Converting Between Three-Month and Six-Month Swap Rates
Summary
The document explains how to approximate a one-year fixed swap rate indexed to three-month Libor from a six-month Libor swap rate and a three-month versus six-month basis spread. At a first pass, subtract the basis spread from the six-month fixed rate to estimate the equivalent three-month fixed rate. The example illustrates the direction and approximate size of that adjustment.
For a more accurate conversion, the basis spread must be translated across payment frequencies and day-count conventions. The example contrasts semiannual 30/360 fixed payments with quarterly Act/360 floating payments; converting the spread’s day-count basis and then matching the quarterly stream to a semiannual equivalent can shift the result. Exact conversion depends on discount factors for the cash flows, so the simple subtraction is only an approximation. The source does not provide those discount factors or a full curve construction, limiting how precisely its example can be reproduced.
Key ideas
- A basis spread can provide a first approximation for converting between swap rates indexed to different tenors.
- Subtract the basis spread from the six-month indexed fixed rate when the quoted spread is paid on the three-month leg in the stated direction.
- Payment frequency and day-count conventions affect the equivalent fixed rate.
- An exact conversion requires discount factors for the relevant cash flows.
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Full text
# Convert 3M rates to 6M rates using Basis Swaps (3M vs 6M) # Convert 3M rates to 6M rates using Basis Swaps (3M vs 6M) How can I convert a 6M Libor rate e.g. 1Y Tenor to a 3M Libor rate using a basis swap 3M vs. 6M? I wanted to know the math and also an example would be great. Update: Example: > `6M Swap 1Y Tenor: 1.925 3M Swap 1Y Tenor: 1.77109 3v6M Basis Swap 1Y Tenor = 15.625 ` ``` 6M Swap 1Y Tenor: 1.925 3M Swap 1Y Tenor: 1.77109 3v6M Basis Swap 1Y Tenor = 15.625 ``` When calculating now the 3N Swap 1Y Tenor based on the 6M Swap and the Basis Swap I receive the following value: `Calc 3M Swap 1Y Tenor: 1.76875` which is a relative differdnce og 0.13%. The day count conventions are the same so I am not sure why I receive this kind of difference. ## Answer by dm63 (score 2, accepted) https://quant.stackexchange.com/a/37312 Let's say 1yr semiannual rate versus 6m Libor is 2.00% and 1yr basis swap is 6m libor = 3m libor + 15bp. Then , to a first approximation 1yr rate versus 3m libor is 2.00-0.15= 1.85%. More precisely , we have to take into account daycount conventions. So, we know that a swap consisting of 2.00% semiannual 30/360 daycount versus 3m libor +15 bp quarterly Act/360 is a fair swap, since both sides are equivalent to 6m libor. So the fixed rate equivalent of 3m libor is actually 2.00% minus the semiannual equivalent of 15bp quarterly Act/360. This conversion is not exactly solvable without having the discount factors for all the cash flows, but an approximation would be to first convert the 15bp to 30/360 daycount by calculating 15*365/360. Then you need to find the semiannual stream equivalent to the above quarterly stream. You might end up with 15.5bp instead of 15bp, so the answer would be 1.845%.
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