Convert 3M rates to 6M rates using Basis Swaps (3M vs 6M)

Question

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

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 1

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%.