978-1259709685 Chapter 24 Solution Manual

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CHAPTER 24 –

CHAPTER 24

WARRANTS AND CONVERTIBLES

Answers to Concepts Review and Critical Thinking Questions

1. A warrant is issued by the company, and when a warrant is exercised, the number of shares

2. a. If the stock price is less than the exercise price of the warrant at expiration, the warrant is

worthless. Prior to expiration, however, the warrant will have value as long as there is some

b. If the stock price is above the exercise price of the warrant, the warrant must be worth at least

the difference between these two prices. If warrants were selling for less than the difference

c. If the warrant is selling for more than the stock, it would be cheaper to purchase the stock than

3. An increase in the stock price volatility increases the bond price. If the stock price becomes more

4. The two components of the value of a convertible bond are the straight bond value and the option

value. An increase in interest rates decreases the straight value component of the convertible bond.

5. When warrants are exercised the number of shares outstanding increases. This results in the value of

the firm being spread out over a larger number of shares, often leading to a decrease in value of each

6. In an efficient capital market the difference between the market value of a convertible bond and the

7. There are three potential reasons: 1) To match cash flows, that is, they issue securities whose cash

flows match those of the firm. 2) To bypass assessing the risk of the company (risk synergy). For

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8. Because the holder of the convertible has the option to wait and perhaps do better than what is

9. Theoretically conversion should be forced as soon as the conversion value reaches the call price

because other conversion policies will reduce shareholder value. If conversion is forced when

10. No, the market price of the warrant will not equal zero. Since there is a chance that the market price

of the stock will rise above the $31 per share exercise price before expiration, the warrant still has

Solutions to Questions and Problems

NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple

steps. Due to space and readability constraints, when these intermediate steps are included in this

solutions manual, rounding may appear to have occurred. However, the final answer for each problem is

found without rounding during any step in the problem.

Basic

1. The conversion price is the par value divided by the conversion ratio, or:

2. The conversion ratio is the par value divided by the conversion price, or:

3. First, we need to find the conversion price, which is the par value divided by the conversion ratio, or:

The conversion premium is the necessary increase in stock price to make the bond convertible. So,

the conversion premium is:

4. a. The conversion ratio is defined as the number of shares that will be issued upon conversion.

b. The conversion price is defined as the face amount of a convertible bond that the holder must

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c. The conversion premium is defined as the percentage difference between the conversion price

of the convertible bonds and the current stock price. So, the conversion premium is:

d. The conversion value is defined as the amount that each convertible bond would be worth if it

were immediately converted into common stock. So, the conversion value is:

eIf the stock price increases by $2, the new conversion value will be:

5. The total exercise price of each warrant is the number of shares each warrant can purchase times the

exercise price, which in this case will be:

Therefore, the warrant effectively gives its owner the right to buy $204 worth of stock for $192. It

follows that the minimum value of the warrant is the difference between these numbers, or:

If the warrant were selling for less than $12, an investor could earn an arbitrage profit by purchasing

6. Since a convertible bond gives its holder the right to a fixed payment plus the right to convert, it

must be worth at least as much as its straight value. Therefore, if the market value of a convertible

bond is less than its straight value, there is an opportunity to make an arbitrage profit by purchasing

7. a. Using the conversion price, we can determine the conversion ratio, which is:

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Therefore, the minimum price the bond should sell for is $877.55. Since the bond price is

b. A convertible bond gives its owner the right to convert his bond into a fixed number of shares.

The market price of a convertible bond includes a premium over the value of immediate

conversion that accounts for the possibility of increases in the price of the firm’s stock before

8. You can convert or tender the bond (i.e., surrender the bond in exchange for the call price). If you

9. a. Since the stock price is currently below the exercise price of the warrant, the lower bound on

the price of the warrant is zero. If there is only a small probability that the firm’s stock price

will rise above the exercise price of the warrant, the warrant has little value. An upper bound on

b. If the stock is trading for $58 per share, the lower bound on the price of the warrant is $3, the

difference between the current stock price and the warrant’s exercise price. If warrants were

selling for less than this amount, an investor could earn an arbitrage profit by purchasing

Intermediate

10. a. The minimum convertible bond value is the greater of the conversion price or the straight bond

price. To find the conversion price of the bond, we need to determine the conversion ratio,

which is:

So, each bond can be exchanged for 10.75 shares of stock. This means the conversion price of

the bond is:

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b. If the stock price were growing by 11 percent per year forever, each share of stock would be

worth approximately $28(1.11)t after t years. Since each bond is convertible into 10.75 shares,

the conversion value of the bond equals ($28)(10.75)(1.11)t after t years. In order to calculate

11. a. The percentage of the company stock currently owned by the CEO is:

b. The conversion price indicates that for every $37 of face value of convertible bonds

outstanding, the company will be obligated to issue a new share upon conversion. So, the new

number of shares the company must issue will be:

12. a. Before the warrant was issued, the firm’s assets were worth:

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b. When the warrant was issued, the firm received $1,650, increasing the total value of the firm’s

assets to $16,500 (= $14,850 + 1,650). If the 8 shares of common stock were the only

outstanding claims on the firm’s assets, each share would be worth $2,062.50 (= $16,500 / 8

c. If the price of platinum is $1,950 per ounce, the total value of the firm’s assets is $19,500 (= 10

oz of platinum × $1,950 per oz). If the warrant is not exercised, the value of the firm’s assets

would remain at $19,500 and there would be 8 shares of common stock outstanding, so the

stock price would be $2,437.50. If the warrant is exercised, the firm would receive the

13. The value of the company’s assets is the combined value of the stock and the warrants. So, the value

of the company’s assets before the warrants are exercised is:

So, the new value of the company is:

This means the new stock price is:

Challenge

14. The straight bond value today is:

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And the conversion value of the bond today is:

We expect the bond to be called when the conversion value increases to $1,250, so we need to find

the number of periods it will take for the current conversion value to reach the expected value at

which the bond will be converted. Doing so, we find:

The bond will be called in 15.55 years.

The bond value is the present value of the expected cash flows. The cash flows will be the annual

15. The value of a single warrant (W) equals:

W = [# / (# + #W)] × Call{S = (V/ #), E = EW}

where:

# = the number of shares of common stock outstanding

Therefore, the value of a single warrant (W) equals:

W = [# / (# + #W)] × Call{S = (V/ #), E = EW}

In order to value the call option, use the Black–Scholes formula. Solving for d1 and d2, we find

d2 = d1 – (2t)1/2

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N(d1) = N(.2486) = .5981

According to the Black–Scholes formula, the price of a European call option (C) on a non-dividend

paying common stock is:

Therefore, the price of a single warrant (W) equals:

16. To calculate the number of warrants that the company should issue in order to pay off $18 million in

six months, we can use the Black–Scholes model to find the price of a single warrant, then divide

this amount into the present value of $18 million to find the number of warrants to be issued. So, the

value of the liability today is:

The company must raise this amount from the warrant issue.

The value of the company’s assets will increase by the amount of the warrant issue after the issue,

but this increase in value from the warrant issue is exactly offset by the bond issue. Since the cash

inflow from the warrants offsets the firm’s debt, the value of the warrants will be exactly the same as

Since the firm must raise $17,468,019.60 as a result of the warrant issue, we know #W × W must

equal $17,468,019.60.

Therefore, it can be stated that:

Using the Black–Scholes formula to value the warrant, which is a call option, we find:

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Next, we need to find N(d1) and N(d2), the area under the normal curve from negative infinity to d1

and negative infinity to d2, respectively.

Using this value in the equation above, we find the number of warrants the company must sell is: