Execution and microstructure arithmetic

Implementation shortfall

The difference between the value of a hypothetical portfolio traded instantly at the decision price and the value actually achieved. It is the only cost measure that captures unfilled quantity, which is why it is the correct benchmark for an order and VWAP is not.

Field	Value
Formula	IS = filled x (P_exec - P_decision) + unfilled x (P_final - P_decision) + fees, signed so that a positive value is a cost for a buy. The first term splits further into filled x (P_arrival - P_decision) for delay and filled x (P_exec - P_arrival) for impact
Worked	Buy 500,000 at a decision price of 50.00, notional 25,000,000. Delay 450,000 x 0.05 = 22,500.00. Impact 450,000 x 0.09 = 40,500.00. Opportunity 50,000 x 0.40 = 20,000.00. Fees 450,000 x 0.0020 = 900.00. Total 83,900.00 = 33.5600 basis points
Cross-check	Execution cost on filled shares alone is 450,000 x (50.14 - 50.00) = 63,000.00, exactly the sum of delay and impact
Component shares	Delay 26.8 percent of the total, impact 48.3 percent, opportunity 23.8 percent, fees 1.1 percent. Fees are the smallest component and receive the most negotiating attention
Opportunity cost is not optional	The 50,000 unfilled shares cost 20,000.00 - 8.00 basis points, more than 20 times the fees. An execution report that omits unfilled quantity omits its largest controllable term

• Trading passively lowers impact and raises opportunity cost, and trading aggressively does the reverse. Any measure that captures one and not the other can be gamed: a trader benchmarked on impact alone can post at the far touch and never fill, and report an excellent number.
• The decision price must be a real, recorded price at a recorded time, not reconstructed later. Reconstructing it is the point at which the measure becomes negotiable, and a shortfall computed against a reconstructed decision price measures nothing.
• Delay cost belongs to the portfolio manager and impact belongs to the trader, which is why the split matters organisationally as well as arithmetically. Here delay is 9.00 basis points and impact 16.20, and they have different owners and different fixes.
• Implementation shortfall is signed against the side. For a sell, every difference reverses sign, and a shortfall measure that does not carry the side explicitly will report a cost as a gain roughly half the time.

Source: Perold 1988

Effective against quoted spread

The quoted spread is what the book shows; the effective spread is what a trade actually paid relative to the midpoint. They differ whenever a trade executes inside the quote, and the ratio of the two is a measure of how much of the displayed spread a trader actually crosses.

Field	Value
Formula	Quoted spread = (ask - bid)/M. Effective half-spread = D*(P - M)/M with D = +1 for a buy and -1 for a sell and M the midpoint at the time of the trade. The doubled convention reports 2*D*(P - M)/M for comparability with the quoted spread
Worked	Bid 49.98, ask 50.02, M = 50.00. Quoted spread 0.04 = 8.0000 basis points. Buy fills at 50.015: effective half-spread (50.015 - 50.00)/50.00 = 3.0000 basis points, doubled 6.0000
Ratio	Effective over quoted = 6.0000/8.0000 = 0.750000. The trade paid three quarters of the displayed spread
Convention hazard	Half-spread 3.0000 and doubled spread 6.0000 are the same measurement. Comparing a half-spread from one source with a doubled spread from another understates by a factor of two
Side inference	D must be known. Where it is inferred from whether the trade printed above or below the midpoint, the inference fails exactly at the midpoint and biases the measured effective spread toward zero

• The quoted spread is a cost only for a trade that crosses it in full at the displayed size. For any trade larger than the displayed depth it understates the cost, and for any trade that executes inside the quote it overstates it. Neither error is small and they run in opposite directions.
• The midpoint used must be the one prevailing at the moment of the trade, and on a fast tape the difference between the midpoint at the order and at the fill is comparable to the spread itself. Stale-midpoint effective spreads are systematically wrong in the direction of the price drift.
• Effective spread measures one trade. An order worked over an hour has an effective spread per fill and an implementation shortfall for the order, and only the second one includes the price drift over the hour.

Realised spread, price impact, and the identity between them

The effective spread decomposes exactly into a realised spread - what the liquidity provider kept - and a price impact - what the trade revealed about value. The decomposition requires a post-trade horizon and it is the whole content of the adverse-selection story in measurable form.

Field	Value
Formula	Effective half = D*(P - M_t)/M_t. Realised half = D*(P - M_{t+delta})/M_t. Impact half = D*(M_{t+delta} - M_t)/M_t. Identity: effective = realised + impact, exactly and by construction
Worked	Buy at 50.015 with M_t = 50.00 and M_{t+5min} = 50.005. Effective half 3.0000 bps, realised half (50.015 - 50.005)/50.00 = 2.0000 bps, impact half (50.005 - 50.00)/50.00 = 1.0000 bps. 3.0000 = 2.0000 + 1.0000
Interpretation	Of the 3 basis points the buyer paid, 1 basis point was information - the market moved permanently - and 2 basis points was compensation to the provider of liquidity
Horizon dependence	The split depends entirely on delta. A longer horizon attributes more of the effective spread to impact and less to the realised spread. The identity holds at every delta; the interpretation changes
Sign of the realised spread	A negative realised spread means the price moved further than the trade paid, so the liquidity provider lost money on the trade. This is the arithmetic definition of adverse selection

• The identity is algebraic and holds for any delta, which means it is not evidence for any model. What the decomposition provides is an accounting split, and the split is only economically meaningful if delta is chosen to match the horizon over which information is incorporated - a modelling choice with no data-driven answer.
• Delta is conventionally set to five minutes or to the end of the day. Those choices are conventions, and comparing a five-minute realised spread with an end-of-day one compares two different quantities.
• The measured impact is contaminated by any market-wide move over delta. Subtracting a benchmark return over the same window is the standard correction and is more important the longer delta is.

Source: Huang and Stoll 1996 for the decomposition; the adverse-selection mechanism is Glosten and Milgrom 1985

The square-root market impact law

Market impact grows approximately with the square root of order size relative to volume rather than linearly. The functional form is stable across studies and venues; the coefficient is not, and must be fitted rather than assumed.

Field	Value
Formula	Impact as a fraction of price = Y * sigma_daily * sqrt(Q/V), with Q the order size, V the average daily volume, sigma_daily the daily volatility, and Y a dimensionless calibration constant
Worked	sigma_daily = 2.00 percent, V = 5,000,000 shares, Q = 500,000 so Q/V = 0.1000. At the stated illustrative Y = 1.0: impact = 0.02 * sqrt(0.10) = 0.00632456 = 63.2456 basis points, or 158,113.88 on a 25,000,000 notional
Sublinearity	Doubling Q multiplies impact per share by sqrt(2) = 1.414214 and total cost by 2*sqrt(2) = 2.828427. Cost per share rises 41.42 percent, not 100 percent
Size ladder at Y = 1.0	250,000 shares: 44.7214 bps. 500,000: 63.2456. 1,000,000: 89.4427. 2,000,000: 126.4911. Four times the size costs twice the basis points
Coefficient	Y is not a universal constant. It depends on the venue, the instrument, the horizon over which the order is worked and the definition of V, and it must be fitted to realised fills. The values used above are stated illustrative choices

• The square-root form is the reason capacity is finite but not sharply bounded. A linear law would imply a hard capacity limit; a square-root law implies costs that rise steadily and can always be paid at the expense of alpha, which is why capacity is a business decision rather than a physical constraint.
• The form is empirical, not derived. It appears consistently in published fits, and the theoretical arguments for the exponent of one half are suggestive rather than conclusive. Treating it as a law of nature and the coefficient as calibratable has the emphasis backwards: both should be fitted, and the exponent checked.
• sigma_daily and V must be measured over a window comparable to the order horizon. Using a one-year average volume for an order worked during an unusual volume day produces an impact estimate that is wrong by whatever factor the volume differs.
• The law describes temporary plus permanent impact together in most formulations. Decomposing it matters for a strategy that trades back out, because only the permanent part is a genuine round-trip cost.

Source: The square-root form appears in Loeb 1983 and is fitted in Almgren, Thum, Hauptmann and Li 2005

Almgren-Chriss optimal execution

The trajectory that minimises expected execution cost plus a risk-aversion multiple of its variance, under linear temporary and permanent impact. The solution is a hyperbolic-sine curve whose shape depends on one dimensionless parameter, and its two limits are TWAP and immediate execution.

Field	Value
Formula	Minimise E[cost] + lambda*Var[cost]. The optimal remaining position is x(t) = X * sinh(kappa*(T-t))/sinh(kappa*T), with kappa^2 = lambda*sigma^2/eta in the continuous limit, eta the temporary impact coefficient and sigma the volatility. lambda to 0 gives x(t) = X*(1 - t/T), the linear TWAP schedule
Worked shape at kappa*T = 2	Remaining fraction at t/T = 0, 0.25, 0.50, 0.75, 1: 1.000000, 0.587086, 0.324027, 0.143677, 0.000000. Half the order is complete by t/T = 0.321528 rather than 0.5
Worked, other urgency levels	kappa*T = 0.5 completes half by t/T = 0.484627, barely different from TWAP. kappa*T = 1.0 by 0.441837. kappa*T = 5.0 by 0.138602, essentially front-loaded
Parameter	Only kappa*T matters for the shape. With lambda = 2 x 10^-6, sigma = 0.60 in price units per square-root day and eta = 1.8 x 10^-7, kappa = sqrt(2e-6 * 0.36 / 1.8e-7) = 2.000000 per day, so a one-day horizon gives kappa*T = 2.000000
Limits	kappa*T to 0 is the risk-neutral limit and gives TWAP exactly. kappa*T to infinity gives immediate execution. Every real schedule sits between them and the position is set by lambda

• The model assumes linear temporary impact, which contradicts the square-root law in the same section. That is a real inconsistency in common practice: the trajectory is taken from a linear-impact model and the cost estimate from a square-root fit. Under square-root impact the optimal trajectory is different and generally less front-loaded.
• The only free parameter with economic content is lambda, and it is a preference rather than a measurement. Presenting an Almgren-Chriss schedule as optimal without stating lambda states nothing: every schedule from TWAP to immediate is optimal for some lambda.
• Permanent impact drops out of the optimisation for a fully executed order under the linear model, because the permanent cost is the same whatever the path. That is a consequence of linearity and does not survive a nonlinear impact function.
• The solution is a static schedule computed at t = 0. It does not respond to price moves during execution, so it is a plan rather than a policy, and comparing it with an adaptive algorithm is comparing two different objects.

Source: Almgren and Chriss 2000

VWAP and TWAP tracking error

A schedule that spreads an order evenly in time matches the market's volume-weighted average price only if volume is evenly spread. The tracking error against VWAP comes entirely from the mismatch between the schedule's weights and the market's realised volume weights, interacting with the price path.

Field	Value
Formula	Execution price = sum_i w_i * P_i for schedule weights w_i. Market VWAP = sum_i v_i * P_i for realised volume weights v_i. Slippage = (sum_i (w_i - v_i)*P_i) / VWAP. A schedule with w_i = v_i has zero slippage identically
Worked setup	Four intervals with average trade prices 50.00, 50.10, 50.30, 50.20 and market volumes 3,000,000, 1,000,000, 1,000,000, 5,000,000 shares. Volume weights 0.3000, 0.1000, 0.1000, 0.5000
Worked result	Market VWAP = (50.00*3 + 50.10*1 + 50.30*1 + 50.20*5)/10 = 501.40/10 = 50.140000. TWAP at 25 percent per interval = 200.60/4 = 50.150000. Buy-side slippage = 0.010000/50.140000 = 1.9944 basis points
Volume-matched schedule	Trading 30, 10, 10 and 50 percent gives exactly 50.140000 and zero slippage by construction
Source of the error	The market traded half its volume in the last interval at 50.20 while TWAP traded a quarter there. The slippage is the covariance between the weight mismatch and the price path and nothing else

• Expected TWAP slippage against VWAP is zero if the price path and the volume profile are uncorrelated. They are not: volume concentrates at the open and close and around news, which is also when prices move, so the covariance is systematically nonzero and its sign varies by market.
• Beating VWAP is not the same as executing well. A trader who fully controls the schedule can match VWAP almost exactly by tracking volume, while an order that should have been done in the first ten minutes accumulates delay cost the VWAP benchmark cannot see. Implementation shortfall can see it.
• The order's own volume is part of the market volume it is benchmarked against. For a large order this makes VWAP partly self-referential and the benchmark easier to beat the larger the order is, which is the wrong incentive.

Kyle's lambda and market depth

In the single-auction model of informed trading, the market maker sets a linear price schedule in total order flow. Its slope is the price impact per unit of net order flow, and its reciprocal is market depth. Both have closed forms in two parameters.

Field	Value
Formula	P = P0 + lambda*(x + u), with x informed demand and u noise-trader flow. Equilibrium: lambda = sigma_0/(2*sigma_u) and informed intensity beta = sigma_u/sigma_0, where sigma_0 is the standard deviation of the terminal value and sigma_u that of noise flow. Depth = 1/lambda
Worked	sigma_0 = 2.00 in price units, sigma_u = 1,000,000 shares. lambda = 2.00/2,000,000 = 0.00000100 per share. Depth 1/lambda = 1,000,000 shares per 1.00 of price move. beta = 1,000,000/2.00 = 500,000 shares per 1.00 of mispricing
Impact by size	100,000 shares moves the price 0.100000, which is 20.0000 basis points at a 50.00 price. 500,000 shares moves it 0.500000 = 100.0000 basis points. 1,000,000 shares moves it 1.000000 = 200.0000 basis points
Informed profit	Expected profit of the informed trader = sigma_0*sigma_u/2 = 2.00 * 1,000,000/2 = 1,000,000.00, which is exactly the aggregate expected loss of the noise traders
Information revealed	Var(value | price) = sigma_0^2/2 = 2.000000, so the standard deviation of the remaining uncertainty falls from 2.0000 to 1.414214. Exactly half the informed trader's private information variance is impounded in the price

• Impact is linear in this model, not square-root. That is a consequence of the normal-linear setup and a single auction; it is the clean benchmark rather than the empirical description, and the two should not be used interchangeably in the same cost model.
• Depth is proportional to noise-trader volume and inversely proportional to the amount of private information. That is the substantive prediction: liquidity dries up when informed trading rises relative to uninformed flow, which is why depth collapses around scheduled announcements.
• The exactly-half information revelation is a specific consequence of the one-period normal model and does not generalise. Its usefulness is as a benchmark: it says that in a well-functioning market a single round of trading impounds a substantial and calculable fraction of private information, not all of it.
• Empirical estimates of lambda are regressions of price change on signed order flow, which is a different object from the equilibrium lambda above. The regression slope confounds the true impact with the endogeneity of order flow, and it is not the model's lambda.

Source: Kyle 1985

Adverse selection as a component of the spread

A liquidity provider loses on average to counterparties who know more, so the spread must be wide enough to cover that loss out of the profit on uninformed flow. The realised-spread decomposition measures the split directly, and the illiquidity measures approximate it from public data.

Field	Value
Formula	Effective half-spread = realised half-spread + price impact half-spread. The impact term is the adverse-selection component. Amihud's illiquidity measure approximates the same quantity from daily data as ILLIQ = mean over days of |return| / dollar volume
Worked from the fill	Effective 3.0000 basis points, realised 2.0000, impact 1.0000. One third of the spread paid was adverse selection and two thirds was compensation for providing liquidity
When the split reverses	A negative realised spread means the impact exceeded the effective spread, so the liquidity provider lost money. That is what a spread being too narrow for the information environment looks like in one number
Amihud measure	ILLIQ has units of return per unit of currency traded. It is a cross-sectional ranking device and its level is not comparable across markets, currencies or periods without normalisation
Kyle correspondence	lambda in the Kyle model is the same idea in a structural form: price impact per unit of order flow. lambda = sigma_0/(2*sigma_u) rises with private information and falls with noise volume, which is exactly the adverse-selection channel

• Adverse selection sets a floor on the spread that no amount of competition among liquidity providers can breach. That is why spreads widen before earnings and around macro releases, and why a cost model calibrated on quiet periods understates event-day costs systematically.
• For a strategy that is itself informed, adverse selection is the cost being paid rather than earned, and its size scales with how much the strategy's signal predicts the next few minutes. A fast signal is expensive to trade for exactly the reason it is valuable.
• The measured impact component includes any market-wide move over the measurement horizon. Without subtracting a benchmark return, adverse selection is overstated on trending days in whichever direction the trades leaned.

Source: Glosten and Milgrom 1985; Amihud 2002 for the illiquidity measure

Turnover, cost drag, and break-even

Cost drag is annual turnover multiplied by cost per unit traded, subtracted directly from gross excess return. Because volatility is unaffected, the entire effect lands on the Sharpe ratio, and the break-even turnover has a one-line closed form.

Field	Value
Formula	Net excess return = gross excess - turnover x cost. Net Sharpe = (gross excess - turnover x cost)/sigma. Break-even turnover = gross excess / cost; break-even cost = gross excess / turnover
Worked	Base series gross annual excess 6.8000 percent, sigma 7.2063 percent, gross Sharpe 0.943622. At 4.0x annual turnover and 15 basis points per unit traded, drag = 60.00 basis points, net excess 620.00 basis points, net Sharpe 0.860361
Cost sensitivity	Same turnover at 30 basis points: drag 120.00, net Sharpe 0.777100. Turnover of 12.0x at 15 basis points: drag 180.00, net Sharpe 0.693840. Turnover of 1.0x at 15 basis points: drag 15.00, net Sharpe 0.922807
Break-even	At 15 basis points the strategy supports 45.33x annual turnover before its excess return is fully consumed. At 30 basis points, 22.67x. At 5 basis points, 136.00x
Break-even cost	At 1.0x turnover the strategy tolerates 680.00 basis points of cost; at 4.0x, 170.00; at 12.0x, 56.67

• Cost reduces the numerator and leaves the denominator alone, so the Sharpe ratio falls exactly in proportion to the return. That makes the drag arithmetic linear and easy, and it also means there is no diversification benefit to costs: they are a deterministic subtraction.
• The break-even turnover figures look generous because the cost is held constant as turnover rises. It is not: higher turnover means larger orders per unit time and, under the square-root law, higher cost per unit traded. The real break-even is materially lower than the linear arithmetic suggests.
• A backtest with no cost model is not a backtest with a small error. At 12x turnover and 30 basis points the drag is 360 basis points on a gross excess of 680, which is over half the return. The cost assumption is a first-order input, not a refinement.

Capacity implied by the impact law

Capacity is the size at which impact consumes the gross alpha. Under the square-root law it has a closed form in the participation rate, and the resulting limit is on participation rather than on absolute size.

Field	Value
Formula	Impact equals gross alpha A when Y*sigma_daily*sqrt(Q/V) = A, so Q/V = (A/(Y*sigma_daily))^2. Capacity in shares per trading event is that participation times V
Worked	sigma_daily = 2.00 percent, illustrative Y = 1.0. A gross alpha of 30 basis points per trade supports Q/V = (0.0030/0.0200)^2 = 0.022500, that is 2.2500 percent of average daily volume
Alpha ladder	A = 30 bps supports 2.2500 percent of volume; A = 60 bps supports 9.0000 percent; A = 100 bps supports 25.0000 percent. Capacity is quadratic in alpha
On a specific name	At V = 5,000,000 shares and 50.00 per share, 2.2500 percent of volume is 112,500 shares or 5,625,000 of notional per trading event
Quadratic scaling	Doubling the gross alpha quadruples the capacity. Halving the volatility also quadruples it. Both are more powerful levers than any execution improvement, which acts only on Y

• Capacity being quadratic in alpha is the structural reason small edges do not scale. A 30 basis point edge and a 60 basis point edge differ by a factor of two in return and a factor of four in capacity, so the larger edge is worth eight times as much in absolute currency terms.
• The calculation gives capacity per trading event, not assets under management. Converting requires the number of positions, the turnover and the holding period, and each of those interacts with impact, so a single capacity number without the whole portfolio specification is not meaningful.
• The result depends entirely on Y, which must be fitted from realised fills. A capacity estimate built on an assumed Y is an assumption dressed as arithmetic, and the quadratic form means an error in Y is squared in the answer.