How do you bootstrap spot rates from the par yield curve step by step?

Bootstrapping is one of the most important quantitative techniques in fixed income. The idea is simple: you use shorter-maturity spot rates (which you already know) to solve for longer-maturity spot rates one step at a time.

The Logic

A par bond (price = 100) pays coupons that should be discounted at individual spot rates, not a single yield. By setting the price to 100 and solving for the unknown longest spot rate, you 'bootstrap' the curve.

Step-by-Step Example

Given par yields from Northgate Federal bonds (fictional):

Step 1: 1-Year Spot Rate

The 1-year par bond has only one cash flow (coupon + principal), so the par yield equals the spot rate:

z1 = 4.00%

Step 2: 2-Year Spot Rate

A 2-year par bond with 4.40% coupon is priced at 100:

100 = 4.40/(1 + z1) + 104.40/(1 + z2)^2

100 = 4.40/1.04 + 104.40/(1 + z2)^2

100 = 4.2308 + 104.40/(1 + z2)^2

95.7692 = 104.40/(1 + z2)^2

(1 + z2)^2 = 104.40/95.7692 = 1.09014

z2 = (1.09014)^0.5 - 1 = 4.407%

Step 3: 3-Year Spot Rate

100 = 4.70/1.04 + 4.70/(1.04407)^2 + 104.70/(1 + z3)^3

100 = 4.5192 + 4.3126 + 104.70/(1 + z3)^3

91.1682 = 104.70/(1 + z3)^3

(1 + z3)^3 = 104.70/91.1682 = 1.14842

z3 = (1.14842)^(1/3) - 1 = 4.722%

Summary of Results

Notice that spot rates are slightly higher than par yields for maturities beyond 1 year when the curve is upward-sloping. This makes intuitive sense: the par yield is a weighted average of spot rates, pulled down by the lower short-term rates.

Why Bootstrapping Matters

Spot rates are the building blocks for:

• Pricing any fixed-income security accurately
• Calculating forward rates (via no-arbitrage)
• Computing the Z-spread (constant spread added to each spot rate)
• Valuing bonds with embedded options using binomial trees

Exam Tip: Bootstrapping questions are heavily computational. Practice until you can do 3-step bootstrapping in under 5 minutes. The exam will typically give you 2-3 par yields and ask for the longest spot rate. Always start from the shortest maturity.

Practice bootstrapping in our CFA Level II question bank.