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Calculating EUR/USD Forward Rates with Interest Parity and Basis

Article Quant Q&A · Author: JerBouma
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Summary

The document explains how to estimate EUR/USD forward rates from spot and interest rates using covered interest rate parity (CIP). Rates must match the forward’s maturity: an overnight rate is not a substitute for a one-year rate, and the calculation needs to account for the time period. For a short maturity, the example converts annualized rates to monthly form and applies them to spot, producing a value close to the quoted market forward.

The difference between a CIP estimate and the market quote reflects cross-currency basis, since parity may not hold exactly. The discussion notes that longer-maturity rates can differ from overnight rates because of expected policy changes. Its numerical illustration is specific to the rates and market conditions cited, so it is not a universal pricing estimate; day-count conventions and curve-based inputs may also matter in practice.

Key ideas

  • Match the interest rates to the forward contract’s maturity.
  • Include the time period when applying interest rates to the spot rate.
  • Covered interest rate parity gives a baseline forward estimate, but market basis can create a difference.
  • Longer-horizon rates may diverge from overnight rates as market expectations change.

Tags

ForexFixed incomeCarryArbitrage
Full text
# Determine forward rates for EUR/USD


# Determine forward rates for EUR/USD












I can't wrap my head around how to determine the interest rates to calculate the forward rates of any currency. At this point, I don't even know if this data is actually available to do the calculation myself.

From Investing.com (link) I wish to determine the 1Y and 1M forward rate for starters. I figured out the formula: `spot rate x (1 + domestic interest rate) / (1 + foreign interest rate)` and I know the spot rate, 1.08 for EUR/USD.

Now when it comes to Domestic and Foreign rates I simply do not understand what I should be using here. First, I figured it must be ESTER and SOFR. The result I get is quite similar but for anything that is not 1Y, any other duration (e.g. 1M) calculation I do just differs greatly. Then I figured Central Bank rates but in the EU it being 0% and in the US 0.5%, that didn't even come near the forward rate.

So now I am confused, how can I calculate these values if that is actually even possible?

## Answer by fes (score 2, accepted)

https://quant.stackexchange.com/a/70548

The ESTR rate should be -0.585 and not 0.585. Converting to monthly form by dividing by 12 and using the CIP formula gives:

$$F=\frac{1+0.0029/12}{1-0.00585/12}\times 1.0810 =1.0818$$

or 8 forward points over the current spot rate. Your website is quoting a market price of roughly 11 forward points. The difference, i.e. the cross currency basis, is merely 3 basis points. For the annual horizon you should be using annual interest rates instead of overnight rates. These are above shorter maturity rates due to expected rate hikes.

## Answer by KT8 (score 1)

https://quant.stackexchange.com/a/70544

I have a couple of comments to add to your question. First, are you taking into account the time component in your formula? It looks like you're computing the forward rate as

$$ S_T = \dfrac{1 + r_d}{1 + r_f} S_0 ,$$ where what it should read is $$ S_T = \dfrac{1 + r_d T}{1 + r_f T} S_0 .$$

Moreover, note that this formula assumes that Covered Interest Rate Parity is satisfied, which we know is not the case (as basis are non-zero).

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.