Swaps, FRAs and interest parity

Interest rate swap, $V_{\text{payer}} = B_{\text{float}} - B_{\text{fixed}}$ — Paying fixed and receiving floating is economically identical to being short a fixed-rate bond and long a floating-rate one, on the same notional and schedule. The notional never changes hands, but since it would cancel on both legs, pretending it does costs nothing and makes the decomposition exact.

On each reset date a floating-rate bond is worth exactly its notional, because the coupon it will pay is precisely the rate used to discount it. That single fact collapses the floating leg to a number and makes the whole swap tractable: at inception the swap is worth zero, so the fixed leg must also be worth par, which determines the swap rate.

Holds when

• Between resets the floating leg is par plus the accrued difference, which is a small correction.
• Modern practice discounts on a separate curve from the one projecting the forwards, which is a refinement rather than a change of principle.

Set the fixed leg equal to par and solve for the coupon.

1. $$1 = s\sum_{i=1}^{n} \tau_i D(t_i) + D(t_n)$$

   Fixed coupons plus the notional at the end, discounted, must equal par.

2. $$s = \frac{1 - D(t_n)}{\sum_i \tau_i D(t_i)}$$

   The denominator is the annuity factor, often written $A$.

Annual discount factors are $D(1) = 0.9709$, $D(2) = 0.9426$ and $D(3) = 0.9151$. What is the three-year swap rate?

FRA — A single-period swap: you agree today the rate for a deposit that starts later. Its fair rate is exactly the forward rate implied by the curve, and a swap is a strip of FRAs — which is the cleanest way to see why a swap rate is an average of forward rates weighted by the discount factors.