# 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

• 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) – noob2 Dec 12 '20 at 10:22
• 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. – Student Dec 12 '20 at 21:55