# 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?

• Discounting of cashflows is most often done with an OIS curve. It does not use IBOR at all. Additionally mixing LIBOR tenors (different LIBOR indexes) e.g. 1M, 3M, 6M, etc, is a very bad and fundamentally incorrect way of discounting, and to extend it via your interpolation scheme is, unfortunately, even worse. I'm sure any book on swaps and many papers will have details. Perhaps someone can suggest something.. – Attack68 Nov 26 '19 at 14:53
• @Attack68 Thank you for your comment. If I am not mistaken, in an OIS, one leg pays an agreed fixed rate and the other a compounded daily reference rate. So, are you saying that this fixed rates agreed in an OIS of a given maturity is the appropriate rate to use for discounting cash flows for that maturity? But aren't OIS collateralized / margined? In that case they do not reflect the same risk premium as a bullet payment on an uncollateralised loan. – Confounded Nov 26 '19 at 15:29

## 1 Answer

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.