# analytical formula for FV of fixed rate of a IRS [closed]

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)? ## 1 Answer 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:

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