CFA Program Level I

Exchange rate quotes, cross rates and % changes: CFA Level I

An exchange rate gives the price of one unit of the base currency in units of the quote currency. A cross rate between two currencies can be built from each one's rate against the US dollar so the dollar cancels. When a rate moves, the base currency's percentage change and the quote currency's are not equal and opposite, because each is measured from a different starting value.

Base and quote

The St. Louis Fed explains that an exchange rate is the value of one currency in terms of another. The base currency is the one being priced and the quote currency prices it. A rate of 0.88 Swiss francs per dollar means 1 USD = 0.88 CHF. The inverse, 1 / 0.88 = 1.14 dollars per franc, is an equally valid rate.

If the rate falls from 0.88 to 0.75 francs per dollar, the dollar buys fewer francs, so the dollar has weakened and the franc has strengthened. Each franc now costs 1 / 0.75 = 1.33 dollars, up from 1.14.

Conventions differ by currency. The Federal Reserve's H.10 release quotes most currencies in units per US dollar, such as yen or Canadian dollars per dollar, but quotes the euro, pound, Australian dollar and New Zealand dollar in US dollars per unit. Check which currency is the base before you calculate.

Cross rates

To get a rate between two non-dollar currencies, arrange the two dollar quotes so the dollar cancels.

Example 1: USD 1.10 per euro and JPY 150 per dollar. JPY per EUR = (USD per EUR) x (JPY per USD) = 1.10 x 150 = 165 yen per euro.

Example 2: USD 1.08 per euro and USD 0.73 per Canadian dollar. Both are in dollars per unit, so divide: CAD per EUR = 1.08 / 0.73 = 1.4795. Check the units: (USD/EUR) / (USD/CAD) = CAD/EUR.

Percentage changes

The base currency's change is the change in the rate: new rate / old rate - 1. The quote currency's change uses the inverse rates: old rate / new rate - 1.

Example: francs per dollar move from 0.90 to 0.80. The dollar's change is 0.80 / 0.90 - 1 = -11.1%. The franc's change is 0.90 / 0.80 - 1 = +12.5%. Check with inverses: 1 / 0.90 = 1.1111 and 1 / 0.80 = 1.25 dollars per franc, and 1.25 / 1.1111 - 1 = 12.5%.

Likewise, if the euro falls 4.00% against the yen, the yen rises 1 / 0.96 - 1 = 4.17% against the euro, not 4.00%.

Where the exam catches people

Believing a rising rate always means the domestic currency strengthened: a rise means the base currency strengthened, whichever currency that is.
Believing one currency's % depreciation equals the other's % appreciation: a 4% fall in one is a 4.17% rise in the other.
Believing a cross rate is found by multiplying any two quotes: arrange them so the common currency cancels, which sometimes means dividing.
Believing every currency is quoted per US dollar: the euro, pound, Australian dollar and New Zealand dollar are usually quoted in dollars per unit.

Written from CFA Institute's documentation

Try 3 questions on this

Pick an answer to see why each option is right or wrong.

An analyst observes that the euro trades at USD1.0800 per euro and the Canadian dollar trades at USD0.7300 per Canadian dollar. The cross rate, expressed as the number of Canadian dollars per euro, is closest to:
Right. In the game that’s Renown for your Economics building.
Not that one: C is right. In the game your Economics building would lose a little Renown, and this question would come back later.
Your answerA.0.67590.7300 / 1.0800 = 0.6759 is the number of euros per Canadian dollar. The question asks for Canadian dollars per euro, so the euro's dollar rate goes on top: 1.0800 / 0.7300 = 1.4795.Your answerB.0.7884Multiplying gives 1.0800 × 0.7300 = 0.7884, in units of US dollars squared per euro-Canadian dollar pair, which is no exchange rate. The cross rate comes from dividing one US dollar rate by the other.The answerC.1.4795Both rates are quoted in US dollars per unit of the other currency, so the cross rate is one divided by the other, letting the dollar legs cancel: (USD1.0800 per EUR) / (USD0.7300 per CAD) = 1.0800 / 0.7300 = 1.4795 Canadian dollars per euro.
Show the full explanation and doc quotes
This mirrors how CME Group divides the EUR/USD fixing by the GBP/USD fixing to obtain EUR/GBP.
Written from www.cmegroup.com/trading/fx/currfixprice.html
The exchange rate between the US dollar and the Swiss franc is quoted as the number of Swiss francs per US dollar. Over one year, the rate moves from 0.9000 to 0.8000. The percentage change in the value of the US dollar relative to the Swiss franc is closest to:
Right. In the game that’s Renown for your Economics building.
Not that one: A is right. In the game your Economics building would lose a little Renown, and this question would come back later.
The answerA.−11.1%The dollar is the base currency and the franc the quote currency, so the rate is the number of francs one dollar buys. One dollar bought 0.9000 francs and now buys 0.8000, so the dollar's value in francs changed by (0.8000 − 0.9000) / 0.9000 = −11.1%.Your answerB.11.1%The magnitude, (0.9000 − 0.8000) / 0.9000 = 11.1%, is right, but the dollar now buys fewer francs. When a base/quote rate falls, the base currency has weakened, so the change is negative, not positive.Your answerC.12.5%This is the franc's change, not the dollar's. Inverting gives 1 / 0.9000 = USD1.1111 and 1 / 0.8000 = USD1.2500 per franc, and 1.2500 / 1.1111 − 1 = +12.5% is how much the franc gained against the dollar.
Show the full explanation and doc quotes
A falling rate means the base currency (the dollar) has weakened against the quote currency (the franc).
Written from www.stlouisfed.org/open-vault/2023/november/worth-of-your-money-abroad
Over one month, the euro depreciates by 4.00% against the Japanese yen. The percentage change in the value of the yen relative to the euro over the month is closest to:
Right. In the game that’s Renown for your Economics building.
Not that one: C is right. In the game your Economics building would lose a little Renown, and this question would come back later.
Your answerA.−4.00%−4.00% is the euro's change: 96/100 − 1 = −4.00%. A fall in the yen-per-euro rate means the yen strengthened, so the yen's change is positive: 1/0.96 − 1 = +4.17%.Your answerB.4.00%The yen's change is measured from the yen's own starting value. Inverting the rates, the euro price of one yen rises from 1/100 to 1/96, an increase of 4.17%, which is larger than the euro's 4.00% fall with its sign flipped.The answerC.4.17%Quote the rate as yen per euro and let it start at 100. A 4.00% fall in the euro takes it to 96. The yen's value in euros is the inverse rate: 1/100 = EUR0.010000 before and 1/96 = EUR0.010417 after. The yen's change = (1/96)/(1/100) − 1 = 1/0.96 − 1 = +4.17%.
Written from www.stlouisfed.org/open-vault/2023/november/worth-of-your-money-abroad