Libor atm is:
3M = -0.54486 % 6M = -0.52514 % 9M = 1Y = -0.47443 %
How to retrieve the 9M Libor rate?
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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.
I would recommend reading:
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.
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.