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E999　 KILL　 Below costsTested · October 2026

Thinly traded share-class twins cost more than they revert

Share-class gaps closed by +11.1 bp gross in 2020-26; the thin-pair edge of 2010-19 (+15.7, +10.1, +8.4 bp) did not repeat (+14.1, +9.5, +11.7), and estimated costs of 72.5 bp make the net -61.4 bp.

share-classes dual-class pairs mean-reversion liquidity spreads us-equities

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Some companies list two share classes that differ only in votes. Their prices usually move together, and an earlier test of three large pairs found a small reversion after unusual gaps. The idea here: the thinner the less-traded class, the larger the reversion should be. On 21 pairs in 2020-26, trading the gap from the open to the close of the next day gained +11.1 bp gross per event, with no consistent liquidity gradient. Estimated costs (spreads of both classes plus fees) averaged 72.5 bp, so the net was -61.4 bp.

Net per event, 2020-26-61.4 bpafter estimated spreads of both legs plus 2 bp fees

Gross per event+11.1 bp3,936 events on 21 pairs; t 4.30; far below the 72.5 bp cost

Average estimated cost72.5 bpper round trip, both classes

Net at a flat 10 bp cost+1.1 bpt 0.6; the pass bar was +10 bp

01 · The idea being tested

The price gap between two share classes of the same company closes, and it closes more in thinly traded pairs where fewer people can fix it.

Why people believe it

Index funds, ETFs and many retail investors can hold only one class, so a class can drift away from its twin for a few days. The less-traded class has thin order books and stale quotes, so the drift should be larger and slower to close there.

Why it would persist The argument, as stated before testing

The two classes cannot be converted into each other, they differ in votes or dividends, and the thin class is too small to absorb large capital. Nobody can lock in the gap with a riskless trade.

02 · How it was tested

We listed 39 pairs of share classes of one company, all traded on US exchanges. Each day we measured how far the ratio of the two prices sat from its own 20-day average. When it was at least 2 standard deviations away, we bought the cheap class and sold the expensive one at the next day's opening auction and closed both in the closing auction. We priced each trade with spreads estimated from daily highs and lows, split the pairs into three liquidity groups, and judged the result on 2020-26.

Pass required

In at least one of four cells (pooled, thin, middle, liquid tercile), both legs trading at least USD 250k a day: mean net at least +10 bp per event, date-clustered t at least 2.5, median net above 0, net above 0 in both halves and after dropping the top 1% of events, and a multiple-comparison adjusted one-sided p below 0.05. Cells with fewer than 100 events or 30 dates cannot pass.

Killed if

Validation mean net at or below 0 in all four cells.

Costs

Per pair round trip: the larger of 4 bp and the 60-day median of a high-low spread estimator (published 2012), for each leg, plus 2 bp fees. Sensitivities: a close-versus-high-low spread estimator (published 2017), a flat 10 bp, double spreads and fees. Borrowing a thin class and market impact are not costed.

Controls

Placebo: the same signal traded one day later. Untradable component: close of the signal day to next open. Machinery check: the three pairs of the earlier test re-run with its statistic gave 1,467 events and +6.93 bp (t 3.88), matching.

03 · What happened

Why it was rejected

Net mean was below zero in all four cells (pooled -61.4 bp, thin -76.5, middle -59.6, liquid -43.4), and the 2010-19 pattern of a gross effect falling with liquidity (+15.7, +10.1, +8.4 bp) did not repeat in 2020-26 (+14.1, +9.5, +11.7 bp).

In 2020-26 the gross reversion was +11.1 bp per event (t 4.30, median +8.6) over 3,936 events on 21 pairs; without the top 1% of events it fell to +4.0 bp, far below the 72.5 bp cost.
Estimated costs (spreads of both classes plus 2 bp fees) averaged 72.5 bp per round trip, so net was -61.4 bp (median -52.6, worst event -1,480.4), and -59.0 bp and -63.3 bp in the two halves of the window.
By liquidity of the thinner class: gross +14.1 bp (thinnest third, t 2.57), +9.5 (middle) and +11.7 (most liquid); average cost 90.6, 69.1 and 55.1 bp; net -76.5, -59.6 and -43.4 bp.
In 2010-19 the gross effect fell steadily with liquidity (+15.7, +10.1, +8.4 bp from thinnest to most liquid); in 2020-26 it did not (+14.1, +9.5, +11.7 bp).
The cost estimator tends to overstate spreads (it gave 37.8 bp for the largest pair, whose real spread is a few bp), so the result was re-run: the close-versus-high-low estimator gives -47.7 bp, and a flat 10 bp gives +1.1 bp (t 0.6); 10 bp is a floor set by the rules, not an estimate of actual cost, the median net there is -1.4 bp, the net is -6.0 bp after dropping the top 1% of events, and no pre-set test passes at it.
Placebo: the signal traded a day late earned +5.0 bp gross (t 2.52, -2.1 bp without the top 1%), about half the real signal.
The three pairs of the earlier test, re-priced with these costs in 2020-26: +8.2 bp gross, 47.7 bp cost, -39.5 bp net.
Only two pairs were positive net (+28.2 and +48.7 bp, on gross +163.4 and +79.5 bp), but on 8 and 12 events, far too few to count (a cell needed at least 100 events to pass); the other 19 were negative, between -27.6 and -297.3 bp.

Cost grows with illiquidity faster than the gap closes

GrossEstimated costNet

2020-26, by liquidity group of the thinner class. Gross reversion is similar in all three groups (+9.5 to +14.1 bp); estimated cost is largest where the class is thinnest, so every net bar is negative.

Chart data (bp per event)

	Gross	Estimated cost	Net
Thinnest third	14.1	90.6	-76.5
Middle third	9.5	69.1	-59.6
Most liquid third	11.7	55.1	-43.4

The liquidity gradient did not carry over

2010-192020-26

Gross reversion by liquidity group, before any cost. 2010-19 falls steadily from thin to liquid; 2020-26 does not.

Chart data (bp gross per event)

	2010-19	2020-26
Thinnest third	15.7	14.1
Middle third	10.1	9.5
Most liquid third	8.4	11.7

Net result under four cost assumptions

Pooled 2020-26 net per event. Only the flat 10 bp cost, the lowest the rules allowed, gets above zero (+1.1 bp, t 0.6), and it is far below the +10 bp pass bar.

Chart data (bp per event)

	Net
High-low spread estimate (primary)	-61.4
Close-high-low spread estimate	-47.7
Flat 10 bp	1.1
High-low estimate, doubled	-134

04 · What it means

Why it failed Interpretation · not established

The reversion is positive on average, but only about +4 bp without the largest 1% of events, and the placebo drops to -2.1 bp on the same trim; either way it is small next to the spreads of the less-traded class. The same thinness that might slow the gap closing also widens the spread, and in these data the spread grew faster. The placebo earning about half the real signal suggests the deviations persist rather than snapping back in one day.

Lesson

An effect found in a few large, easy-to-trade pairs does not scale to the rest of the universe: the extra pairs bring the similar gross effect and higher costs. Whatever is said about illiquid corners of the market has to be priced with the spread of that corner, not with the cost that held in the liquid cases.

Scope of this result

Covers 39 hand-selected US-listed common share-class pairs, current listings only, daily signals traded from the next open to the next close, 2000 to 2026 with the verdict taken on 2020-26. Spreads are estimated from daily highs and lows, not measured from quotes; borrowing costs and market impact are not included. Pairs that no longer exist are not in the sample.

A rejected experiment applies to the tested conditions and does not prove an idea can never work. bp = basis point, 0.01%.

1. Test design

Pre-registered

Yes: rule and criteria fixed before the test data was examined

Data

Daily open, high, low, close, adjusted close and volume of US-listed common stock share classes — public consolidated daily price history for US-listed equities; daily; 2000-01..2026-10; public download, no login

Universe

39 pairs of economically identical common share classes of one issuer (3 from the earlier test and 36 new), both legs US-exchange listed. The list was built from company ticker groups and issuer names, not from returns, by a hand selection that cannot be audited by a formal rule. Preferred shares, notes, warrants, depositary-receipt-versus-ordinary pairs and over-the-counter legs are excluded. Current listings only, so classes that were merged or delisted are missing. Primary cell: the 36 new pairs.

Signal

Log ratio of the two adjusted closes, minus its mean over the previous 20 days. An event when the absolute deviation is at least 2 times the sample standard deviation of that deviation over the same 20 days (minimum 15 observations). Direction: short the expensive class, long the cheap class. The signal is known before the next open.

Execution

Opening auction to closing auction of the next common trading day, USD 1 on each leg. Pairs whose ratio jumped more than 20% in a single day within 10 days before to 11 days after the signal (mergers, unifications, data errors) are dropped: 3,889 pair-days, 698 events.

Costs

Controls

Sample split

Selection 2010-2019, used only to fix the liquidity cut-offs (USD 1.57M and USD 31.0M of 20-day median dollar volume of the thinner class) and look at the gradient. Validation 2020-01 to 2026-10 decides the verdict, with halves split at 2023-06.

Pass criterion

Kill criterion

Validation mean net at or below 0 in all four cells.

Power

No power calculation was done. The pooled gross effect had a standard error of about 2.6 bp and the thinnest third about 5.5 bp, so an effect of +10 bp would be seen in the pooled test but not reliably in the thin third alone. The kill came from the cost, not from lack of power.

2. Result

Observations

3,936

Instruments

Net mean

-61.4

95% interval

Not reported

Median

-52.6

Worst

-1,480.4

Finding 1

In 2020-26 the gross reversion was +11.1 bp per event (t 4.30, median +8.6) over 3,936 events on 21 pairs; without the top 1% of events it fell to +4.0 bp, far below the 72.5 bp cost.

Finding 2

Estimated costs (spreads of both classes plus 2 bp fees) averaged 72.5 bp per round trip, so net was -61.4 bp (median -52.6, worst event -1,480.4), and -59.0 bp and -63.3 bp in the two halves of the window.

Finding 3

By liquidity of the thinner class: gross +14.1 bp (thinnest third, t 2.57), +9.5 (middle) and +11.7 (most liquid); average cost 90.6, 69.1 and 55.1 bp; net -76.5, -59.6 and -43.4 bp.

Finding 4

In 2010-19 the gross effect fell steadily with liquidity (+15.7, +10.1, +8.4 bp from thinnest to most liquid); in 2020-26 it did not (+14.1, +9.5, +11.7 bp).

Finding 5

The cost estimator tends to overstate spreads (it gave 37.8 bp for the largest pair, whose real spread is a few bp), so the result was re-run: the close-versus-high-low estimator gives -47.7 bp, and a flat 10 bp gives +1.1 bp (t 0.6); 10 bp is a floor set by the rules, not an estimate of actual cost, the median net there is -1.4 bp, the net is -6.0 bp after dropping the top 1% of events, and no pre-set test passes at it.

Finding 6

Placebo: the signal traded a day late earned +5.0 bp gross (t 2.52, -2.1 bp without the top 1%), about half the real signal.

Finding 7

The three pairs of the earlier test, re-priced with these costs in 2020-26: +8.2 bp gross, 47.7 bp cost, -39.5 bp net.

Finding 8

Only two pairs were positive net (+28.2 and +48.7 bp, on gross +163.4 and +79.5 bp), but on 8 and 12 events, far too few to count (a cell needed at least 100 events to pass); the other 19 were negative, between -27.6 and -297.3 bp.

3. Interpretation

Why it failed

Lesson

Scope

4. Reproducibility

Steps

1) Download daily open, high, low, close, adjusted close and volume for each class of the pairs. 2) Compute the log ratio of adjusted closes, its deviation from the previous 20-day mean, and the 20-day standard deviation of that deviation; mark an event when the absolute deviation is at least 2 standard deviations. 3) Trade short the expensive and long the cheap class from the next open to the next close; gross return is the difference of simple returns of the two legs. 4) Cost: per leg the larger of 4 bp and the 60-day median of the daily high-low spread estimate published in 2012 (negatives set to 0), summed over both legs, plus 2 bp. 5) Drop pair-days with a ratio jump above 20% in the window from 10 days before to 11 days after. 6) Fix liquidity terciles on 2010-19 data of the thinner leg's 20-day median dollar volume and apply them unchanged to 2020-26. 7) Date-clustered t, multiple-comparison adjustment over the four cells, placebo and sensitivities as listed.

Grade

COMPLETE