CFA Exchange Rate Calculations: Base vs Price Currency and Cross Rates

Currency questions are among the most reliably mishandled calculations in the CFA curriculum, and almost never because the maths is hard. The arithmetic is multiplication and division. The difficulty is entirely in notation — knowing which currency you are pricing and which you are pricing it in.

The notation

An exchange rate quoted as A/B means:

• A is the price currency (also called the quote currency)
• B is the base currency
• The number tells you how many units of A it takes to buy one unit of B

So USD/EUR = 1.10 means one euro costs 1.10 US dollars. The euro is the base; the dollar is the price.

Read it as "price per base", the same way you would read a share price: the base currency is the thing being bought, the price currency is what you pay in.

The direction of movement follows from this. If USD/EUR rises from 1.10 to 1.15, one euro now costs more dollars, so the euro has appreciated and the dollar has depreciated. A rising quote always means the base currency is strengthening.

Candidates who lose these marks almost always do so by reading the first currency as the one being priced rather than the one doing the pricing.

Direct and indirect quotes

These terms are relative to a stated home country, which the question will specify.

• Direct quote: domestic currency is the price currency. For a US investor, USD/EUR is direct.
• Indirect quote: domestic currency is the base currency. For a US investor, EUR/USD is indirect.

They are simply reciprocals of each other. If a question specifies a home country, it is signalling that this distinction is being tested.

Percentage changes

The trap here is that the appreciation of one currency is not the mirror-image depreciation of the other.

Example. USD/EUR moves from 1.25 to 1.30.

Euro appreciation: (1.30 − 1.25) / 1.25 = +4.00%

For the dollar, invert first: 1/1.25 = 0.8000 and 1/1.30 = 0.7692.

Dollar depreciation: (0.7692 − 0.8000) / 0.8000 = −3.85%

Not −4.00%. Percentage changes are not symmetric because the denominators differ. Any question that gives you a percentage change in one currency and asks for the other is testing exactly this, and −4.00% will be sitting in the answer choices.

Cross rates

A cross rate is derived from two other rates sharing a common currency. The method is to arrange the quotes so the common currency cancels.

Example. Given JPY/USD = 150 and USD/EUR = 1.10, find JPY/EUR.

JPY/USD × USD/EUR = JPY/EUR
150 × 1.10 = 165

The USD cancels because it appears once as a price currency and once as a base currency. If instead you are given EUR/USD, invert it first so the shared currency appears on opposite sides.

Treat the notation as fractions and cancel them algebraically. That single habit removes almost all cross-rate errors.

Bid-ask spreads

Dealers quote two prices, and the convention is stated from the dealer's perspective:

• Bid: the price at which the dealer buys the base currency from you
• Ask (offer): the price at which the dealer sells the base currency to you

The ask always exceeds the bid, and you always transact on the worse side. Selling the base currency, you receive the bid. Buying it, you pay the ask.

Inverting a two-sided quote

This is the highest-value technical point in the topic, and it appears regularly.

To invert a bid-ask quote, take reciprocals and swap the sides:

New bid = 1 / old ask  |  New ask = 1 / old bid

Example. USD/EUR is quoted 1.1000 – 1.1020. What is EUR/USD?

New bid = 1 / 1.1020 = 0.9074
New ask = 1 / 1.1000 = 0.9091
So EUR/USD = 0.9074 – 0.9091

The spread survives the inversion — it must, or the reciprocal quote would let you trade back and forth at a profit.

Cross rates with spreads

When computing a cross rate from two two-sided quotes, use the combination that is worst for you. In practice: multiply the two bids to get the cross bid, and multiply the two asks to get the cross ask, having first arranged the quotes so the common currency cancels. The resulting cross spread is wider than either input spread, which is why crossing through a third currency costs more than trading a direct pair.

Forward rates and forward points

Forward rates are usually quoted as points added to or subtracted from the spot rate, scaled by the decimal convention of the pair. For a pair quoted to four decimal places, points are divided by 10,000.

Example. Spot USD/EUR = 1.1000, three-month forward points = +45. Forward rate = 1.1000 + 45/10,000 = 1.1045.

Positive points mean the base currency trades at a forward premium; negative points mean a forward discount.

What determines the points

Covered interest rate parity, and it is an arbitrage relationship rather than a forecast:

F / S = (1 + i_price) / (1 + i_base)

Adjusting the rates for the period in question. The consequence is a rule worth memorising outright:

The currency with the higher interest rate trades at a forward discount.

The intuition: if you could earn a higher rate in one currency and lock in an unchanged exchange rate forward, you would have riskless profit. The forward price must move to eliminate that, which means the high-yield currency must be worth less forward than spot.

Note that this is an arbitrage-enforced relationship, not a prediction of where spot will go. Questions often present a forward rate and ask what it implies about future spot; the answer is that under covered interest parity it implies nothing beyond the interest differential.

Common errors

• Reading the base currency backwards. The base is the second currency in A/B notation. Everything else depends on this.
• Assuming symmetric percentage changes. A 4% appreciation is never exactly a 4% depreciation the other way.
• Forgetting to swap sides when inverting a bid-ask quote.
• Using the wrong side of the spread. You always transact at the price that favours the dealer.
• Getting the forward premium the wrong way round. Higher interest rate means forward discount, which feels counterintuitive and is therefore tested.
• Misapplying the points scaling. Check the decimal convention of the pair before dividing.

Where this reappears

The notation established here carries through the whole programme. It underpins the exchange rate material in Level 1 Economics, the currency management and hedging decisions at Level 3, and any multinational valuation question where cash flows arrive in more than one currency.

Candidates who leave Level 1 still hesitating over which currency is the base pay for it repeatedly. Ten minutes spent fixing the convention is one of the better returns available in the curriculum.