A quote of 1.0850 dollars per euro is a direct quote in New York and an indirect quote in Frankfurt. It is the same number and the same market, and the only thing that changed is where the person reading it is standing.
That is the whole of the definition, and it takes about a minute to learn. The marks are lost afterwards, on the consequences: that the two quotes are reciprocals but the percentage changes are not, that the curriculum notation and the trading screen convention are written in opposite orders, and that inverting a two-way price means crossing the bid and the ask rather than inverting each one where it stands.
Key Takeaways
- A direct quote is units of domestic currency per one unit of foreign currency, so direct and indirect depend entirely on where the observer is standing.
- The indirect quote is the reciprocal of the direct quote: 1 / 1.0850 = 0.9217 euros per dollar.
- In a quote written price over base, USD/EUR = 1.0850 means 1.0850 dollars buy one euro; a dealing screen writes the identical rate as EUR/USD.
- Percentage changes are not symmetric: a move from 1.0850 to 1.1500 dollars per euro is a 5.99% euro appreciation and a 5.65% dollar depreciation.
- If one currency appreciates by x, the other depreciates by x / (1 + x), so the depreciation figure is always the smaller of the two.
- To measure a currency, put it in the base position first; inverting a percentage change after computing it does not work.
- Inverting a two-way price crosses the sides: the bid of the inverse is one divided by the ask of the original.
What a direct quote is
A direct quote is the price of one unit of the foreign currency expressed in units of the domestic currency. It answers the question a traveller asks at an airport counter: how much of my money does one unit of theirs cost?
Direct quote = units of domestic currency per one unit of foreign currency
domestic and foreign are defined by where the person asking is standing, not by any property of the currencies themselves
For an investor in the United States, a quote of 1.0850 dollars per euro is direct. For an investor in the euro area, that same number is an indirect quote, because it prices the domestic currency in foreign terms. Neither investor is looking at a different market. They are looking at the same rate from opposite sides.
That is the first thing to fix, and it is the thing most often left vague. Direct and indirect are not properties of a quote. They are properties of a quote plus a location. A question that does not say where the observer is standing has not asked a complete question, which is why examinable questions always say it, usually in the first clause and usually in a way that is easy to read past.
| Observer | Direct quote | Indirect quote |
|---|---|---|
| Based in the United States | 1.0850 dollars per euro | 0.9217 euros per dollar |
| Based in the euro area | 0.9217 euros per dollar | 1.0850 dollars per euro |
The indirect quote and the reciprocal
The indirect quote is the direct quote turned upside down, and the arithmetic is a single division.
Indirect quote = 1 / direct quote
so 1 / 1.0850 = 0.9217 euros per dollar, and 1 / 0.9217 = 1.0850 dollars per euro
Notice that the reciprocal relationship holds exactly, while the corresponding relationship between percentage changes does not hold at all. That single fact generates most of the marks available on this topic, and it is covered in section four.
Base and price currency, and the notation problem
Direct and indirect depend on where the observer stands, which makes them unusable in a market where the two parties are in different countries. The professional convention removes the observer entirely and names the two currencies by their role.
In any quote, one currency is the base currency, the thing being priced, and the other is the price currency, the units the price is expressed in. A quote is the number of units of the price currency required to buy one unit of the base currency. The exchange rate is a price like any other, and the base currency is what sits on the shelf.
| Written as | Read as | Base currency | Price currency |
|---|---|---|---|
| USD/EUR = 1.0850 | 1.0850 dollars buy one euro | EUR | USD |
| EUR/USD = 0.9217 | 0.9217 euros buy one dollar | USD | EUR |
| INR/USD = 88.40 | 88.40 rupees buy one dollar | USD | INR |
Assuming the curriculum notation and the trading floor convention agree. They do not. In the price over base convention used in the readings, dollars per euro is written USD/EUR. On a dealing screen the same rate is conventionally labelled EUR/USD, base first. The number 1.0850 means dollars per euro in both cases, but the label is reversed. In an exam, follow the price over base convention. In practice, sanity check the magnitude: a euro has been worth roughly one dollar and a rupee has not, so a quote of 88.40 on a dollar and rupee pair can only be rupees per dollar whatever the label says.
Appreciation and depreciation are not symmetric
If the euro appreciates against the dollar by 6%, the dollar has not depreciated against the euro by 6%. The two percentages describe the same movement measured against different bases, and they differ by more than rounding.
The rate moves from 1.0850 dollars per euro to 1.1500 dollars per euro. By how much has the euro appreciated against the dollar, and by how much has the dollar depreciated against the euro?
- Measuring the euro: the euro is the base currency in this quote, so the change is read directly. 1.1500 / 1.0850 − 1 = 0.0599, an appreciation of 5.99%.
- Measuring the dollar requires the dollar to be the base, so invert both quotes. 1 / 1.0850 = 0.9217 euros per dollar before, and 1 / 1.1500 = 0.8696 euros per dollar after.
- 0.8696 / 0.9217 − 1 = −0.0565, a depreciation of 5.65%.
Answer: the euro appreciated 5.99% and the dollar depreciated 5.65%. Both figures are correct and they describe one movement. The euro percentage is larger because it is measured against the smaller starting value, and the gap widens as the move gets bigger. On a 50% appreciation of the euro, the dollar depreciation is 33.3%, not 50%.
The procedure that avoids the error every time is mechanical. Put the currency whose movement is being measured in the base position first, then compute the percentage change on the quote. Measuring the euro means working with dollars per euro. Measuring the dollar means working with euros per dollar. Inverting after computing the percentage does not work, because the reciprocal of a ratio is not the reciprocal of a percentage change.
If the currency being measured appreciates by x, the other currency depreciates by x / (1 + x). A 5.99% appreciation gives 0.0599 / 1.0599 = 5.65%, which is exactly the figure computed the long way. The relationship also explains the direction of the error: the depreciation percentage is always the smaller of the two, so an answer where both are equal is wrong on inspection.
Cross rates
Most currency pairs are not quoted directly against each other. They are quoted against the dollar, and the rate between them is constructed. The rule is that the units have to cancel, and if the arithmetic is set up so that they do, the multiplication or division takes care of itself.
A dealer shows 1.0850 dollars per euro and 1.2600 dollars per pound. What is the euro per pound rate?
- The answer is required in euros per pound, so pounds must be in the denominator and euros in the numerator.
- Dividing dollars per pound by dollars per euro cancels the dollars: (USD per GBP) / (USD per EUR).
- 1.2600 / 1.0850 = 1.1613 euros per pound.
Answer: 1.1613 euros per pound. The check is to read the result back as a sentence. One pound buys 1.1613 euros, which is consistent with a pound being worth more than a euro, since a pound buys 1.2600 dollars and a euro buys only 1.0850. If the arithmetic had been inverted the result would have been 0.8611, and reading that back as one pound buying 0.86 euros fails the same check immediately.
Reading the answer back as a sentence is worth more than any memorised rule about when to multiply and when to divide. A cross rate that has been inverted always produces a statement that is obviously false about which currency is worth more, and that takes two seconds to notice.
Bid, ask, and which side you get
Real quotes come in pairs. The bid is the price at which the dealer will buy the base currency, the ask or offer is the price at which the dealer will sell it, and the ask is the higher of the two. The client always transacts on the side that is worse for the client, which is the definition of a spread rather than a comment about dealers.
| Quote | Bid | Ask |
|---|---|---|
| Dollars per euro | 1.0846 | 1.0854 |
| Euros per dollar | 0.92132 | 0.92200 |
The inverse quote is not obtained by inverting bid to bid. It is obtained by crossing them: the bid of the inverted quote is one divided by the ask of the original, and the ask of the inverted quote is one divided by the bid.
bidinverse = 1 / askoriginal and askinverse = 1 / bidoriginal
1 / 1.0854 = 0.92132 and 1 / 1.0846 = 0.92200, which preserves the ordering that the ask must exceed the bid
The reason is worth one line, because it makes the rule impossible to forget. Buying euros and selling dollars are the same transaction described twice. The side of the market does not change when the description of the trade is reversed, so the price that was an ask in one direction is a bid in the other.
Three question shapes recur. Convert between a direct and an indirect quote, where the only trap is identifying the observer. Compute an appreciation and the matching depreciation, where the distractor is the same percentage with the sign reversed. Or build a cross rate, where the distractor is the reciprocal of the correct answer. In all three, the reliable defence is to write the units next to every number and check that they cancel.
The short version
- Direct means domestic per foreign, indirect means foreign per domestic, and which is which follows the observer rather than the currencies.
- The two quotes are exact reciprocals. The percentage changes derived from them are not.
- A 5.99% appreciation of one currency is a 5.65% depreciation of the other, because the two are measured against different starting values.
- Measure a currency by putting it in the base position, then take the percentage change on that quote.
- Cross rates are built so that the common currency cancels, and the answer is checked by reading it back as a sentence about which currency is worth more.
- Inverting a bid and ask pair crosses the sides, because buying one currency and selling the other are the same transaction described twice.
Frequently Asked Questions
What is a direct quote in foreign exchange?
A direct quote is the price of one unit of the foreign currency expressed in units of the domestic currency. For an investor in the United States, 1.0850 dollars per euro is a direct quote. For an investor in the euro area, the same number is an indirect quote. Direct and indirect are therefore properties of a quote plus a location, not of the quote alone.
How do you convert a direct quote into an indirect quote?
Take the reciprocal. If the direct quote is 1.0850 dollars per euro, the indirect quote is 1 divided by 1.0850, which is 0.9217 euros per dollar. The relationship holds exactly in both directions, which is why it is the one part of this topic that never causes trouble.
Why is a 6% appreciation not a 6% depreciation the other way?
Because the two percentages are measured against different starting values. A move from 1.0850 to 1.1500 dollars per euro is an appreciation of 1.1500 / 1.0850 minus 1, which is 5.99%. Inverting both quotes gives 0.9217 and 0.8696 euros per dollar, a change of minus 5.65%. If one currency appreciates by x, the other depreciates by x divided by (1 + x), so the depreciation is always the smaller number.
Does USD/EUR mean dollars per euro or euros per dollar?
In the price over base convention used in the readings, USD/EUR means 1.0850 dollars buy one euro, with EUR as the base currency. On a dealing screen the same rate is conventionally labelled EUR/USD, base first, so the label is reversed while the number means the same thing. In an exam, follow the price over base convention, and in practice check the magnitude, since a quote of 88.40 on a dollar and rupee pair can only be rupees per dollar.
How do you calculate a cross rate?
Set the calculation up so that the common currency cancels. Given 1.0850 dollars per euro and 1.2600 dollars per pound, the euro per pound rate is 1.2600 divided by 1.0850, which is 1.1613 euros per pound. Then read the answer back as a sentence: one pound buys 1.1613 euros, which is consistent with a pound being worth more than a euro. An inverted cross rate always fails that check immediately.
How do you invert a bid and ask quote?
Cross the two sides. The bid of the inverted quote is one divided by the ask of the original, and the ask of the inverted quote is one divided by the bid. From 1.0846 bid and 1.0854 ask in dollars per euro, the euros per dollar quote is 0.92132 bid and 0.92200 ask. The reason is that buying euros and selling dollars are the same transaction described twice, so the side of the market does not change when the description is reversed.
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