25: Chapter Warrants and Convertibles
25.1 a. A warrant is a security that gives its holder the right, but not the obligation, to buy shares of common stock directly from a company at a fixed price for a given period of time. Each warrant specifies the number of shares of stock that the holder can buy, the exercise price, and the expiration date. b. A convertible bond is a bond that may be converted into another form of security, typically common stock, at the option of the holder at a specified price for a specified period of time. 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 bound on the price of the warrant is zero. 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 between the current stock price and the exercise price, an investor could earn an arbitrage profit (i.e. an immediate cash inflow) by purchasing warrants, exercising them immediately, and selling the stock. c. If the warrant is selling for more than the stock, it would be cheaper to purchase the stock than to purchase the warrant, which gives its owner the right to buy the stock. Therefore, an upper bound on the price of any warrant is the firms current stock price. a. The primary difference between warrants and call options is that, when warrants are exercised, the firm issues new shares. Both the purchase price and the exercise price of a warrant are received by the firm and increase the value of its assets. Unless a firm is selling calls on its own shares, this does not hold true for options. b. When call options are exercised, the number of shares the firm has outstanding remains unchanged. Shares of the companys stock are simply transferred from one individual to another. When warrants are exercised, however, 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 individual share. The decrease in the per-share price of a companys stock due to a greater number of shares outstanding is known as dilution. a. Before the warrant was issued, Survivors assets were worth $3,500 (= 7 oz of platinum * $500 per oz). Since there are only two shares of common stock outstanding, each share is worth $1,750 (= $3,500 / 2 shares). b. When the warrant was issued, the firm received $500 from Tina, increasing the total value of the firms assets to $4,000 (= $3,500 + $500). If the two shares of common stock were the only outstanding claims on the firms assets, each share would be worth $2,000 (= $4,000 / 2 shares). However, since the warrant gives Tina a claim on the firms assets worth $500, the value of the firms assets available to stockholders is only $3,500 (= $4,000 - $500). Since there are two shares outstanding, Survivors value per share remains at $1,750 (= $3,500 / 2 shares) after the warrant issue. Note that the firm uses Tinas $500 to purchase one more ounce of platinum. c. If the price of platinum is $520 per ounce, the total value of the firms assets is $4,160 (= 8 oz of platinum * $520 per oz). If Tina does not exercise her warrant, the value of the firms assets would remain at $4,160 and there would be two shares of common stock outstanding. If Tina exercises her warrant, the firm would receive the warrants $1,800 striking price and issue Tina one share. The total value of the firms assets would increase to $5,960 (= $4,160
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25.2
25.3
25.4
Answers to End-of-Chapter Problems
+ $1,800). Since there would now be 3 shares outstanding and no warrants, Survivors price per share would be $1,986.67 (= $5,960 / 3 shares). Since the $1,986.67 value of the share that she will receive is greater than the $1,800 exercise price of the warrant, investors will expect Tina to exercise. The firms stock price will reflect this information and rise to $1,986.67 per share on the warrants expiration date. 25.5 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 firms 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 $8, the current price of General Modems common stock. One would never pay more than $8 to receive the right to purchase a share of the companys stock if the firms stock were only worth $8. b. If General Modems stock is trading for $12 per share, the lower bound on the price of the warrant is $2, the difference between the current stock price and the warrants exercise price. If warrants were selling for less than this amount, an investor could earn an arbitrage profit by purchasing warrants, exercising them immediately, and selling the stock. As always, the upper bound on the price of a warrant is the current stock price. In this case, one would never pay more than $12 for the right to buy a single share of General Modems stock when he could purchase a share outright for $12. Ricketti currently has 10 million shares of common stock outstanding that sell for $17 per share and 1 million warrants outstanding worth $3 each. Therefore, the value of the firms assets before the warrants are exercised is $173 million [= (10 million shares * $17 per share) + (1 million warrants * $3 per warrant)]. Once the warrants are exercised, the total value of the firms assets increases by $15 million (= 1 million warrants * $15 per warrant). Since each warrant gives its owner the right to receive one share, the number of shares of common stock outstanding increases by 1,000,000. Therefore, once the warrants have been exercised, the value of Rickettis assets is $188 million (= $173 million + $15 million) and there are 11 million (= 10 million + 1 million) shares of common stock outstanding. The price per share of Rickettis common stock after the warrants have been exercised is $17.09 (= $188 million / 11 million shares). Note that since the warrants were exercised when the price per warrant ($3) was above the exercise value of each warrant ($2 = $17 - $15), the stockholders gain and the warrant holders lose. 25.7 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 $21 per share exercise price before expiration, the warrant still has some value. Its market price will be greater than zero. (As a practical matter, warrants that are way out-of-the-money may sell at 0, due to transaction costs.) Since Warrant X gives its owner the right to purchase 3 shares for $20 each, the total exercise price of each warrant is $60 (= 3 * $20). Each share of Firm Y is currently selling for $25 per share. The value of three shares of the firm is $75 (= 3 * $25). Therefore, Warrant X effectively gives its owner the right to buy $75 worth of stock for $60. It follows that the minimum value of Warrant X is $15 (= $75 - $60), the difference between the exercise price of the warrant and the value of the stock received from the warrant exercise. If Warrant X were selling for less than $15, an investor could earn an arbitrage profit by purchasing the warrant, exercising it immediately,
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25.6
25.8
Answers to End-of-Chapter Problems
and selling the stock. Here, the warrant holder pays less than $15 while receiving the $15 difference between the price of 3 shares and the exercise price. 25.9 The value of a single warrant (W) equals: W = [# / (# + #W)] * Call{S = (V/ #), K = KW} where # = the number of shares of common stock outstanding #W = the number of warrants outstanding Call{S, K} = a call option on an underlying asset worth S with a strike price K V = the firms value net of debt KW = the strike price of each warrant In this problem: # #W V KW = 4,000,000 = 500,000 = $88,000,000 = $20
Therefore, the value of a single warrant (W) equals: W = [# / (# + #W)] * Call{S = (V/ #), K = KW} = [4,000,000 / (4,000,000 + 500,000) * Call{S = ($88,000,000 / 4,000,000), K = $20} = (8/9)*Call(S = $22, K = $20) In order to value the call option, use the Black-Scholes formula. The inputs to the Black-Scholes formula are: S= $22 K =$20 t=1 s2 = 0.04 r = 0.07
After identifying the inputs, solve for d1 and d2: d1 = [ln(S/K) + (r + s2)(t) ] / (s2t)1/2 = [ln(22/20) + {0.07 + (0.04)}(1) ] / (0.04*1)1/2 = 0.9266 d2 = d1 - (s2t)1/2 = 0.9266 - (0.04*1)1/2 = 0.7266 Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative infinity to d2, respectively. N(d1) = N(0.9266) = 0.8229 N(d2) = N(0.7266) = 0.7662
Answers to End-of-Chapter Problems
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According to the Black-Scholes formula, the price of a European call option (C) on a nondividend paying common stock is: C = SN(d1) Ke-rtN(d2) = (22)(0.8229) (20)e-(0.07)(1) (0.7662) = $3.82 The Black-Scholes price of the call option is $3.82. Therefore, the price of a single warrant (W) equals: W = (8/9)*Call(S = $22, K = $20) = (8/9)($3.82) = $3.40 Therefore, the value of each of Superior Clamps warrants is $3.40. 25.10 To calculate the number of warrants that Omega should issue in order to pay off $10 million in six months, use the Black-Scholes model to find the price of a single warrant, then divide this amount into the present value of $10 million to find the number of warrants to be issued. Since Omega owes $10 million in 6 months and the current yield on Treasury bills that mature in six months is 10% per annum (continuously-compounded), Omega must raise $9,512,294 [= $10,000,000 / (e(0.10*0.5))] from the warrant issue today in order to meet its debt obligation of $10 million in six months. Since the value of Omegas assets is $150 million after the announcement, the value of the firms assets will rise to $159.5 million (= $150 million + $9.5 million in proceeds) after the warrants are issued. Since the cash inflow from the warrants offsets the firms $9.5 million in debt, the value of the warrants will be exactly the same as if the cash from the warrants were used to immediately pay off the debt. In this case, the value of the firms assets after the warrant issue would be $150 million (= $159.5 million - $9.5 million cash to pay off debt). Use $150 million as the firms value net of debt (V) in the Black-Scholes formula. The firm has 1.5 million shares of common stock outstanding and wishes to issue warrants with a strike price of $95. The value of a single warrant (W) equals: W = [# / (# + #W)] Call{S * = (V/ #), K = KW} where # = the number of shares of common stock outstanding #W = the number of warrants outstanding Call{S, K} = a call option on an underlying asset worth S with a strike price K V = the firms value net of debt KW = the strike price of each warrant In this problem: # = 1,500,000 V = $150,000,000
Answers to End-of-Chapter Problems B-376
KW = $95 Therefore, the value of a single warrant (W) equals: W = [# / (# + #W)] * Call{S = (V/ #), K = KW} = [1,500,000 / (1,500,000 +#W)] * Call{S = ($150,000,000/ 1,500,000), K = $95} = [1,500,000 / (1,500,000 +#W)] * Call(S = $100, K = $95) Since the firm must raise $9,512,294 as a result of the warrant issue, we know #W * W must equal $9,512,294. Therefore, it can be stated that: $9,512,294 = (#W)(W) $9,512,294 = (#W)([1,500,000 / (1,500,000 +#W)] * Call(S = $100, K = $95) In order to value the call option, use the Black-Scholes formula. s2 = 0.5625 r = 0.10
The inputs to the Black-Scholes formula are:
S= $100 K =$95 t = 0.5
After identifying the inputs, solve for d1 and d2: d1 = [ln(S/K) + (r + s2)(t) ] / (s2t)1/2 = [ln(100/95) + {0.10 + (0.5625)}(0.50) ] / (0.5625*0.50)1/2 = 0.4562 d2 = d1 - (s2t)1/2 = 0.4562 - (0.5625*0.50)1/2 = -0.0742
Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative infinity to d2, respectively. N(d1) N(d2) = N(0.4562) = 0.6758 = N(-0.0742) = 0.4704
According to the Black-Scholes formula, the price of a European call option (C) on a nondividend paying common stock is: C = SN(d1) Ke-rtN(d2) = (100)(0.6758) (95)e-(0.10)(0.50) (0.4704) = $25.07 The Black-Scholes price of the call option is $25.07. Inserting this value into the equation above:
Answers to End-of-Chapter Problems B-377
$9,512,294 = (#W) [1,500,000 / (1,500,000 +#W)] *Call(S = $100, K = $95) $9,512,294 = (#W) [1,500,000 / (1,500,000 +#W)]*($25.07) #W = 507,906 Therefore, in order to pay off $10 million worth of debt in 6 months, Omega should issue 507,906 warrants today. 25.11 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 the bond and holding it until expiration. In Scenario 1, the market value of the convertible bond is $1,000. Since this amount is greater than the convertibles straight value ($900), Scenario A is feasible. In Scenario 2, the market value of the convertible bond is $900. Since this amount is less than the convertibles straight value ($950), Scenario B is not feasible. Scenario 1 is more likely. 25.12 a. The conversion price indicates that for each $25 of face value of the bond, the convertible bondholder can receive 1 share. Since the $25 conversion price divides into the $1,000 face value of the bond 40 times (= $1,000 / $25), each convertible bond can be exchanged for 40 shares of Sportimes stock. Since each share is currently trading for $24, the value of immediate conversion of a single convertible bond is $960 (= $24 per share * 40 shares). Therefore, the minimum value that each convertible bond should sell for is $960. 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 firms stock before the maturity of the bond. If the stock price rises, a convertible bondholder will convert and receive valuable shares of equity. If the stock price decreases, the convertible bondholder holds the bond and retains his right to a fixed interest and principal payments. 25.13 a. Rob Stevens currently owns 500,000 of Isners 4,000,000 shares. Therefore, he owns 12.5% (= 500,000 / 4,000,000) of the firms common stock. b. The conversion price indicates that for every $20 of face value of convertible bonds outstanding, Isner will be obligated to issue a new share upon conversion. Since there is currently $20 million worth of convertible bonds (face value) outstanding, Isner will issue 1,000,000 (= $20,000,000 / $20) new shares when it calls the convertible bonds and forces conversion. This increases the number of Isners outstanding shares to 5,000,000 (= 4,000,000 + 1,000,000). After conversion, Rob Stevens will only own 10% (= 500,000 / 5,000,000) of the firms common stock. 25.14 a. The conversion ratio is defined as the number of shares that will be issued upon conversion. Since each bond is convertible into 28 shares of Hannons common stock, the conversion ratio of the convertible bonds is 28.
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Answers to End-of-Chapter Problems
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 28 shares and each convertible bond has a face value of $1,000, one must surrender $35.71 (= $1,000 face value per bond / 28 shares per bond) in order to receive one share of Hannons common stock. c. The conversion premium is defined as the percentage difference between the conversion price of the convertible bonds and the current stock price. Since Hannons common stock is trading for $31.25 per share and the conversion price of each of its convertible bonds is $35.71, the conversion premium is 14.27% [= ($35.71 / $31.25) 1]. d. The conversion value is defined as the amount that each convertible bond would be worth if it were immediately converted into common stock. Since each convertible bond gives its owner the right to 28 shares of Hannons common stock, currently worth $31.25 per share, the conversion value of the each bond is $875 (= 28 shares * $31.25 per share). e. If Hannons common stock price increases by $2, the new conversion value of the
bonds will be $931 (= 28 shares * 33.25 per share).
25.15 a. The straight value of a convertible bond is the bonds value if it were not convertible into common stock. Since the bond will pay $1,000 in 10 years and the appropriate discount rate is 10%, the present value of $1,000, discounted at 10% per annum, equals the straight value of this convertible bond. Straight Value = = $1,000 / (1.10)10 $385.54
Therefore, the straight value of the convertible bond is $385.54. b. The conversion value is defined as the amount that the convertible bond would be worth if it were immediately converted into common stock. Since the convertible bond gives its owner the right to 25 shares of MGHs common stock, currently worth $12 per share, the conversion value of the bond is $300 (= 25 shares * $12 per share). Therefore, the conversion value of the convertible bond is $300. c. The option value of a convertible bond is defined as the difference between the market value of the bond and the maximum of its straight value and conversion value. In this problem, the bonds market value is $400, its straight value is $385.54, and its conversion value is $300. Option Value = Market Value - max[Straight Value, Conversion Value] = $400 max[$385.54, $300 ] = $400 - $385.54 = $14.46 Therefore, the option value of the convertible bond is $14.46.
Answers to End-of-Chapter Problems
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25.16
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 of stock. In this problem, the conversion price is $180. Since the bond has a face value of $1,000, it is convertible into 5.55556 (= $1,000 / $180) shares. The conversion value is defined as the amount that the convertible bond would be worth if it were immediately converted into common stock. Since the convertible bond gives its owner the right to 5.55556 shares of common stock, currently worth $60 per share, the conversion value of the bond is $333.33 (= 5.55556 shares * $60 per share). Therefore, the conversion value of this convertible bond is $333.33.
25.17
a. The straight value of a convertible bond is the bonds value if it were not convertible into common stock. The bond makes annual coupon payments of $60 (= 0.06 * $1,000) at the end of each year for 30 years. In addition, the owner will receive the bonds face value of $1,000 when the bond matures in 30 years. The straight value of the bond equals the present value of its cash flows. Since the bond makes annual coupon payments of $60 (= 0.06 * $1,000) for 30 years, the present value of the coupon payments can be found by calculating the present value of an annuity that makes payments of $60 for 30 years, discounted at 12%. PV(Coupon Payments) = $60A300.12 = $483.31 Since the repayment of principal occurs in 30 years, the present value of the principal payment can be found by discounting the $1,000 face value of the bond by 12% for 30 years. PV(Principal Payment) = $1,000 / (1.12)30 = $33.38 Straight Value = PV(Coupon Payments) + PV(Principal Payment) = $483.31 + $33.38 = $516.69 Therefore, the straight value of the convertible bond is $516.69. 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. In this problem, the conversion price is $125. Since the bond has a face value of $1,000, it is convertible into 8 (= $1,000 / $125) shares. The conversion value is defined as the amount that the convertible bond would be worth if it were immediately converted into common stock. Since the convertible bond gives its owner the right to 8 shares of common stock, currently worth $35 per share, the conversion value of the bond is $280 (= 8 shares * $35 per share). Therefore, the conversion value of this convertible bond is $280. c. If Firm As stock price were growing by 15% per year forever, each share of its stock would be worth approximately $35(1.15)t after t years. Since each bond is convertible into 8 shares, the conversion value of the bond equals ($35*8)(1.15)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:
Answers to End-of-Chapter Problems
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($35*8)(1.15)t = $1,100 t = 9.79
Therefore, it will take 9.79 years for the conversion value of the convertible bond to exceed $1,100.
25.18
This practice would generally be used for risky companies. Simply increasing the interest rate might cause these companies to forgo positive net present value projects in order to pay the interest. In keeping the interest rate lower, the company can gain value by investing in high growth projects, and the financial institutions can then be rewarded with higher returns when they exercise the warrants or call options on the shares of the firm.
Answers to End-of-Chapter Problems
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