Say I'm looking to bootstrap two zero curves based on two swap curves with different underlying currencies and, consequently, two different pay structures in the swap contracts. For example, say I want to use vanilla EUR interest rate swaps that have an annual pay frequency for the fixed leg (e.g., EUSA1 ICPL, EUSA2 ICPL, etc). On the other hand, I want to use vanilla USD interest rate swaps that have a semiannual pay frequency (e.g., USSW1 CMPN, USSW2 CMPN, etc). If I want the resultant bootstrapped zero curves to be "comparable", is there any other way to do this other than annualizing the USD swap rates into effective rates? That is, convert the base USD swap rates into effective rates using:
s(t;n,1) = (1 + s(t;n,m)/m)^m) - 1
wehre s(t;n,m) is the n-year swap rate with coupon frequency m at time t
In this case, USD swap rates would have m=2 and then bootstrapping would be applied. Is there a huge issue with assuming different pay frequencies when constructing the zero curves? Can I just assume the semiannual pay frequency of the USD swaps?