trying to have a back of the enveloppe way of working out generic forward starting swap rates like 2y2y or 5y3y to put in a spreadsheet without too much loss of accuracy. Whats a good way to look at it / work it out for exple 3y2y? Thanks all!
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Swap rates are essentially weighted averages of forwards. If you have a swap curve ready at hand then you can calculate the forwards as follows.
$$term = head \cdot tail \Longrightarrow (1+\frac{r_1}{f})^{f t_1} = (1+\frac{r_{t_2}}{f})^{ft_2} \cdot (1+\frac{\color{red}{r_{t_3}}}{f})^{ft_3} $$
where $f$ is the fixed leg payment frequency, $t_i$ the time to maturity for each component and $r_i$ the fixed interest rate.
So if we take your 5y3y example, we would have $t_1 = 8$ (term), $t_2 = 5$ (head) and $t_3 = 3$ (tail, i.e. our forward rate). Since you have the spot starting swaps you only have one unknown variables, $r_3$ which is the forward rate starting in 5y and maturing in 8y from today. Approximately, for EUR we'd have an 8y rate of -3.2bp, 5y at -1.6bp and payment frequency is annual. The rest is just simple algebra.
Here's an example in Excel:
=100*(((1-0.00032)^8/(1-0.0016)^5)^(1/3)-1)
