Determination of Fair Fixed Swap Rate

Scenario

In an interest rate swap (IRS), party A pays a fixed coupon $R$ and receives floating SOFR/LIBOR rates on payment dates $T_1, T_2, \dots, T_N$ with accrual tenors $\tau_i = T_i - T_{i-1}$.

Your Task

1. By utilizing the par bond telescoping property of the floating leg, express the present value of the floating leg strictly in terms of the terminal discount factor $P(0, T_N)$.
2. Express the present value of the fixed leg in terms of $R$ and the discount factors $P(0, T_i)$.
3. Equate both legs to derive the fair fixed swap rate $R^*$: $R^* = \frac{1 - P(0, T_N)}{\sum_{i=1}^N \tau_i P(0, T_i)}$
4. Given semi-annual tenors $\tau_i = 0.5$ for a 2-year swap ($N = 4$) with discount factors: $P(0, 0.5) = 0.9800, P(0, 1.0) = 0.9580, P(0, 1.5) = 0.9350, P(0, 2.0) = 0.9100$, compute the annualized fair swap rate.

Constraints

• Standard plain-vanilla interest rate swap with notional $N_0 = 1$.
• Zero initial value at inception.

Examples Floating leg value equals $1 - P(0, T_N)$. Annuity factor equals $\sum \tau_i P(0, T_i)$.