Introduction

The bootstrap method is a fundamental technique used in finance to estimate zero-coupon rates from observed market yields. Zero-coupon rates represent the implied interest rates for investments that don’t pay periodic interest but instead compound all interest until maturity. In this chapter, we will delve into the bootstrap method, exploring its step-by-step process to calculate zero-coupon rates. This method is essential for constructing the yield curve, a critical tool in financial analysis and decision-making.

Bootstrap Method Overview

The bootstrap method involves constructing the yield curve by iteratively solving for zero-coupon rates at various maturities using observed market yields for bonds of different maturities. The basic idea is to use the yields of coupon-bearing bonds to infer the zero-coupon rates. This process ensures that the calculated rates are consistent with the observed market prices.

Step-by-Step Calculation:

• Select Benchmark Securities: Begin by selecting a set of benchmark securities that have observable market yields. These securities should cover a range of maturities.

• Initial Estimates: Start with an initial estimate for the zero-coupon rate at the shortest maturity. This estimate can be obtained from a risk-free rate or any relevant source.

• Iterative Calculation: Use the initial estimate to calculate the present value of the benchmark security’s cash flows. The cash flows are discounted using the initial estimate to obtain the security’s price.

• Iterative Refinement: Compare the calculated price with the actual market price of the benchmark security. Adjust the zero-coupon rate estimate iteratively until the calculated price matches the market price.

• Zero-Coupon Rate Extraction: Once the zero-coupon rate is found for the shortest maturity, move on to the next benchmark security with a longer maturity. Use the previously calculated zero-coupon rates to estimate the next rate iteratively.

• Repeat the Process: Continue this process for all selected benchmark securities, gradually building the yield curve by deriving zero-coupon rates for different maturities.

Example: Suppose we have two benchmark securities:

• Security A with a maturity of 1 year, market yield = 3%
• Security B with a maturity of 3 years, market yield = 4%

Assuming an initial estimate of the zero-coupon rate for a 1-year maturity is 2.5%, we can use iterative calculations to refine the estimate until the calculated price of Security A matches its market price.

Once the zero-coupon rate for 1 year is established, we can move on to Security B and use the previously calculated rate to iteratively find the zero-coupon rate that matches its market price.