Basis points, the unit of tolls

Where you are. Two lessons have priced the border crossing, in two shapes that refuse to add. Lesson 6’s corridor simulator charges flat fees: a fixed amount at each of the canonical corridor’s three hops, indifferent to what the payment carries. Lesson 7 found the bigger toll hiding inside the exchange rate itself: the FX desk’s half-spread, a cut proportional to whatever you convert, measured in hundredths of a percent. A fixed amount and a buried rate are different kinds of number - one is currency, one is a fraction - so the corridor’s real price is still two figures on two lines. This lesson supplies the unit that puts them on one line: the unit finance actually argues in, and the course’s toll unit from here to the end.

The syllable worth a salary

Listen in on two banks agreeing an exchange rate, a single stylised trade of 500 million. The first desk quotes a price. The second does not haggle in percent; nobody at this size does. It asks for the rate “three tighter”, and the first desk winces before agreeing. Three what? Three hundredths of one percent - a difference so far down the decimal places that a spreadsheet would round it away. Multiply it out instead: three hundredths of a percent of 500 million is 150,000. One syllable moved a sum most people work a year for, and neither desk found the conversation strange. At these volumes the second decimal of a percentage is a fortune, so the people who argue over second decimals long ago built a unit that makes those decimals whole numbers. This lesson is that unit.

The idea in one paragraph

A basis point is one hundredth of one percent - equivalently, one part in ten thousand - so 100 basis points make one percent and 50 basis points are half a percent. Finance quotes fees, spreads and yields in basis points for the same reason your code stores money in integer cents rather than floating-point dollars: choose a unit fine enough and the numbers people argue about become whole and unambiguous. This lesson makes it the course’s toll unit and puts it straight to work. Re-express the corridor’s whole stack - flat fees that ignore size, an FX margin that scales with it - in basis points of the notional, the face amount the payment moves, and the corridor’s real price appears. It is not one price: the stylised corridor that charges a 10,000 payment 65 basis points charges a 200 payment 800, and nothing about the corridor changed between those two sentences except the size of the payment crossing it.

One part in ten thousand

Percent already compresses: “per hundred” turns 0.07 into 7. A basis point compresses a hundred times further: one part in ten thousand, a plain rate of 0.0001. The three ways of writing a rate line up like this:

The last column is the mnemonic worth keeping: on a payment of ten thousand, one basis point is exactly one unit of currency. That is the whole conversion rule, and it is why every formula in this lesson multiplies or divides by 10,000.

The toll stack, priced twice

Now put the corridor’s two tolls in one place. From lesson 6, the flat part: each of the canonical corridor’s three hops charges a flat 5, stylised, so 15 leaves the payment whatever its size. From lesson 7, the proportional part: the desk’s margin, the half-spread kept as a rate of 50 bps, taking 50 parts in ten thousand of whatever converts. In currency the stack is one addition:

$\text{total} = 3 \times 5 + \text{notional} \times \frac{50}{10\,000}$

In words: the hops charge what they charge, and the desk takes its rate of the notional. To see what the crossing really costs, divide the total by the notional and re-express it in the new unit, which means multiplying by 10,000:

$\text{toll in bps} = \frac{15 \times 10\,000}{\text{notional}} + 50$

Read it term by term, because the whole lesson lives in this line. The margin passes straight through: 50 bps in, 50 bps out, identical at every size. The flat fees arrive divided by the payment they sit on: a shrinking nuisance for a large payment, a monster for a small one. Every corridor’s price in basis points is a constant floor plus a blow-up term, and only the blow-up cares about size. Run the two payments the exercise runs:

Ask who paid more and the currency row answers: the 10,000 payment, 65 against 16, four times as much, because its margin scales with size. Ask who paid a larger share - the only question the sender cares about - and the ranking flips hard: 800 basis points against 65, more than a twelvefold multiple. Decompose the 800 and the culprit confesses. The flat 15 is 750 bps of a 200 notional but only 15 bps of 10,000, while the margin contributes the same 50 bps to both stacks: 750 plus 50 against 15 plus 50. Eight percent of the small payment is gone before the recipient sees a single unit.

The floor and the cliff

The figure freezes the same story across every size at once, for three stylised corridors: a bank-style corridor with heavy flat fees of 25 and a 250 bps margin, a digital-style corridor at 2 and 90 bps, and one in between at 10 and 150 bps.

The vertical axis is percent; multiply by a hundred to read it in bps. Every corridor draws the same curve. On the right each line lies flat on its own floor - 2.5%, 1.5% and 0.9%, which are just the three margins of 250, 150 and 90 bps with the flat fees diluted to nothing. On the left every line turns up a cliff, the cheap corridor’s included: its flat fee of 2 is 1% of a 200 payment but 2% of a payment of 100, and the shape wins over the numbers every time. The digital-style corridor is cheapest at every size here, and even it cannot escape the geometry.

Check yourself

Do this

Twenty minutes, from module-03-across-borders. Open code/toll_booth.py - the arithmetic behind this lesson’s toll booth widget, with constants where the widget has sliders, and the constants are the module’s own: the canonical corridor’s three hops at a flat 5 each from lesson 6, and the desk’s cut from lesson 7 kept as FX_MARGIN_BPS = 50. Your work is toll_stack, four lines in the TODO(you): the flat part (HOPS * FLAT_FEE), the FX part (notional * FX_MARGIN_BPS / 10_000), their total, and the total re-expressed in basis points of the payment (total * 10_000 / notional). Keep the arithmetic in exactly that shape; the assertions compare with == and every result lands exactly.

python3 code/toll_booth.py

Green prints both stacks - 16.00 in total on the 200 payment, 65.00 on the 10,000 - then four assertions walk the lesson’s argument in order: the flat part identical for both, the currency ranking, the bps flip, the twelvefold multiple. The run ends with the final line verbatim:

same corridor, same tolls: the 200 payment pays 800 bps and the 10,000 payment pays 65 bps - flat fees shrink for no one, so the smallest payments pay the steepest rate

If the bps assertion fires with 8.0 where 800 was expected, you converted to percent rather than basis points: the factor is 10,000, not 100. If the FX margin prints 500,000 on the large payment, you multiplied by the bps and skipped the division - a rate in basis points is that number over ten thousand, always. The completed version is solutions/toll_booth.py; compare after you are green.

What you can now do. You can quote any toll in the unit the trade is actually argued in, converting between basis points, percent and currency without pausing, and you can decompose any corridor’s price into the two terms that govern it: a floor that is the margin and passes through at every size, and a flat-fee term that divides by the notional and punishes the small. You proved the flip on one corridor: the same three hops and the same desk charge the 10,000 payment 65 bps and the 200 payment 800, a twelvefold gap manufactured by arithmetic alone. From here on the course prices every toll this way. The next lesson points the unit at the corridors where the 200 payment lives - remittances - and puts real, dated cost statistics beside the stylised fees; the shape you built today is the shape of that story.