Equivalency of FX forwards and FX fixed for fixed swaps? Are they still the same under multiple curves environment?

Question

I am encountering two approaches for valuation of FX swaps (fixed for fixed, e.g. fixed USD payments for fixed EUR payments) which seem to result into different values although in theory they should be the same.

1. Bloomberg's SWPM

   Bloomberg's SWPM takes EUR cash flows, discounts them at USD discount curve adjusted by EURUSD basis curve and then converts the EUR NPV into USD using EURUSD spot.

2. Using FX forwards some choose to convert EUR cash flows into USD cash flows using EURUSD forward rates. Then discount the converted USD cash flows using a USD curve and this gives the NPV of the EUR leg in USD. Subtracting the other leg (USD) gives the NPV of the entire swap.

I always thought these two approaches should result into the same value, but they don't. Is there a reason for this? Is one better than the other?

Answer 1

Yes I guess in theory they should be the same value, but only because in practise there is no arbitrage between the 2 approaches presuming you end up with the same instrument. I guess you mean that.

1. getting a EUR net present value from USD curves adjusted for EURUSD basis is perhaps 1 way of calculating interest rate differentials. Presumably it gives a correct differential for EUR NPV which you then convert to USD at spot.

2. Alternatively, getting a USD net present value from FX forward prices that have the interest rate differential already calculated, and then calculating an NPV from a USD curve.

Yeah, so the use of the different curves is like, priced under competition, and should be non arbitrageable otherwise we'd all be working that trade. So how much difference are you seeing in the values? Can you give 2 worked examples?