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What is an exchange rate?
The price of one currency in another: where that number comes from, and why its direction of reading catches everyone out.
price in $ = price in € × rate ($/€)Problem / motivation
On an airport board: “€1 = $1.14”. Three questions hide behind that small number — what it measures, who set it, and which way it reads. It is the third that loses people money.
Let's start with the word. A FOREIGN CURRENCY is simply somebody else's money: the dollar is a foreign currency for us, the euro is one for an American. (And beware a widespread shortcut: it is not “one country, one currency” — the euro is the currency of twenty-one countries at once, Bulgaria having joined on 1 January 2026.) The exchange rate then answers a conversion question: “€1, how many dollars is that?”
But an exchange rate is not a conversion table like kilometres into miles, fixed once and for all by convention. It is a PRICE: the price of one currency, expressed in another — and like any price, it forms on a market, it moves, and it has a UNIT. Those two ideas, price and unit, are the whole course: the first explains why the number changes constantly, the second avoids the commonest error on the subject. The 1.14 used here is that of late July 2026; it will have moved by the time you read these lines, and that is exactly the point.
Assumptions
In your view, which assumptions are needed for this model to hold? Jot down your ideas — no lead is wrong, this is your worksheet.
Your worksheet is still empty. Go for it: propose at least one idea.
0 idea(s) proposedFormalization
Back to the board: €1 = $1.14. Put differently, one euro costs 1.14 dollars — so the rate is 1.14 $/€, which reads “1.14 dollars per euro”, exactly as you read “€3 per kilo” on a market stall. That unit is no notational flourish: it carries all the information. It says which currency is BOUGHT (the euro, in the denominator) and in which currency you pay (the dollar, in the numerator).
Let's convert €200 into dollars. You multiply by the price of one euro: €200 × 1.14 $/€ = $228. Look at what happens in the units: the € of the amount cancels with the € in the denominator, and only $ remain. The result is indeed in dollars, so the operation was the right one. And if we had divided? We would have obtained €200 ÷ 1.14 $/€ = 175.4 €²/$ — “square euros per dollar”, a unit that does not exist. The error denounces itself. That is why there is no point learning by heart “euros to dollars, you multiply”: it is enough to write the units and check that they cancel.
How much is $1 worth in euros? You invert: 1 ÷ 1.14 = 0.877, and the unit inverts too — 0.877 €/$. Converting an American price into euros therefore means multiplying by 0.877 €/$… or dividing by 1.14 $/€, which is rigorously the same operation. A computational warning here: the inverse rate rarely comes out neatly. 1/1.14 = 0.877193…; if you round it to 0.88, then 0.88 × 1.14 = 1.0032, and a round trip euros → dollars → euros manufactures 0.32% of money that does not exist. On a large amount, rounding is no longer a detail.
This is the trap almost nobody sees. If the rate goes from 1.14 to 1.25 $/€, the euro gains 0.11/1.14 = 9.65%. How much does the dollar lose? Not 9.65%: it goes from €0.877 to €0.800, that is −8.80% exactly. The reason fits in one line: the two changes are linked by (1 + x) × (1 + y) = 1, since one rate is the inverse of the other. And it works both ways: if the rate FALLS, it is the euro that loses and the dollar that gains — always by two different percentages (the solving step puts figures on it, exercise 5 has you compute it). Vocabulary, finally: when a currency rises or falls like this on a market, it is said to APPRECIATE or DEPRECIATE. The words “devaluation” and “revaluation” mean something quite different and are reserved for fixed exchange rate regimes (interpretation, point 2). Click each term:
Solving / calculation
Two conversions in both directions, on the real rate of late July 2026 (1.14 $/€) — then the question everyone cares about: what happens when the euro falls?
- Direction 1 — a purchase in euros, seen in dollars
€200 × 1.14 $/€$228 (the € cancel: the result really is in dollars) - The rate turned over
1 ÷ 1.140.877 €/$ — one dollar is worth 88 euro cents - Direction 2 — a purchase in dollars, seen in euros: take a barrel of oil at $80
$80 ÷ 1.14 $/€ (or 80 × 0.877 €/$)€70.18 — the division gives 70.18, multiplying by the rounded inverse rate gives 70.16: 2 cents apart, that is the rounding of beat 3 - The euro depreciates: the rate falls to 1.05 $/€
$80 ÷ 1.05 $/€€76.19 for the SAME $80 barrel - By how much has the bill risen?
(76.19 − 70.18) / 70.18+8.57%, without the price of the barrel moving by a cent - And the euro, by how much has it fallen? (the asymmetry of beat 4)
(1.05 − 1.14) / 1.14−7.89% — and not −8.57%: check (1 − 0.0789) × (1 + 0.0857) = 1 to within rounding, and exactly 1 on the unrounded values (−7.8947% and +8.5714%)
Keep both directions, and above all the second: when the euro depreciates by 7.89%, everything France buys in dollars — oil, gas, raw materials, microprocessors — costs 8.57% more in euros, without any seller having raised a price. This is the mechanism the course “The causes of inflation” calls the imported shock, and it is also the left-hand half of the J-curve of the course “The trade balance”: the import bill grows heavier at once, while exports take months to benefit from a weaker currency. And the eventual recovery is not even guaranteed: it assumes the quantities sold end up reacting strongly enough to prices. Economists made a precise condition of it, the MARSHALL-LERNER condition, which the course “The trade balance” sets out — when it is not met, a weaker currency worsens the balance durably, and not only for the length of the dip. So an exchange rate is never good or bad in itself: it makes winners and losers at the same time.
price in $ = price in € × rate; inverse rate = 1 / rate; round trip = amount × (1 − margin)²Three things to try. Bring the rate below 1.14 and watch the import bill rise. Compare the two percentage changes: they are never equal. And raise the bank's margin to see what a round trip euros → dollars → euros really costs — a round trip that, on paper, should cost nothing at all.
Economic interpretation
Three things the computation does not say: where this price comes from, how its movements are named, and why its LEVEL means nothing.
From a market, the foreign exchange market, open continuously. Who buys euros there? An American who wants to buy a German car, an investor who wants to place money in the euro area, a company repatriating profits. Who sells them? The same people, the other way round. The rate is the price that balances those supplies and demands (course “The law of supply and demand”). Hence an essential lever: when a central bank raises its interest rates, placing money in its currency pays more, capital flows in, demand for that currency rises — and it appreciates (course “The policy rate and transmission”). Central banks can also intervene directly, buying or selling their own currency with their foreign exchange reserves.
It all depends on the exchange rate REGIME. In a FLOATING regime — the case of the euro against the dollar — the rate is free: a currency is said to APPRECIATE or DEPRECIATE, nobody decided it. In a FIXED regime, an authority announces a parity and undertakes to defend it: the CFA franc is pegged to the euro, the Danish krone too, the Hong Kong dollar to the American dollar. Only there do the words DEVALUATION and REVALUATION mean anything: they denote the official decision to change the parity. Saying “the euro has been devalued” after a fall on the market is therefore a frequent misuse — the euro floats, it has no official parity to devalue.
You often read that a currency is “strong” because its rate is a large number. That is wrong: $1 is worth about 150 yen, and that says nothing whatever about the yen's soundness — the figure depends only on the unit chosen when the currency was created. What does mean something is the MOVEMENT (appreciating, depreciating) and the comparison of prices between countries, which the course “The real exchange rate and PPP” deals with. There remains the question of who gains: a depreciating euro lightens the load for French exporters (their products become cheaper seen from the United States) and adds to the bill for importers and outbound tourists. An appreciating euro does the reverse. Nobody can want both at once.
Limits / critiques
Exercises
The rate is 1.20 $/€. How many dollars do you get with €150 (in $)?
The rate is 1.25 $/€. How much is $1 worth, in euros (two decimals)?
The rate goes from 1.25 $/€ to 1.00 $/€. By how much does the DOLLAR appreciate against the euro (in %)?
The euro depreciates against the dollar. Who benefits most?
The euro/sterling rate is 0.85 £/€. How many pounds do you get with €200 (in £)?
A barrel of oil costs $90. The rate is 1.20 $/€. How much does it cost in euros (in €)?
The euro goes from 1.14 $/€ to 1.05 $/€ on the foreign exchange market. How do you say it?
True or false: since $1 is worth about 150 yen, the yen is a weak currency.