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Multicurve Discounting and the OIS–LIBOR Term Basis

Article Quant Q&A · Author: Confounded
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Summary

The document asks how to discount a cash flow at a tenor beyond the longest available LIBOR tenor, and whether interpolating quoted rates is appropriate when those rates reflect different payment periods. The answer frames the problem through the distinction between overnight lending rolled over time and lending at a fixed term. An overnight indexed swap rate reflects the compounded cost of rolling overnight exposure, while a term LIBOR rate includes the value of committing funds for a fixed period without the same opportunity to withdraw or reallocate them.

The difference is described as the OIS–LIBOR basis, associated with the term value of money and the flexibility embedded in rolling overnight loans. The suggested practical approach is to calculate the compounded OIS rate for the target horizon and add an appropriate term basis, potentially extrapolated from observed relationships across shorter tenors. The explanation is conceptual; it does not specify a curve-building procedure, interpolation method, or calibration evidence, so the basis estimate remains a modeling choice.

Key ideas

  • Overnight indexed swap rates reflect overnight lending exposure rolled across the horizon.
  • A fixed-term loan differs because the lender cannot freely withdraw or reallocate funds during that term.
  • The OIS–LIBOR basis captures a difference between rolling overnight rates and fixed-tenor term rates.
  • For a longer discount horizon, the answer proposes combining a compounded OIS rate with an estimated term basis.
  • The document does not provide a quantitative calibration or validated extrapolation method.

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Fixed incomeDerivatives pricingStatistics
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# Discounting in multicurve environment


# Discounting in multicurve environment












Given that the longest tenor for LIBOR in 12M, what rate do we use to discount a cash flow due in, say, 18 months? Some suggest that we use the 12M rates for 1 year and 2 year to interpolate, but these rates assume that there will be a payment at the 12 months point, whereas our cash flow doesn't.

It seems to me that we need to model how the risk premium / spread changes as a function of Tenor (and possibly maturity) and then apply this to calculate appropriate 18M rate. For example, say the difference between 6M rate at 6 month and 12M rate at 1 year is 50bp, then (assuming additive relationship for simplicity of exposition) we would then add this 50bp to the 12M rate at 1 year to arrive at a rate for discounting the cash flow at 18M. (In effect, this assumes that extending the liquidity horizon by 6 months from 1 year to 18 months cost the same a extending it by 6 months from 6 month to 1 year).

However, I haven't really seen this being done / discussed in the literature. How is this done in practice?

## Answer by Attack68 (score 1)

https://quant.stackexchange.com/a/49964

OIS rates, and the OIS fixing, reflect unsecured lending on an overnight basis.

OIS rates compounded for 1Y reflect unsecured lending for a 1Y period via rolling overnight loans for a 1 year period. If, on one any of these days, the counterparty defaulted this would result in a default of the loan.

The difference between this and unsecured lending for a fixed period of 1Y (within which the counterparty also might default) is that you, as lender, cannot choose to reallocate your funds at any point within the 1Y, if choosing fixed tenor, via prohibiting rolling over of the funds to the next day.

Therefore, financially speaking, you might say that opting to loan money for a 1Y period with overnight rolling is the same as lending for a fixed 1Y period and purchasing an option to withdraw the loan at any time.

The "purchase of this option" is reflected in the fact that you will technically accrue more interest by lending for 1Y fixed without the option that lending for 1y on a rolling basis. I.e. this is the OIS-LIBOR basis.

If there is a relationship (which there is) between the OIS-LIBOR basis for 3M and for 6M and for 12M, then you might well be able to extrapolate the expectation for the OIS-LIBOR basis for 18M based on the same fundamental concept.

I.e. you calculate the OIS compounded rate for 18M (annualised) and add the appropriate basis reflecting the term value of money.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.