How do you bootstrap spot rates from the par yield curve step by step?
CFA Level II fixed income. I understand that the par curve gives yields for coupon bonds priced at par, but I need to extract spot (zero-coupon) rates from it. The bootstrapping process confuses me -- can someone show the method with a clear numerical example?
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):
| Maturity | Par Yield |
|---|---|
| 1 year | 4.00% |
| 2 year | 4.40% |
| 3 year | 4.70% |
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
| Maturity | Par Yield | Spot Rate | Spread |
|---|---|---|---|
| 1 year | 4.00% | 4.000% | 0 bps |
| 2 year | 4.40% | 4.407% | +0.7 bps |
| 3 year | 4.70% | 4.722% | +2.2 bps |
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.
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