Currency Cross-Rates

Updated for the 2026-2027 CFA® Level I curriculum.

A cross-rate is an exchange rate between two currencies that is not quoted directly. You build it from two quotes that share a common third currency. Level I tests this as a calculation skill, and most errors come from inverting the wrong quote or losing track of units. This note teaches the setup that avoids both mistakes.

Key Takeaways

• A cross-rate is calculated, not directly quoted. It comes from two rates that share a common currency.

• Every quote needs units: price currency per one unit of base currency.

• The setup must cancel the common currency, either by multiplying or dividing, depending on where that currency sits in each quote.

• If the common currency sits on the same side of both quotes, divide. If it sits on opposite sides, multiply (or invert one quote first).

• The reciprocal of your answer should make sense. If you solved for A/B, then 1 divided by your answer should equal B/A.

• Interpret the number in plain terms: how many units of the price currency buy one unit of the base currency.

• Bid-ask cross-rates follow the same cancellation logic, but you must pair the correct bid or ask side for each leg of the calculation.

What You Need to Know for CFA Level I

• Rewrite each given quote in words before doing any math (for example, "1.10 USD per 1 EUR").

• Recognize when a quote needs to be inverted before it can be used in the calculation.

• Build a chain of quotes in which the common currency appears once in a numerator and once in a denominator, so it cancels.

• State your final answer with explicit units and identify which currency is the base and which is the price currency.

• Extend to bid-ask cross-rates only when a question specifies bid and ask sides. The same cancellation logic applies, but you calculate the bid and ask separately.

What a Currency Cross-Rate Is

Picture three currencies as the corners of a triangle: USD, EUR, and GBP. Two sides of the triangle are quoted directly in the market. and are both observable spot rates. The third side, EUR/GBP, is not quoted directly. It is implied by the other two. That implied rate is the cross-rate.

A cross-rate exists because currency markets do not quote every possible pair against every other pair. Instead, most currencies are quoted against a small number of major currencies, often the US dollar. When you need a rate between two non-dollar currencies, you derive it using the dollar as the common link.

The orientation matters. "" and "" are reciprocals of each other, not the same number. EUR/GBP states how many euros it takes to buy one pound. states how many pounds it takes to buy one euro. Getting the orientation backward is one of the most common Level I errors on this topic.

A properly derived cross-rate should also be internally consistent. If you calculate EUR/GBP from and , then invert your answer, you should get back without needing separate inputs. This consistency check is a simplified form of no-arbitrage reasoning. It does not involve interest rates. The full interest-rate version of no-arbitrage consistency belongs to spot-forward parity, covered in the next note.

A Unit-Cancellation Method for Cross-Rates

The safest way to calculate a cross-rate is to treat each quote as a fraction and cancel units the way you would in any dimensional analysis problem.

Start by writing each quote as , meaning units of the price currency per one unit of the base currency . If a quote is given in the wrong direction for your calculation, invert it first: .

For two quotes expressed against a common currency X, and , the cross-rate between Y and Z is:

Where:

• is the common currency shared by both quotes.

• is the currency you want as the price currency in the final answer.

• is the currency you want as the base currency in the final answer.

• and are the given spot quotes, both stated as units of X per one unit of the other currency.

This formula works because appears in both the numerator and denominator when you divide the fractions, and it cancels out, leaving .

Do not memorize "multiply for this case, divide for that case" without checking units. Candidates who skip writing units are the ones who multiply when they should divide, or divide in the wrong order.

Cross-Rate Calculation Patterns

The formula above covers the most common setup, but the common currency can appear in three different positions across the two quotes.

Pattern 1: Both quotes share the same price currency

Example: and , both stated as USD per one unit of the other currency.

Since USD sits in the same position in both quotes, divide directly: .

Pattern 2: Both quotes share the same base currency. Example: CHF/USD = 1.10 and JPY/USD = 149.50, both stated as units per one USD. Since USD is the base in both quotes, divide the two rates directly in their given orientation: .

Pattern 3: The common currency is the price currency in one quote and the base currency in the other. Example: (USD is price) and (USD is base). Here, invert one quote so the common currency lines up on the same side as in Pattern 1 or Pattern 2, then apply the same division.

Recognizing which pattern you are facing before you start calculating prevents the most common setup error: applying the formula to quotes that are not yet aligned.

Cross-rates can also be quoted with bid and ask sides. The same cancellation logic applies, but you calculate the cross bid using the bid side of each leg and the cross ask using the ask side of each leg. Because bid-ask spreads compound, the calculated cross spread is typically wider than either individual quoted spread. This extension matters mainly for recognizing that a cross-rate spread is not invented independently. It follows from the two underlying quotes.

How to Check and Interpret the Result

Before finalizing a cross-rate answer, run through this checklist:

1. Units. Confirm the final answer's units match what was asked (price currency per one unit of base currency).

2. Reciprocal. Invert your answer and confirm it produces a sensible rate for the opposite orientation.

3. Magnitude. Check that the size of the number makes sense given the relative strength of the two currencies against the common currency.

4. Economic meaning. State what the number means in plain language: how many units of one currency it takes to buy one unit of the other.

5. Rounding. Carry enough decimal places through intermediate steps. Rounding too early distorts the final answer, especially when one quote has several decimal places.

Do not call the result a forward rate. A cross-rate calculated from two spot quotes is still a spot rate. It becomes a forward-related concept only when interest rate differentials and time to maturity enter the calculation, which is the subject of the next two notes.

Worked Example

A candidate is given two spot quotes against the US dollar:

• = 1/10 (1.10 US dollars buy 1 euro)

• = 1.25 (1.25 US dollars buy 1 pound)

Question: Calculate the EUR/GBP cross-rate, stated as euros per 1 British pound.

Step 1: Write the quotes in words.

= 1.10 means 1 euro costs 1.10 US dollars.

= 1.25 means 1 pound costs 1.25 US dollars.

Step 2: Identify the orientation requested. The question asks for EUR/GBP, meaning euros per 1 pound.

Step 3: Set up the calculation. USD is the price currency in both quotes, so this is Pattern 1. Divide:

Step 4: Verify with unit cancellation.

.

First invert to get = 1 / 1.10 = 0.9091.

Then multiply: 0.9091 × 1.25 = 1.1364. USD cancels in this chain, leaving . The result matches Step 3.

Step 5: Check the reciprocal.

= 1 / 1.1364 = 0.8800.

This says 1 euro buys 0.88 pounds. Since 1 pound costs more dollars than 1 euro (1.25 versus 1.10), it makes sense that 1 pound is worth more than 1 euro, so should be greater than 1. It is.

Interpretation: At these spot rates, it takes 1.1364 euros to buy 1 British pound.

Common Exam Traps

Inverting the wrong quote

Candidates sometimes invert the quote that is already in the correct orientation, instead of the one that needs flipping. Always check which currency needs to end up in the numerator before inverting anything.

Multiplying or dividing from memory without unit cancellation

Skipping the units step is the single biggest source of errors on this topic. Write "USD per EUR" instead of just "1.10" and the correct operation becomes visible.

Giving the reciprocal of the requested rate

Solving for GBP/EUR when the question asks for produces a technically correct calculation with the wrong final answer. Reread the requested orientation before finalizing.

Dropping currency units from intermediate steps. Carrying only numbers through a multi-step calculation makes it easy to lose track of which currency cancels and which remains.

Using a forward-rate or interest-parity formula. A cross-rate calculated from spot quotes does not involve interest rates or time. Bringing in a forward premium or discount formula here answers a different question.

Practice Question

A trader observes the following spot quotes:

= 0.90 (0.90 US dollars per 1 Swiss franc)

= 0.10 (0.10 US dollars per 1 Swedish krona)

What is the cross-rate, expressed as Swiss francs per 1 Swedish krona?

1. 0.111 CHF per SEK

2. 9.00 CHF per SEK

3. 0.090 CHF per SEK

FAQs About Currency Cross-Rates