# Does a 1Y swap depend on zero curve beyond the 1Y point?

When using market swap rates to calibrate a discount curve, it seems that the PV of a 1Y swap depends on the zero curve at points beyond the 1Y mark.

For example, a USD 1Y swap with trade date today (Wed 16th March 2016) and a spot lag of 2 days will have an effective date of Fri 18th March 2016, and the floating leg accrual periods (after adjusting using modified following convention) are

Period  Start Date    End Date
---------------------------------
1       18-Mar-2016   20-Jun-2016
2       20-Jun-2016   19-Sep-2016
3       19-Sep-2016   19-Dec-2016
4       19-Dec-2016   20-Mar-2017


The final payment clearly depends on today's LIBOR curve for 20th March 2017, which is more than one year after today (in fact it is 1.011 years in ACT/365 day count).

It seems odd to me that the PV of a 1Y swap would depend on points on the zero curve beyond the one year mark, but that seems to be the case if you correctly compute the start and end dates.

Is it the case? If not, how do institutions normally deal with this?

The problem is that the definition of "one year" depends on the market convention:

1. Roll Day: What's one year after February 28, 2015? Is it February 28, 2016 or February 29, 2016? According to the standard swap roll-day convention, it would be 2/28/16. But in the US government bond market, it would be 2/29/16. Now you can see why it's problematic: the difference from 2/28/15 to both 2/28/16 and 2/29/16 can be considered "1-year," depending on the convention in question.

2. Effective Start Date: Is it the trade date, or something else? As you said, the effective start date is usually 2 business days after the trade date for a standard swap. The maturity date is 1-year after the effective date, not the trade date. So whether the swap in question is a "1-year" or "1-year 2-day" swap depends on the perspective.

For pricing, what you need is a unique discount factor for each date. It shouldn't matter how you define "year" internally within your model. In other words, the final cash flow should be discounted with $d(\text{3/20/2017})$. How you convert that 3/20/2017 into a year fraction is entire up to you. In a discount curve setting, Actual/365 is the most prevalent choice.

• Great, thank you. That's a little clearer. Commented Mar 16, 2016 at 21:09

This all depends how you build the swap curve. If you are using just annual zero rates to build the curve, the answer may be that a one year par rate (paid quarterly or semi annually) is not the same as a one year zero rate. Specifically, it has slightly shorter duration. Therefore the hedge for a one year par rate, in terms of zero rates, should contain a small amount of 2 year swaps (and possibly longer swaps) in the opposite direction to the one year swap. There's nothing wrong with that.

Dealers don't use zero swaps to build curves. They use eurodollar futures and par swap rates to build curves, then imply the zero swaps, just fyi.

• Thanks for the advice. I'm not using zero swaps (I don't even have this data) but rather par swap rates and eventually eurodollar futures. What do dealers do in currencies where there isn't a liquid futures market btw (eg JPY)? Commented Mar 16, 2016 at 21:04
• In JPY there are liquid futures on TIBOR (Tokyo Interbank Offered Rate) which are used at the short end of the curve. If there is a par swaps market (say based on 3 month rates), but NO futures market at all, then you would have to use the current 3 month fixing at the short end of the curve, then "bootstrap" your way out the curve using your par rates and making sure the implied forward curve is smooth
– dm63
Commented Mar 18, 2016 at 16:51

Depends mostly on the curve interpolation method, not on the instruments you build it from. It is a desired property that dependency is localized (this is the jargon for what you describe), but for the curve to be smooth, monotone and/or convex, one may have to accept some spillover.

Google "convex monotone" or search for other works on the curve interpolation, to which Pat Hagan has contributed.