Why Forward Swap Rates Differ from Same-Tenor Forward Rates
Summary
The document distinguishes a forward rate over a future period from the fixed rate on a swap that begins in the future and spans the same tenor. The forward rate is derived from discount factors at the period’s endpoints, while a forward starting swap rate is a discounted average of the forward rates for its payment periods. Because the swap rate incorporates discount factors across the swap’s payment dates, the two quantities can differ even when their start and end dates match. The discussion notes that a flat curve is an obvious case in which they coincide.
It also clarifies the underlying of a swaption: a swaption gives the right to enter a swap, so its underlying rate is the rate on the corresponding forward starting swap, rather than a single-period forward rate. The explanation is conceptual and uses a simplified curve setup. It does not develop valuation formulas for swaptions or address market conventions, curve construction, or complications such as multiple discounting curves.
Key ideas
- A forward rate and a same-tenor forward starting swap rate are different quantities.
- A forward swap rate reflects discounted rates across the swap’s payment periods.
- A flat rate curve is an example where the forward rate and swap rate coincide.
- A swaption’s underlying rate is the rate on the forward starting swap it gives the holder a right to enter.
- The discussion does not cover market conventions or full swaption valuation.
Tags
Full text
# Relationship Between Forward Starting Swap Rates and Same-Tenor Forward Rates
# Relationship Between Forward Starting Swap Rates and Same-Tenor Forward Rates
New to all this. I'm playing around with a very basic swap pricer (in Excel, whose only input is the swap rate curve), and I noticed that although the 2y2y forward rate $f_{2,4}=\left(\frac{DF_2}{DF_4}\right)^{\frac{1}{4-2}}-1$ and the 2y forward starting 2y swap rate $R_{2,4}=\frac{\sum_{i=3}^{4}DF_{i}f_{i-1,i}}{\sum_{i=3}^{4}DF_{i}}$ were quite similar, they were off by a bit. I figured it was a rounding matter, so I tried doing it again with $f_{1,3}$ and $R_{1,3}$ and it seemed to persist. Clearly, they don't seem to be the same. But why? I'm leaning towards the fact that the swap rate depends on $DF_3$ (I assume more generally on the segment between Year 2 and Year 4) but the forward rate doesn't. Is there a specific shape of the swap curve that would make $f_{2,4}-R_{2,4}=0$?
Extension: This also made me think about swaptions. I understand that the underlying of an option is a forward (e.g., underlying of FX forward for an FX option). However, I feel a bit confused in rates space, which has to do with my question above: is the underlying of a swaption (e.g., 1y10y) the 1y10y forward rate $f_{1,10}$ or the rate on a 1y forward starting 10y swap $R_{1,10}$? Leaning towards the latter since a swaption is an option on... a swap.
## Answer by dm63 (score 2, accepted)
https://quant.stackexchange.com/a/81335
"I'm leaning towards the fact that the swap rate depends on DF3 (I assume more generally on the segment between Year 2 and Year 4) but the forward rate doesn't" --Correct
"Is there a specific shape of the swap curve that would make f2,4−R2,4=0 ?"- the most obvious example is a flat curve where all rates are the same
"is the underlying of a swaption (e.g., 1y10y) the 1y10y forward rate f1,10 or the rate on a 1y forward starting 10y swap R1,10 ? Leaning towards the latter since a swaption is an option on... a swap." -- CorrectShown 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.