FRA Discounting and the Choice Between LIBOR and OIS Curves
Summary
The document explains why the value of a forward rate agreement can include a discount factor based on the underlying forward LIBOR rate, and asks whether another money-market rate should be used instead. It frames discounting as a choice tied to the market’s assumed fundamental rates. Before the financial crisis, LIBOR was often treated as a base rate; recognition of its credit risk helped drive broader use of overnight indexed swap curves for discounting.
In a multi-curve framework, LIBOR can still determine the forward rate while a separate curve, such as OIS, discounts the cash flow. The response offers collateralization as a practical guide: collateralized transactions commonly use OIS or an equivalent curve, while LIBOR discounting may be used for non-collateralized transactions. This is a rule of thumb rather than a complete valuation framework; the appropriate curves depend on contract terms and market conventions.
Key ideas
- The FRA payoff formula discounts the rate difference over the accrual period.
- The discount curve depends on the market framework and the transaction’s funding and collateral assumptions.
- Post-crisis practice increasingly separated LIBOR forwarding from OIS discounting.
- A multi-curve approach uses one curve to project rates and another to discount cash flows.
- Collateralization provides a useful rule of thumb for curve choice, but it does not replace contract-specific conventions.
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Full text
# meaning of discount term in FRA value
# meaning of discount term in FRA value
Consider a forward rate agreement on LIBOR (say), which starts 2 months from now, expires after 3 months and has strike $K$, and is based on $3M$ LIBOR -- $FRA_{2\times 5}$. Now the present value of this contract is,
\begin{equation} \frac{\alpha\cdot (L_{3m}(2m) - K)}{1 + \alpha\cdot L_{3m}(2m)} \end{equation}
where $\alpha$ is the relevant day count fraction, $L_{3m}(2m)$ refers to the 3 month LIBOR rate in 2 months time (assume unit notional for ease).
I see the term $1 + \alpha\cdot L_{3m}(2m)$ in the above equation as a discount rate applied to the pay out $L_{3m}(2m) - K$.
Why do we apply the $L_{3m}(2m)$ rate for the discounting? Why not apply a short term money market rate or any other rate for that matter -- I guess I am also asking how we choose the rate that we apply for the discounting?
## Answer by Probilitator (score 3, accepted)
https://quant.stackexchange.com/a/11493
A very good and up-to-date question.
Whether you use the LIBOR-rate or any other rate for discounting depends on what you decide to be the fundamental rates in the market.
Before the crisis LIBOR-rates were mostly seen as the fundamental market rates (or the "risk-neutral" rates). After the crisis it turned out that these rates were not completely free of default risk. This is why noawadys an increasing number of banks has started to use the OIS-curve as the base curve.
In the paper LIBOR vs. OIS: The Derivatives Discounting Dilemma Hull and White present a solid case in favor of OIS discounting.
For the concrete application to forward-contracts this entails that one might end up using a different rate than LIBOR for discounting - e.g. ones base on overnight indexed swaps (OIS). Thus you will have to work with two curves instead of one (LIBOR + the curve you use for discounting)
As always Fabio Mercurio already did most of the work again. You will find an approah on how to deal with forward-contracts in a multi-curve setting in his paper: LIBOR Market Models with Stochastic Basis
> An easy rule of thumb when to use LIBOR and when the e.g. OIS-curve: When the transactions involved in the product are collaterized $\to$ use OIS or an equivalent. For non-collaterized transaction discounting with LIBOR can be used.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.