How OIS Discounting Changes Swap Rate Weights
Summary
A swap rate can be expressed as a weighted average of forward Libor rates, with discount factors determining the relative weights. The discussion corrects the accrual-period indexing in the question: each six-month forward rate applies to the period ending on its payment date. It then compares the weights produced by OIS discounting with those from a non-OIS discount curve.
Assuming no convexity adjustment between the forward rates used in the two calculations, the difference between swap rates comes from the changed weights. If non-OIS rates exceed OIS rates, the OIS discount factors can put relatively more weight on later payments. On an upward-sloping forward curve, that can make the OIS-discounted swap rate higher. This is a conditional explanation, not a universal ranking: the result depends on the curve shape, discount factors, and the simplifying assumption. The excerpt provides no market data or numerical example to quantify the effect.
Key ideas
- A par swap rate is a discount-factor-weighted average of forward rates.
- OIS and non-OIS discounting can produce different relative weights across payment dates.
- With an upward-sloping forward curve, greater weight on later rates can raise the OIS-discounted rate.
- The comparison assumes no convexity adjustment and depends on market curve conditions.
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Full text
# Swap rates comparison
# Swap rates comparison
I'd like to compare the swap rates using OIS discounting $S_{a,b}^{OIS, 6M}$ and a swap rate $S_{a,b}$ not using OIS discounting: $$S_{a,b}^{OIS, 6M}=\sum_{a+1}^b L_{6M}(0,T_i,T_{i+6M})\times df^{OIS}(O,T_i)/\sum_{a+1}^b df^{OIS}(O,T_i)$$ and: $$S_{a,b}=\sum_{a+1}^b L(0,T_i,T_{i+6M})\times df(O,T_i)/\sum_{a+1}^b df(O,T_i)$$
Let's suppose for simplification that the rates are positive. Is there a way to compare both in general? Thanks you in advance.
## Answer by Antoine Conze (score 0, accepted)
https://quant.stackexchange.com/a/34441
Please note that there is a slight indexing error in your formulas: for standard upfront swaps the libor rate paid on $T_i$ covers the period $[T_{i-6M}, T_i]$ so the correct formula for the swap rate is $$ S_{a,b}^{OIS,6M} = \sum_{i=a+1}^b L_{6M}(0, T_{i-6M}, T_{i}) \times df^{OIS}(0, T_i)/\sum_{i=a+1}^b df^{OIS}(0, T_i) $$
As can be seen from this formula, the swap rate is a weighted average of forward libor rates $L_{6M}(0, T_{i-6M}, T_{i})$, the weights being the discount factors divided by the fixed leg PV01: $$ w_i^{OIS} = df^{OIS}(0, T_i)/\sum_{k=a+1}^b df^{OIS}(0, T_k)$$ $$ w_i = df(0, T_i)/\sum_{k=a+1}^b df(0, T_k) $$
Assuming there is no convexity adjustment between OIS discounting forward libor rates and non OIS discounting forward libor rates, the difference between the OIS discounting swap rate and the non OIS discounting swap rate will result from different relative weights.
For instance if your non OIS rates are above OIS rates, then libor rates for short maturities will have a smaller weight in the OIS case than in the non OIS case. In a market configuration where the libor curve is increasing with maturity then the OIS discounting swap rate will be above the non OIS discounting swap rate.
## Answer by dm63 (score 0)
https://quant.stackexchange.com/a/34440
As you can see the swap rate is a weighted average of the forward Libors. Since Libor>OIS, we typically have df(OIS)>df, so the OIS discounted swap rate is more heavily weighted towards the back end of the swap. In an upward sloping yield curve which we currently have, this means that the OIS discounted swap rate is slightly higher than the non OIS.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.