Calculate the “ten year zero rate” given two bonds with two prices

Question

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 1

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 0.357 \% \\ \end{align*}