CHAPTER 24
WARRANTS AND CONVERTIBLES
Answers to Concepts Review and Critical Thinking Questions
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
probability that the stock price will rise above the exercise price in the time remaining until
expiration. Therefore, if the stock price is below the exercise price of the warrant, the lower
3. An increase in the stock price volatility increases the bond price. If the stock price becomes more
volatile, the conversion option on the stock becomes more valuable.
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
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
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.
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:
Conversion price = $1,000/13.80
4. a. The conversion ratio is defined as the number of shares that will be issued upon conversion. Since
each bond is convertible into 21.5 shares of common stock, the conversion ratio of the convertible
bonds is 21.5.
b. The conversion price is defined as the face amount of a convertible bond that the holder must
surrender in order to receive a single share. Since the conversion ratio indicates that each bond is
convertible into 21.5 shares, the conversion price is:
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:
e If 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:
Exercise price = 3($58)
Exercise price = $174
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
7. a. Using the conversion price, we can determine the conversion ratio, which is:
Conversion ratio = $1,000/$75
Conversion ratio = 13.33
So, each bond can be exchanged for 13.33 shares of stock. This means the conversion price of
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 the price
of the warrant is $62, the current price of the common stock. One would never pay more than
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:
Conversion ratio = $1,000/$89
Conversion ratio = 11.24
b. If the stock price were growing by 10 percent per year forever, each share of stock would be
worth approximately $26(1.10)t after t years. Since each bond is convertible into 11.24 shares,
the conversion value of the bond equals ($26)(11.24)(1.10)t after t years. In order to calculate the
number of years that it will take for the conversion value to equal $1,100, set up the following
equation:
11. a. The percentage of the company stock currently owned by the CEO is:
b. The conversion price indicates that for every $36 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:
Value of assets = 13 oz of platinum($890 per oz)
Value of assets = $11,570
So, the price per share is:
b. When the warrant was issued, the firm received $890, increasing the total value of the firm’s
assets to $12,460 (= $11,570 + 890). If the 12 shares of common stock were the only outstanding
claims on the firm’s assets, each share would be worth $1,038.33 (= $12,460/12 shares).
c. If the price of platinum is $950 per ounce, the total value of the firm’s assets is $13,300 (= 14 oz
of platinum × $950 per oz). If the warrant is not exercised, the value of the firm’s assets would
remain at $13,300 and there would be 12 shares of common stock outstanding, so the stock price
would be $1,108.33. If the warrant is exercised, the firm would receive the warrant’s $1,000
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:
Company value = 27,000,000($32) + 1,800,000($7)
Company value = $876,600,000
14. The straight bond value today is:
Straight bond value = $49(PVIFA9%,20) + $1,000/1.0920
Straight bond value = $625.73
$291.60(1.12)t = $1,250
t = 12.84 years
The bond will be called in 12.84 years.
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
#W = the number of warrants outstanding
N(d2) = N(–.1987) = .4213
According to the Black-Scholes formula, the price of a European call option (C) on a non-dividend
paying common stock is:
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:
PV of liability = $18,000,000e–.06(6/12)
W = [#/(# + #W)] × Call(S, K)
W = [2,700,000/(2,700,000 + #W)] × Call($88.89, $95)
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:
N(d1) = N(.0736) = .5293
N(d2) = N(–.2800) = .3897
According to the Black-Scholes formula, the price of a European call option (C) on a non-dividend
paying common stock is: