Cross Exchange Rate Calculator
From two currencies' rates against a common quote currency, derive the cross rate between them — in both directions.
Input Data
Results
At a glance:Cross rate = indirect rate between two currencies via a common intermediary B (usually USD). A-to-C = (A-to-B) / (C-to-B); C-to-A is its reciprocal (product = 1). Used when no direct quote exists. Example: USD/HKD 7.8, EUR/HKD 8.5 → USD/EUR = 7.8/8.5 ≈ 0.9176. WARNING: Pure mid-rate math; real FX adds bid-ask spreads and fees, rates move; reference only, not advice.
Formula
A-to-C rate = (A-to-B rate) ÷ (C-to-B rate).
C-to-A rate = (C-to-B rate) ÷ (A-to-B rate).
The two rates are reciprocals: rate(AC) × rate(CA) = 1.
$$A C $Cross_{A/C}=\\dfrac{Rate_{A/B}}{Rate_{C/B}}$$$$$C A $Cross_{C/A}=\\dfrac{Rate_{C/B}}{Rate_{A/B}}$$$$$$Cross_{A/C}\\times Cross_{C/A}=1$$$How to Use
- Enter A's rate against common B.
- Enter C's rate against the same B.
- View the A-to-C and C-to-A cross rates.
FAQ
What is a cross rate and why do we need it?
A cross rate is the rate between two currencies derived via a common intermediary, rather than directly quoted. Most FX volume is against the USD; for non-USD pairs (EUR/JPY, GBP/AUD) there is often no active direct quote, so you derive it from each versus USD. In daily life, if you track foreign currencies only against HKD, you can still compare USD vs EUR via USD/HKD and EUR/HKD.
How is the cross rate computed?
Use the same quote currency as the bridge and divide. With rateAB = 1 A in B and rateCB = 1 C in B, A-to-C = rateAB / rateCB and C-to-A = rateCB / rateAB. Intuition: convert 1 A to rateAB of B, then B back to C at rateCB per B. The two results are reciprocals. Ensure both inputs are quoted against the same B (take reciprocals to align directions first).
Why does my result differ from the bank / money changer?
This is the theoretical mid-rate. Real exchange applies a bid-ask spread (they buy low, sell high), fees, and possibly two rounds of spread for a true cross trade, plus constant rate movement and cash-vs-wire price gaps. Use it to estimate and compare, not as your executable rate.
What is triangular arbitrage and its link to cross rates?
Triangular arbitrage profits from a brief inconsistency among three currencies by cycling A→B→C→A. The cross rate is exactly the consistency relationship that should hold; when the direct A/C quote diverges from the cross, arbitrageurs close the gap almost instantly. So cross rates hold in liquid markets, and this calculator shows the theoretical rate — not a trade recommendation.
Why does my cross rate differ from the bank's?
Because this computes the theoretical mid-rate (pure division), while banks/changers quote a bid-ask spread and may charge fees; cross trades effectively take two spreads. Rates also move between quote and execution, and cash rates differ from wire. Use the result as a sanity baseline to judge whether a quote is reasonable, not as the rate you will get.
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References
Content review: Calculatorism Finance Team. Results are for reference only; please refer to the relevant authorities for the official figures.