# Interpolating Libor 9M rate?

Libor atm is:

3M = -0.54486 %
6M = -0.52514 %
9M =
1Y = -0.47443 %


How to retrieve the 9M Libor rate?

• For starters, I suggest a simple linear interpolation between 3M and 12M should suffice. Effectively, an interpolation between 3M/12M and 6M/12M is only off by 0.2bps, so linear interpolation should do, no? – Kermittfrog Nov 6 '20 at 11:40
• I meant 6M/12M. Sorry – Kermittfrog Nov 6 '20 at 12:59

It depends on what you want to do with the interpolated 9M rate.

For example, I encountered this practical problem once. Desk loaned some money to an agricultural firm that, for liquidity reasons, wanted to pay interest like this:

• a coupon with 9 months worth of interest, reset from 9M USD LIBOR + spread

• 3 monthly coupons reset from 1M USD LIBOR + spread

• another 9-month holiday, followed by 4 floaters - repeated for several years.

Everyone was happy until, a few years into this, they unexpectedy stopped publishing 9 months tenor (circa 2013). The language in the loan documents literally meant the latest available 9M LIBOR would be re-used until the maturity. Neither party liked that. The lawyers spoke and agreed that for this loan, a synthetic 9M LIBOR would be linearly interpolated from 6M and 12M tenors (that were still being published) - just the arithmetic mean, 1/2 of each of the 2 observable tenors. No one cared whether this linear interolation, if used in other contexts, might admit arbitrage or lead to other problems not relevant to this loan. Any more complicated interpolation would add no value to either party and would confuse the lawyers and the customer.

• Always nice to get a glimpse at your treasure trove of experiences... – Kermittfrog Nov 6 '20 at 16:59
• Thank you! I'll try to remember more cool tales to share. – Dimitri Vulis Nov 6 '20 at 17:37

Erik Schlogl, Arbitrate-free Interpolation in Models of Market Observable Interest Rates.

Andersen and Piterbarg, Interest Rate Modeling, Chapter 15.

Finally, this Masters Thesis is really nice.

• Wow the supervisor was my lecturer for the course stochastic integration – simsalabim Nov 6 '20 at 15:48
• That's a cool coincidence! The answer to your question is there in the three references that I pointed out! Thanks! – rvignolo Nov 6 '20 at 15:59

Speaking for USD, people stay away from tenors other than 1M, 3M, 6M, and somehow 12M.

And if any client has a need for a different tenor, almost always will a linear interpolation between the closest tenors be used.

12M libor is still published but not a lot of new contracts are indexed on it. Someday some 12M libor swap might trade in the interdealer markets but it is more inventory-management related than flow related.

• I've seen 12Mo libor used in (floater) consumer mortgages. Monthly payments reset from 12Mo libor. Consumers don't realize that they may not be getting a good dea. – Dimitri Vulis Nov 6 '20 at 14:19
• I am not expert here but I guess it might make sense for some consumers whose income cycle is yearly ? – ZelliZello Nov 6 '20 at 14:25
• @DimitriVulis why wouldn't monthly payments not be good for clients? – simsalabim Nov 6 '20 at 15:51
• The cheat is: unsiphsticated retail customers don"t realize that their monthly payments reset not from 1mo Libor, but from 12mo Libor, usually a tiny bit higher. – Dimitri Vulis Nov 6 '20 at 16:08
• I thought you meant 12M libor every 12 months. 12Mo libor payments every month is weird indeed. – ZelliZello Nov 6 '20 at 16:15