What OIS Curves Tell You About Libor and Swap Discounting

# OIS rate to build Term structure


# 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.

Thanks for your time.

## Answer by Dom (score 3)

https://quant.stackexchange.com/a/59047

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

## Answer by Jan Stuller (score 1)

https://quant.stackexchange.com/a/59046

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.