How do I bootstrap a swap curve from par swap rates?

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AcadiFi · Accepted Answer

Bootstrapping converts par swap quotes into zero-coupon discount factors sequentially, shortest to longest. A par swap has PV of zero at inception...

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Example with quoted rates: 1Y = 3.20%, 2Y = 3.45%, 3Y = 3.60%. Assume annual payments, notional $1.

Step 1 — 1Y zero: Z(1) = 1 / (1 + 0.0320) = 0.96899.

Step 2 — 2Y zero: solve 0.0345 × Z(1) + (1 + 0.0345) × Z(2) = 1. Plugging Z(1): 0.03343 + 1.0345 × Z(2) = 1, so Z(2) = 0.93442. Convert to zero rate: (1/0.93442)^(1/2) − 1 = 3.455%.

Step 3 — 3Y zero: 0.0360 × [Z(1) + Z(2)] + 1.0360 × Z(3) = 1, giving Z(3) ≈ 0.89851, zero rate ≈ 3.615%.

For missing tenors (e.g., 4Y between 3Y and 5Y), interpolate linearly on zero rates or use cubic splines. Modern practice uses OIS discounting with separate forecasting curves — we cover this multi-curve framework in the Derivatives module.

```mermaid
flowchart LR
    A[Par Rates] --> B[Bootstrap 1Y]
    B --> C[Bootstrap 2Y]
    C --> D[Bootstrap 3Y+]
    D --> E[Zero Curve]
```

How do I bootstrap a swap curve from par swap rates?

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