Chapter 3, Problem 20PS

a)

Summary Introduction

To discuss: Discount factor.

Explanation of Solution

Computation of discount factors for each year is as follows:

Formula to compute of discount factors for each year is as follows:

b)

Summary Introduction

To discuss: Present values.

Explanation of Solution

Compute present values:

(i)

5%, 2-year bond

$\begin{array}{c}\text{PV=}\frac{\text{CF}}{{\left(1+r\right)}^n}\\ =\frac{\$50}{1.05}+\frac{\$1,050}{{1.054}^2}\\ =992.80\end{array}$

(ii)

5%, 5-year bond

$\begin{array}{c}\text{PV=}\frac{\text{CF}}{{\left(1+r\right)}^n}\\ =\frac{\$50}{1.05}+\frac{\$50}{{1.054}^2}+\frac{\$50}{{1.057}^3}+\frac{\$50}{{1.059}^4}+\frac{\$1,050}{{1.060}^5}\\ =959.34\end{array}$

 (iii)

10%, 5-year bond

$\begin{array}{c}\text{PV=}\frac{\text{CF}}{{\left(1+r\right)}^n}\\ =\frac{\$100}{1.05}+\frac{\$100}{{1.054}^2}+\frac{\$100}{{1.057}^3}+\frac{\$100}{{1.059}^4}+\frac{\$1,100}{{1.060}^5}\\ =\$1,171.43\end{array}$

c)

Summary Introduction

To discuss: The reason why 10% bond is lower than 5% bond.

Explanation of Solution

Fist compute the yield for these two bonds.

For the 5% bond:

The computation of r using the equation of present value is as follows:

$\begin{array}{c}\text{PV}=\frac{\text{CF}}{{\left(1+r\right)}^n}\\ \$959.34=\frac{\$50}{\left(1+r\right)}+\frac{\$50}{{\left(1+r\right)}^2}+\frac{\$50}{{\left(1+r\right)}^3}+\\ \text{=}\frac{\$50}{{\left(1+r\right)}^4}+\frac{\$1,050}{{\left(1+r\right)}^5}\\ =.05964,\text{or 5}\text{.964\%}\text{.}\end{array}$

For the 10 % bond:

$\begin{array}{c}\text{PV}=\frac{\text{CF}}{{\left(1+r\right)}^n}\\ \$959.34=\frac{\$100}{\left(1+r\right)}+\frac{\$100}{{\left(1+r\right)}^2}+\frac{\$100}{{\left(1+r\right)}^3}+\\ \text{=}\frac{\$100}{{\left(1+r\right)}^4}+\frac{\$1,100}{{\left(1+r\right)}^5}\\ =.05937,\text{or 5}\text{.937\%}\text{.}\end{array}$

The yield of a bond is generally based on the current rate at the time of payment and the coupon payment. The 10% bond has a little better amount of its total payments, if the rate of interest is lesser than 5% of the bond Therefore, the yield of 10% bond is marginally lower.

d)

Summary Introduction

To discuss: The yield to maturity on 5 years zero coupon bond.

Explanation of Solution

The computation of yield to maturity on 5 years zero coupon bond is as follows:

Hence the yield to maturity is 6.10%.

e)

Summary Introduction

To discuss: The yield to maturity on a 5-year annuity.

Explanation of Solution

The calculation of yield to maturity on 5 years’ annuity is as follows:

Formula to compute yield to maturity on 5 years annuity is as follows:

Hence the yield to maturity on 5years annuity is 5.845%.

f)

Summary Introduction

To discuss: Whether the 5-year bond lie between the yield of five years’ annuity and five years zero coupon bond.

Explanation of Solution

The yield on the five-year note lies amongst yield on a five-year annuity and the yield on a five-year zero-coupon bond. It is because the treasury bond cash flows lie amongst the cash flows of two financial instruments through a period of increasing rate of interest. That is, the annuity has stable equivalent payments; the zero-coupon bond has one payment at the expiration; and the bond’s payments are a mixture of these.