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

  • $\begingroup$ Where do your interest rates 0.1% and 0.3% come from? Are you taking into account the Cross Currency Basis? It has to be added to USD Libor: (1+r_USD+basis)/(1+r_EUR) $\endgroup$ – noob2 Dec 12 '20 at 10:22
  • $\begingroup$ 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

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.

  • $\begingroup$ 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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