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)?


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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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