Term Structure of Interest Rates › Bootstrapping the Zero Curve

Direct single-payment prices do not cover every maturity needed for a curve. Long-dated zeros do exist, including Treasury STRIPS, but coupon-bearing instruments often supply additional pricing information. Bootstrapping is one way to extract maturity-specific discount factors from those prices.

A coupon bond's yield to maturity is the single internal rate of return that discounts its payment schedule to its price under a stated convention. It summarizes several cash-flow dates through a nonlinear pricing equation; it is not a simple average of their zero rates. We can work backward from coupon bond prices sequentially, or use a fitted curve model.

With suitable instruments and a local interpolation rule, each step can leave one new scalar unknown. Earlier coupon-date discounts must already be known or supplied by that rule. When interpolation depends on the candidate maturity rate, solve the resulting scalar equation rather than treating those intermediate discounts as fixed.

The algorithm works as follows:

1. Start with an instrument that isolates one discount factor, such as a zero-coupon bond. The shortest coupon bond alone may not identify discounts for all its payment dates.
2. For each successive maturity, discount earlier cash flows using solved nodes and the chosen interpolation rule.
3. Solve for the new rate that reproduces the instrument's price, checking that the implied discount factor is positive.
4. Repeat for the remaining instruments, with consistent currency, credit and quoting conventions.

Consider a 2-year annual coupon bond with face value $100, annual coupon payment $c$ dollars, and price $P$. Here $c$ is a cash amount: a 5% annual coupon on $100 face gives $c=5$. If we know the 1-year zero rate $z(1)$, we can solve for $z(2)$:

$$P = c \cdot e^{-z(1) \cdot 1} + (100 + c) \cdot e^{-z(2) \cdot 2}$$

This equation says that the bond's price equals the present value of all its cash flows. The first coupon payment at year 1 is discounted at the known 1-year rate. The final payment (coupon plus principal) at year 2 is discounted at the unknown 2-year rate.

Rearranging step by step to isolate the unknown rate:

$$\begin{aligned} P &= c \cdot e^{-z(1) \cdot 1} + (100 + c) \cdot e^{-z(2) \cdot 2} && \text{(bond pricing equation)} \\ P - c \cdot e^{-z(1)} &= (100 + c) \cdot e^{-z(2) \cdot 2} && \text{(isolate unknown term)} \\ e^{-z(2) \cdot 2} &= \frac{P - c \cdot e^{-z(1)}}{100 + c} && \text{(solve for discount factor)} \\ -z(2) \cdot 2 &= \ln\left(\frac{P - c \cdot e^{-z(1)}}{100 + c}\right) && \text{(take natural log)} \\ z(2) &= -\frac{1}{2} \ln\left(\frac{P - c \cdot e^{-z(1)}}{100 + c}\right) && \text{(solve for } z(2) \text{)} \end{aligned}$$

where:

• $P$: market price of the 2-year bond
• $c$: annual coupon payment
• $z(1)$: previously bootstrapped 1-year zero rate
• $z(2)$: unknown 2-year zero rate we're solving for

The numerator is the residual price after subtracting the first coupon's present value. It must be positive for a finite real zero rate when the final cash flow is positive. Dividing by that final payment gives its discount factor; the logarithm then recovers the rate. If the residual is nonpositive, check the data, assumptions and instrument suitability rather than taking its logarithm.

Let's bootstrap a zero curve from the following bonds, all with $100 face value and annual coupon payments:

Step 1: 1-year zero rate

The 1-year bond is a zero-coupon bond, so we can extract its rate directly using the fundamental zero-rate formula:

$$\begin{aligned} z(1) &= -\frac{1}{1} \ln\left(\frac{96.15}{100}\right) && \text{(zero rate formula)} \\ &= -\ln(0.9615) && \text{(simplify)} \\ &\approx 0.0392607 \approx 3.93\% && \text{(rounded result)} \end{aligned}$$

Step 2: 2-year zero rate

The 2-year bond pays $5 at year 1 and $105 at year 2. We know the 1-year rate, so we can discount the first coupon and solve for the 2-year rate:

$$101.50 = 5D(1) + 105 \cdot e^{-z(2) \cdot 2}$$ $$101.50 = 5 \cdot 0.9615 + 105 \cdot e^{-z(2) \cdot 2}$$ $$101.50 = 4.8075 + 105 \cdot e^{-z(2) \cdot 2}$$ $$D(2)=e^{-z(2) \cdot 2} = \frac{96.6925}{105} \approx 0.9208809524$$ $$z(2) = -\frac{1}{2} \ln\left(\frac{96.6925}{105}\right) \approx 0.0412123 \approx 4.12\%$$

Step 3: 3-year zero rate

The 3-year bond pays $6 at years 1 and 2, and $106 at year 3. Using the rates already determined, we can discount the intermediate coupons and solve for the remaining unknown:

$$103.00 = 6D(1) + 6D(2) + 106 \cdot e^{-z(3) \cdot 3}$$ $$103.00 = 6 \cdot 0.9615 + 6 \cdot \frac{96.6925}{105} + 106 \cdot e^{-z(3) \cdot 3}$$ $$6D(1)+6D(2) \approx 5.769 + 5.5252857143 = 11.2942857143$$ $$D(3)=e^{-z(3) \cdot 3} = \frac{103-6D(1)-6D(2)}{106} \approx \frac{91.7057142857}{106} \approx 0.8651482480$$ $$z(3) = -\frac{1}{3} \ln D(3) \approx 0.0482848 \approx 4.83\%$$

We retain unrounded discount factors throughout the calculation and round only the displayed results. The bootstrapped zero curve is:

The bootstrapped rates (3.93%, 4.12%, 4.83%) match our manual calculations exactly. The discount factors (0.9615, 0.9209, 0.8651) show how $1 received in the future is worth progressively less as the maturity increases. A dollar in 3 years is worth only about 87 cents today.

Real-world bootstrapping must handle bonds that don't mature at convenient annual intervals. When you need a discount factor for a maturity that falls between bootstrapped points, you interpolate. Common interpolation methods include:

• Linear interpolation on zero rates: Simple but can produce non-smooth forward rates
• Log-linear interpolation on discount factors: Preserves positive factors between positive endpoints; the implied interval forward rate is positive, zero, or negative according to whether the factors decline, stay constant, or increase
• Cubic spline interpolation: Produces smooth curves but may oscillate