IRS plain vanilla - expiry in 5 years - principal is 1$ - semianual payment

How could the analytical formula be derived for the fair value of the fixed rate (initially no value of the swap)?


The key inputs to this calculation are two yield curves obtained from market data: $\{v_i\}$ the discounting factors (value today of \$1 received at time i) and $\{r_i\}$ the forecasting curve (forward semiannual rates for period i to i+1).

The calculation itself proceeds as follows. There are two legs to a fixed/floating interest rate swap.

The fixed leg, which has a present value (PV) equal to the sum of its cashflows discounted based on their payment date:

$PV = N R \sum_i d_i v_i $

for $N$ the notiional, and $d$ the day count fraction and $v$ the discount period for period $i$.

The floating leg has a present value:

$PV = N \sum_j d_j v_j r_j $

for $r$ the floating rate forecast for the period $j$. $i$ and $j$ differ if the schedules are not aligned.

The to derive the mid-maket rate you set these equal to each other:

$ \implies R = \frac{\sum_j v_j d_j r_j}{\sum_i v_i d_i} $


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