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Estimating a EUR/USD Forward from Rates and Cross-Currency Basis

Article Quant Q&A · Author: Daniel
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Summary

The post asks whether a later EUR/USD forward rate can be estimated from a quoted one-year forward and assumed flat short rates. The proposed method applies a spot-forward relationship: infer the spot exchange rate from the known forward and the two currencies’ rates, then use that spot estimate with the same rates and the target maturity to estimate another forward. The example assumes the cross-currency basis is zero.

This is an approximation, not a forecast of where the market will actually quote the forward. The answer emphasizes that a real calculation would use the relevant basis and forward term structure, which are not supplied. It also interprets “flat” rates as applying across maturities. The post provides no market-data check or worked numerical result for the requested future date, so its value is mainly in showing the assumptions required for a simplified rate-based calculation.

Key ideas

  • A quoted forward and interest rates can be used to infer an implied spot rate.
  • That inferred spot can be combined with rates and maturity to estimate another forward.
  • The example assumes flat rates and a zero cross-currency basis.
  • Missing basis and forward-curve information limit the estimate’s accuracy.

Tags

ForexDerivatives pricing
Full text
# Forward EURUSD exchange rate for a Future date


# Forward EURUSD exchange rate for a Future date












Assume that on today's date as of `11/22/2020` the 1 year `forward exchange rate` for `EUR/USD` is 1.5 for maturity `11/22/2021`.

Current `RFR` for EUR and USD is flat `0.5% and 0.1%`.

With this information, can I calculate the expected Forward exchange rate for same maturity for some future date, say `03/22/2021`?

Thanks for your time.

## Answer by Jan Stuller (score 3, accepted)

https://quant.stackexchange.com/a/59517

I think it's possible. When you say the RFRs are flat, I think we can interpret that as flat for all maturities, including the 1y. So from the 1y Forward rate, we can back out the spot rate via the following relationship between the Spot, Forwards and the RFRs:

$$S_{EUR/USD}(1+r_{USD})^n=(1+r_{EUR}+r_{Basis})^nF_{EUR/USD}$$

Above, $r_{Basis}$ stands for the Cross-Currency basis between EUR and USD. In a normal market, you would have the Spot, the Forward, the RFRs, and so the basis term would be the term that you could back out from the equation above.

Given the data you have provided, let's assume that the Xccy basis term is zero. For the 1-year case, $n=1$. You can plug in the value for the Forward and the RFRs and back out the spot rate $S_{EUR/USD}$.

Then you can plug it back into the equation above again, use the flat RFRs, but adjust the value of $n$ to scale it for the maturity you need (so half year would be $n=0.5$, two years would be $n=2$).

It's only an approximation, but without further info (i.e. the Xccy basis, the full term-structure of the Forwards, etc.) probably as good as we can do.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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