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