How is a plain vanilla fixed-for-floating interest rate swap priced at inception?

Question

AcadiFi · Accepted Answer

A plain vanilla IRS is structured so present value of fixed leg equals PV of floating leg at inception — giving zero NPV. Harborview Asset Management enters a 3-year $50M swap with Crestwood Securities. Using OIS discount factors: D(0.5)=0.9802, D(1.0)=0.9604, D(1.5)=0.9404, D(2.0)=0.9203, D(2.5)=0.9001, D(3.0)=0.8800.

The floating leg PV equals notional × (1 − D(T)) = $50M × (1 − 0.8800) = $6,000,000 (assuming par floater assumption).

The fixed rate K solves: K × notional × Σ τᵢ × D(tᵢ) = floating PV.

Sum of discount factors × day fractions (0.5 each) = 0.5 × (0.9802 + 0.9604 + 0.9404 + 0.9203 + 0.9001 + 0.8800) = 2.7907.

K = $6,000,000 / ($50M × 2.7907) = 4.3001%.

This is the at-market swap rate — Harborview pays 4.30% fixed, receives floating. Zero value at t=0.

```mermaid
flowchart TD
  A[Build OIS curve] --> B[Compute D_t]
  B --> C[Float PV = N times 1 minus D_T]
  B --> D[Annuity = sum tau D]
  C --> E[Swap rate K]
  D --> E
```

Post-inception, as rates move, the swap develops positive or negative MTM to each party. The key insight: a floating-rate bond trades at par on reset dates, so valuing the floating leg reduces to notional minus its PV at maturity.

How is a plain vanilla fixed-for-floating interest rate swap priced at inception?

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