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.