The price of one currency in another. Read quote units, translate cash amounts and measure appreciation without confusing reciprocal percentage changes.
Foreign exchange rate
A foreign exchange rate is the price of one currency expressed in another currency. It converts a cash amount in one denomination into its value in the other. For a dollar-funded investor holding euros, a rise in the number of dollars per euro raises the dollar value of that holding. The same move raises the dollar cost of a fixed euro payment.
An exchange rate needs a direction. “The exchange rate rose” is incomplete until the currencies and their roles are stated.
Write the units before calculating
Under the usual market currency-pair notation, EUR/USD =1.20 means 1.20 USD per EUR. EUR is the base currency; USD is the quote currency. The market pair label and a dimensional fraction read in different orders: EUR/USD names the pair, while the conversion factor has units USD/EUR.
For clarity, define St explicitly as USD per EUR. Then
Names such as “direct” and “indirect” quotation depend on which currency the user regards as domestic. Explicit units keep the calculation meaningful across desks, data feeds and reporting currencies. CME also cautions that futures quotation conventions can differ from those used in spot markets.
Appreciation is relative—and percentages are asymmetric
Consider fictional, comparable rates at two times, ignoring interest, spreads and fees:
Quotation
Before
After
Percentage change
USD per EUR
1.20
1.26
+5%
EUR per USD
5/6
50/63
−1/21≈−4.7619%
The euro appreciates against the dollar because each euro buys more dollars. The dollar depreciates against the euro because each dollar buys fewer euros. This says nothing by itself about either currency against a third currency.
For the first row,
r=1.201.26−1=0.05.
Reversing the quotation means taking reciprocals of the rates, then calculating a percentage change using the new starting denominator:
rinverse=1/S01/S1−1=1+r1−1=−1+rr,r>−1.
Thus a 5% euro appreciation corresponds to an approximately 4.7619% dollar depreciation. Equal and opposite simple percentage changes are only a small-move approximation, not an exact rule. The two rows describe the same relative-price change.
Fictional USD per EUR rises from 1.20 to 1.26, a 5% appreciation of EUR. The inverse EUR per USD falls from 0.833333 to 0.793651, approximately 4.7619%. EUR 10,000 translates from USD 12,000 to USD 12,600.
Original fictional comparison. The same currency move has different percentage magnitudes when the quote is reversed. Reciprocals and the inverse percentage are rounded; valuations exclude interest, spread and fees.
A fixed EUR 10,000 holding translates from USD 12,000 to USD 12,600: a USD 600 increase. A fixed EUR 10,000 liability becomes USD 600 more expensive to fund. Direction of exposure matters as much as direction of quotation. This is currency translation alone; an investment's total return may also include interest, asset-price movements and trading costs.
Which rate is usable for the task?
A spot FX quotation relates to near settlement; a forward quotation relates to an agreed future exchange. State the settlement basis when comparing them. A forward quotation is not interchangeable with a current spot rate or automatically a forecast of the eventual spot rate.
A dealer's bid and ask identify different executable sides: the dealer buys the base currency at the bid and sells it at the ask. A midpoint or published reference rate serves a different purpose. The ECB explicitly publishes its euro reference rates for information and discourages using them for transactions. An actual conversion also depends on quote availability, size and charges.
For practical quote-side and cross-rate calculations, the SelfMadeQuant FX lesson has a public preview; the full lesson is available with a signed-in free account.
John C. Hull, Options, Futures, and Other Derivatives, 9th Global Edition, Pearson, 2018, §1.3, pp. 28–29: explicit currency units, bid/offer quotations and spot versus forward exchange.
