The Arithmetic of Selectivity: Expectancy, Costs and How Often You Trade.
Every trade pays its cost for certain while its edge is an estimate, so cutting trades whose edge falls short of their cost can raise net profit while lowering activity; each trader's ledger shows whether it does. Here is what the science actually says, and what to do with it.
01The Opening
The total cost of activity depends on the count of trades, not only the price of one
Trading became much cheaper per trade, and investors traded more. Across US equity, the aggregate cost of trading fell from 0.55% of total market value a year at the start of the 1980s to 0.21% by the mid-2000s, while annual turnover, the value traded in a year as a share of the value held, rose from 42% to 173%.[5] Total spending on active investing, fees included, stayed between 61 and 74 basis points of market value in 24 of those 27 years;[5] a basis point is a hundredth of a percentage point, the difference between two percentages. In one century-long study of US stock trading costs, commissions roughly halved every seven or eight years from the mid-1970s.[6]
These are aggregates, not the cost of one trade, and they show that a cheaper trade does not make frequent trading cheap. Net profit over a period is roughly the number of trades taken times the edge per trade, the average gross result a trade is expected to earn, minus the cost of every trade taken. Every trade is certain to pay its cost; its edge is an estimate. Cutting trades whose expected edge falls short of their cost can therefore raise net profit and lower activity at once.
The claim is fewer sub-threshold trades, not fewer trades. A skipped trade that meets the written criteria gives up its expected value; the guide to strategic inaction treats not acting as a deliberate decision. Whether cutting sub-threshold trades raises net profit for a given trader is an empirical question for that trader's records.
A risk unit, written R, is the amount a trader plans to lose if a trade reaches its exit for a loss. Results in R make trades of different sizes comparable: a gain of twice the planned loss is +2R, and a full planned loss is -1R.
Each trade got cheaper and investors bought more; the total cost of activity depends on the count of trades, not only the price of one.
The strongest field evidence comes from complete trade records and describes populations, not individuals. A review of research on individual investors finds that, in the samples studied, they underperform standard benchmarks on average.[7] The largest samples cover every trade on one exchange,[1] accounts at US discount brokers[2][4] and a cohort of day traders in Brazilian equity futures.[8]
Averages hide a mixed population. In one study, individual investors in the top tenth of performance in the first half of the sample went on to earn about 6% a year, risk-adjusted.[9] NYSE stocks earned positive excess returns in the month after intense buying by individuals;[10] US stocks with net retail buying beat those with net retail selling over the next week, with only suggestive evidence of information in the orders;[11] and one commission-free app's users, combined, showed good timing and positive risk-adjusted returns over two years spanning the end of the 2010s,[12] although their intense buying of the most-bought stocks preceded negative returns.[13] The sign depends on the market, the horizon and the measure. These studies measured turnover, switching between securities, order flow and day trading; none measured trades below a trader's written threshold.
The field studies describe averages over mixed populations in particular markets and periods; none measured sub-threshold trades.
A trade log, the Selection Ledger, records each setup and whether it met the trader's written criteria. A trade that meets them is graded PASS; a sub-threshold trade, taken although it fails them, is graded FAIL. This article adds two columns, Result in R and Costs, so the ledger can compare the mean result, net of costs, of PASS trades against FAIL trades.
The two columns, the PASS against FAIL comparison and the sample rule that governs it are this article's method, and no study has tested them as a whole. The arithmetic beneath them is exact. What is open is whether one trader's FAIL trades carry less edge than they cost.
Because the grade is fixed before the outcome, the comparison asks whether the criteria separate trades that pay for themselves from trades that do not. If FAIL trades lose after costs, cutting them builds profit while cutting activity; if they net less than PASS trades but above zero, they still add profit; if the groups net the same, the criteria are not capturing edge.
The ledger turns the arithmetic into a test: PASS against FAIL, net of costs, on the trader's own trades.
02The Mechanism
Trading Expectancy Net of Costs: The Arithmetic of Selectivity
Costs arrive on every trade. Edge arrives, if at all, on average.
Expectancy is win rate times average win minus loss rate times average loss, less cost; costs grow with trade count, and edge is hardest to know near the threshold.
Expectancy, the average result a trade would produce if it were repeated many times, is the expected value of one trade. In R it is the win rate times the average win, minus the loss rate times the average loss. A setup that wins 40% of the time, gains 2R on average when it wins and loses 1R when it loses has a gross expectancy of 0.40 × 2 - 0.60 × 1 = +0.20R per trade. Every worked example here uses invented round numbers for a generic setup.
Costs come off every trade. Cost per trade covers commissions and fees, the bid-ask spread, the gap between the buying and selling price, and slippage, the gap between the price a plan named and the fill. If 1R is 200 in the account's currency and these come to 10 on a round trip, entry and exit together, cost per trade is 10 ÷ 200 = 0.05R. Net expectancy is then 0.20 - 0.05 = +0.15R; one trade's result less its costs is its net R. The cost share, average cost over average gross result, is 0.05 ÷ 0.20 = 25%.
Expectancy is an average, not a promise. The setup above loses 60% of the time, and over a few trades its observed average can sit far from the true one.
Net expectancy is win rate times average win, minus loss rate times average loss, minus cost, all in R.
Costs scale with the number of trades; edge does not. At 0.05R a trade, 100 trades cost 5R and 200 trades cost 10R, whatever they earn. An added trade raises net profit only if its expected gross edge exceeds its cost.
Each part of cost can be measured. Total execution cost is the implementation shortfall, the gap between a paper portfolio traded at decision prices and the real one.[14] The effective spread, paid relative to the midpoint price, can be estimated from price changes[15] or daily data,[16] and asset-pricing research treats the spread as a cost investors require to be paid for.[17] Market impact, the price move a trade causes, grows with size and speed.[18] Costs differ by trading style even among institutions.[19]
Cost levels belong to their periods. In US discount-broker accounts of the late 1980s and early 1990s, commissions averaged 2.23% of trade value on purchases and 2.76% on sales;[4] Taiwan in the late 1990s capped commissions at 0.1425% and taxed sales at 0.3%.[1] None of these is a current cost.
Cost is linear in the count of trades; gross edge has to be earned again by every trade added.
A written threshold splits the trades a trader takes into PASS and FAIL. The figure below follows 80 trades, 40 of each, at the same cost of 0.05R. If PASS trades carry +0.20R gross and FAIL trades +0.02R, the PASS group nets 40 × 0.15 = +6.0R and the FAIL group 40 × (-0.03) = -1.2R, together +4.8R. Without the FAIL trades the trader nets +6.0R from half the activity; costs on the FAIL trades were 0.05 ÷ 0.02 = 250% of their gross edge.
The same arithmetic can point the other way. If FAIL trades carry +0.08R gross, they net 40 × 0.03 = +1.2R, and cutting them gives up 1.2R. Signal detection theory describes every threshold as a trade between misses, qualifying chances passed up, and false alarms, actions that should not have been taken;[20][21] the signal detection deep dive sets out the framework.
The trades nearest the threshold are where expectancy is hardest to know. Where expected gross edge roughly equals cost, a trade just below the line has net expectancy near zero. Its cost is certain to be paid; its edge is an estimate from a small group of trades whose single results vary widely. The sign of its net result is the least reliable number in the ledger.
Cutting FAIL trades raises net profit only when their gross edge is below their cost; when it is above, cutting them loses money.
PASS trades clear their cost. +0.20R gross, 0.05R cost, +0.15R net.
The threshold assigns the group. Grades are fixed before the outcome.
FAIL trades can sit below the cost bar. At +0.02R gross they lose 1.2R; at +0.08R they add 1.2R.
Arithmetic, not data: each arrow is a step in the sum, not a cause. Net result is the number of trades times the net per trade, where net is gross edge less the cost every trade pays. Invented round numbers for a generic setup and 80 trades, 40 of each group at the same cost of 0.05R: PASS trades carry +0.20R gross and net 40 × 0.15 = +6.0R; FAIL trades carry +0.02R gross and net 40 × (-0.03) = -1.2R; together +4.8R. Cost is certain; edge is an estimate.
Diagram
66 % of individual investors' measured annual performance gap on the Taiwan Stock Exchange in the late 1990s was commissions (32%) plus a transaction tax on sales (34%), in a record of every trade on the exchange[1]
Complete-population field data · one exchange, five years · share of a measured gap; selection and timing made up the rest
That share belongs to one market in one era. What carries over is its structure: commissions on both sides and the tax on every sale were paid whether or not the trade worked, while the gross result depended on which trades were chosen.[1] Unit costs fell sharply in later decades;[5][6] the arithmetic did not change.
Gross edge is small next to the spread of single results. In the worked setup, a 40% win rate with wins of 2R and losses of 1R gives a mean of +0.20R and a standard deviation, the typical distance of one result from the mean, of √(0.40 × 2² + 0.60 × 1² - 0.20²) = √2.16 = 1.47R, more than seven times the edge. A trader's own spread will differ, but for any plausible edge it is several times the mean.
An observed mean sharpens only with the square root of the count. Its standard error, s ÷ √n, is 1.47 ÷ 10 = 0.147R after 100 trades, so a real edge of +0.20R sits only 0.20 ÷ 0.147 = 1.36 standard errors from zero.[22] For 40 FAIL trades with the same spread it is 1.47 ÷ √40 = 0.23R, several times the 0.03R by which their edge falls short of cost.
With single results spread several times wider than the edge, the sign of a small group's expectancy is the last thing a short log reveals.
03Evidence
What Trade Records Show About Activity and Costs
01At a glance
The Evidence at a Glance · 04 Findings
What the Trade Records Show
Four field studies, each from one market and period: what investors earned against a reference and, where a study separated them, what went to costs.
Individual Investors, Taiwan · 3.8 pp a year. In a record of every trade on the Taiwan Stock Exchange over five years in the late 1990s, individual investors' trades earned 3.8 percentage points a year less than a benchmark, as measured, mostly through commissions and a tax on sales.[1]
Heaviest Traders · 11.4 % a year. Among 66,465 households at one large US discount broker over six years in the 1990s, those that traded most earned an annual net return of 11.4%, while the market returned 17.9%.[2]
Gross and Net Skill · 37.9 bps a day, after fees. Among Taiwanese day traders from the early 1990s to the mid-2000s, the 500 ranked top on past performance earned 61.3 basis points a day before fees the following year and 37.9 after the fees the study applied.[3]
Buying Versus Selling · 3.3 pp over one year. In 10,000 accounts at one US discount broker from the late 1980s to the early 1990s, securities bought returned 3.3 percentage points less over the next year than securities sold (3.2 market-adjusted), before round-trip costs averaging about 5.9% in that sample.[4]
Sources, including two working papers
50
Evidence signal. Individuals' trades in Taiwan trailed a benchmark, the most active US households trailed the market, and US investors' purchases trailed their sales; in Taiwan, commissions and a tax on sales made up most of individuals' shortfall. Editorial judgment. These studies set the stakes; the test of one trader's sub-threshold trades comes from that trader's ledger.
02The ranking
Ranked by methodological weight
No points score was assigned.
Ranked by completeness of the record, separation of costs, causal clarity and replication.
Completeness of the record. Every trade in a market or cohort ranks above a sample from one broker. Costs separated. Studies that separate gross results from costs rank above those reporting net results only. Causal clarity. Designs that rule out more alternative explanations, such as forward ranking or within-person comparison, rank higher. Replication value. Findings repeated in other markets or periods rank higher.
Ranking criteria, in order
03The design
What the records can show
Every study here is observational.
All five studies ranked below are observational: they record what traders did and earned, and no one assigned anyone to trade more or less, so they cannot show that trading less would have helped a given trader. Heavy traders self-select, choosing their own activity, and may differ in other ways, such as risk.[2]
A gross return is earned before costs and a net return after them; an abnormal return is measured against a benchmark for risk. Three of the five separate costs from gross results: one splits the result into cost and selection, and two report gross and net side by side.[1][2][3]
04Order fills
What was measured
Measured outcomes, not proof that each decision to trade was the error.
How orders fill also shapes what is measured. In records of all individual investors in Finland, one study found that individuals' poor post-trade returns arise largely and mechanically from limit orders, which fill only at a set price and suffer adverse selection, the tendency of a resting order to fill when the price is about to move against it.[23] The studies below describe measured outcomes, not proof that each decision to trade was the error.
Every study here is observational, so each describes what traders earned, not what trading less would have earned them.
The 5 Strongest Studies on Trading Activity and Costs
Each study measured something different: shortfalls against a benchmark, the market or the securities sold, gross against net results for ranked day traders, and the share of persistent day traders who lost money.
01Anchor
Record of every trade on the Taiwan Stock Exchange over five years, with losses split into cost and selection
3.8 percentage points a year by which individual investors' trades earned less than a benchmark, as measured, in the late 1990s.
Using the exchange's full trade and order data, the study split individuals' measured shortfall into trading losses (27%), commissions (32%), transaction taxes on sales (34%) and market-timing losses (7%).[1] Institutions trading the same market gained 1.5 percentage points a year after commissions and taxes, so costs alone did not decide who lost.[1]
What this proves: In one complete market, most of individuals' measured shortfall was the cost layer. Why it ranks first: The most complete sample, and the only study here that splits the result into cost and selection. Limitation: One market, with a tax on sales, in one era.
Rubric breakdown
The five studies, in rank order.
Ranked by completeness of the record, separation of costs, causal clarity and replication.
02
Taiwanese day traders ranked on past performance and followed the next year, early 1990s to mid-2000s[3]
The 500 top-ranked day traders earned 61.3 basis points a day before fees the following year and 37.9 after the fees the study applied, as the published abstract reports.[3] Proves: real gross skill existed in a small group, and fees took more than a third of it. Why ranked here: Ranking in one year and testing in the next limits luck and hindsight, but it covers day traders only, with fees applied by assumption.
Not scored
03
10,000 accounts at one US discount broker, securities bought compared with securities sold[4]
From the late 1980s to the early 1990s, securities bought returned 3.3 percentage points less over the next year than those sold, or 3.2 market-adjusted, while round trips cost about 5.9%.[4] Proves: here, what investors bought did worse than what they sold, even before switching costs. Why ranked here: Comparing each investor's purchases with their own sales removes market-wide effects, but the data come from one broker in a far costlier commission era.
Not scored
04
Panel of 66,465 households at one US discount broker, returns by turnover[2]
Over six years in the 1990s, the households that traded most earned 11.4% a year net against 17.9% for the market; the average household earned 16.4% and turned over 75% of its portfolio a year.[2] Proves: across turnover groups in this panel, the group that traded most netted the least. Why ranked here: A large panel with gross and net returns by turnover group, but heavy traders self-select and differ in risk tilt.
Not scored
05
Complete cohort of new day traders in Brazilian equity futures, working paper[8]
Of those who began day trading these contracts in the mid-2010s and persisted for more than 300 sessions, 97% lost money, according to the working paper's abstract; a summary of the paper puts the persisters at 1,551 of 19,646 starters (7.9%).[8] Proves: among the traders who kept going in this market, almost all lost money. Why ranked here: A complete cohort, but a working paper whose figures differ between versions, covering one market.
Not scored
04Stakes
Costs Compound, and Short Records Persuade
Four ways the arithmetic shows up in a trader's results.
The same noise misleads research on trading rules. Testing many rules on one price history and keeping the best overstates its performance unless the number of trials is counted,[34][35][36] so backtested results need a discount for the search[37] and a higher bar for significance.[38] Of 452 published US equity anomalies, 65% failed the usual significance hurdle in one replication once microcaps were mitigated, and 82% failed a stricter one.[39]
Small samples also exaggerate the results that pass. In noisy, small-sample settings a statistically significant result tends to overstate the true effect, a Type M error, and can even have the wrong sign, a Type S error.[40] Studies with low statistical power, the chance of detecting a real effect of a given size, produce significant findings that are less often true and effects that are inflated.[41][42]
A drawdown, a fall from a previous peak, is a property of any process with variance, including one with positive expectancy, and its expected size has a mathematics of its own.[43] Limiting drawdown through position size is a separate question for the sizing article in this series.[44]
Noise produces streaks, drawdowns and false confidence in short records; none of them, alone, says whether an edge exists.
Costs Grow With Count
Cost per trade times the number of trades is a bill paid whatever the trades earn. Doubling the count doubles the bill. In the US household panel, the households that traded most netted the least.[2]
What it looks like in the ledger · a Costs total growing faster than the Result total, a cost share above 100% in the FAIL group
Gross Edge Turns Negative
Among Taiwanese day traders from the early 1990s to the mid-2000s, the 500 ranked bottom on past performance earned -11.5 basis points a day before fees the following year and -28.9 after, as the published abstract reports.[3] A small gross loss became a net loss more than twice as large.
What it looks like in the ledger · trades that roughly break even before costs and lose after them
Short Records Persuade
Asked about a significant result, 84 psychologists at professional meetings gave a median of .85 as the chance that a smaller replication would again be significant, where a reasonable estimate was .48; only 9 answered between .40 and .60.[31] Trained researchers expected small samples to mirror large ones. A separate formal model describes a different error, underweighting what large samples reveal.[32]
What it looks like in the ledger · a criterion rewritten after one good or bad week
Losing Streaks Are Normal
For independent trades, where one result does not change the odds of the next, with a 50% win rate, the chance of at least one run of five or more losses within 100 trades is 81.0%; at a 40% win rate it is 97.6%, and a run of eight or more has a 49.0% chance. A run of losses does not by itself show that an edge is gone. Streaks in short sequences are also easy to misread.[33]
What it looks like in the ledger · five or more losses in a row in a log whose mean net R is positive
05Protocol
A Four-Step Ledger Test: PASS Against FAIL, Net of Costs
Add two columns, fix each grade before the outcome, wait for enough trades, then compare. Every threshold is the reader's own.
The protocol, as a sequence.
Every trade → Before the outcome → Until the count is reached → At the planned review
Log Result and Costs
Record each taken trade's gross result in R and, in a separate column, its costs in R.
Total execution cost is the gap between a paper trade at decision prices and the real one;[14] the two columns give the cost share.
Logging only net profit and loss, which hides what went to costs.
Grade Before You Know
Grade each trade PASS or FAIL against the written criteria at entry, before its result is known.
A grade written after the result lets the outcome choose the group, so the comparison measures luck.
Re-grading a FAIL trade as PASS after it wins.
Wait for the Sample
Do not compare PASS and FAIL until each group holds at least 15.7 × (s ÷ δ)² trades.
Here s is the standard deviation of R across a group's trades and δ the smallest gap in mean R worth acting on; the count follows from the standard error of each mean.[22][45]
Comparing after 20 trades, or re-running the comparison monthly on a growing sample.[37]
Compare the Gap With Its Error
Change a criterion only if the gap between the two mean net R values exceeds about twice its standard error; otherwise record "cannot tell".
The standard error of the gap is √(s1² ÷ n1 + s2² ÷ n2), which allows the two groups different spreads.[46]
Reading "no clear difference" as "no difference", or trusting the size of a gap that clears the bar in a small sample.[40]
06Verdict
The verdict.
Log result and cost in R, grade before the outcome, and change criteria only when PASS and FAIL differ by more than twice the gap's standard error.
Bottom line
Cut the trades that cannot pay their cost, keep the ones that can, and let enough trades decide which is which.
Costs Are Certain, Edge Is an Estimate, and the Ledger Decides Which Trades to Cut
Every trade pays its cost; only some carry enough edge to cover it. Net profit is the sum of gross edge minus cost over the trades taken, so cutting trades whose edge falls below their cost can raise net profit while lowering activity. A skipped trade that meets the criteria gives up its expectancy, so the aim is fewer sub-threshold trades, never fewer trades. In the field records, individuals' trades in Taiwan in the late 1990s and the most active US households in the 1990s earned less than their benchmarks; of Brazilian equity-futures day traders who began in the mid-2010s, 97% of those who traded more than 300 sessions lost money; and among Taiwanese day traders from the early 1990s to the mid-2000s, the top-ranked group stayed positive after fees, but fewer than 1% of the whole day-trader population earned positive net returns predictably and reliably. A trader's own ledger, read at the sample rule, decides the rest.
The arithmetic of selectivity is exact; applying it to one trader is a measurement. Net expectancy subtracts from gross expectancy a cost that arrives on every trade. Whether a threshold separates trades that clear their cost from those that do not is a fact held only in one trader's records.
Three results carry over. In Taiwan, costs made up most of individuals' measured shortfall and took more than a third of the top day traders' gross result, so the cost column deserves the attention the result column gets.[1][3] Some individual investors showed persistent skill, and the top-ranked Taiwanese day traders stayed positive after fees, so selectivity is not inactivity.[3][9] Short records mislead even trained researchers, so the comparison waits for its count.[31][40]
The goal is not fewer trades or more. It is a threshold that admits trades whose edge clears their cost and turns away the rest, checked by a planned comparison, one test fixed in advance, rather than by a feeling about the last few trades.
The most active households trailed the market.
Costs Are Certain
Net profit is the sum of gross edge minus cost over the trades taken. A trade whose gross edge is below its cost lowers net profit, however small the cost.
Short Records Mislead
Single results spread several times wider than the edge, so a small group's mean can carry either sign. A handful of trades says nothing about the criteria.
Run the Ledger Test
Log result and costs in R, grade before the outcome, wait for 15.7 × (s ÷ δ)² trades per group, then compare the gap with twice its standard error. Below the count, change nothing.
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