# Back of the enveloppe forward irs pricing

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!

$$term = head \cdot tail \Longrightarrow (1+\frac{r_1}{f})^{f t_1} = (1+\frac{r_{t_2}}{f})^{ft_2} \cdot (1+\frac{\color{red}{r_{t_3}}}{f})^{ft_3}$$
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.
=100*(((1-0.00032)^8/(1-0.0016)^5)^(1/3)-1)