Question

A 17-year, \(1,000 par value zero-coupon rate bond is to be issued to yield 7 percent.

a. What should be the initial price of the bond? (Take the present value of \)1,000 for 17 years at 7 percent.)

b. If immediately upon issue, interest rates dropped to 6 percent, what would be the value of the zero-coupon rate bond?

c. If immediately upon issue, interest rates increased to 9 percent, what would be the value of the zero-coupon rate bond?

Short answer

(a) The initial price of the bond is computed as $316.57

(b) The price of the bond at the interest of 6% is $371.36

(c) The price of the bond at the interest of 9% is $231.07

Step-by-step solution

Computation of Initial Price of bond

$\begin{array}{rcl}\mathbf{Initial}\mathbf{}\mathbf{Price}\mathbf{}\mathbf{of}\mathbf{}\mathbf{bond}& \mathbf{=}& \mathbf{Par}\mathbf{}\mathbf{Value}\mathbf{\times }\frac{\mathbf{1}}{{\left(1+r\right)}^{\mathbf{t}}}\\ & \mathbf{=}& \mathbf{1}\mathbf{,}\mathbf{000}\mathbf{\times }\frac{\mathbf{1}}{{\left(1+0.07\right)}^{\mathbf{17}}}\\ & \mathbf{=}& \mathbf{\$}\mathbf{316}\mathbf{.}\mathbf{57}\end{array}$

Computation of price of the bond at interest rate of 6%

$\begin{array}{rcl}\mathbf{Price}\mathbf{}\mathbf{of}\mathbf{}\mathbf{bond}& \mathbf{=}& \mathbf{Par}\mathbf{}\mathbf{Value}\mathbf{\times }\frac{\mathbf{1}}{{\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{r}\mathbf{)}}^{\mathbf{t}}}\\ & \mathbf{=}& \mathbf{1}\mathbf{,}\mathbf{000}\mathbf{\times }\frac{\mathbf{1}}{{\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{06}\mathbf{)}}^{\mathbf{17}}}\\ & \mathbf{=}& \mathbf{\$}\mathbf{371}\mathbf{.}\mathbf{36}\end{array}$

Computation of price of the bond at an interest rate of 6%

$\begin{array}{rcl}\mathbf{Price}\mathbf{}\mathbf{of}\mathbf{}\mathbf{bond}& \mathbf{=}& \mathbf{Par}\mathbf{}\mathbf{Value}\mathbf{\times }\frac{\mathbf{1}}{{\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{r}\mathbf{)}}^{\mathbf{t}}}\\ & \mathbf{=}& \mathbf{1}\mathbf{,}\mathbf{000}\mathbf{\times }\frac{\mathbf{1}}{{\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{09}\mathbf{)}}^{\mathbf{17}}}\\ & \mathbf{=}& \mathbf{\$}\mathbf{231}\mathbf{.}\mathbf{07}\end{array}$