# Compute forward swap rate from spot swap rate?

I am pretty new to interest rate swap and this question might sound silly.

Why does 3y2y (3yr forward and 2yr tenor) swap rate roughly equal to (5*5y swap - 3*3y swap) / (5-3)?

The 5y swap is effectively the weighted average of the 2y and 3y2y swap if discount rates are low enough and maturities are short enough. Lets suppose your discount curve is flat and zero, so all we have is a set of forecast libors from which we calculate fair swap rates. Then we know the following: $$S_{5y} = \frac{1}{4\times5}\sum_{i=0}^{i=4\times5}F_i$$ $$S_{3y} = \frac{1}{4\times3}\sum_{i=0}^{i=4\times3}F_i$$ $$S_{3y2y} = \frac{1}{4\times2}\sum_{i=4\times3+1}^{i=4\times5}F_i$$

Its easy to see then that $$2 \times S_{3y2y} = 5 \times S_{5y} - 3 \times S_{3y}$$ and so the closer our discount curve is to flat and zero the closer this relationship is to true.

Let's have a look at this in a single curve world. The fair rate for a swap starting in $$t$$ and ending in $$T$$, $$s_{t,T}$$ is set such that the present values of the floating and the fixed leg are equal, i.e.:

$$s_{t,T}=\frac{\sum_{u=t}^T \tilde{\Delta_u} F_{u-\tilde{\Delta_u}, u}D_u}{\sum_{u=t}^{T}\Delta_uD_u}$$ where $$\tilde{\Delta}$$ and $$\Delta$$ are the legs' accrual period factors and $$D_u$$ is the discount factor for a cashflow at time $$u$$.

To a first order of approximation, for small $$u$$ or in a low interest rate environment, $$D_u\approx 1$$, i.e. $$\sum_{u=t}^{T}\Delta_uD_u\approx T-t$$.

Hence,

\begin{align} s_{t,T}&\approx \frac{Ts_{0,T}-ts_{0,t}}{T-t}\\ &=\frac{T\frac{\sum_{u=0}^T \tilde{\Delta_u} F_{u-\tilde{\Delta_u},u}D_u}{\sum_{u=0}^{T}\Delta_uD_u}-t\frac{\sum_{u=0}^t \tilde{\Delta_u} F_{u-\tilde{\Delta_u},u}D_u}{\sum_{u=0}^{t}\Delta_uD_u}}{T-t}\\ &\approx \frac{T\frac{\sum_{u=0}^T \tilde{\Delta_u} F_{u-\tilde{\Delta_u},u}D_u}{T}-t\frac{\sum_{u=0}^t \tilde{\Delta_u} F_{u-\tilde{\Delta_u},u}D_u}{t}}{T-t}\\ &= \frac{\sum_{u=0}^T \tilde{\Delta_u} F_{u-\tilde{\Delta_u},u}D_u-\sum_{u=0}^t \tilde{\Delta_u} F_{u-\tilde{\Delta_u},u}D_u}{T-t}\\ &=\frac{\sum_{u=t}^T \tilde{\Delta_u} F_{u-\tilde{\Delta_u},u}D_u}{T-t}\\ &\approx \frac{\sum_{u=t}^T \tilde{\Delta_u} F_{u-\tilde{\Delta_u},u}D_u}{{\sum_{u=t}^{T}\Delta_uD_u}} \end{align}

In the first step, we apply your formula, and in the third step we use the approximation $$D_u\approx 1$$. In the last step, we apply that approximation again, but this time in the other direction "$$1\approx D_u$$".

HTH?