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Inferring EUR/USD Changes from EUR/JPY and USD/JPY

Article Quant Q&A · Author: emcor
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Summary

The document shows how to infer a cross exchange rate from two rates quoted against the same currency. Under a no-arbitrage assumption, EUR/USD is obtained by dividing EUR/JPY by USD/JPY. If EUR/JPY rises by 20% while USD/JPY rises by 25%, the new ratio is 1.2 divided by 1.25 times its original level, so EUR/USD falls by 4%.

The calculation illustrates why percentage moves in related currency pairs should not simply be subtracted: the implied cross rate depends on their ratio. The result assumes consistent quote conventions and no arbitrage across the three pairs. It is a short algebraic example, not an analysis of transaction costs, bid-ask spreads, timing differences, or whether the stated exchange-rate moves can be captured in live markets.

Key ideas

  • A cross exchange rate can be derived by dividing two rates that share a quote currency.
  • The calculation assumes no arbitrage and consistent exchange-rate quote conventions.
  • The stated 20% and 25% rises imply a 4% decline in EUR/USD.
  • The example omits spreads, transaction costs, and timing differences.

Tags

ForexArbitrage
Full text
# Implied exchange rate


# Implied exchange rate












The EUR/JPY exchange rate increased by 20%.

The USD/JPY exchange rate increased by 25%.

By how much will the EUR/USD exchange rate decrease?

## Answer by AFK (score 1, accepted)

https://quant.stackexchange.com/a/21025

This seems pretty basic, assuming no arbitrage: $$X^{EUR}_{USD}(t+1) = X^{EUR}_{JPY}(t+1)/X^{USD}_{JPY}(t+1) = (1.2*X^{EUR}_{JPY}(t))/(1.25 * X^{USD}_{JPY}(t)) = 0.96 * X^{EUR}_{JPY}(t)$$ So EUR/USD decreases by 4%.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.