Currency Cross Rate Calculator
Calculate the implied cross rate between two currencies via USD and measure net triangular arbitrage profit after transaction costs. Ideal for traders spotting mispricing across three currency pairs simultaneously.
About this calculator
A cross rate is the exchange rate between two currencies derived indirectly through a third currency — typically the US dollar. The implied cross rate is: Cross = Base/USD ÷ Quote/USD. If this implied rate differs from the direct market cross rate, a triangular arbitrage profit (or loss) exists. The net profit formula used here is: Profit = Amount × |( BaseCurrencyRate / QuoteCurrencyRate ) − DirectCrossRate| − Amount × (TransactionCost% / 100) × 3. The factor of 3 reflects transaction costs on each of the three legs of the triangular trade. A positive result after subtracting these costs indicates a genuine arbitrage opportunity. In efficient markets the implied and direct cross rates converge rapidly, so mispricings are typically fractions of a pip.
How to use
Suppose EUR/USD = 1.0820, GBP/USD = 1.2650, and the direct EUR/GBP market rate = 0.8555. Transaction cost = 0.02%, arbitrage amount = $500,000. Implied EUR/GBP = 1.0820 / 1.2650 ≈ 0.8554. Deviation = |0.8554 − 0.8555| = 0.0001. Gross profit = $500,000 × 0.0001 = $50. Transaction cost = $500,000 × (0.02/100) × 3 = $300. Net result = $50 − $300 = −$250. The opportunity is unprofitable after costs. A trader would need a larger deviation or lower transaction costs to profit.
Frequently asked questions
How do you calculate a currency cross rate without using USD as the base currency?
To find the cross rate between Currency A and Currency B when both are quoted against USD, divide the A/USD rate by the B/USD rate: Cross(A/B) = Rate(A/USD) / Rate(B/USD). For example, if EUR/USD = 1.0820 and CHF/USD = 1.1100, then EUR/CHF = 1.0820 / 1.1100 ≈ 0.9748. If one currency is quoted as USD/X rather than X/USD, you first need to invert it. Most professional platforms calculate crosses automatically, but understanding the underlying arithmetic helps you verify quotes and spot data errors.
What is triangular arbitrage and how does it exploit cross rate discrepancies?
Triangular arbitrage is a three-step trade that converts Currency A into Currency B, Currency B into Currency C, and Currency C back into Currency A, profiting when the round-trip yields more than the starting amount. For example, if USD → EUR → GBP → USD produces $1.0005 from every $1.00, the $0.0005 gain is the arbitrage profit before costs. The discrepancy arises when a market-maker's direct quote for a cross pair does not exactly match the rate implied by two separate USD pairs. Electronic trading has made such opportunities extremely short-lived — often lasting only milliseconds — and available only to participants with low-latency access and tight spreads.
Why do transaction costs make most retail triangular arbitrage attempts unprofitable?
Each leg of a triangular arbitrage trade incurs a bid-ask spread, and three spreads combined typically exceed the small mispricing being exploited. Retail forex brokers often charge 1–3 pips per trade; a three-leg arbitrage therefore needs the cross-rate deviation to exceed 3–9 pips just to break even. In liquid major-pair markets, deviations rarely reach even 1 pip before algorithms correct them. Institutional traders with direct market access, co-located servers, and near-zero spreads can sometimes capture these opportunities, but for most retail traders triangular arbitrage is a theoretical exercise rather than a practical strategy.