Chapter 15, Problem 11QP

a.

Summary Introduction

To determine: What will be the market value of the bonds if they are non callable.

Introduction:

Callable bond is a bond type that let the bond issuer to retain the privilege of redeeming the bond at some point before the bonds attains its maturity date. The bond issuer has an option to pay for by providing a higher coupon rate.

Answer to Problem 11QP

Solution: The price of the bond today is $1,121.48.

Explanation of Solution

Calculate the price of the bond in one year when the interest rate increases:

It is given that coupon rate is 6.50%, interest rate is 8%, and future value is $1,000.

$\begin{array}{c}\text{Price of the bond=}\left(\text{Coupon rate}\times \text{Future value}\right)+\frac{\left(\text{Coupon rate}\times \text{Future value}\right)}{\text{Rate of interest}}\\ =\left(6.50\%\times \$1,000\right)+\frac{\left(6.50\%\times \$1,000\right)}{8\%}\\ =65+\frac{65}{8\%}\\ =65+812.5\\ =\$877.5\end{array}$

Therefore, the price of the bond is $877.5.

Calculate the price of the bond in one year when the interest rate decreases:

It is given that coupon rate is 6.50%, interest rate is 5%, and the future value is $1,000.

$\begin{array}{c}\text{Price of the bond}=\left(\text{Coupon rate}\times \text{Future value}\right)+\frac{\left(\text{Coupon rate}\times \text{Future value}\right)}{\text{Rate of interest}}\\ =\left(6.50\%\times \$1,000\right)+\frac{\left(6.50\%\times \$1,000\right)}{5\%}\\ =65+\frac{65}{5\%}\\ =65+1,300\\ =\$1,365\end{array}$

Therefore, the price of the bond in one year is $1,365.

Determine the price of the bond today:

It is given that profitability in one year is 35% probability that interest rate increases to 8% and there is 65% probability that it will decrease by 5%.

$\begin{array}{c}\text{Market value of the bond}=\frac{\begin{array}{l}\left(\begin{array}{l}{\text{Price of the bond}}_{\text{Increase in interest rate}}\times \\ {\text{Probability}}_{\text{8\%}\text{interest}\text{rate}}\end{array}\right)+\\ \left(\begin{array}{l}{\text{Price of the bond}}_{\text{Decrease in interest rate}}\times \\ {\text{Probability}}_{\text{5\%}\text{interest}\text{rate}}\end{array}\right)\end{array}}{\left(1+\text{One year interest rate}\right)}\\ =\frac{\left(\$877.5\times 35\%\right)+\left(\$1,365\times 65\%\right)}{\left(1+6.5\%\right)}\\ =\frac{\$307.125+\$887.25}{1.065}\\ =\frac{\$1,194.375}{1.065}\\ =\$1,121.48\end{array}$

Therefore, the market value of the bond is $1,121.48.

b.

Summary Introduction

To determine: The value of the call provision to the company.

Explanation of Solution

If the rate of interest increases, the bond prices will decrease. If the bonds price decreases, the organization will not be able call them. In this situation, the bondholders will obtain the coupon payment, C, plus the current worth of the outstanding payments. So, if interest rate increases, the bonds rate in one year will be:

${\text{P}}_1=\text{C}+\frac{\text{C}}{\text{0}\text{.08}}$

If interest rates decrease, the supposition is that the bonds are callable.  In this situation, the bondholders will obtain the call rate, added to the coupon payment, C. Hence, the rate of the bonds if interest rates decreases will be:

$\begin{array}{c}{\text{P}}_1=\left(\$1,000+C\right)+C\\ =\$1,000+2C\end{array}$

The current selling rate of the bond is the present value of the anticipated payoff to the bondholders. To determine the coupon rate, the desired issue price should be equal to the present value of the anticipated value at end of year payoffs, and solve for C.

$\begin{array}{c}\text{Coupon rate}=\left[\frac{\left(\begin{array}{l}\left(\text{Future value}\times \left(1+\text{Interest rate}\right)\right)-\\ {\text{Probability}}_{5\%\text{Interest rate}}\times \text{Future value}\end{array}\right)}{\begin{array}{l}{\text{Probability}}_{8\%\text{Interest rate}}+\frac{{\text{Probability}}_{8\%\text{Interest rate}}}{\text{Interest rate}}+\\ \left({\text{Probability}}_{8\%\text{Interest rate}}\times 2\right)\end{array}}\right]\\ =\frac{\$1,000\times \left(1+6.5\%\right)-\left(65\%\times \$1,000\right)}{35\%+\left(\frac{35\%}{8\%}\right)+\left(65\%\times 2\right)}\\ =\$68.88\end{array}$

Therefore, the present value coupon rate is $68.88.

Calculate the coupon rate required to sell the bonds at par value:

$\begin{array}{c}\text{Coupon rate=}\frac{\text{Present value coupon rate}}{\text{Future value}}\\ =\frac{\$68.88}{\$1,000}\\ =0.069\text{or}6.89\%\end{array}$

Therefore, the coupon rate required to sell the bonds at par value is 6.89%.

c.

Summary Introduction

To determine: The value of the call provision to the company.

Explanation of Solution

Determine the value of the call provision:

The call provision value will be given by the difference between the value of an outstanding, call provision and non-callable bond.

Determine the value of a non callable bond with the similar coupon rate:

It is given that interest rate will be 5% and present value of coupon rate is $68.88.

$\begin{array}{c}\text{Non}\text{callable bond value=}\frac{\text{Coupon rate}}{\text{Interest rate}}\\ =\frac{\$68.88}{0.05}\\ =\$1,377.59\end{array}$

Therefore, the non callable bond value is $1,377.59.

Calculate the value of call provision to the company:

$\begin{array}{c}\text{Value of call provision}=\frac{{\text{Probability}}_{\text{Interest rate 5\%}}\times \left(\begin{array}{l}\text{Non-callable bond value}-\\ \left(\text{Future value+Coupon rate}\right)\end{array}\right)}{\left(1+\text{Interest rate}\right)}\\ =\frac{65\%\times \left(\$1,377.59-\left(\$1,000+\$\text{68}\text{.88}\right)\right)}{\left(1+6.5\%\right)}\\ =\$188.42\end{array}$

Therefore, the value of call provision is $188.42.