5.1f}")
# T= 0.25d  impact= 44.7  risk=  50.0  objective= 49.7
# T=  0.5d  impact= 31.6  risk=  70.7  objective= 38.7
# T=    1d  impact= 22.4  risk= 100.0  objective= 32.4
# T=    2d  impact= 15.8  risk= 141.4  objective= 30.0   <- minimum
# T=    5d  impact= 10.0  risk= 223.6  objective= 32.4
# T=   10d  impact=  7.1  risk= 316.2  objective= 38.7

Impact falls monotonically with horizon and risk rises, exactly as claimed. The objective has an interior minimum, here near two days. Setting the derivative to zero gives it in closed form, and raising risk aversion moves it earlier.

The risk-aversion parameter looks like a free choice and is not. Three observable quantities determine it, and only one of them is known to the execution desk.

Volatility and order size relative to volume are both measurable from market data. They pull in opposite directions: higher volatility argues for speed, larger size argues for patience.

Signal decay is the third, and it is the one that usually dominates. It is a property of why the trade is being made, and it lives with the portfolio manager rather than the trader.

This is the practical failure mode in most institutions. An execution desk optimising impact without knowing the signal horizon will systematically trade too slowly for fast signals and too quickly for slow ones. The information needed to set urgency correctly sits on the other side of an organisational boundary, and the number that would reveal the error, opportunity cost, is the one most often left out of the report.

The optimisation above produces a trajectory chosen at the start and followed regardless of what happens. That is a real limitation, and the fix has a cost of its own.

A static schedule is decided up front. It is predictable, easy to monitor, and immune to being talked out of. It also ignores information: if liquidity turns out to be unusually deep, the schedule does not take advantage, and if the price moves sharply in your favour, it does not accelerate.

An adaptive schedule responds to conditions. It trades more when the book is deep and spreads are tight, less when they are not.

Adaptation is better in expectation and introduces two problems. It makes the order's behaviour a function of observable market state, which means other participants can infer it, and inferring a large buyer's reaction function is worth money. And it opens the door to discretion drifting into a directional bet: an algorithm that slows down when the price rises is, in effect, deciding it does not like the trade any more.

The usual compromise is bounded adaptation: a static baseline with limited deviation, so behaviour stays roughly predictable to its owner and only roughly predictable to everyone else.

The framework is standard because it is tractable, and its assumptions are worth naming because each fails in a recognisable way.

The second row is the most interesting. Timing risk is modelled as symmetric variance, but a trader buying because they expect the price to rise faces an asymmetric distribution: waiting is more likely to hurt than help, by assumption.

When the reason for the trade is a directional view, the model understates the cost of patience. That reinforces the earlier point: the urgency parameter is where the view enters, and choosing it well requires knowing something the schedule optimiser is not told.

The optimisation answers one question: over what horizon, and with what profile, to work a decided order.

It does not decide whether to trade, and it does not decide the size. Those are upstream, and the previous cursus on risk covers the second. It also does not decide how each individual slice reaches the market, which turns out to matter as much as the schedule.

That second gap is substantial. A schedule specifying 20,000 shares in the next fifteen minutes is silent on whether they are sent as one order or fifty, whether they rest passively or cross the spread, which venues receive them, and whether they are displayed.

Those choices determine a large part of the realised cost, and they interact with the schedule: a passive placement policy will not complete an aggressive schedule, and an aggressive one wastes a patient schedule's advantage.

The next lesson is about that layer, where the trade-off from this one reappears in a different form at every child order.