The cross-rate identity, derived.
Why exchange rates are conventionally quoted via a single base currency, and how to derive any pair from any two direct rates against that base.
Why a base currency?
The interbank FX market quotes most pairs against the US dollar. There are roughly 180 ISO 4217 currencies; quoting all 32,220 pairs (n × (n−1) for n=180) is impractical. Quoting all 179 USD pairs and computing the rest by cross-rate is tractable and consistent.
The convention reduces the number of independently maintained rates by two orders of magnitude. The trade-off is a small loss of precision when both legs are illiquid currencies whose USD rates are themselves noisy.
The identity, formally
For currencies A, B, and base X, with rate rAX meaning “1 unit of A buys rAX units of X”:
rAB = rAX · (1 / rBX) = rAX / rBX
The identity holds because the conversion A → X → B is mathematically equivalent to A → B; arbitrage in liquid markets enforces the identity to within transaction-cost tolerance.
Worked example: EUR/JPY
Direct rates from the bundled table:
- EUR/USD = 1.077586 (1 EUR = 1.0776 USD)
- USD/JPY = 154.20 (1 USD = 154.20 JPY)
Cross-rate: EUR/JPY = 1.077586 × 154.20 ≈ 166.16. Verify by inverse: JPY/EUR = 1 / 166.16 ≈ 0.00602.
Worked example: GBP/CHF
Direct rates:
- GBP/USD = 1.273885
- USD/CHF = 0.890
Cross-rate: GBP/CHF = 1.273885 × 0.890 ≈ 1.1338. The inverse: CHF/GBP ≈ 0.8821.
Triangular arbitrage and why the identity holds
If the cross-rate quoted by a bank diverges from the implied cross-rate, an arbitrageur can buy at one and sell at the other for a risk-free profit. In liquid markets this opportunity closes within milliseconds; in illiquid markets it can persist for minutes or hours but is bounded by transaction costs. For reference purposes, treating the identity as exact is appropriate.
Where the identity breaks down
- Illiquid currencies. Pairs against currencies with restricted convertibility (Argentine peso black market, sanctioned currencies) often trade at rates that don't reconcile via USD.
- Stale rates. A USD rate that hasn't been refreshed for an hour will produce a stale cross-rate — fine for reference, dangerous for execution.
- Asymmetric quote conventions. Some pairs (EUR/USD, GBP/USD) are quoted “EUR-base” or “GBP-base” even when USD is the conventional base for other pairs. Watch the direction of the rate.