How do you bootstrap spot rates from a par yield curve? I keep getting the wrong numbers.

Bootstrapping is mechanical once you see the pattern. Let me walk through it carefully.

The Setup

Suppose the par yield curve for annual-pay government bonds is:

Maturity	Par Yield
1 year	4.00%
2 year	4.40%
3 year	4.70%

Each par yield means a bond with that coupon rate trades at exactly 100 (par). We need to find the zero-coupon spot rates z_1, z_2, z_3.

Step 1: 1-Year Spot Rate

The 1-year par bond has a single cash flow: 104 at year-end.

100 = 104 / (1 + z_1) z_1 = 4.00%

The 1-year spot rate equals the 1-year par yield. Always.

Step 2: 2-Year Spot Rate

The 2-year par bond pays a 4.40 coupon at Year 1 and 104.40 at Year 2:

100 = 4.40 / (1.04) + 104.40 / (1 + z_2)^2 100 = 4.2308 + 104.40 / (1 + z_2)^2 95.7692 = 104.40 / (1 + z_2)^2 (1 + z_2)^2 = 104.40 / 95.7692 = 1.09013 z_2 = (1.09013)^{0.5} - 1 = 4.4049%

Notice z_2 (4.4049%) is slightly above the par yield (4.40%) — this is because the upward-sloping curve means earlier cash flows are discounted at a lower rate.

Step 3: 3-Year Spot Rate

The 3-year par bond pays 4.70 at Year 1, 4.70 at Year 2, and 104.70 at Year 3:

100 = 4.70 / (1.04) + 4.70 / (1.044049)^2 + 104.70 / (1 + z_3)^3 100 = 4.5192 + 4.3113 + 104.70 / (1 + z_3)^3 91.1695 = 104.70 / (1 + z_3)^3 (1 + z_3)^3 = 104.70 / 91.1695 = 1.14839 z_3 = (1.14839)^{1/3} - 1 = 4.7199%

Summary of Bootstrapped Spot Rates

Maturity	Par Yield	Spot Rate
1 year	4.00%	4.0000%
2 year	4.40%	4.4049%
3 year	4.70%	4.7199%

Deriving Forward Rates

The 1-year forward rate one year from now, f(1,1):

(1 + z_2)^2 = (1 + z_1)(1 + f(1,1)) (1.044049)^2 = (1.04)(1 + f(1,1)) 1.09013 = 1.04 x (1 + f(1,1)) f(1,1) = 1.09013 / 1.04 - 1 = 4.8125%

The 1-year forward rate two years from now, f(2,1):

(1 + z_3)^3 = (1 + z_2)^2 x (1 + f(2,1)) 1.14839 = 1.09013 x (1 + f(2,1)) f(2,1) = 1.14839 / 1.09013 - 1 = 5.3462%

Common Mistakes

• Rounding intermediate spot rates too early — carry at least 4 decimal places through the chain.
• Forgetting that each step requires all previously computed spot rates, not just the par yields.

For more fixed income term structure problems and interactive calculators, visit AcadiFi's CFA Level II Fixed Income section.