Effective and realized spreads

Reconcile signed post-trade spreads, volume weights and missing-horizon samples.

36 minutes · intermediate · Original Poker4Quant lesson

What you will learn

Derive E=R+I for signed executions, preserve it under volume weighting, and construct and repair a missing-horizon sample mismatch.

Prerequisites: Quotes, spreads and reference prices · Resting exposure and future fills · Implementation shortfall and unfilled orders · Expected value as a weighted average

Fix the clocks, side and units

An execution price measures a transaction; a later midpoint measures a different observation. A buyer may pay above the midpoint when trading and still receive an asset whose later reference value rises further. A liquidity supplier can collect an initial price concession yet face a negative subsequent marked result. We need a ledger that separates the initial concession from the intervening reference movement. This lesson constructs that ledger, then asks when averages of its components can legitimately be added. The difficult operation is preserving the same population and weights when some later observations are missing.

Assume fictional fully executed trades in one asset with prices in chips per unit. Receipt and execution are simultaneous, so their initial midpoint m0 is the same. Every trade has a known initiator sign d: +1 for a buyer taking liquidity and −1 for a seller taking liquidity. Quantity v is a positive integer. There are no fees, rebates, financing costs, hedges, partial fills or additional trades by the supplier. Record midpoint mH at a declared horizon H after each execution. These midpoints are reference valuations, not executable liquidation prices. We know trade direction rather than inferring it from prints. Quotes and executions are synchronized by assumption.

Derive a signed decomposition

Define effective spread E=2d(p−m0), realized spread R(H)=2d(p−mH), and signed reference movement I(H)=2d(mH−m0). All three use chips per unit. The factor two makes a one-sided midpoint distance comparable in scale to a full quoted spread; it does not mean two trades occurred. Positive E means an initiator paid a concession relative to the initial midpoint. Positive I means the reference moved in the initiator’s direction. For a sell, a falling midpoint therefore gives positive I. Positive R is favorable to the supplier’s marked comparison, although it remains a proxy rather than audited profit.

Insert and subtract mH inside the effective spread: 2d(p−m0)=2d[(p−mH)+(mH−m0)]=R(H)+I(H). This identity is pathwise algebra. It needs neither independence nor a probability model, and it does not assert that trading caused midpoint movement. The supplier’s marked gain on v units under our simplified inventory comparison is vR(H)/2. A positive realized spread on a sell means the supplier bought below the later midpoint. A negative realized spread on a buy means the supplier sold below that later midpoint. Both interpretations follow from the same sign convention rather than separate rules memorized for each side.

Reconcile an original three-trade tape

Take initial quotes 98/102, so m0=100 for every row. Trade A is a buy of one at 102; its short-horizon midpoint is 101 and long-horizon midpoint is 104. Trade B is a sell of three at 99; its short midpoint is 98 and long midpoint is 100. Trade C is a buy of two at 101; its short midpoint is 100 and long midpoint is 103. These are separate trades with their own relative clocks, not successive events in one evolving quote tape. Prices inside the initial quotes are allowed as supplied executions. No matching-engine inference is required.

At the short horizon, A has E=4, R=2, I=2. B has E=2 because −2(99−100)=2; R=−2 because −2(99−98)=−2; I=4 because −2(98−100)=4. C has E=2, R=2, I=0. Six total units give weighted E=(1×4+3×2+2×2)/6=14/6=7/3. Weighted R=(2−6+4)/6=0, while weighted I=(2+12+0)/6=7/3. Thus the weighted identity holds. The unweighted trade average of R would be 2/3, answering a different question about a uniformly chosen trade rather than a uniformly chosen executed unit.

Check independently with supplier cash and marked inventory. Against A the supplier receives one times 102, or 102 chips. Its short marked obligation is 101, giving gain one. Against B the supplier pays 3×99=297 for inventory marked 3×98=294, giving loss three. Against C it receives 2×101=202 against value 2×100=200, giving gain two. Total gain is zero, matching six times weighted R divided by two. Keeping cash amounts explicit prevents accidentally treating a doubled spread as actual money earned.

Move the horizon without moving the execution

At the long horizon, A has R=−4 and I=8; B has R=2 and I=0; C has R=−4 and I=6. E remains 4,2,2 because neither initial midpoint nor execution changed. Weighted R=(−4+6−8)/6=−1 and weighted I=(8+0+12)/6=10/3. Their sum is again 7/3. Independently, supplier gains are −2, +3 and −4, totaling −3; six times −1 divided by two also gives −3. The same execution tape can therefore show zero short-horizon marked gain and a negative longer-horizon marked gain without any accounting contradiction.

This decomposition is often used to discuss adverse selection: a supplier may trade just before reference value moves against its inventory. But the I term also includes public news, broad market changes, timing effects and other sources of movement. Our tape contains no counterfactual market without those executions. Calling I a signed movement proxy is supported by its definition; claiming it proves private information or causal price impact is not. A negative I is also admissible. If the midpoint moves against the initiator, R can exceed E while their sum identity still holds. Absolute values would destroy this signed relationship.

Repair a missing-data comparison

Now hide B’s short-horizon midpoint. An analyst keeps E over all six units but computes R and I only over observed A and C. On those three observed units, R=(2+4)/3=2 and I=(2+0)/3=2/3, whose sum is 8/3. That differs from full-tape E=7/3 by 1/3. The identity has not failed: the analyst combined different populations. E over the same observed units is (4+4)/3=8/3 and reconciles exactly. Alternatively restore B’s quote and retain the full tape. Neither repair establishes that a complete-case statistic estimates the missing population without selection bias.

In the workshop change the horizon, hide or restore B’s reference, and choose which trades belong in the effective-spread comparison. Construct a common positive-volume sample before entering weighted E, R and I and the actual supplier marked gain. The zero-row boundary is deliberately available: an empty sample has no per-unit averages, so enter none for those three fields and zero for total marked gain. Do not fill a missing midpoint with zero or silently carry a quote from another horizon. The activity invalidates answers after every sample or horizon edit. Compare the short and long horizons on an unchanged sample, then deliberately break and repair the sample alignment.

The historical SEC report linked here motivates doubled midpoint comparisons and a later-reference profit proxy. Its five-minute convention is historical context, not a claim that one horizon exhausts modern reporting. The separately checked SEC staff FAQ lists several horizons. Our labels short and long intentionally denote supplied teaching snapshots, not regulatory clocks or compliance calculations. Before applying the method to data, declare event timestamps, direction classification, horizon, missingness rules, quantity weighting and fees. Arithmetic consistency is a necessary audit, but it cannot by itself validate those measurement choices.

Worked example

Reconcile the three supplied trades at both horizons and after hiding B’s short quote.

Short rows (E,R,I): A=(4,2,2), B=(2,−2,4), C=(2,2,0).
Volume-weighted short values: E=14/6=7/3, R=0/6=0, I=14/6=7/3.
Independent supplier gain: (102−101)+3(98−99)+2(101−100)=0.
Long weighted values: E=7/3, R=−6/6=−1, I=20/6=10/3. Gain=(102−104)+3(100−99)+2(101−103)=−3.
Hide B: observed A,C give R=2, I=2/3. Restrict E to A,C: E=8/3, restoring equality.

A horizon changes marked attribution; a sample change changes the estimand. Declare both before comparing.

Hands-on: Repair a spread comparison sample

Construct matching volume-weighted samples, move the horizon and repair missing-quote mismatches.

Repair a spread comparison sample

Receipt and execution share midpoint 100. Known initiator signs, no fees or hedges. Choose exactly the trades with observed horizon quotes for a common-volume comparison. Missing quotes are not zero. Enter none for averages when no units remain. Visit-only draft; no check credit.

Horizon: short. Selected volume: 6. Initial midpoint: 100.

Construct the comparison population

Toggle trade inclusion. A missing row remains visible to expose selection. After hiding B, try auditing the original full sample before repairing it.

Reconcile a common-volume sample.

Common mistakes

Using buyer signs for seller executions.
Treating doubled per-unit spreads as total cash profit.
Averaging trade rows when the target weights units.
Combining effective spreads from all trades with later components from complete cases.
Calling a signed movement term proof of causal adverse selection.
Reporting zero averages when the comparison sample is empty.

Keep these ideas

E=2d(p−m0), R=2d(p−mH), I=2d(mH−m0); E=R+I.
Common nonnegative volume weights preserve the pathwise identity.
Supplier marked gain equals total volume times weighted R/2 in this model.
Horizon and missingness rules are part of the measurement.