Who Owns the Fraction?

1The layer below one cent

Money settles in cents. Calculations do not. A bank computing daily interest applies a daily rate carried to five or more decimal places S10. A broker rounds fractional share holdings to the fifth or sixth decimal S21. A remittance provider may show you an exchange rate rounded to two decimals while computing your amount with the full rate S11. Only at the end, when the number must become money, does it get cut down to the cent. The ACH file that moves the money has room for dollars and cents and nothing smaller S18.

So there are three precisions in every system: the one used to compute, the one shown to the customer, and the one that settles. The gaps between them are where the fractions live. Most of the time nobody looks there, because each gap is worth less than a cent.

2Why rounding is economically meaningful

Take a 3% fee on a $10.17 purchase. The exact fee is $0.3051. Nobody can pay $0.3051, so it becomes $0.31, and the effective rate is 3.05%. On a $10.15 purchase the exact fee is $0.3045, it becomes $0.30, and the effective rate is 2.96%. One customer overpays by half a cent; the next underpays by half a cent.

For this fee the arithmetic is unusually clean. Because 3 and 100 share no factors, any 100 consecutive cent amounts produce every possible sub-cent leftover exactly once. Under nearest-cent rounding, 50 of them round up, 49 round down and 1 is exact. The average absolute move is a quarter of a cent ($0.0025). The average net move depends on one tiny detail, the tie at exactly half a cent:

• Round half up (the usual school rule) sends the tie up every time. Net drift: +$0.00005 per fee.
• Round half to even (banker's rounding) sends the tie up half the time. Net drift: $0 in expectation (exactly zero over any 200 consecutive cent amounts).
• Always up net drift: +$0.00495 per fee, nearly a hundred times the half-up figure.

Truncation and always-down are the same for positive amounts. For refunds and credits they differ: −$0.0157 truncates to −$0.01 but floors to −$0.02.

All of this assumes every leftover is equally likely, which holds for synthetic amounts and not for real prices. For a 3% fee the whole-dollar part of a price always produces whole cents, so the leftover depends only on the price's cents ending:

Retail prices cluster at .99, .95 and .00. A fee on prices ending in .99 rounds up every time, by three hundredths of a cent each time. So neutrality is not a property of the rule alone. It is a property of the rule and the data together.

A single event is trivial. What matters at scale is the statistical behavior of the rule on real data, and the rule is decided by whoever writes the code. Figure 1 shows the other face of the same arithmetic. On small amounts the effective rate swings widely around the nominal one, from 0% (fees under half a cent round away) to 5.9%. The swing fades as amounts grow.

2bWhen the fraction never ends

Some leftovers are not a few extra decimals. They never stop. Split $100 into three payments and each one is $33.333…, forever. Pay 5% a year on $1,000 and one day of simple interest is 10/73 of a dollar, $0.136986301369863… repeating. Inside a computer it gets stranger: most systems count in binary, where even one cent has no exact form. Adding one cent ten thousand times in ordinary floating point gives 100.00000000001425, not 100.

An infinite decimal does not create an infinite loss. The leftover of any single payment is still less than a cent. The loss comes from rounding each piece on its own and forgetting what was cut off. There are only a few honest ways to handle it:

1. Keep the exact fraction. Store 100/3 or 10/73 as a ratio, compute with it, and round once at the end. Nothing is lost along the way.
2. Split so the total survives. $33.33 + $33.33 + $33.34 = $100.00. Someone gets the extra cent, and a written rule decides who: the first payment, the last, the largest remainder, or each party in turn.
3. Carry the leftover forward. Remember what was not paid and add it to the next calculation. The running error never exceeds half a cent, however many events there are.
4. Cut at a fixed decimal, then apply a rule. NASD's 2004 guidance did this: truncate at the fifth decimal, then alternate up and down S1.
5. Round every piece and forget the rest. This is where bias is born.

CALCULATED with exact rational arithmetic. Simple interest, 365-day year, monthly crediting on calendar months; real deposit accounts compound, so this illustrates the rounding step, not any bank's yield.

The same $50 of interest becomes $51.10 if each day is rounded and credited on its own, a 2.2% overpayment, or $47.45 if each day is truncated, a 5.1% underpayment. Accruing exactly and crediting monthly lands at $50.03. Carrying the leftover forward lands at exactly $50.00. The published rules we found sit on the safe side of this table: Regulation DD's examples carry daily rates to five or more decimals S10, and Chase rounds once per interest period, not per day S8.

So the never-ending fraction is a solved mathematical problem. The unsolved part is the cent it leaves behind, and who is named to receive it. That choice is a business rule, and it is almost never published.

3The law does not use one rule

We found no general federal rule on rounding neutrality. What exists is a patchwork. Each regime governs the precision it cares about, usually disclosure, and leaves the leftover to contract.

Two of these are worth noticing because they are deliberate engineering. Regulation E separates display precision from computational precision: the rate you see may be rounded, but the amount you receive must be computed before rounding S11. And the IRS reconciles payroll tax rounding on Form 941 line 7, where the adjustment may be a positive or a negative adjustment S12. Both are designs that stop a fraction from leaking. Neither is general.

On unfairness, the CFPB's examiner guidance says an act or practice that causes a small amount of harm to a large number of people may be deemed to cause substantial injury S14. That is one of three elements; the injury must also be not reasonably avoidable and not outweighed by benefits. We found no case applying it to rounding. This report draws no UDAAP conclusion about any practice or firm.

4A documented case: broker pass-through fees

When you sell stock, the exchanges and FINRA owe the SEC a fee on the dollar volume of covered sales, currently $20.60 per million dollars, a rate that stays in force until 60 days after the next appropriation S3. The SEC collects it from the exchanges and FINRA, which recover it from their member firms S2. Firms usually pass it on to customers, trade by trade. FINRA's own Trading Activity Fee works the same way, and FINRA states plainly that rounding rules do not apply to it because it is calculated and reported in aggregate S4. FINRA also reconciles at its own level: it paused the TAF for October to December 2026 because 2026 revenues were projected to exceed expectations S26.

So the liability is computed on totals, while the customer charge is computed per trade and must land on a cent. That mismatch guarantees a residual. Its sign depends on the rounding rule.

In August 2004 NASD, FINRA's predecessor, wrote that it understood these charges had resulted in overages at some broker-dealer firms, in part due to the practice of routinely rounding up to the next cent. Its guidance was precise: firms should truncate at the fifth decimal and, if there is any remainder, alternate between rounding up and rounding down to the next cent. Firms that charge more than the fee should refrain from calling it a "Section 31 fee" or "SEC fee" S1. This is the clearest regulator statement on rounding overages we found. It is from 2004, it is advisory, and it is NASD's understanding rather than an SEC finding. We located no SEC finding on the point.

What firms publish today:

• Robinhood says the SEC fee is rounded up to the nearest penny, and that it does not pass the fee on for equity sales of $500 or less. Its TAF charge is rounded to the nearest penny, waived for 50 shares or less, and paused for now. For options it says its regulatory fee may differ from or exceed the actual fee we paid S5.
• Fidelity says its "Additional Assessment", typically $0.01 to $0.03 per $1,000 of sales, is designed to offset the charges imposed on Fidelity by the SROs, but may differ from or exceed them. It lists the rounding methodology used by Fidelity among the causes and makes no representation that the two will be equal S6.
• Schwab says its Industry Fee may differ from or exceed the actual fees properly paid by Schwab, lists the rounding methodology used among the causes, and does not notify clients when it changes the fee S27.

Three of the largest retail brokers reserve in writing the room for the customer charge to exceed the regulatory cost, and name rounding as a reason. None publishes how large the difference is, or in which direction it ran last year. That is a disclosure fact, not evidence of an excess. Methods also differ from the 2004 guidance, which was advisory; we draw no conclusion about compliance.

Rounding up hits small trades hardest. On a $1,000 sale the exact fee is $0.0206. Rounded up, the customer pays $0.03, 46% more than the fee. On a $50,000 sale the difference disappears.

Does the firm keep the difference? Public evidence cannot say. Waivers push the other way: when a broker absorbs the fee on small sales, it pays regulators money it never collected. We simulated the same published rule (round up, waive at or below $500) on three synthetic customer populations:

SIMULATION. 100,000 synthetic sell orders per mix, log-normal, seed 20261019. Per-trade net = charged minus exact fee owed on that sale.

With mostly small sellers, the same rule leaves the broker short. With mostly large sellers, it leaves the broker ahead by about half a cent per trade. The direction of a rule does not tell you the direction of the money. Only the firm's own reconciliation of collections against liability can, and none is published.

5The control case: Schertzer v. Bank of America

A cardholder sued Bank of America over its 3% foreign transaction fee. Her complaint listed 18 purchases where the fee, after rounding, came to more than 3%, and claimed the bank should have rounded down. In March 2020 the court dismissed the contract claims with leave to amend. It reconstructed the exact fees itself and wrote: In essence, Schertzer is ignoring all of the times BofA rounded down and she received the benefit of BofA's policy. It added that without rounding she would have owed a fraction of a penny, which cannot be paid, and that nothing suggested the policy was not applied neutrally. It also called the $0.18 at stake de minimis. The related unfair-competition claim was dismissed without leave, and the rounding theory was not pleaded again; the case continued on other fees S7. The court's remarks on neutrality were made on the pleadings, not on data.

We use the case as a methodological control, because the same 18 purchases produce three different numbers depending on the baseline:

• $0.18: rounded-up fees compared with rounded-down fees, the plaintiff's baseline.
• $0.0577: rounded-up fees compared with the exact fees, the economic residual on those purchases.
• Unknown: the net across all her purchases, including the ones that rounded down. Neither side put it in the record.

Exact fees and charged fees follow the court's reconstructed table. The court's own table shows $0.04 and $0.02 for the $13.22 and $6.61 purchases, evidently typos: its pre-rounding amounts ($0.3966 and $0.1983) imply $0.40 and $0.20, which we use.

The lesson cuts both ways. A list of round-ups proves nothing about bias. But a court accepting neutrality on the pleadings does not prove neutrality in the data either. Only the full population answers the question.

6The architectures

We classified every system we could document into seven classes. R4 (provider-favorable) requires a demonstrated beneficiary. Direction alone earns R5.

R0

Exact. No residual at this layer.

R1

Neutral rounding, expected to balance.

R2

Carry-forward. The residual is kept and settled later.

R3

Consumer-favorable by rule.

R4

Provider-favorable, beneficiary shown.

R5

A residual exists; its destination is not public.

R6

A regulator or SRO reconciles it explicitly.

Rounding rules differ in more than their averages. Figure 3 runs the same 200,000 synthetic fees through five rules. Nearest-cent rules net out within dollars per million fees. Directional rules net out at about $4,950 per million, up or down. Carry-forward keeps a running total and nets out at zero.

Figure 4 shows why the distinction matters over time. Directional rules grow in a straight line with volume. Nearest-cent rules wander but stay small. Carry-forward is pinned to zero by construction. Half of that design exists in the wild: Goldman Sachs Bank USA's Apple Savings agreement deducts interest paid but not earned from the next month, but does not say whether a fraction rounded away is ever paid back S9. Reconciling only one side is not carry-forward.

7What happens at scale

The populations are large and well measured. In 2024 U.S. core noncash payments excluding wires numbered 236.6 billion; card payments numbered 187.7 billion, of which 120.6 billion were debit and 67.1 billion credit; ACH payments numbered 39.7 billion and checks 9.2 billion S16. PayPal alone completed 25.4 billion payment transactions worldwide in 2025 S17.

What is not measured is how many of those payments carry a sub-cent calculation at all, and under which rule. Most card payments carry no percentage fee to the consumer. The percentage fees fall on merchants, under contracts that are not public. A national model built by multiplying a rounding assumption across all payments would violate the first rule of this research: do not multiply a rounding event across transactions where it does not occur.

What we can do is state the arithmetic of a rule as a hypothetical. If every U.S. card payment carried one nearest-cent rounding step with uniformly spread leftovers, which neither happens nor fits real price endings:

• Gross movement in both directions: $469.2 million a year.
• Net drift under round-half-even: $0.
• Net drift under round-half-up: $9.4 million.
• Net drift under always-up: $929.1 million.

SCENARIO ANALYSIS, NOT OBSERVED PROFIT. Hypothetical rule on a hypothetical population. Nothing here is money anyone took.

8The billion-dollar sensitivity question

How big does a bias have to be before it matters? Figure 6 multiplies a range of average per-event biases by the observed counts. At one ten-thousandth of a dollar per card payment the total is about $19 million a year. At half a cent it approaches a billion. The table gives every step.

SCENARIO ANALYSIS, NOT OBSERVED PROFIT. Bias × count. No step in this table is claimed to occur.

9Where the thesis fails

The hypothesis was that fragmented rules leave room for systematic bias at scale. Much of the evidence pushes the other way:

• Settlement is exact. ACH carries only dollars and cents S18. The rails themselves create no fractions.
• Regulators already built reconciliation. Form 941 line 7 S12, Regulation E's unrounded-rate rule S11, NASD's alternating rule S1, FINRA's aggregate TAF S4.
• Published consumer rules are neutral. Chase rounds interest half-up at the end of each period S8. Stripe rounds its fee to the nearest cent S19, and so did Square, which also chose banker's rounding for tax because standard rounding, in its words, results in a bias towards a higher amount S28. IRS whole-dollar rounding is optional and symmetric S13.
• The fee-setter returns its own excess. FINRA paused its trading fee for a quarter when revenue ran ahead of plan S26.
• Some rules favor the customer. Fees under half a cent round to zero. Robinhood absorbs the SEC fee on sales of $500 or less and rounds CAT fees under a cent to zero S5.
• A court rejected a one-sided sample and called the amount de minimis. Schertzer S7.

10What remains opaque

We searched for a published sub-cent rule and found none for: PayPal fees and currency conversion; interchange and network assessments at Visa, Mastercard and American Express; and deposit interest in the current Bank of America and Wells Fargo agreements, which say only that interest is paid in whole cents (Citi was checked only against a 2023 document). Visa's installment interface returns a separate amount for the last installment, which suggests the remainder lands in the final payment, but no rule is stated S24. Square's 2024 rounding page has disappeared from the live site S28. Fidelity, Schwab and Robinhood disclose that their fees may exceed regulatory costs but not by how much S6S27S5. Robinhood rounds fractional-share values and dividends to the cent without saying where the leftover goes S21. Coinbase lets customers clear crypto "dust" under about two cents for no proceeds S20; that is a different mechanism from fiat rounding and we keep it apart.

Absence of a published rule is not evidence of bias. It is evidence that nobody outside can check.

11The real systemic issue

Five candidate problems, ranked by what the evidence supports:

1. No reconciliation between per-customer charges and aggregate liability. Wherever a provider recovers a cost computed on totals by charging customers in rounded pieces, a residual is guaranteed. The 2004 notice shows it accumulated once. Nothing public shows whether it does today.
2. Disclosure. Most rounding rules are not published. Where they are, they often sit in support pages that change without notice, not in contracts. The brokers that do put it in contracts disclose the possibility of a difference, never its size.
3. No standardized reporting. No regulator collects rounding residuals. FINRA's own fee avoids them by staying in aggregate.
4. Fragmented regulation. Real, but the pieces that exist mostly work.
5. Residual capture. Possible in principle, described once by a regulator in 2004, not demonstrated today.

12Operational Truth

A policy review asks what a rule says. This case asks what the system does when the rule, the code and billions of transactions meet. Answering it took four layers that are rarely joined: the law (who may charge what), the implementation (which rounding function, at which step), the arithmetic (the exact leftover per event and its distribution) and the evidence (how strongly each link is supported). Zircular's work is that join, applied as a standing check rather than a one-time study.

For each rule, a residue register would hold: jurisdiction, sector, entity, product, governing authority, formula, computational precision, display precision, settlement precision, rounding function, rounding step, exact amount, charged amount, residual, beneficiary, reconciliation mechanism, disclosure status, event volume, gross and net residual, evidence grade, source and effective date. The files behind this report are a first version of that register.

Run continuously, it would catch what a policy review misses:

• A fee schedule that keeps the old rate after a regulator changes it. Fidelity names differences between the dates of changes to rates as a cause of divergence S6; the Section 31 rate went from $27.80 to zero to $20.60 per million within a year S3.
• A shift in customer mix that flips the sign of a rule that never changed, as in section 4. Or a shift in price endings, as in section 2.
• Display precision leaking into money, the error Regulation E was written to prevent.
• A finer price grid. A half-penny minimum tick for some stocks, adopted in 2024, is now due by November 2027 after two exemptive orders S22S25. When it arrives it will put more sub-cent amounts into every fee calculation downstream.

Methods and reproducibility

All money arithmetic uses Python's decimal module; no floating point touches a monetary value. Simulations use a fixed seed (20261019). Every figure is labeled Observed, Calculated, Simulation or Scenario. The model, registers and chart data are in the files model.py, registers.py, source_register.csv, evidence_matrix.csv, entity_rounding_matrix.csv and chart_data/. Before publication an independent adversarial reviewer re-checked every source, quote, date and calculation against live sources. It found and we corrected: a compliance date that had moved a year (Rule 612, now November 2027); a fee paused since October (FINRA's TAF); a mis-stated court outcome in Schertzer and an unverifiable reporter citation; a missing sentence in Fidelity's disclosure ("designed to offset"); two firms we had wrongly listed as publishing nothing (Square, Schwab); two over-strong classifications (Fidelity, Goldman Sachs); and a headline that multiplied a fee rounding across all card payments under an assumption real prices break. Its recomputation of every figure matched.

This is research, not legal advice. Nothing here alleges that any named firm acted unlawfully.

Findings and grades

Sources

1. S1 · A NASD (now FINRA). Notice to Members 04-63: Section 31 fees, guidance on charging customers. 2004-08-20.
2. S2 · A SEC. Release 34-49928, Collection Practices Under Section 31 of the Exchange Act (Rule 31, Form R31). 2004-06-28.
3. S3 · A FINRA. Information Notice: New rate for fees paid under Section 31 ($20.60 per million from Apr 4 2026, in effect until 60 days after the FY2027 appropriation). 2026-03-17.
4. S4 · A FINRA. Trading Activity Fee FAQ (Q100.10: rounding rules do not apply; aggregate reporting). retrieved 2026-10-19.
5. S5 · D Robinhood support. Trading fees on Robinhood (SEC fee, TAF, CAT fee rounding). retrieved 2026-10-19.
6. S6 · A Fidelity Brokerage Services. Brokerage Commission and Fee Schedule (Additional Assessment; Options Regulatory Fee). PDF modified 2026-09-09; retrieved 2026-10-19.
7. S7 · B U.S. District Court, S.D. Cal., via govinfo. Schertzer v. Bank of America, N.A., No. 19cv264 JM(MSB), 2020 WL 1046890, 2020 U.S. Dist. LEXIS 37503 (S.D. Cal. Mar. 4, 2020) (Miller, J.). 2020-03-04.
8. S8 · A JPMorgan Chase Bank, N.A.. Deposit Account Agreement (interest paid in whole cents; half-up at end of each interest period). effective 2026-06-14.
9. S9 · A Goldman Sachs Bank USA, Salt Lake City Branch. Apple Savings Deposit Account Agreement (rounding and interest calculation). effective 2026-03-19.
10. S10 · A CFPB. Regulation DD, 12 CFR 1030.3(f)(1) and Appendix A with Official Interpretation. current eCFR.
11. S11 · A CFPB. Regulation E, Official Interpretation of 12 CFR 1005.31 (comments 31(b)(1)-3, 31(b)(1)(iv)-2). current.
12. S12 · A IRS. Instructions for Form 941, line 7 (fractions of cents). current revision.
13. S13 · A IRS. Instructions for Schedule D (Form 1040) 2025, rounding off to whole dollars. 2025 tax year.
14. S14 · A CFPB. CFPB Supervision and Examination Manual, UDAAP procedures (Manual V.2). Oct 2012 text, file dated 2023-09.
15. S15 · A CFPB. Regulation Z, 12 CFR 1026.22(a)(2)-(3) APR accuracy tolerances (closed-end credit). current.
16. S16 · A Federal Reserve Board. Federal Reserve Payments Study, 2024 data (core noncash payments, excluding wires). last updated 2026-07-01.
17. S17 · A PayPal / SEC EDGAR. PayPal Holdings Form 10-K FY2025 (payment transactions 25.4 billion). FY2025.
18. S18 · D Nacha. ACH Guide for Developers, file details (amount field $$$$$$$$cc). retrieved 2026-10-19.
19. S19 · D Stripe support. Rounding rules for Stripe fees. retrieved 2026-10-19.
20. S20 · D Coinbase help. Remove dust (sell balances under $0.02 for no proceeds). retrieved 2026-10-19.
21. S21 · A Robinhood Financial / Robinhood Securities. Robinhood Customer Agreement (fractional shares, dividends). rev. 2026-10-01.
22. S22 · A SEC. Press release 2024-137: tick sizes, access fees, round lots (Rule 612 amendments). 2024-09-18.
23. S23 · A SEC. Press release 2025-130: exemptive order on compliance with certain Regulation NMS rules. 2025.
24. S24 · D Visa Developer. Installment Transaction Service, how-to (returns installment amount and a separate last installment amount). retrieved 2026-10-19.
25. S25 · A SEC. Release 34-105656: temporary exemptive relief from Rules 610(c) and 612 until the first business day of November 2027. 2026-06-11.
26. S26 · A SEC / FINRA. Release 34-106409 (SR-FINRA-2026-021): temporary pause of the Trading Activity Fee, Oct 1 to Dec 31 2026. 2026-09-18.
27. S27 · A Charles Schwab & Co.. Schwab Pricing Guide for Individual Investors (Industry Fee). April 2026; retrieved 2026-10-19.
28. S28 · D Square Support Center (Internet Archive copy). Understanding Fee Rounding (archived). page modified 2024-05-14; captured 2024-06-20.