Chapter 10, Problem 38PS

a(i)

Summary Introduction

To calculate:

The bond price if yield to maturity is$8\%$on zero coupon bonds where bond pays$\$1000$at maturity.

Introduction:

Price at which investor can be purchased a bond, that be called bond price or the price at which the bond can be redeemed on maturity.

Answer to Problem 38PS

Price of Zero coupon bond having yield to maturity is$8\%$which is equal to$\$463.19$.

Explanation of Solution

Given:

Coupon rate$\left(r\right)$is$0\%$

Face value$\left(\text{FV}\right)$is$\$1000$

Price of the bond

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

Where,

PV is present value on which bond can be purchased,

PMT is coupon payment,

$n$is no. of year i.e.$\text{10 years}$,

  $\begin{array}{c}\text{PV}={\displaystyle \sum _{\text{t=1}}^{10}\frac{\$0}{{\left(1.08\right)}^1}}+\frac{\$1000}{{\left(1.08\right)}^{10}}\\ =\$463.19\end{array}$

a(ii)

Summary Introduction

To calculate:

The bond price if yield to maturity is$8\%$on 8% coupon bond where bond pays$\$1000$and$\$80$coupon once per year at maturity.

Introduction:

Price at which investor can be purchased a bond, that be called bond price or the price at which the bond can be redeemed on maturity.

Answer to Problem 38PS

Price of$8\%$coupon bond having yield to maturity is$8\%$which is equal to$\$1000$

Explanation of Solution

Given:

Coupon rate$\left(r\right)$is$8\%$

Face value$\left(\text{FV}\right)$is$\$1000$

Price of the bond

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

Where

PV is present value on which bond can be purchased

PMT is coupon payment i.e. $80

$n$is no. of year i.e.$\text{10 years}$

  $\begin{array}{c}\text{PV}={\displaystyle \sum _{\text{t=1}}^{10}\frac{\$80}{{\left(1.08\right)}^1}}+\frac{\$1000}{{\left(1.08\right)}^{10}}\\ =\frac{\$80}{{\left(1.08\right)}^1}+\frac{\$80}{{\left(1.08\right)}^2}+\frac{\$80}{{\left(1.08\right)}^3}+\frac{\$80}{{\left(1.08\right)}^4}+\frac{\$80}{{\left(1.08\right)}^5}+\frac{\$80}{{\left(1.08\right)}^6}+\frac{\$80}{{\left(1.08\right)}^7}+\frac{\$80}{{\left(1.08\right)}^8}+\frac{\$80}{{\left(1.08\right)}^9}+\frac{\$80}{{\left(1.08\right)}^{10}}+\frac{\$1000}{{\left(1.08\right)}^{10}}\\ =\$74.07+\$68.58+\$63.51+\$58.80+\$54.45+\$50.41+\$46.68+\$43.22+\$40.02+\$37.06+\$463.19\\ =\$1000\end{array}$

a(iii)

Summary Introduction

To calculate:

The bond price if yield to maturity is$8\%$on$10\%$coupon bond where bond pays$\$1000$and$\$100$coupon once per year at maturity.

Introduction:

Price at which investor can be purchased a bond, that be called bond price or the price at which the bond can be redeemed on maturity.

Answer to Problem 38PS

Price of$10\%$coupon bond having yield to maturity is$8\%$which is equal to$\$1134.19$

Explanation of Solution

Given:

Coupon rate$\left(r\right)$is$10\%$

Face value$\left(\text{FV}\right)$is$\$1000$

Price of the bond

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

Where,

PV is present value on which bond can be purchased

PMT is coupon payment i.e. $100

$n$is no. of year i.e.$\text{10 years}$

  $\begin{array}{c}\text{PV}={\displaystyle \sum _{\text{t=1}}^{10}\frac{\$100}{{\left(1.08\right)}^1}}+\frac{\$1000}{{\left(1.08\right)}^{10}}\\ =\frac{\$100}{{\left(1.08\right)}^1}+\frac{\$100}{{\left(1.08\right)}^2}+\frac{\$100}{{\left(1.08\right)}^3}+\frac{\$100}{{\left(1.08\right)}^4}+\frac{\$100}{{\left(1.08\right)}^5}+\frac{\$100}{{\left(1.08\right)}^6}+\frac{\$100}{{\left(1.08\right)}^7}+\frac{\$100}{{\left(1.08\right)}^8}+\frac{\$100}{{\left(1.08\right)}^9}+\frac{\$100}{{\left(1.08\right)}^{10}}+\frac{\$1000}{{\left(1.08\right)}^{10}}\\ =\$92.59+\$85.73+\$79.38+\$73.50+\$68.06+\$63.02+\$58.35+\$54.03+\$50.02+\$46.32+\$463.19\\ =\$1134.19\end{array}$

b(i)

Summary Introduction

To calculate:

The bond price if yield to maturity is$8\%$in beginning of next year on zero coupon bond where bond pays$\$1000$.

Introduction:

Price at which investor can be purchased a bond, that be called bond price or the price at which the bond can be redeemed on maturity.

Answer to Problem 38PS

Price of zero coupon bonds having yield to maturity is$8\%$in beginning of next year which is equal to$\$500.25$

Explanation of Solution

Given:

Coupon rate$\left(r\right)$is$0\%$

Face value$\left(\text{FV}\right)$is$\$1000$

We stand on next year so total year of maturity on the beginning of next year is$9$years

Price of the bond

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

Where,

PV is present value on which bond can be purchased

PMT is coupon payment i.e. $100

$n$is no. of year i.e.$\text{10 years}$

  $\begin{array}{c}\text{PV}={\displaystyle \sum _{\text{t=1}}^9\frac{\$0}{{\left(1.08\right)}^1}}+\frac{\$1000}{{\left(1.08\right)}^9}\\ =\$500.25\end{array}$

b(ii)

Summary Introduction

To calculate:

The bond price if yield to maturity is$8\%$in beginning of next year on$8\%$coupon bond where bond pays$\$1000$and$\$80$coupon once per year at maturity.

Introduction:

Price at which investor can be purchased a bond, that be called bond price or the price at which the bond can be redeemed on maturity.

Answer to Problem 38PS

Price of$8\%$zero coupon bonds having yield to maturity is$8\%$in beginning of next year which is equal to$\$1000$.

Explanation of Solution

Given:

Coupon rate$\left(r\right)$is$0\%$

Face value$\left(\text{FV}\right)$is$\$1000$

We stand on next year so total year of maturity on the beginning of next year is$9$years

Price of the bond

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

Where,

PV is present value on which bond can be purchased

PMT is coupon payment i.e.$\$80$

$n$is no. of year i.e.$\text{10 years}$

  $\begin{array}{c}\text{PV}={\displaystyle \sum _{\text{t=1}}^9\frac{\$80}{{\left(1.08\right)}^1}}+\frac{\$1000}{{\left(1.08\right)}^9}\\ =\frac{\$80}{{\left(1.08\right)}^1}+\frac{\$80}{{\left(1.08\right)}^2}+\frac{\$80}{{\left(1.08\right)}^3}+\frac{\$80}{{\left(1.08\right)}^4}+\frac{\$80}{{\left(1.08\right)}^5}+\frac{\$80}{{\left(1.08\right)}^6}+\frac{\$80}{{\left(1.08\right)}^7}+\frac{\$80}{{\left(1.08\right)}^8}+\frac{\$80}{{\left(1.08\right)}^9}+\frac{\$1000}{{\left(1.08\right)}^9}\\ =\$74.07+\$68.58+\$63.51+\$58.80+\$54.45+\$50.41+\$46.68+\$43.22+\$40.02+\$500.25\\ =\$1000\end{array}$

b(iii)

Summary Introduction

To calculate:

The bond price if yield to maturity is$8\%$in beginning of next year on$8\%$coupon bond where bond pays$\$1000$and$\$100$coupon once per year at maturity.

Introduction:

Price at which investor can be purchased a bond, that be called bond price or the price at which the bond can be redeemed on maturity.

Answer to Problem 38PS

Price of$10\%$coupon bond having yield to maturity is$8\%$in beginning of next year which is equal to$\$1124.93$

Explanation of Solution

Given:

Coupon rate$\left(r\right)$is$0\%$

Face value$\left(\text{FV}\right)$is$\$1000$

We stand on next year so total year of maturity on the beginning of next year is$9$years

Price of the bond

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

Where,

PV is present value on which bond can be purchased

PMT is coupon payment i.e.$\$100$

$n$is no. of year i.e.$\text{10 years}$

  $\begin{array}{c}\text{PV}={\displaystyle \sum _{\text{t=1}}^9\frac{\$100}{{\left(1.08\right)}^1}}+\frac{\$1000}{{\left(1.08\right)}^9}\\ =\frac{\$100}{{\left(1.08\right)}^1}+\frac{\$100}{{\left(1.08\right)}^2}+\frac{\$100}{{\left(1.08\right)}^3}+\frac{\$100}{{\left(1.08\right)}^4}+\frac{\$100}{{\left(1.08\right)}^5}+\frac{\$100}{{\left(1.08\right)}^6}+\frac{\$100}{{\left(1.08\right)}^7}+\frac{\$100}{{\left(1.08\right)}^8}+\frac{\$100}{{\left(1.08\right)}^9}+\frac{\$1000}{{\left(1.08\right)}^9}\\ =\$92.59+\$85.73+\$79.38+\$73.50+\$68.06+\$63.02+\$58.35+\$54.03+\$50.02+\$500.25\\ =\$1124.93\end{array}$

c(i)

Summary Introduction

To calculate:

The rate of return on zero coupon bond during one year holding period where yield to maturity is$8\%$.

Introduction:

Return is a profit on an investment. So rate of return is a profit over a period of time of investment. It expressed as percentage of the original investment.

Answer to Problem 38PS

Rate of return on zero coupon bonds having yield to maturity is$8\%$if it is holding for$1$year is which is equal to$8\%$

Explanation of Solution

Given:

Coupon rate$\left(r\right)$is$0\%$

Face value$\left(\text{FV}\right)$is$\$1000$

PV as per${10}^{\text{th}}$year is$\$463.19$[as per a(i)]

PV as per$9^{\text{th}}$year is$\$500.25$[as per b(i)]

Total Income on bond including coupon income is

  $\begin{array}{c}\text{Total Income}=\left(\$500.25-\$463.19\right)+\$0\\ =\$37.06\end{array}$

Rate of return on the bond

  $\begin{array}{c}\text{Rate of Return= }\frac{\text{Total Income}}{{\text{PV for 10}}^{\text{th}}\text{year}}\times 100\\ =\frac{\$37.06}{\$463.19}\times 100\\ =8\%\end{array}$

c(ii)

Summary Introduction

To calculate:

The rate of return on$8\%$coupon bond during one year holding period where yield to maturity is$8\%$.

Introduction:

Return is a profit on an investment. So rate of return is a profit over a period of time of investment. It expressed as percentage of the original investment.

Answer to Problem 38PS

Rate of return on$8\%$coupon bond having yield to maturity is$8\%$if it is holding for$1$year is nil.

Explanation of Solution

Given:

Coupon rate$\left(r\right)$is$8\%$

Face value$\left(\text{FV}\right)$is$\$1000$

PV as per${10}^{\text{th}}$year is$\$1000$[as per a(ii)]

PV as per$9^{\text{th}}$year is$\$1000$[as per b(ii)]

Here, formula to calculate and given are added followed by a detailed explanation / working out the complete problem.

Total Income on bond including coupon income is

  $\begin{array}{c}\text{Total Income}=\left(\$1000-\$1000\right)+\$80\\ =\$80\end{array}$

Rate of return on the bond

  $\begin{array}{c}\text{Rate of Return= }\frac{\text{Total Income}}{{\text{PV for 10}}^{\text{th}}\text{year}}\times 100\\ =\frac{\$80}{\$1000}\times 100\\ =8\%\end{array}$

c(iii)

Summary Introduction

To calculate:

The rate of return on$10\%$coupon bond having one-year holding period where yield to maturity is$8\%$.

Introduction:

Return is a profit on an investment. So rate of return is a profit over a period of time of investment. It expressed as a percentage of the original investment.

Answer to Problem 38PS

Rate of return on$10\%$coupon bond having yield to maturity is$8\%$if it is holding for$1$year is nil.

Explanation of Solution

Given,

Coupon rate$\left(r\right)$is$10\%$

Face value$\left(\text{FV}\right)$is$\$1000$

PV as per${10}^{\text{th}}$year is$\$1134.19$[as per a(iii)]

PV as per$9^{\text{th}}$year is$\$1124.93$[as per b(iii)]

Total Income on bond including coupon income is

  $\begin{array}{c}\text{Total Income}=\left(\$1124.93-\$1134.19\right)+\$100\\ =\$90.74\end{array}$

Rate of return on the bond

  $\begin{array}{c}\text{Rate of Return= }\frac{\text{Total Income}}{{\text{PV for 10}}^{\text{th}}\text{year}}\times 100\\ =\frac{\$90.74}{\$1134.19}\times 100\\ =8\%\end{array}$