Chapter 24, Problem 19QP

Convertible Calculations [LO6] Starset, Inc., has a $1,000 face value convertible bond issue that is currently selling in the market for $960. Each bond is exchangeable at any time for 18 shares of the company’s stock. The convertible bond has a 4.9 percent coupon, payable semiannually. Similar nonconvertible bonds are priced to yield 7.4 percent. The bond matures in 20 years. Stock in the company sells for $45 per share.

a. What are the conversion ratio, conversion price, and conversion premium?

b. What is the straight bond value? The conversion value?

c. In part (b), what would the stock price have to be for the conversion value and the straight bond value to be equal?

d. What is the option value of the bond?

Summary Introduction

To find: The conversion ratio, conversion price, and conversion premium.

Introduction:

The price of a share at which the convertible securities like the preferred shares or bonds can be exchanged for a general stock is the conversion price. The number of general shares that are obtained at the time of conversion for every convertible security is the conversion ratio. The amount where the cost of the convertible security is over the value of the current market’s general stock is the conversion premium.

Answer to Problem 19QP

The conversion ratio is 18, the conversion price is $55.56, and the conversion premium is 23.46%.

Explanation of Solution

Given information:

Company S has a face value of $1,000 for the convertible bond issue that at present sells for $960 in the market. Every bond is exchangeable at any time for eighteen shares of the company’s stock. The coupon rate of the convertible bond is 4.9% and it can be paid semiannually. The same type of the convertible bonds are priced to yield at 7.4%. The maturity of the bond is at 20 years. The company’s stock sells for $45 per share.

The conversion ratio is given at 18. The conversion price is calculated as follows:

Formula to compute the conversion price:

$\text{Conversion price}=\frac{\text{Par value}}{\text{Conversion ratio}}$

Computation of the conversion price:

$\begin{array}{c}\text{Conversion price}=\frac{\text{Par value}}{\text{Conversion ratio}}\\ =\frac{\$1,000}{18}\\ =\$55.56\end{array}$

Hence, the conversion price is $55.56.

Formula to calculate the conversion premium:

$\text{Conversion premium}=\frac{\left(\text{Conversion price}-\text{Current stock price}\right)}{\text{Current stock price}}$

Computation of the conversion premium:

$\begin{array}{c}\text{Conversion premium}=\frac{\left(\text{Conversion price}-\text{Current stock price}\right)}{\text{Current stock price}}\\ =\frac{\left(\$55.56-\$45\right)}{\$45}\\ =\$0.2346\end{array}$

Hence, the conversion premium is 0.2346 or 23.46%.

Summary Introduction

To find: the straight bond value and the conversion value.

Introduction:

The bond that makes the payment of interest at regular intervals, and at the time of maturity that pays back the original principal amount that is invested is the straight bonds. The securities financial worth that is got by exchanging a convertible security for its underlying asset is a conversion value.

Answer to Problem 19QP

The straight bond value is $747.15 and the conversion value is $810.

Explanation of Solution

Formula to compute the straight bond value:

$\text{Straight bond value}=C\times \left[\frac{1-\frac{1}{{\left(1+r\right)}^t}}{r}\right]+\frac{F}{{\left(1+r\right)}^t}$

Note: C is the coupon paid at each period, r is the rate for a period, t is the number of periods, F is the face value of the bond.

Computation of the straight bond value:

$\begin{array}{c}\text{Straight bond value}=C\times \left[\frac{1-\frac{1}{{\left(1+r\right)}^t}}{r}\right]+\frac{F}{{\left(1+r\right)}^t}\\ =\$24.50\times \left[\frac{1-\frac{1}{{\left(1+0.037\right)}^{40}}}{0.037}\right]+\frac{1,000}{{\left(1+0.037\right)}^{40}}\\ =\$24.50\times \left[\frac{1-0.233802819}{0.037}\right]+233.8028197\\ =\$507.346782+233.8028197\end{array}$

$=\$741.15$

Note:

• The given coupon rate is 4.9%, thus, the annual coupon rate is $49 and the semiannual coupon rate is $24.50
• The annual rate for the period is 7.4% and the semiannual rate for the period is 3.7%
• The number of periods is 40 years as the maturity period is 20 years

Hence, the straight bond value is $741.15.

Formula to compute the conversion value:

$\text{Conversion value}=\text{Conversion ratio times}\times \text{Price of the stock}$

Computation of the conversion value:

$\begin{array}{c}\text{Conversion value}=\text{Conversion ratio times}\times \text{Price of the stock}\\ \text{=18}\times \text{\$45}\\ \text{=\$810}\end{array}$

Hence, the conversion value is $810.

Summary Introduction

To find: The price of the stock to have the straight bond value and the conversion value to be equal (from the part b).

Introduction:

The bond that makes the payment of interest at regular intervals, and at the time of maturity that pays back the original principal amount that is invested is the straight bonds. The securities financial worth that is got by exchanging a convertible security for its underlying asset is a conversion value.

Answer to Problem 19QP

The price of the stock is $41.17.

Explanation of Solution

It is essential to set the straight bond value equivalent to the stock price conversion ratio times and by doing this, the stock price can be found as follows:

$\begin{array}{c}\$741.15=18S\\ S=\$41.17\end{array}$

Hence, the price of the stock is $41.17.

Summary Introduction

To find: The option value of the bond.

Introduction:

The value that is placed on the private willingness to make payment for the maintenance and preservation of the public asset is the option value.

Answer to Problem 19QP

The option value of the bond is $218.85.

Explanation of Solution

There are 2 option value that must be taken in considering a convertible bond. The conversion option value is the market value minus the floor value, and the speculative value of an option thus, it is defined as the floor value minus the straight bond value. In the case of the conversion value lesser than the straight bond value, then the speculative option is zero.

$\begin{array}{c}\text{Conversion value}=\$960-810\\ =\$150\end{array}$

$\begin{array}{c}\text{Speculative option value}=\$810-741.15\\ =\$68.85\end{array}$

$\begin{array}{c}\text{Total option value}=\text{Conversion value}+\text{Speculative option value}\\ \text{=\$150+\$68}\text{.85}\\ \text{=\$218}\text{.85}\end{array}$

Hence, the total option value is $218.85.