# OIS rate to build Term structure

There are some discussions (e.g. Difference between OIS Rate and Fed Funds Rate) on usage of OIS rate to build the Libor term structure, but I still failed to grasp the reason behind it.

As far as I know, an OIS rate is the Interest rate swap's rate for which the floating leg's payout is based on the Overnight lending rate among banks.

Also, typically, that floating leg's payment is based on Geometric average of daily Overnight rates within respective leg's tenor. For example, if for a Swap having frequency of floating leg as 3months, then the payout will be based on the Geometric average of daily Overnight rates among banks for that 3 months.

So, my question is using such information how can I exactly calculate the Libor rate for term, say, 10 years etc?

Any insight on this subject will be highly appreciated.

• Ps: the answer here may also be helpful with regard to how forward Libor rates can be constructed. Oct 31 '20 at 13:58
• You might want to start with some background reading, e.g. jdawiseman.com/books/pricing-money/…. OIS and LIBOR are two separate but related interest rates. You don't use OIS rates to construct a LIBOR curve, you use LIBOR rates. OIS comes into the picture for discounting the cashflows on a swap where the floating leg is LIBOR. HTH. Oct 31 '20 at 17:03

By definition, the overnight rate is the rate at which banks lend to each other overnight. Overnight index swaps (OIS) allow banks to 'lock in' the cost of funding overnight for a specific term. They exchange a predetermined OIS rate for a payoff equal to the growth of the notional amount of money lent at the overnight rate for a specific term.

The overnight rate is equal to the rate at the shortest maturity on the Libor curve. However it says nothing about longer term Libor rates. The OIS rate just tells you about the market's expectations with regard to future changes in the overnight rate. There is therefore no term structure information in the OIS rate and hence it cannot help you determine Libors beyond a one day term as these embed a term structure premium which is absent from the OIS rate.

However, while OIS rates do not tell you about Libor, they do play a role in determining expectations of future Libor. This is because interest rate swaps (IRS) which exchange a fixed rate against future Libor payments, and which therefore embed information about future expectations of Libor, are now discounted using OIS rates. This is because OTC contracts like IRS are now subject to daily variation margin and so the counterparty risk is essentially an overnight risk.

This means that if you are to extract the "index curve" of a particular Libor rate from the prices of interest rate swaps, your valuation model needs to use OIS discounting if it is to be consistent with market practice. The difference between the implied Libor rates using Libor discounting and those using OIS discounting may be a few basis points, which can be considered significant for long maturity trades.

If you want to explore this using a model, I have built some analytics in FinancePy, a python finance library, and here is an example Jupyter notebook.

https://github.com/domokane/FinancePy/blob/master/notebooks/products/funding/FINIBORDUALCURVE_BuildingEuriborDualCurveFollowingQLExample.ipynb

• Thanks for this insight. However still I don't understand how really OIS rates can be used to derive the Libor for longer terms like 4y etc. Any workaround towards that derivation would be helpful. Oct 31 '20 at 16:59
• They can't by themselves. That's my main point.
– Dom
Nov 1 '20 at 12:10

This is my most up-to-date understanding of the matter:

(i) OIS Swaps are here to stay. Already today, in the US, there two types of OIS Swaps, ones indexed to the Effective Federal Funds Rate (EFFR) and ones indexed to the Secured Overnight Financing Rate (SOFR). Both these swaps work the way you have described (floating based on Geometric Average of daily overnight realized rates, etc.)

(ii) Currently, Libor rates and Swaps indexed to Libor rates live their own independent life. Libor rates are quoted daily based on a survey of Libor-contributing banks: these provide an estimate of "the rate they believe they could borrow at from other banks" (IBOR stands for Interbank Offer Rate).

(iii) Libor rates as we know them are planned to cease to exist in 2021 (see for example here) Instead, in the US, the current USD Libor will be replaced by SOFR rate + a fixed spread.

That's where the connection between OIS Swaps and Libor swaps arises: in the future, both USD Libor and USD OIS Swaps based on SOFR (not EFFR) will be indexed to the same underlying rate: i.e. the SOFR rate.

However, the mechanics of the USD Libor Swaps (based on SOFR) and USD OIS Swaps (based on SOFR) will still remain different: the OIS Swaps based on SOFR will still compound based on Geometric Average of overnight realized rates, whilst the USD Libor Swaps will work the same way as today's Libor swaps: just the underlying rate will be SOFR + fixed spread.

• Just a brief comment on the very last paragraph of your answer: the "Libor" swaps in the future will not exactly work as the Libor swaps work today. The main reason is that once they start referencing the SOFR, they will become backward looking by defintion (as you describe in (i) above), i.e., exact cashflows will only become known at the END of the accrual period, and not in advance (as is the case today). [1/2] Nov 2 '20 at 12:54
• NB: This is mainly because there is no "forward looking" SOFR -- but some people are asking for that (mainly derived by expectations priced in by the SOFR futures market). It's not sufficiently liquid yet I believe but I think ARRC plans to start publishing something like this in 2021. Moreover, this is more a consideration for trades conducted in the future. From a practical POV you might be right: the existing, legacy LIBOR trades in the book, might simply be "re-struck" using SOFR+spread instead of LIBOR, ignoring the timing discrepancy. 2021 will bring some more clarity here... [2/2] Nov 2 '20 at 12:58
• @KevinT: I am not sure I agree: SOFR is indeed backward looking in the sense that it is the average Secured Overnight Funding Rate "realized last night". But it is published every morning by the New York Fed. Similarly, the LIBOR is published every morning (based on a survey of banks, who estimate their forward-looking borrowing cost). But that's the entire point: the market wants to move away from forward-looking estimate, they want an undisputable "realized" rate. Nov 6 '20 at 7:23
• @KevinT As far as the mechanics of the swaps: that will be identical, in the sense that you just take the published "SOFR" every morning, instead of teh published Libor every morning: and you will use that published SOFR as your swap fixing (just like we do with the published Libor): the only difference is that a constant spread will be added... so I think the mechanics of the Libor swaps, once we switch the SOFR, will be just the same... Nov 6 '20 at 7:25
• I'd be interested to hear other people comment on this (might even open a new question); I think that in most working papers the calculation of the spread was referring to the compounded version of the overnight rate (e.g. isda.org/2020/07/21/…). That being said: Once the spread is fixed, your proposition (using a single but properly term-scaled version of the individual reset) sounds feasible, indeed. Nov 16 '20 at 16:55