FX Swap Par Value Under Interest Rate Parity
Summary
The document explains why an FX swap should have zero value at inception under idealized interest rate parity. It distinguishes the spot exchange, which is agreed at the prevailing spot rate, from the forward exchange, whose fair rate reflects the interest rate difference between the two currencies. A numerical EUR/USD example shows how investing in euros or converting to dollars, earning the dollar yield, and exchanging back at the forward rate leads to equivalent outcomes.
The discussion also notes that a traded swap may not have exactly zero value after costs such as bid-ask spreads, dealer compensation, hedging costs, and cross-currency basis. The displayed question's simple subtraction compares dollar amounts at different dates without consistently discounting or expressing cash flows in a common currency, so those figures alone do not establish a value mismatch. The excerpt points toward an explicit cash-flow explanation but does not include that final answer, and it assumes simplified rates and no basis for its parity illustration.
Key ideas
- Interest rate parity determines the fair FX forward rate from spot and the two currency interest rates.
- The spot leg at the agreed market spot rate has zero present value at inception.
- The forward leg is fair when the two currency investment paths produce equivalent outcomes.
- Bid-ask spreads, dealer costs, hedging costs, and cross-currency basis can affect actual trade value.
Tags
Full text
# FX swap par value
# FX swap par value
What is the relationship to apply so that an FX swap value is 0 at inception?
For example, for a short 1y EURUSD swap with 1mm euro notional, at inception spot = 1.1000 and 12m fwd = 1.1022, EUR 1y yield = 0.1%, USD 1y yield = 0.3%. I am assuming I am short the swap so I am long on the spot leg and short of the fwd leg. Note that 12m fwd = spot*(1+r_USD)/(1+r_EUR) = 1.100*(1.003/1.001) = 1.1022
at inception below equivalence should apply, but the 2 legs don't match:
swap value = spot_leg - fwd_leg = 0
spot_leg = N*spot = 1mm * 1.100 = $1.100mm
fwd_leg = PV(N*fwd) = (1mm * 1.1022)/(1.003) = $1.098mm
## Answer by AlRacoon (score 1)
https://quant.stackexchange.com/a/59916
Theoretically the two legs at inception are of equal value. In practice, they will not be. There are transactions costs associated with every trade. The most obvious is the bid ask spread. Any dealer that will trade with you will be looking to make a profit for making a market and taking interim risk/hedging costs.
## Answer by KevinT (score 1)
https://quant.stackexchange.com/a/63902
All the other answers/comments are correct by pointing out XCCY basis and/or bid-ask. Let's assume these are non-existent for the sake of the example.
So using your data, we have spot $S = 1.10$, $r_{EUR} = 0.1\%$, and $r_{USD} = 0.3\%$. This implies that forward $F$ is equal to $F = S \frac{1+r_{USD}}{1+r_{EUR}} = 1.102198$.
Now the FX Swap is just the sum of FX Spot + FX Forward. The FX Spot exchange executed at the current rate $S$ is a zero PV transaction, so to make the FX Swap fair at initiation, we only need to look at the FX Forward. This one is fair in the sense of "interest-rate-parity": investing 100 EUR at the EUR-rates today delivers you 100.10 EUR in 1 year. This must be the same as converting the EUR into USD today, then investing these dollars at the USD yield, and then after one year convert this amount back to Euros. Hence, 100 EUR delivers you 110 USD, and these grow at 0.3% so to return you 110.33 USD in 1y. Converting these 110.33 USD to EUR at the 1y-forward rate gives you 110.33/1.102198 = 100.10 EUR in 1y -- so exactly the same as if you'd invested in the EUR markets directly. Hence, also the forward (and thus the swap) are "fair".
## Answer by user35980 (score 1)
https://quant.stackexchange.com/a/78361
Already a well answered question, but I think the question is best addressed by illustrating the cashflows of the FX swap explicitly (using numbers provided in the question) to see why the "swap value=0":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.