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.

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.

Introduction:

The future value of money refers to 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

The value of the bond in 2025 will be $50.5.

Explanation of Solution

Given information:

Person X will purchase a bond worth $50 in 2015. He plans to hold it until 2025 or for 10 years. The rate of interest on the bond is 0.10 percent per year.

The formula to calculate 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:

$\begin{array}{c}\text{FV}=\text{PV}\times {\left(1+r\right)}^t\\ =\$50\times {\left(1+0.001\right)}^{10}\\ =\$50\times 1.0100\\ =\$50.5\end{array}$

Hence, the future value of investment after 10 years is $50.5.

c)

Summary Introduction

To calculate: The rate of return.

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.

Answer to Problem 16QP

The bond has to fetch a return of 7.18% per year to double its value in 2035.

Explanation of Solution

Given information:

Person X purchased a bond worth $50 in 2015. He plans to hold it until 2025 or for 10 years. The rate of interest on the bond is 0.10 percent per year. The future value of this bond in 2025 will be $50.50 (Refer to Part (b) of the solution). Instead of cashing the bond at $50.50, he plans to reinvest the bond value until the value of the bond doubles in 2035.

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 $50.50 in 2025. He plans to invest for 10 years until the value of the bond doubles in 2035. Hence, the future value of the bond will be $101$\left(\$50.50\times 2\right)$.

$\begin{array}{c}\text{P}\times {\left(1+r\right)}^t=\text{FV}\\ \$50.50\times {\left(1+r\right)}^{10}=\$101\\ {\left(1+r\right)}^{10}=\frac{\$101}{\$50.50}\\ {\left(1+r\right)}^{10}=2\end{array}$

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

Hence, the bond has to fetch a return of 7.18% per year to double its value in 2033.