Chapter 4, Problem 16QP

a)

Summary Introduction

To calculate: The rate of return of a bond

Introduction:

Rate of return refers to the gain or loss on the investment. It also refers to the increase or decrease in the capital value of an investment.

The future value of money is the amount of dollars that an investment grows over a definite period at a particular rate of interest rate. In other words, it refers to the future value of present cash investments.

Answer to Problem 16QP

A bond with a face value of $50 would double its value in 20 years, if the rate of return is 3.53%.

Explanation of Solution

Given information:

Person X purchased a bond worth $50. He plans to hold it for 20 years until the value of the bond, doubles.

The formula to calculate the rate of return:

$\text{P}\times {\left(1+r\right)}^t=\text{FV}$

Where,

“P” refers to the principal amount invested,

“r” refers to the simple rate of interest,

“t” refers to the number of years or periods of investment,

“FV” refers to the future value or the current market value.

Compute the rate of return:

The present value of the bond is $50. He plans to invest for 20 years until the value of the bond, doubles. Hence, the future value of the bond will be $100 $\left(\$50\times 2\right)$.

$\begin{array}{c}\text{P}\times {\left(1+r\right)}^t=\text{FV}\\ \$50\times {\left(1+r\right)}^{20}=\$100\\ {\left(1+r\right)}^{20}=\frac{\$100}{\$50}\\ {\left(1+r\right)}^{20}=2\end{array}$

$\begin{array}{c}1+r=2^{\frac{1}{20}}\\ r=1.0353-1\\ r=0.0353\text{ or }3.53\%\end{array}$

Hence, the rate of return is 3.53% per year.

b)

Summary Introduction

To calculate: The future value of the bond

Answer to Problem 16QP

The value of the bond in 2022 will be $53.08.

Explanation of Solution

Given information:

Person X will purchase a bond worth $50 in 2012. He plans to hold it until 2022 that is for 10 years. The rate of interest on the bond is 0.60 percent per year.

The formula to compute the future value:

$\text{FV}=\text{PV}\times {\left(1+r\right)}^t$

Where,

“FV” refers to the future value or the current market value,

 “PV” refers to the present value,

“r” refers to the simple rate of interest,

“t” refers to the number of years or periods of investment.

Compute the future value:

$\begin{array}{c}\text{FV}=\text{PV}\times {\left(1+r\right)}^t\\ =\$50\times {\left(1+0.0060\right)}^{10}\\ =\$50\times 1.061646\\ =\$53.08\end{array}$

Hence, the future value of investment after 10 years is $53.08.

c)

Summary Introduction

To calculate: The rate of return

Answer to Problem 16QP

The bond has to fetch a return of 7.18% per year to double its value in 2032.

Explanation of Solution

Given information:

Person X purchased a bond worth $50 in 2012. He plans to hold it until 2022 that is for 10 years. The rate of interest on the bond is 0.60 percent per year. The future value of this bond in 2022 will be $53.08 (Refer Part (b) of the Answer). Instead of cashing the bond at $53.08, he plans to reinvest the bond value until the value of the bond doubles in 2032.

The formula to calculate the rate of return:

$\text{P}\times {\left(1+r\right)}^t=\text{FV}$

Where,

 “P” refers to the principal amount invested,

“r” refers to the simple rate of interest,

“t” refers to the number of years or periods of investment,

“FV” refers to the future value or the current market value.

Compute the rate of return:

The present value of the bond is $53.08 in 2022. He plans to invest for 10 years until the value of the bond doubles in 2032. Hence, the future value of the bond will be $106.16 $\left(\$53.08\times 2\right)$.

$\begin{array}{c}\text{P}\times {\left(1+r\right)}^t=\text{FV}\\ \$53.08\times {\left(1+r\right)}^{10}=\$106.16\\ {\left(1+r\right)}^{10}=\frac{\$106.16}{\$53.08}\\ {\left(1+r\right)}^{10}=2\end{array}$

$\begin{array}{c}1+r=2^{\frac{1}{10}}\\ r=1.0718-1\\ r=0.0718\text{ or }7.18\%\end{array}$

Hence, the bond has to fetch a return of 7.18% per year to double its value in 2032.