Bootstrapping Zero Rates

Bootstrapping zero rates from coupon-bearing instruments is one of the methods for constructing a discount curve, which is essential for computing the present value of future cashflows. Bootstrapping is an iterative process that is best illustrated with an example.

Consider the following 3 instruments:

Principal ($)Maturity (years)Coupon ($)Price ($)
1000.5094.9
1001090
1001.5896

The first two instruments have no coupon payments, and all value is realized at maturity. This means we can easily calculate the zero rates by solving these equations for RR:

100=94.9eR⋅0.5100 = 94.9 e^{R \cdot 0.5}
100=90eR⋅1.0100 = 90 e^{R \cdot 1.0}

Solving these equations gives us the 6 month and 1 year zero rates of 10.469% and 10.536% respectively. These initial zero coupon instruments are often treasury bills, and serve as the foundation for bootstrapping the rest of the curve.

Calculating the 1.5 year zero rate using the third instrument is a bit more involved, as it has an annual coupon payment of $8 (i.e. 2 semi-annual coupon payments of $4). To compute the 1.5 year zero rate using this instrument, we need to discount the coupon payments using the zero rates we have already calculated, and then solve for RR like so:

4e−0.10469⋅0.5+4e−0.10536⋅1.0+104e−R⋅1.5=964e^{-0.10469 \cdot 0.5} + 4e^{-0.10536 \cdot 1.0} + 104e^{-R \cdot 1.5} = 96

Using some algebra, we can solve for the unknown 1.5 year zero rate RR:

e−1.5R=0.85196e^{-1.5R} = 0.85196
R=−ln⁡(0.85196)1.5=0.10681R = -\frac{\ln(0.85196)}{1.5} = 0.10681

You could then repeat this process to compute the 2 year zero rate, and so on. This is the essence of the bootstrap method. Here is a calculator to help you further explore this concept:

Time to Maturity (Years)
Annual Coupon ($)
Price ($)
☰
☰
☰
☰
Assumptions
  • All coupon payments are semiannual.
  • The face value of each instrument is $100.

Modern Practice

Since the financial crisis of 2008, the industry has moved towards using overnight index swaps (OIS) as the foundation for constructing discount curves. Previously, discount curves were often constructed using the bootstrapping method applied to instruments derived from LIBOR rates, such as interest rate swaps or forward rate agreements. However, LIBOR was found to be susceptible to manipulation, prompting the shift to more robust OIS-based curves. While the simplified bootstrapping method discussed earlier provides a foundational understanding, the modern approach incorporates more complex instruments and methodologies