How do I back out the swap rate from a spot rate curve?

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AcadiFi · Accepted Answer

The par swap rate C for an N-period swap is the coupon that would price a fixed-rate bond to par using the zero curve. The formula is C = (1 - Z_N) / sum(Z_i) where Z_i are discount factors derived from spot rates. Consider a 3-year semiannual swap with semiannual spot rates: 1.00%, 1.05%, 1.10%, 1.15%, 1.20%, 1.25% (periodic, already halved). Discount factors are 1/(1.01), 1/(1.0105)^2, 1/(1.0110)^3, 1/(1.0115)^4, 1/(1.0120)^5, 1/(1.0125)^6 = 0.9901, 0.9793, 0.9677, 0.9552, 0.9419, 0.9279. Sum = 5.7621. Numerator = 1 - 0.9279 = 0.0721. Semiannual swap rate = 0.0721 / 5.7621 = 1.2513%. Annualized swap rate ≈ 2.50%. ```mermaid flowchart LR A[Spot
Curve] --> B[Discount
Factors] B --> C[Annuity
Sum] B --> D[Final
DF] C --> E[Swap
Rate] D --> E ``` Key intuition: the swap rate is a weighted average of forward rates, with weights equal to each period's discount factor. When the curve is upward sloping, the swap rate sits between the short spot and the long spot but closer to the long end because the larger forwards pull it up.

How do I back out the swap rate from a spot rate curve?

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