Chapter 5, Problem 16QP

a)

Summary Introduction

To determine: The rate of return on the zero coupon bond

Introduction:

A bond refers to the debt securities issued by the governments or corporations for raising capital. The borrower does not return the face value until maturity. However, the investor receives the coupons every year until the date of maturity.

Bond price or bond value refers to the present value of the future cash inflows of the bond after discounting at the required rate of return.

Zero coupon bonds refer to the bond that does not make any coupon payments during the life of the bond. However, the issuer issues the bond at a very high discount.

Answer to Problem 16QP

The company was raising zero coupon bonds at 4.86 percent per year.

Explanation of Solution

Given information:

Company TC is a subsidiary of Company TM. Company TC announced the sale of securities to the public on March 28, 2008. As per the deal, the company promised to pay the owners of the securities $100,000 on March 28, 2038.

However, the investors would not receive any dividend or interest before the maturity period. The investors paid $24,099 to buy the security. The investors sacrificed $24,099 on March 28, 2008 to receive $100,000 on March 28, 2038.

The formula to calculate the current price of the zero coupon bond:

$\text{Bond value}=\frac{F}{{\left(1+r\right)}^t}$

Where,

“F” refers to the face value paid at maturity

“r” refers to the yield to maturity

“t” refers to the periods to maturity

Compute the rate of return as follows:

The remaining time to maturity is 30 years $(2038-2008)$. The future value of the bond in $100,000. The present value is $24,099. Assume that the bond makes annual coupon payments.

$\begin{array}{c}\text{Bond value}=\frac{F}{{\left(1+r\right)}^t}\\ \$24,099=\frac{\$100,000}{{\left(1+r\right)}^{30}}\\ r={\left(\frac{\$100,000}{\$24,099}\right)}^{\frac{1}{30}}-1\\ =0.0486\text{ or }4.86\%\end{array}$

Hence, the rate of return on the bond is 4.86 percent.

b)

Summary Introduction

To calculate: The annual rate of return from 2008 to 2019

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 investor received an annual rate of return of 5.27 percent on the bond from 2008 to 2019.

Explanation of Solution

Given information:

Company TC is a subsidiary of Company TM. Company TC announced the sale of securities to the public on March 28, 2008. As per the deal, the company promised to pay the owners of the securities $100,000 on March 28, 2038.

However, the investors would not receive any dividend or interest before the maturity period. The investors paid $24,099 to buy the security. The investors sacrificed $24,099 on March 28, 2008 to receive $100,000 on March 28, 2038.

One investor who bought the bond on March 28, 2008 for $24,099 decided to sell the bond on March 28, 2019 for $42,380.

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 annual rate of return from 2008 to 2019:

The time between the sale and purchase of the bond is 11 years $\left(2019-2008\right)$. Hence, the number of years of investment (t) is 11.

$\begin{array}{c}\text{P}\times {\left(1+r\right)}^t=\text{FV}\\ \$24,099\times {\left(1+r\right)}^{11}=\$42,380\\ {\left(1+r\right)}^{11}=\frac{\$42,380}{\$24,099}\\ {\left(1+r\right)}^{11}=1.7586\end{array}$

$\begin{array}{c}1+r={1.7586}^{\frac{1}{11}}\\ r=1.0527-1\\ r=0.0527\text{ or }5.27\%\end{array}$

Hence, the investor earned an annual return of 5.27% per year from 2008 to 2019.

c)

Summary Introduction

To calculate: The annual rate of return from 2019 to 2038.

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 investor would receive an annual rate of return of 4.62 percent on the bond from 2019 to 2038.

Explanation of Solution

Given information:

Company TC is a subsidiary of Company TM. Company TC announced the sale of securities to the public on March 28, 2008. As per the deal, the company promised to pay the owners of the securities $100,000 on March 28, 2038.

However, the investors would not receive any dividend or interest before the maturity period. The investors paid $24,099 to buy the security. The investors sacrificed $24,099 on March 28, 2008 to receive $100,000 on March 28, 2038.

The value of the bond on March 28, 2019 was $42,380. An investor purchased the bond at $42,380 on March 28, 2019 and held it until maturity.

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 annual rate of return from 2019 to 2038:

The time between the sale and purchase of the bond is 19 years $\left(2038-2019\right)$. Hence, the number of years of investment (t) is 19.

$\begin{array}{c}\text{P}\times {\left(1+r\right)}^t=\text{FV}\\ \$42,380\times {\left(1+r\right)}^{19}=\$100,000\\ {\left(1+r\right)}^{19}=\frac{\$100,000}{\$42,380}\\ {\left(1+r\right)}^{19}=2.35961+r={2.3596}^{\frac{1}{19}}\\ r=1.0462-1\\ r=0.0462\text{ or}4.62\%\end{array}$

Hence, the investor would have earned an annual return of 4.62% per year from 2019 to 2038.