How do you bootstrap zero-coupon rates from coupon bond prices step by step?

Question

AcadiFi · Accepted Answer

Bootstrapping the zero-coupon yield curve is a core FRM Part I skill. The process extracts implied spot rates from observed prices of coupon-bearing bonds, working from the shortest maturity to the longest. Each step uses previously derived rates to solve for the next unknown rate.

**The Bootstrapping Process:**

```mermaid
graph TD
  A[Start: Observe market prices of coupon bonds] --> B[Step 1: 1-Year Bond]
  B --> B1[Price = Face / 1+z1 -- Solve for z1]
  B1 --> C[Step 2: 2-Year Bond]
  C --> C1[Price = C1/1+z1 + C2+Face / 1+z2 squared -- Solve for z2]
  C1 --> D[Step 3: 3-Year Bond]
  D --> D1[Price = C1/1+z1 + C2/1+z2 squared + C3+Face / 1+z3 cubed -- Solve for z3]
  D1 --> E[Continue for each maturity...]
  E --> F[Complete Zero Curve: z1, z2, z3, ...]
  style A fill:#c9a84c,color:#0a0a0f
  style B1 fill:#1e3a5f,color:#fff
  style C1 fill:#1e3a5f,color:#fff
  style D1 fill:#1e3a5f,color:#fff
  style F fill:#1a3c2a,color:#fff
```

**Worked Example:**

Suppose we observe the following bonds issued by the Sovereign Treasury, each with annual coupon payments and $100 face value:

| Maturity | Coupon Rate | Market Price |
|---|---|---|
| 1 Year | 3.00% | $99.52 |
| 2 Year | 4.00% | $99.10 |
| 3 Year | 4.50% | $98.27 |

**Step 1 — Solve for z1 (1-year spot rate):**
The 1-year bond pays $103 at maturity (100 face + 3 coupon).
$99.52 = $103 / (1 + z1)
1 + z1 = 103 / 99.52 = 1.03497
**z1 = 3.497%**

**Step 2 — Solve for z2 (2-year spot rate):**
The 2-year bond pays $4 coupon at year 1 and $104 at year 2.
$99.10 = $4 / (1.03497) + $104 / (1 + z2)^2
$99.10 = $3.8648 + $104 / (1 + z2)^2
$104 / (1 + z2)^2 = $95.2352
(1 + z2)^2 = 104 / 95.2352 = 1.09194
1 + z2 = 1.04496
**z2 = 4.496%**

**Step 3 — Solve for z3 (3-year spot rate):**
The 3-year bond pays $4.50 at year 1, $4.50 at year 2, and $104.50 at year 3.
$98.27 = $4.50 / (1.03497) + $4.50 / (1.04496)^2 + $104.50 / (1 + z3)^3
$98.27 = $4.3480 + $4.1203 + $104.50 / (1 + z3)^3
$104.50 / (1 + z3)^3 = $89.8017
(1 + z3)^3 = 104.50 / 89.8017 = 1.16369
1 + z3 = 1.05178
**z3 = 5.178%**

**Resulting Zero Curve:**

| Maturity | Zero Rate (Spot Rate) |
|---|---|
| 1 Year | 3.497% |
| 2 Year | 4.496% |
| 3 Year | 5.178% |

Notice the zero curve is upward-sloping and steeper than the coupon rates, which is typical when the yield curve slopes upward — this is because coupon bonds blend rates across all maturities.

**Forward Rates from Spot Rates:**
Once you have spot rates, you can derive implied forward rates. The 1-year forward rate one year from now:
(1 + z2)^2 = (1 + z1)(1 + f(1,2))
(1.04496)^2 = (1.03497)(1 + f(1,2))
f(1,2) = 1.09194 / 1.03497 - 1 = **5.503%**

**Exam Tips:**
1. Always start with the shortest maturity and work forward
2. Use previously solved spot rates as inputs — one error propagates through all subsequent rates
3. If given a zero-coupon bond, the spot rate comes directly from the price without bootstrapping

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How do you bootstrap zero-coupon rates from coupon bond prices step by step?

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