# Annualizing the pay frequency of underlying swaps when bootstrapping the zero curve?

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?

• It’s not the fixed frequency that drives the basis, but the relevant tenor of the underlying index, eg 3M, 6M. The swap is commonly quoted under collateralization; hence no need to annualized the fixed quote. If the underlying index frequencies (and their credit risk) don’t match up:that’s a different story. HTH? Nov 19, 2021 at 16:51
• what do you want to compare? par swap rates? for example 6m EUR par swap rate vs 6m USD par swap rate? and your problem is that the EUR swaps used to build your zero curve are 1Y and you would like to compare it with the USD swap rate that is 6M? so you are trying to make both swap rates comparable, is that what you are asking? Nov 19, 2021 at 16:54
• if we want to compare par swap rates, forward swap rates, etc, we don't annualize anything before building the curve. The curve must be built so it reprices properly the market instruments used as inputs. Once we have the zero curves, then we can (using the discount factors) build par swap rates with any payment frequency, day count convention, etc we want, so we can compare. Nov 19, 2021 at 17:07
• @Sebastian Okay, maybe I wasn't as clear as I should have been. The swap rates used, for both USD and EUR, are against the 3M LIBOR and 6M EURIBOR, respectively. I suppose, in this sense, there is a discrepancy between the underlying tenors. Anyway. The terms of the swaps are the same, e.g., starting with the 1Y there is a 1-to-1 equivalence of the terms for both EUR and USD. Any points needed that are not directly sourced are filled via standard interpolation. My question: There is a fundamental difference in both the tenor of the underlying and the pay frequency.
– QVC
Nov 21, 2021 at 19:59
• @Sebastian It was my thought that, if I am looking to "equate" the two bootstrapped curves, then there should be some transformation concerning either the pay frequency (e.g. annualizing the "coupons") or the underlying tenors. Is this nor a correct assumption? For instance, if I am looking at a 5% Corporate bond that pays coupons quarterly vs a 5% bond that pays coupons annually, there is going to be a difference in the effective rate. How do I reconcile this with swaps of different underlying tenors/pay frequencies when bootstrapping the zero curve? Or maybe this is a non-issue?
– QVC
Nov 21, 2021 at 20:03