Chapter 10, Problem 38PS

Summary Introduction

To calculate the yield, prices and rate of return from the given information.

Answer to Problem 38PS

The price of 10% coupon bond is$\$1,134.20$

Explanation of Solution

The following information applies to both the parts of the question: Table showing the information related to the three bonds mentioned in the question

	Zero coupon bond	8% bond	10% bond
Maturity	10Years	10Years	10Years
Coupon rate	0%	8%	10%
Face Value	Value paid on maturity = $1,000	$1,000	$1,000

1. Calculate the price of all three bonds using the following formula:

$PV={\displaystyle \sum _{t=1}^n\frac{PMT}{{\left(1+r\right)}^t}}+\frac{FV}{{\left(1+r\right)}^n}$

Here,

'PV' is the price of the bond at which it can be purchased today.

'PMT' is the semiannual coupon payment of the bond.

'FV' is the value of the bond at the time of redemption in future.

'n' is the number of times the payment is made of the interest.

Calculate the price of zero-coupon bond.

$\begin{array}{l}r=0\%\\ FV=\$1,000\\ n=10years\\ PV={\displaystyle \sum _{t=1}^n\frac{PMT}{{\left(1+r\right)}^t}}+\frac{FV}{{\left(1+r\right)}^n}\\ ={\displaystyle \sum _{t=1}^n\frac{\$0}{\left(1+0.08\right)}}+\frac{\$1,000}{{\left(1+0.08\right)}^{10}}\\ =\frac{\$1,000}{2.16}\\ =\$463.19\end{array}$

The price of zero- coupon bond is$\$463.19$ .

Calculate the price 8% coupon bond.

Substitute

The price of 8% coupon bond is$\$1000.00$

Calculate the price 10% coupon bond.

Substitute $\begin{array}{l}r=8\%\\ PMT=\left(0.10\times 1000\right)\\ =\$100\\ FV=\$1,000\\ n=10years\\ PV={\displaystyle \sum _{t=1}^n\frac{PMT}{{\left(1+r\right)}^1}}+\frac{FV}{{\left(1+r\right)}^n}\\ ={\displaystyle \sum _{t=1}^{10}\frac{\$100}{{\left(1+0.08\right)}^1}}+\frac{\$1,000}{{\left(1+0.08\right)}^{10}}\\ PV=\frac{\$100}{{\left(1.08\right)}^1}+\frac{\$100}{{\left(1.08\right)}^2}+\frac{\$100}{{\left(1.08\right)}^3}+\frac{\$100}{{\left(1.08\right)}^4}+\frac{\$100}{{\left(1.08\right)}^5}+\\ \frac{\$100}{{\left(1.08\right)}^6}+\frac{\$100}{{\left(1.08\right)}^7}+\frac{\$100}{{\left(1.08\right)}^9}+\frac{\$100}{{\left(1.08\right)}^{10}}\\ =\frac{\$100}{1.08}+\frac{\$100}{1.17}+\frac{\$100}{1.26}+\frac{\$100}{1.36}+\frac{\$100}{1.47}+\frac{\$100}{1.59}+\\ \frac{\$100}{1.71}+\frac{\$100}{1.85}+\frac{\$100}{2.00}+\frac{\$100}{2.16}+\frac{\$1,000}{2.16}\end{array}$

$\begin{array}{l}=92.59+85.73+79.38+68.06+63.02+58.35+\\ 54.03+50.02+46.32+463.19\\ =\$1,134.20\end{array}$

The price of 10% coupon bond is$\$1,134.20$.

Following is the table representing the prices of all the three types of bonds:

Particulars	Zero	8% coupon	10% coupon
Current price	$463.19	$1,000	$1,134.20
Price one year from now	$500.25	$1,000	$1,124.94
Price increase	$37.06	$0.00	($9.26)
Coupon income	$0.00	$80.00	$100.00
Income	$37.06	$80.00	$90.74
Rate of Return	8.00%	8.00%	8.00%

Summary Introduction

(B)

To calculate the yield, prices and rate of return from the given information.

Answer to Problem 38PS

The return of all the bonds is calculated as 8%.

Explanation of Solution

The prices of the bonds after one year are calculated using the same formula as in part (a), only the value of the n changes in the calculation. It changes from 10 years to 9 years now and the calculation is as shown above.

The price of the zero coupon bond after one year is $500.25, 8% coupon bond is $1,000 and the price of the 10% coupon bond is $1,134.20.

The return on the three bonds is calculated using the following formula:

$\begin{array}{l}\mathrm{Re}turn\left(Zerocoupon\right)=\frac{\left(Interest+Endoftheperiodprice-Beginningprice\right)}{Beginningprice}\times 100\\ =\frac{0+\$500.25-\$463.19}{\$463.19}\times 100\\ =\frac{\$37.06}{\$463.19}\times 100\\ =8.00\%\end{array}$

$\begin{array}{l}\mathrm{Re}turn\left(10\%coupon\right)=\frac{\left(Interest+Endoftheperiodprice-Beginningprice\right)}{Beginningprice}\times 100\\ =\frac{\$100+\$1,124.94-\$1,134.20}{\$1,134.20}\times 100\\ =\frac{\$90.74}{\$1,134.20}\times 100\\ =8.00\%\end{array}$

$\begin{array}{l}\mathrm{Re}turn\left(8\%coupon\right)=\frac{\left(Interest+Endoftheperiodprice-Beginningprice\right)}{Beginningprice}\times 100\\ =\frac{\$80+\$1,000-\$1,000}{\$1,000}\times 100\\ =\frac{\$80}{\$1,000}\times 100\\ =8.00\%\end{array}$

The return of all the bonds is calculated as 8%.