Currency crosses: uncovering the exposures hidden in two legs

Cancel the units

A cross rate can be derived from two exchange rates that share a currency. If EUR/USD gives dollars per euro and USD/JPY gives yen per dollar, multiplication gives yen per euro. The dollar units cancel. If two quotes have the same denominator instead, division may be required. Writing units makes the operation unambiguous and prevents a plausible-looking but incorrect number.

This arithmetic is an identity for consistent simultaneous prices before transaction costs. It is not an automatic arbitrage opportunity. Actual quotes have separate bid and ask prices, finite depth, timestamps and settlement conventions. A synthetic route that looks cheaper at midpoints can become more expensive once both legs are executable.

For two quotes expressed as dollars per currency, division is the right operation. If EUR/USD is 1.10 and GBP/USD is 1.25, EUR/GBP is 1.10 ÷ 1.25 = 0.88 pounds per euro. The dollars cancel only after one quote is inverted. Writing '$ per € divided by $ per £' is often faster and safer than memorizing a list of pair-specific rules, especially when a data vendor uses an unfamiliar quote direction.

Match the intermediate notional

A long EUR/USD exposure combined with a long USD/JPY exposure can resemble long EUR/JPY only when the intermediate dollar quantities match. Equal lot counts do not establish this. The euro leg has a dollar value that depends on its entry exchange rate, while the dollar leg is denominated directly in dollars. An unmatched remainder leaves a separate dollar exposure.

For a cash illustration, buying 100,000 euros at 1.10 dollars per euro requires 110,000 dollars. Obtaining those dollars with yen at 150 yen per dollar costs 16.5 million yen. If the second transaction instead buys only 100,000 dollars, the missing 10,000 dollars must come from somewhere else. The combined position is not the clean cross originally described.

The distinction between a cash inventory and a rolling hedge matters after the initial match. Exchanging €100,000 against ¥16.5 million through a perfectly matched dollar leg leaves no dollar principal balance in the cash example. Two derivatives may instead realize gains, losses or financing in dollars along the way. Their initial spot equivalence does not promise identical interim cash flows or margin requirements. Check what each contract actually settles before treating the two-leg position as operationally identical to a direct cross.

Use the correct side of each quote

To buy euros with yen through dollars, you buy dollars using the USD/JPY ask and buy euros using the EUR/USD ask. The synthetic ask is the product of the two asks under those quote conventions. To sell euros into yen, both bid sides apply. If either currency quote is inverted, the relevant side also inverts: the reciprocal of an ask becomes a bid.

This exercise is useful even without executing a synthetic route. It teaches why direct and synthetic midpoint comparisons are incomplete. The all-in comparison should also include commissions, financing, minimum order increments, settlement differences and the risk that the second leg executes after the market has moved.

For EUR/GBP derived by division, the executable sides are also different from a simple midpoint ratio. Selling euros for pounds means selling EUR/USD at its bid and buying pounds at the GBP/USD ask: synthetic EUR/GBP bid = EUR/USD bid ÷ GBP/USD ask. Buying euros with pounds gives the opposite side: EUR/USD ask ÷ GBP/USD bid. The rule follows the transactions; using bid divided by bid would quietly assume an unavailable conversion rate on one leg.

Correlation follows the currency triangle

A cross is related to its component returns. In logarithmic returns, the change in EUR/JPY equals the change in EUR/USD plus the change in USD/JPY. Its variability therefore depends on both component variabilities and their correlation. Two volatile legs can partially offset when they tend to move in opposite directions; they can amplify each other when they move together.

Historical correlation is an estimate, not a stable law. A policy surprise or funding event may change the relationship precisely when positions are largest. Evaluate the combined exposure under scenarios in which one leg moves alone, both move together and the historical relationship breaks. Counting two orders as diversification says nothing about the economic currencies held.

Ordinary percentage returns contain an interaction term. If EUR/USD rises 2% and USD/JPY rises 3%, EUR/JPY rises 1.02 × 1.03 − 1 = 5.06%, not exactly 5%. Log returns add exactly because logarithms turn the price multiplication into addition. Either convention can work, but a spreadsheet must not combine simple returns in one column with log returns in another and then label the total exact.

Document the actual instrument

Spot balances, rolling leveraged contracts and futures do not necessarily settle or finance in the same way. A direct cross futures position may have different contract increments or margin treatment from two component contracts. A hedge assembled from mismatched maturities introduces another source of residual exposure even if the initial spot notionals match.

In a paper exercise, write a currency inventory after each leg: euros owned or owed, dollars owned or owed, and yen owned or owed. Then mark each inventory at a hypothetical later rate. This accounting check is more reliable than relying on a platform's pair labels. It also exposes accidental financing needs and makes the intended hedge easier to audit.

As a verification step, reverse the hypothetical transaction using the corresponding bid prices. The round trip should reveal costs even when midpoints are unchanged. If your worksheet produces a riskless gain from identical unchanged quotes, check quote inversion, intermediate notional and which side of each market was used before trusting the result.

A triangular-arbitrage calculation needs three executable prices that coexist for the required size. If the first leg fills and the next quote disappears, the position has become an open currency exposure rather than a locked arbitrage. Commissions, minimum sizes and rounding can leave a small residual even when the algebra is correct. The right paper exercise therefore includes the sequence of fills and leftover balances, not just the product of three midpoint exchange rates.