IR Yield Curve and Fixing Dates

Question

Consider two FRAs.

1. 3x6 , Effective 3 months from now, terminates in 6 months. The floating leg payer pays 3-month LIBOR. Fixing date for LIBOR 40 business days. To price this at par, the fixed leg payer will pay r40.

2. 3x6 , Effective 3 months from now, terminates in 6 months. The floating leg payer pays 3-month LIBOR. Fixing date for LIBOR 2 business days. To price this at par, the fixed leg payer will pay r2.

Will r2 and r40 be different? I would think so. These FRAs represent different risks. How will I determine r2 and r40 ? What is the exact relationship between r2 and r40 ?

Now, suppose Instead of FRAs, I want to construct two 5 year fixed for floating swaps, both swaps have floating rate semi-annual payments of 6 month LIBOR but again have 2 business day and 40 business day fixings. Will I construct two different Yield Curves to price these guys? Will the fixing dates of my instruments used to bootstrap the curve have to match the fixing dates of my swap floating rate? How would I price these two floating legs?

Answer 1

Instrument 2 looks to me like the standard regular definition of a 3x6 FRA. This is a relatively liquid instrument, so that forward rate r2 is just the price of the FRA and is available on Bloomberg, etc. If you have a yield curve model and associated suite of functions there will certainly be a function to return that forward rate, because it's vanilla.

The forward rate r40 will not have the same value as r2, as you correctly suggest. However, you would not calculate r40 starting from r2, because the only relevant thing those two instruments have in common is the payment date. Instead you would calculate rf, which is the forward rate for a vanilla FRA (ie fixing = 2 days) that fixes on the same date as Instrument 1. Then r40 equals rf multiplied by an exponent involving the volatility of the numeraire. Hull's book (Options, Futures and Other Derivatives) is where I read about this, and it gives the exact formula you want and explains the assumptions.

No, you would not need two yield curves. You build your yield curve out of the vanilla liquid instruments you have available. You then query this curve to get your vanilla forward rates for the fixing dates you require, then if needed (if your payment date is non-standard) you multiply by the measure adjustment term as described above. It's the same logic as for pricing Libor-in-Arrears.

Answer 2

1. I wouldn't contstruct two curves, i would just adjust my futures cupons. Assumming that the par swap quotes are already in a 2 spot days convention for most markets. That means that when calculating the FRA for a Libor Cupon you actually capture the value a -2 days fixing.
2. If one of your swaps have a 40 business days, i assume the quote might not be 40 spot days. So when you actually use the quotes for bootstrapping you wouldn't be capturing the 40 days buisness days lag in your curve.
3. You just need to adjust the FRA calculation to the actual date you need to interpolate for each cupon, by adding or substracting the fixing days to the FRA dates. And then discount cashflows like a regular swap.