Compute and explain what happens when one currency appreciates or depreciates against another.
The idea
A percentage change in an exchange rate depends on which currency is the base. Take a rate quoted as domestic currency per unit of foreign currency, moving from S0 at the start to S1 at the end. The foreign currency is the base, so its percentage change is: Foreign change = (S1 − S0) ÷ S0. The domestic currency's change is the reciprocal calculation: Domestic change = S0 ÷ S1 − 1. The two are not equal in size: if the euro rises 10% against the dollar, the dollar falls only 9.09% against the euro. A higher rate means the base currency has appreciated and the quote currency has depreciated against it. Cross changes compound: if A appreciates 3% against C and B depreciates 2% against C, then A appreciates by 1.03 ÷ 0.98 − 1, which is 5.10%, against B. For returns on foreign assets, Domestic return = (1 + local return) × (1 + foreign change) − 1. The usual mistakes are dividing by the wrong base, adding percentages and getting the direction backwards.
Formulas
Percentage change in the base (foreign) currency
foreign change = (S1 − S0) ÷ S0
- S0
- the starting rate, domestic currency per unit of foreign currency
- S1
- the ending rate, quoted the same way
The foreign currency is the base of the quote, so divide by the starting rate. A positive result means the foreign currency appreciated.
Percentage change in the quote (domestic) currency
domestic change = S0 ÷ S1 − 1
- S0
- the starting rate, domestic currency per unit of foreign currency
- S1
- the ending rate, quoted the same way
The reciprocal of the base currency's move, so the two are not equal in size: a 10% rise in the euro is only a 9.09% fall in the dollar.
Home-currency return on a foreign asset
domestic return = (1 + local return) × (1 + foreign change) − 1
- local return
- the asset's return in its own currency
- foreign change
- the percentage change in the foreign currency against the home currency
Returns compound, they do not add: 4% local with a 3% currency fall gives 0.88%. The same compounding applies to cross-currency changes.
A practice question
GBP/USD (USD per GBP) rises from 1.2500 to 1.3000. The percentage change in the GBP against the USD is closest to:
- +3.85%, an appreciation of the GBPWhy not: Divides the change by the new rate; the base for the GBP's percentage change is the old rate, 1.2500.
- -4.00%, a depreciation of the GBPWhy not: Wrong direction: a rise in USD per GBP means the GBP buys more USD, so it appreciates.
- +5.00%, an appreciation of the GBPWhy not: Treats the 0.05 move as 5 percent without dividing by the starting rate.
- +4.00%, an appreciation of the GBP - correct
The GBP is the base currency, so its change is (1.3000 - 1.2500)/1.2500 = +4.00%. It buys more USD, so it has appreciated.
A harder one
EUR/USD (USD per EUR) falls from 1.0800 to 1.0260. By what percentage has the USD appreciated against the EUR, closest to?
- 5.00%Why not: Reports the EUR's depreciation (-5.00%) as the USD's appreciation; the two percentages differ.
- 5.26% - correct
- -5.26%Why not: Right size, wrong sign: the USD appreciates, it does not depreciate.
- 5.40%Why not: Treats the 0.0540 fall in the rate as a percentage.
The USD's value in EUR is the reciprocal: 1/1.0260 against 1/1.0800, a change of 1.0800/1.0260 - 1 = +5.26%. The EUR fell 5.00% (1.0260/1.0800 - 1); the USD's appreciation is larger.
The hardest tier
A US investor holds a EUR-denominated bond that earns a 4% return in EUR. Over the year, the EUR depreciates 3% against the USD. What is the investor's return in USD, closest to?
- 0.88% - correct
- 1.00%Why not: Subtracts the currency loss from the local return, ignoring that the 3% loss applies to the grown value (the compounded product).
- 7.12%Why not: Applies a 3% gain instead of a loss; the EUR depreciated, which lowers the USD value.
- -3.00%Why not: Ignores the 4% EUR return and reports only the currency loss.
USD return = (1 + local return) x (1 + currency change) - 1 = 1.04 x 0.97 - 1 = 0.88%. The currency loss offsets most of the bond's local return.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
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