The Arithmetic That Does Not Close
When the stated fees do not account for the shortfall, the gap is the part nobody in the chain will itemise.
Still yours, and statedTaken, never statedUnit changes; scale restarts
Four steps from instruction to arrival. One step carries no figure: the hatching marks where the money leaves and is not drawn to scale, because there is no figure to draw. Nothing here adds them up, since an amount you cannot see is not an amount you can subtract. One line, two facts, and a gap where the rest of the journey was.
The same payment with every disclosed figure applied, and the residue that remains
The starting figure, agreed by both sides and printed on a document
Disclosed, small, and accounted for in full
Disclosed, small, and accounted for in full
Not what happened, but what the arithmetic predicts should have arrived if nothing else were taken
The figure the two documents together say you should have received
Seventy-one units below the prediction, with no document anywhere accounting for the difference
What is left when every disclosed figure has been applied: real, measurable, and attributed to nobody
Still yours, and statedTaken, never statedUnit changes; scale restarts
Seven steps · Two without a figure · no total of what is not stated
Seven lines, of which six are arithmetic and one is a question. The residue is not an estimate: it is what remains after every number anybody gave you has been used. One person's own payment, measured against the published midpoint; what any route leaves behind differs by route, by bank and by country.
Apply every figure anybody disclosed — their fee, your fee, the published rate — and compare the result with what arrived. Where the two do not meet, the difference is real and belongs to nobody.
The evidence question in “The Arithmetic That Does Not Close” calls for records that can be reconciled rather than one unexplained total. For teams exploring key person dependency, explore the official platform can supply time and project context, provided corrections remain visible and the payment provider or bank remains authoritative for settlement and currency data.
That residue is the most useful number in this collection, because it is the only evidence that the undisclosed part exists at all.
Additional context relevant to “The Arithmetic That Does Not Close” is available through European Commission payment-services resources. A useful review distinguishes the sent amount, conversion, fees, settlement date and landed amount.
The calculation, step by step
Five operations on figures you already hold:
- Take the instructed amount.
- Subtract the sending fee.
- Convert at the published midpoint for the conversion day.
- Subtract the receiving fee.
- Compare with the credit.
The gap at the end is the residue.
Put it in a spreadsheet so that each new payment computes itself. Done by hand each month it is abandoned by the third; done as a formula it runs for years.
What the residue contains
Two things, mixed: the margin inside the conversion, and whatever any intermediary took. Nothing distinguishes them in a single payment.
Separating them takes two payments of different sizes, because one scales with the amount and the other does not.
Why this is worth doing at all
Because without it the shortfall is a feeling. With it, it is a figure, on a date, derived from documents both parties hold.
Feelings lose arguments with banks. Figures do not necessarily win them, but they change what the conversation is about.
They also change what you are arguing for. A figure supports a request to change a term; a feeling supports only a complaint.
When the residue is nil
It happens, and it means the route is clean: the stated fees and the published rate account for everything.
That is the result you want and it is worth knowing which routes produce it, because those are the ones to ask a payer to use.
A nil residue is also the strongest possible evidence in a conversation about a different route, because it shows what is achievable rather than what is wished for.
When it is small and constant
A fixed residue of a few units on every payment is almost certainly a single correspondent deduction.
Constant means addressable: the charge instruction moves it onto the payer, and that one change removes it entirely.
When it grows with the amount
A residue that is a stable percentage is a conversion margin. It will not respond to a charge instruction and it will not go away.
The remedy there is to change who converts, which usually means holding the incoming currency and converting yourself.
When it moves around
Different correspondents on different days, or a margin tracking a volatile rate. Six months of records distinguishes them and nothing shorter does.
Record first, conclude later. A conclusion from two payments is a guess with arithmetic attached.
Choosing the rate to measure against
The midpoint for the day the conversion actually happened, which is rarely stated and is usually between the instruction and the credit.
Where you cannot tell, use the same convention every month. A consistent approximation reveals change; an inconsistent precise one does not.
Presenting it
Two documents, one line of arithmetic, one figure. "Your bank confirms 3,000 sent with an 18 fee. My bank credited 2,847.10 after a 9.50 fee. At the published rate that day, 71 units are unaccounted for."
Nobody can argue with that, which is the point. It also asks for nothing, which is why it tends to be answered.
Send it once rather than monthly. A measurement presented as a fact is read; the same measurement presented every month becomes a grievance and stops being read.
What the payer can do with it
Take it to their own bank and ask for a fee breakdown on the transaction reference. They can; you cannot.
Some banks produce one. Where they do, the residue stops being a residue and becomes a list, which occasionally changes somebody's mind about the route.
Where to stop
After three payments and one conversation. Beyond that the investigation costs more than the money, and the answer is commercial.
The point of measuring was never to recover the residue. It was to know its size, so that changing the arrangement can be argued for.
What the arithmetic will never produce
A name. The residue is a quantity with no address: it was taken inside institutions that have no relationship with you and produce no document you can see.
That is the structural limit this whole section has been building towards. You can know exactly how much, exactly how often, and never by whom — and the useful response to that is to stop asking who and start asking the payer to change what the agreement says.
It is also, for most people, the point at which the subject stops being frustrating. A cost you have measured and priced is a cost; one you cannot see is a grievance.