Par rate of Interest Rate Swap

Question

I'm interested in deriving the par rate of an interest rate swap priced under the single curve framework. Let's follow the corresponding Wikipedia article for the sake of notation simplicity.

The present value of the fixed leg can be calculated as $$PV_{fixed} = NR\sum_{i=1}^{n_1}d_i x_i,$$ where $N$ is the notional, $R$ is the fixed rate, $n_1$ is the number of payments of the fixed leg, $d_1$ is the decimalised day count fraction of the accrual in the $i$'th period and $x_i$ is the corresponding discounting factor.

Similarly the present value of the floating leg is given by $$PV_{float} = N\sum_{j=1}^{n_2}r_j d_j x_j,$$ where $n_2$ is the number of payments of the floating leg and $r_j$ are the forecasting (forward) rates.

In order to find the par rate we set $PV_{fixed} - PV_{float} = 0$ and solve for $R$, the resulting expression is $$R = \frac{\sum_{j=1}^{n_2} r_j d_j x_j}{\sum_{i=1}^{n_1} d_i x_i}.$$

However, Wikipedia claims that under the single curve framework this expression can be simplified further to $$R = \frac{x_0 - x_{n_2}}{\sum_{i=1}^{n_1} d_i x_i}$$

The above expression isn't obvious to me. How do we conclude that $\sum_{j=1}^{n_2} r_j d_j x_j = x_0 - x_{n_2}$?

Answer 1

It follows from the fact that, under the single-curve framework, the projected rate $r_j$ is found via:

$$r_j = \frac{\frac{x_{j-1}}{x_j} - 1}{d_j}$$

If you plug in this expression for $r_j$ in the summation then all terms will cancel out except for $x_0$ and $x_{n_2}$.