Skip to content
All library documents

Bootstrapping a Three-Year Zero Rate from Coupon Bonds

Article Quant Q&A · Author: Barto_Wynne12
Read the original

Summary

This question asks how to bootstrap a three-year zero rate from annual coupon bonds when one-year and two-year zero rates are already known. The central point is that each earlier coupon payment must be discounted using the zero rate for its own maturity: the one-year rate for the first payment and the two-year rate for the second. The remaining present value, together with the bond’s price and final cash flow, can then be used to solve for the three-year rate.

The response emphasizes that the three-year bond price is needed to determine that rate; the coupon rate alone is not enough. The document gives no worked calculation or broader discussion of curve construction, market conventions, or how to obtain bond prices. It is a concise clarification of the bootstrapping setup rather than a complete pricing guide.

Key ideas

  • Discount each coupon cash flow using the zero rate matching that payment’s maturity.
  • The three-year zero rate is the remaining unknown after earlier cash flows are valued.
  • A three-year bond price is needed to solve for its zero rate.
  • The coupon rate alone does not determine the zero rate.

Tags

Fixed incomeDerivatives pricingArbitrage
Full text
# Zero Rate Bootstrapping Question


# Zero Rate Bootstrapping Question












I'm just starting to read about Arbitrage Trading and am currently looking at zero coupon rates as it's in the textbook I am using and had a question about bootstrapping.

Say that I have the maturities and and coupon rates for a 1-year bond, a 2-year bond, and a 3-year bond. In constructing the 1-year zero rate, I can just use the 1-year bond, and in constructing the 2-year zero rate, I use the 1-year zero rate to deal with the first annual payment and use the leftover amount to deal construct the 2-year zero rate.

When constructing the 3-year zero rate, do I need to handle the two prior annual payments using their respective zero rates, or can I use the 2-year zero rate for both?

For example, if I know that my zero rates for year 1 and year 2 are $5$% and $5.75$%, and that 3-year coupon rate is $6.10$, should I subtract out $\frac{6.10}{1.0575^1} + \frac{6.10}{1.0575^2}$ or $\frac{6.10}{1.05^1} + \frac{6.10}{1.0575^2}$ before solving for my 3-year zero rate?

Any answer/explanation would be welcome!

## Answer by user68819 (score 1)

https://quant.stackexchange.com/a/82489

You need a 3y bond price in order to solve for the 3y zero rate (given the 1y and 2y zeroes).

Given this price you have any equation with only 1 unknown, the 3y zero rate.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.