Chapter 16, Problem 11PS

a.

Summary Introduction

To calculate: The price of the bond today.

Introduction:

Bond Price: It is supposed to be the present discounted value of future cash stream that is created by a bond. In simple words, it is sum of all the present value of coupon payments along with the present value of the par value of the time of maturity.

Answer to Problem 11PS

The price of the bond today is $103.90.

Explanation of Solution

Given information:

1-year zero-coupon bonds yields 7%

2-year zero-coupon bonds yields 8%

3-year and longer maturity bonds yields 9%

Payment of coupons annually is 8%

Let us calculate the price of the bond as on today. The formulas to be used is as follows:

  $PV=\frac{C}{{(1+y)}^t}+\frac{FV}{{(1+y)}^T}$

Where

C= periodic coupon payment

Y= YTM

FV= the bond’s par value or face value

t=time

T=the number of periods until the bond’s maturity date

Since the par value is given to us, let us assume the par value to be $100. Let us now calculate the coupon payment.

Calculation of coupon payment:

  $\begin{array}{l}=\$100\times 8\%\\ =\$100\times \frac{8}{100}\\ =\$100\times 0.08\\ =\$8\end{array}$

The value of $8 has to be used for further calculations.

  $PV=\frac{\$8}{{(1+7\%)}^1}+\frac{\$8}{{(1+8\%)}^2}+\frac{\$8}{{(1+9\%)}^3}+\frac{\$108}{{(1+9\%)}^3}$

Let us convert the percentages into decimals for easy calculations.

  $\begin{array}{l}PV=\frac{\$8}{{(1+0.07)}^1}+\frac{\$8}{{(1+0.08)}^2}+\frac{\$8}{{(1+0.09)}^3}+\frac{\$108}{{(1+0.09)}^3}\\ PV=\frac{\$8}{(1+0.07)}+\frac{\$8}{(1+0.1664)}+\frac{\$8}{(1+0.295029)}+\frac{\$108}{(1+0.295029)}\\ PV=\frac{\$8}{(1.07)}+\frac{\$8}{(1.1664)}+\frac{\$8}{(1.295029)}+\frac{\$108}{(1.295029)}\\ PV=7.4766+6.8587+6.1774+83.3958\\ =\$103.9086 or\$103.90\end{array}$

Therefore, the price of the bond today is $103.90.

b.

Summary Introduction

To calculate: The price of each bond in one year if yield curve is flat at 9% at that time.

Introduction:

Bond Price: It is supposed to be the present discounted value of future cash stream that is created by a bond. In simple words, it is sum of all the present value of coupon payments along with the present value of the par value of the time of maturity.

Answer to Problem 11PS

The price of the bond when the yield is flat at 9% will be $103.64.

Explanation of Solution

Given information:

1-year zero-coupon bonds yields 7%

2-year zero-coupon bonds yields 8%

3-year and longer maturity bonds yields 9%

Payment of coupons annually is 8%

A yield curve is said to flat when the interest rates remain stable over a said period.

We are told that the yield curve is flat at 9%. So, in all the three bonds, we will consider the yield as 9%. Let us now calculate the price of bond.

Calculation of coupon payment:

  $\begin{array}{l}=\$100\times 8\%\\ =\$100\times \frac{8}{100}\\ =\$100\times 0.08\\ =\$8\end{array}$

The value of $8 has to be used for further calculations.

  $\begin{array}{l}PV=\frac{\$8}{{(1+9\%)}^1}+\frac{\$8}{{(1+9\%)}^2}+\frac{\$8}{{(1+9\%)}^3}+\frac{\$108}{{(1+9\%)}^3}\\ PV=\frac{\$8}{{(1+0.09)}^1}+\frac{\$8}{{(1+0.09)}^2}+\frac{\$8}{{(1+0.09)}^3}+\frac{\$108}{{(1+0.09)}^3}\\ PV=\frac{\$8}{(1+0.09)}+\frac{\$8}{(1+0.1881)}+\frac{\$8}{(1+0.295029)}+\frac{\$108}{(1+0.295029)}\\ PV=\frac{\$8}{(1.09)}+\frac{\$8}{(1.1881)}+\frac{\$8}{(1.295029)}+\frac{\$108}{(1.295029)}\\ PV=7.3394+6.7334+6.1774+83.3958\\ =\$103.64\end{array}$

Therefore, the price of the bond when the yield is flat at 9% will be $103.64.

c.

Summary Introduction

To calculate: Rate of return on each bond.

Introduction:

Bond Price: It is supposed to be the present discounted value of future cash stream that is created by a bond. In simple words, it is sum of all the present value of coupon payments along with the present value of the par value of the time of maturity.

Answer to Problem 11PS

The rate of return on 1-year bond will be 9.81% and on 2-year bond it will be 8%, and on 3-year bond it will be 4.76%

Explanation of Solution

Given information:

1-year zero-coupon bonds yields 7%

2-year zero-coupon bonds yields 8%

3-year and longer maturity bonds yields 9%

Payment of coupons annually is 8%

Calculation of rate of return on 1-year bond:

  $Rateofreturn=Currentyieldatpurchasedate+capitalgain$

This formula can further be simplified.

  $\begin{array}{l}Rateofreturn=\frac{Couponpayment}{Bond’spurchaseprice}+\frac{Bond’spricewhensold-Bond’spurchaseprice}{Bond’spurchaseprice}\\ =\frac{\$8}{\$100}+\frac{Bond’spricewhensold-\$100}{\$100}\end{array}$

Since we are not aware of the bond’s price when it is sold, let us calculate it now.

Calculation of bond’s selling price:

Years to maturity =2

Coupon payment=$8

Interest=7%

Face value=$100

Since Excel application simplifies our calculations let us make use of it here.

  $\begin{array}{l}=PV\left(rate,nper,pmt,fv\right)\\ =PV(0.07,2,8,100)\\ =-\$101.81\end{array}$

If we ignore the ‘-‘ sign, our selling price of the bond will be $101.81.

Let us now calculate the rate of return.

Let us now substitute the required values in the formula.

  $\begin{array}{l}=\frac{\$8}{\$100}+\frac{\$101.81-\$100}{\$100}\\ =\frac{\$8}{\$100}+\frac{\$1.81}{\$100}\\ =0.08+0.0181\\ =0.0981\end{array}$

or 9.81 % when converted into percentages.

Therefore, the rate of return on 1-year bond will be 9.81%.

Calculation of rate of return on 2-year bond:

  $\begin{array}{l}Rateofreturn=\frac{Couponpayment}{Bond’spurchaseprice}+\frac{Bond’spricewhensold-Bond’spurchaseprice}{Bond’spurchaseprice}\\ =\frac{\$8}{\$100}+\frac{Bond’spricewhensold-\$100}{\$100}\end{array}$

Since we are not aware of the bond’s price when it is sold, let us calculate it now.

Calculation of bond’s selling price:

Years to maturity =3

Coupon payment=$8

Interest=8%

Face value=$100

Since Excel application simplifies our calculations let us make use of it here.

  $\begin{array}{l}=PV\left(rate,nper,pmt,fv\right)\\ =PV(0.08,3,8,100)\\ =-\$100\end{array}$

If we ignore the ‘-‘ sign, our selling price of the bond will be $100

Let us now calculate the rate of return.

  $\begin{array}{l}=\frac{\$8}{\$100}+\frac{\$100-\$100}{\$100}\\ =\frac{\$8}{\$100}+\frac{\$0}{\$100}\\ =0.08+0\\ =0.08\end{array}$

or 8%

Therefore, the rate of return on 2-year bond will be 8%.

Calculation of rate of return on 3-year bond:

  $\begin{array}{l}Rateofreturn=\frac{Couponpayment}{Bond’spurchaseprice}+\frac{Bond’spricewhensold-Bond’spurchaseprice}{Bond’spurchaseprice}\\ =\frac{\$8}{\$100}+\frac{Bond’spricewhensold-\$100}{\$100}\end{array}$

Since we are not aware of the bond’s price when it is sold, let us calculate it now.

Calculation of bond’s selling price:

Years to maturity =4

Coupon payment=$8

Interest=9%

Face value=$100

Since Excel application simplifies our calculations let us make use of it here.

  $\begin{array}{l}=PV\left(rate,nper,pmt,fv\right)\\ =PV(0.09,4,8,100)\\ =-\$96.76\end{array}$

If we ignore the ‘-‘ sign, our selling price of the bond will be $96.76.

Let us now calculate the rate of return.

  $\begin{array}{l}=\frac{\$8}{\$100}+\frac{\$96.76-\$100}{\$100}\\ =\frac{\$8}{\$100}+\frac{(-3.24)}{\$100}\\ =0.08-0.0324\\ =0.0476\end{array}$

or 4.76%

Therefore, the rate of return on 3-year bond will be 4.76%