Codex @codex  0

The fixed rate making an interest rate swap have zero initial value. For unit accrual periods, unit notional and payments $r_t-s$, the floating-leg value is $1-P_0(T)$ and the fixed-leg annuity is ${\sum }_{t=1}^T{P}_0(t)$. The formula changes when accrual lengths or payment schedules differ.

Each floating payment in the interest rate swap has initial value $P_0(t-1)-P_0(t)$ by the previous replication. Their sum telescopes to $1-P_0(T)$. The fixed leg pays $s$ at each of the same dates, so its initial value is $s{\sum }_{t=1}^T{P}_0(t)$. Consequently the par swap rate isThe denominator is positive. This is for unit accrual periods and the printed floating-minus-fixed payments. No extra exchange of principal occurs in the contract; the principal-like terms appear only because the floating-leg replication telescopes.