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Deriving a Ten-Year Zero Rate from Two Coupon Bond Prices

Article Quant Q&A · Author: Marie. P.
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Summary

The document explains how to extract a ten-year zero-coupon discount value from prices of two bonds with the same maturity but different coupon rates. Because the 8% coupon stream is exactly twice the 4% stream, subtracting twice the lower-coupon bond price from the higher-coupon price cancels the coupon cash flows, regardless of rates at earlier maturities. The remaining amount isolates the value of the principal repayment at year ten.

The example then treats that value as the discounted value of the face amount and takes a logarithm to solve for the continuously compounded zero rate, obtaining approximately 3.57%. The key lesson is that the coupon cash flows need not be valued individually or assigned a full yield curve. The calculation depends on the bonds having the same maturity, face value, and coupon timing, and on the stated continuous discounting convention; it does not establish rates at other maturities.

Key ideas

  • Matching maturity bonds with proportional coupons can be combined to cancel coupon cash flows.
  • The residual price difference isolates the discounted principal repayment at maturity.
  • A logarithm converts the isolated discount factor into a continuously compounded zero rate.
  • This calculation identifies only the ten-year rate under the stated bond and discounting assumptions.

Tags

Fixed incomeStatistics
Full text
# Calculate the "ten year zero rate"  given two bonds with two prices


# Calculate the "ten year zero rate"  given two bonds with two prices












I have a little question and need some help with the notation. So, the question goes as follows:

> A bond with a maturity of ten years that pays annual coupons of 8% has a price of \$90. A bond with a maturity of ten years and annual coupons of 4% has a price of \$80. What is the ten year zero rate?

I don't actually know what the ten-year zero rate is. I set up a system of equations with the information that is given: Is it just all about finding $y$?

\begin{align*} \$90 =& \sum_{i=1}^{10} e^{-yi} (0.08 Z) &+& e^{-y\cdot 10}Z\\ \$80 =& \sum_{i=1}^{10} e^{-yi} (0.04 Z) &+& e^{-y \cdot 10}Z\\ \Leftrightarrow \$10 =& \sum_{i=1}^{10} e^{-yi} (0.04 Z)\\ \Rightarrow \$70 =& e^{-y\cdot 10}Z \end{align*}

And is there any way to solve for the zero rate by hand? (This is the reason why I'm wondering; to solve this system, a calculator is necessary, but all the other homework problems were solvable by hand!)

## Answer by David Nehme (score 2, accepted)

https://quant.stackexchange.com/a/4130

The way you are trying to solve these equations makes assumptions about the rates less than 10 years and therefore the shape of the yield curve. \$90 is the value of 8% coupons plus a 10-year zero-coupon bond. \$80 is the value of the 4% coupons plus a 10-year zero-coupon bond. 8% coupons are worth twice 4% coupons over the same period, regardless of the interest rates.

So 90 - 2*80 = value of 8% coupons - 2( value of 4% coupons) - 10-year zero-coupon bond

and

\begin{align*} \$70 &= e^{-10y} \cdot 100 \\ 0.7 &= e^{-10y} \\ \ln(0.7) &= -10 y \\ 0.35667 &= 10y \\ y &\approx 3.57 \% \quad\quad\\ \end{align*}

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