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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

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  • $\begingroup$ I used random values for the rates but the equivalence (swap value = spot_leg - fwd_leg = 0) should hold given that I use these 2 rates to derive the fwd? I have not included the xccy basis. This latter has an impact of the market price but in this example I am just calculating the theoretical value of the swap. $\endgroup$
    – Student
    Dec 12 '20 at 21:55
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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.

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  • $\begingroup$ I agree transaction costs/ccy availability impact the market price of a swap.Here I am just calculating the theoretical value of the swap. Is my formula/calculation wrong? I guess the spot leg and fwd leg should match at least theoretically? $\endgroup$
    – Student
    Dec 12 '20 at 21:58
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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".

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