Where you are. Six lessons have built the relay. The message travels apart from the money (lesson 2); the payment hops through correspondents, and every hop adds a fee, a delay and a sanctions screen (lesson 3); each hop lands on one account with two names, mirrored to the digit (lesson 4); the yen that pays out in Tokyo was parked there in advance, one corridor’s share of a worldwide float of idle pre-funding (lesson 5); and lesson 6 wired all of it into a simulator that turns days-not-seconds into an output you can read. That prices two of the three costs lesson 1 promised: the relay’s fees and delays, and the pile of parked money. The third is the currency toll. Somewhere in the chain your payment stopped being dollars and became yen, and nothing yet has asked what that conversion cost. This lesson prices it, and the price is on no receipt: it hides inside the exchange rate itself.
Zero commission, in letters this tall
The arrivals hall of any international airport has the same shop. The sign is the biggest thing in it: ZERO COMMISSION, NO FEES, in letters taller than your forearm. Below the sign, in figures a tenth the size, a board with two columns: WE BUY and WE SELL. The rate on your phone, the one the news quoted this morning, sits somewhere between those two columns and appears nowhere on the board.
You hand over 1,000 dollars and receive 149,250 yen. Your phone says 1,000 dollars is 150,000 yen. The receipt lists the notes handed over and the notes handed back, and no third line. Every word on the sign is true, and 750 yen is missing. This lesson is about where it went.
The idea in one paragraph
At any instant a currency pair has a mid-market rate: the midpoint of what dealers are currently paying and charging for it, the number the news quotes and your phone shows. Nobody converts at it; it is a reference, not an offer. A dealer quotes two prices around it instead: a bid, slightly below the mid, at which it buys your dollars, and an ask, slightly above, at which it sells you dollars. The gap between bid and ask is the FX spread, one percent of the mid in this lesson’s deliberately fat, stylised quote, and the spread is the price. One conversion crosses from the honest midpoint to one side of the quote, so its toll is the notional times half the spread: no fee line, no receipt entry, just a rate that was never the mid.
Two prices surround every rate
Start where the honest number comes from. Dealers trade dollars for yen with each other all day in the interbank market, the wholesale tier of the two-tier map you drew in module 1, where the counterparties are banks and the sizes are millions. That market has its own two prices, a sliver apart, and the midpoint between them is the mid-market rate: 150.00 yen per dollar in this lesson’s stylised quote. It is the fairest single estimate of what a dollar is worth in yen right now, which is exactly why the news quotes it. It is also not an offer, any more than a shop’s cost price is a thing you can pay at the till.
The counter quotes two prices around it. Its bid, 149.25, is what it pays you per dollar when you sell dollars. Its ask, 150.75, is what it charges per dollar when you buy them. Both are real offers; the mid is not. The bank buys below the midpoint and sells above it, and the 1.50 yen between its two prices, one percent of the mid, is the FX spread. One percent is kiosk-sized on purpose: between dealers on a heavily traded pair the spread is a small fraction of that, and the exercise names its quote stylised for the same reason module 2 named its payment days stylised. The width varies enormously; the structure never does.
Wider than the screen; scroll it sideways.
One conversion pays half the spread
Now price the hook’s conversion. You sold 1,000 dollars, so the bank bought them at its bid: 149,250 yen. The mid would have given 150,000. The gap is 750 yen, and it has a clean formula, because the bid sits exactly half the spread below the mid:
In words: one conversion costs the amount converted times half the spread, because a single conversion crosses from the honest midpoint to one side of the quote, and each side sits half the spread away. Here that is 1,000 dollars times 0.75 yen of half-spread: the missing 750 yen, half a percent of what the mid would have given.
Now walk straight back to the counter. Your 149,250 yen buys dollars at the ask, 150.75, and returns just over 990 of them. Nothing happened, two conversions, zero fees charged, and just under ten dollars is gone. A round trip crosses the quote twice, once through each side, and pays the half-spread both times: the full spread, collected in two silent halves.
Why the booth can charge it
The spread is not paperwork, and it is not quite a swindle either. A dealer quoting both sides stands ready to buy currency it may struggle to resell and to sell currency it must restock, and the rate can move against its inventory while it waits; the spread is the price of bearing that, plus whatever the competition allows. That last clause does the real work. Where dealers are many and customers compare, in the interbank tier, spreads are hair-thin. Where the customer is captive, in an airport with a taxi to catch and yen needed tonight, the spread widens to whatever the tall letters can distract from. The width is set by your alternatives.
Review
The mid is a reference, not an offer
At any instant a currency pair has a mid-market rate: the midpoint of what dealers are currently paying and charging, the number the news quotes and your phone shows. Nobody converts at it. It is a reference, not an offer, any more than a shop’s cost price is a thing you can pay at the till. A dealer quotes two prices around it instead. Its bid sits slightly below the mid and is what it pays you when you sell; its ask sits slightly above and is what it charges when you buy. Both are real offers and the mid is not. The gap between bid and ask is the spread, and the spread is the price.
One conversion pays half the spread
A single conversion crosses from the honest midpoint to one side of the quote, so its toll is the notional times half the spread. There is no fee line and no receipt entry; there is just a rate that was never the mid. That is how zero commission in letters this tall can be literally true and still expensive. The width varies enormously - a kiosk quote is a whole percent, and between dealers on a heavily traded pair it is a small fraction of that - but the structure never varies. Two prices straddle a midpoint, you get the worse one, and the difference is collected silently inside the exchange rate.
Check yourself
1. The kiosk charged, in the most literal sense, nothing: no commission, no fee, and every word on its sign was true. Account for the missing 750 yen.
It never left the rate. The bank bought 1,000 dollars at 149.25 when the honest midpoint was 150.00, so every dollar handed over fetched 0.75 yen less than its fair value, and a thousand of them left 750 yen behind the counter. The toll is not added to the conversion; it is the conversion, done at a rate that was never the mid. That is why no receipt can show it: there is no second transaction to print.
2. The full spread is 1.50 yen, yet the exercise asserts that one conversion costs exactly the notional times 0.75. Why half?
The quote straddles the mid symmetrically: bid and ask each sit half the spread from the midpoint, on opposite sides. A single conversion crosses from the mid to one side only, so it pays one half. Selling and immediately rebuying crosses both sides, which is why the exercise’s round trip loses money in each direction and pays the full spread across its two legs.
3. You will never once in your life convert at the mid-market rate. What is it for?
Measurement. It is the midpoint of the wholesale market’s own quote, the fairest available estimate of the currency’s worth, and every retail quote straddles it. Against the mid, any offered rate becomes a number: distance times notional equals toll. Without it, “a good rate” is a feeling; with it, comparing the kiosk, your bank and a transfer app is arithmetic on one yardstick.
4. One counter advertises zero commission and quotes this lesson’s one-percent spread. Another charges a printed 5-dollar fee but quotes a spread a fifth as wide. You are converting 1,000 dollars. Which is cheaper, and what did you have to do to find out?
Convert both to total cost against the mid. The fee-free counter costs half its spread: half a percent of 1,000 dollars, about 5 dollars. The fee-charging counter costs its printed 5 dollars plus half of its narrow spread, about 1 dollar more: roughly 6 in all. At this notional the tall letters genuinely win. At 10,000 dollars the spread cost grows tenfold and the printed fee does not, and the answer flips hard the other way. The procedure was the same both times: mid, distance, notional, plus every printed fee. Lesson 9 builds that procedure into a full cost stack.
Do this
Fifteen minutes, from module-03-across-borders, and for once the ledgers sit this lesson out: the file is pure arithmetic, no imports. Open code/fx_spread.py. The constants pin the stylised quote: a mid of 150, a full spread of one percent of it, so 0.75 yen of half-spread each way, bid 149.25, ask 150.75. Your work is spread_cost, four numbers in order: the yen a 1,000-dollar conversion actually receives at the bid; the yen the mid would have given; the gap between them; and that gap restated as a share of the mid-rate value, counted in hundredths of a percent, fifty of them here. (The next lesson gives that counting unit its industry name.) Keep the arithmetic in exactly the shape the comments dictate; the assertions compare with ==.
python3 code/fx_spread.pyThe harness checks the geometry before your function runs, that the quote straddles the mid symmetrically and spans the full quoted spread, and the toll after it: one conversion costs exactly the notional times the half-spread, and converting straight back loses money too. Green ends with the final line verbatim:
no fee appears on any statement, yet 750 yen stayed with the bank: the FX toll is collected inside the rate itself, and every conversion pays half the spreadIf the toll comes out negative, you subtracted the mid conversion from the bid conversion instead of the other way round. If the half-spread assertion fires, you converted at the ask: you are selling dollars, and the bank buys what you sell at its low price, the bid. The completed version is solutions/fx_spread.py; compare after you are green.
What you can now do. You can price a currency conversion the way the market does: find the mid, read the offered rate, and the distance between them, times the notional, is the toll, whatever the sign above the counter says. You can say why one conversion pays half the spread and a round trip the whole of it, and why the toll can never appear on a receipt: it is not a charge on the conversion but the rate the conversion used. Two refinements follow. The half-percents this lesson wrote out in words are clumsy at wholesale sizes, so lesson 8 introduces the unit the industry counts tolls in; and the spread never travels alone, so lesson 9 stacks it on top of the relay’s fixed fees from lesson 3 and finds whom the combined bill lands on hardest. The booth is priced; next, its unit.