Ross公司理财(第七版)答案 Ch024

Chapter 24: Warrants and Convertibles24.1 a. A warrant is a security that gives its holder the right, but not the obligation, to buy shares of commonstock directly from a company at a fixed price for a given period of time. Each warrant specifies thenumber 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 commonstock, at the option of the holder at a specified price for a specified period of time.24.2 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 thestock price will rise above the exercise price in the time remaining until expiration. Therefore, if thestock price is below the exercise price of the warrant, the lower bound on the price of the warrant iszero.b.If the stock price is above the exercise price of the warrant, the warrant must be worth at least thedifference between these two prices. If warrants were selling for less than the difference between thecurrent stock price and the exercise price, an investor could earn an arbitrage profit (i.e. an immediatecash 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 topurchase the warrant, which gives its owner the right to buy the stock. Therefore, an upper bound onthe price of any warrant is the firm’s current stock price.24.3 a. The primary difference between warrants and call options is that, when warrants are exercised, the firmissues new shares. Both the purchase price and the exercise price of a warrant are received by the firmand increase the value of its assets. Unless a firm is selling calls on its own shares, this does not holdtrue for options.b.When call options are exercised, the number of shares the firm has outstanding remains unchanged.Shares of the company’s stock are simply transferred from one individual to another. When warrantsare exercised, however, the number of shares outstanding increases. This results in the value of thefirm being spread out over a larger number of shares, often leading to a decrease in value of eachindividual share. The decrease in the per-share price of a company’s stock due to a greater number ofshares outstanding is known as dilution.24.4 a. Before the warra nt was issued, Survivor’s assets were worth $3,500 (= 7 oz of platinum * $500 peroz). 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 fr om Tina, increasing the total value of the firm’sassets to $4,000 (= $3,500 + $500). If the two shares of common stock were the only outstandingclaims on the firm’s assets, each share would be worth $2,000 (= $4,000 / 2 shares). However, sincethe warr ant gives Tina a claim on the firm’s assets worth $500, the value of the firm’s assets availableto stockholders is only $3,500 (= $4,000 - $500). Since there are two shares outstanding, Survivor’svalue per share remains at $1,750 (= $3,500 / 2 shares) after the warrant issue. Note that the firm usesTina’s $500 to purchase one more ounce of platinum.c.If the price of platinum is $520 per ounce, the total value of the firm’s assets is $4,160 (= 8 oz ofplatinum * $520 per oz). If Tina does not exercise h er warrant, the value of the firm’s assets wouldremain at $4,160 and there would be two shares of common stock outstanding. If Tina exercises herwarrant, the firm would receive the warrant’s $1,800 striking price and issue Tina one share. The totalval ue of the firm’s assets would increase to $5,960 (= $4,160 + $1,800). Since there would now be 3shares outstanding and no warrants, Survivor’s price per share would be $1,986.67 (= $5,960 / 3shares). Since the $1,986.67 value of the share that she will receive is greater than the $1,800 exerciseprice of the warrant, investors will expect Tina to exercise. The firm’s stock price will reflect thisinformation and rise to $1,986.67per share on the warrant’s expiration date.24.5 a. Since the stock price is currently below the exercise price of the warrant, the lower bound on the priceof the warrant is zero. If there is only a small probability that the firm’s stock price will rise above theexercise price of the warrant, the warrant has little value. An upper bound on the price of the warrantis $8, the current price of General Modem’s common stock. One would never pay more than $8 toreceive the right to purchase a share of the company’s stock if the firm’s stock were only worth $8.b.If General Mod em’s 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 warrant’s exercise price. If warrants wereselling 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 awarrant is the current stock price. In this case, one would never pay more than $12 for the right to buya single share of General Modem’s stock when he could purchase a share outright for $12.24.6 Ricketti currently has 10 million shares of common stock outstanding that sell for $17 per share and 1million warrants outstanding worth $3 each. Therefore, the value of the f irm’s assets before the warrantsare 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 firm’s assets increases by $15 million (= 1 millionwarrants * $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 Ricketti’s assets is $188 m illion (= $173million + $15 million) and there are 11 million (= 10 million + 1 million) shares of common stockoutstanding.The price per share of Ricketti’s 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 valueof each warrant ($2 = $17 - $15), the stockholders gain and the warrant holders lose.24.7 No, the market price of the warrant will not equal zero. Since there is a chance that the market price of thestock 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.)24.8 Since Warrant X gives its owner the right to purchase 3 shares for $20 each, the total exercise price of eachwarrant 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 $90 worth of stock for $75. It follows that the minimum value of Warrant X is $15 (= $90 - $75), the difference between the exercise price of the warrant and the value of the stock received from the warrant exercise. IfWarrant X were selling for less than $15, an investor could earn an arbitrage profit by purchasing thewarrant, exercising it immediately, 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.24.9 The value of a single warrant (W) equals:W = [# / (# + #W)] * Call{S = (V/ #), K = K W}where # = the number of shares of common stock outstanding#W= the number of warrants outstandingCall{S, K} = a call option on an underlying asset worth S with a strike price KV = the firm’s value net of debtK W = the strike price of each warrantIn this problem:# = 4,000,000#W= 500,000V = $88,000,000K W= $20Therefore, the value of a single warrant (W) equals:W = [# / (# + #W)] * Call{S = (V/ #), K = K W}= [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 σ2 = 0.04K =$20 r = 0.07t = 1After identifying the inputs, solve for d1 and d2:d1= [ln(S/K) + (r + ½σ2)(t) ] / (σ2t)1/2= [ln(22/20) + {0.07 + ½(0.04)}(1) ] / (0.04*1)1/2= 0.9266d2= d1 - (σ2t)1/2= 0.9266 - (0.04*1)1/2= 0.7266Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative infinityto d2, respectively.N(d1) = N(0.9266) = 0.8229N(d2) = N(0.7266) = 0.7663According to the Black-Scholes formula, the price of a European call option (C) on a non-dividendpaying common stock is:C = SN(d1) – Ke-rt N(d2)= (22)(0.8229) – (20)e-(0.07)(1) (0.7663)= $3.81The Black-Scholes Price of the call option is $3.81.Therefore, the price of a single warrant (W) equals:W = (8/9)*Call(S = $22, K = $20)= (8/9)($3.81)= $3.39Therefore, the value of each of Superior Clamp’s warrants is $3.39.24.10 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 sixmonths 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 val ue of Omega’s assets is $150 million after the announcement, the value of the firm’s 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 firm’s $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 firm’s assets after the warrant issue would be $150 million (= $159.5 millio n - $9.5 million cash to pay off debt). Use $150 million as the firm’s 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 = K W}where # = the number of shares of common stock outstanding#W= the number of warrants outstandingCall{S, K} = a call option on an underlying asset worth S with a strike price KV = the firm’s value net of debtK W = the strike price of each warrantIn this problem:# = 1,500,000V = $150,000,000K W= $95Therefore, the value of a single warrant (W) equals:W = [# / (# + #W)] * Call{S = (V/ #), K = K W}= [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.The inputs to the Black-Scholes formula are: S= $100 σ2 = 0.5625K =$95 r = 0.10t = 0.5After identifying the inputs, solve for d1 and d2:d1= [ln(S/K) + (r + ½σ2)(t) ] / (σ2t)1/2= [ln(100/95) + {0.10 + ½(0.5625)}(0.50) ] / (0.5625*0.50)1/2= 0.4562d2= d1 - ( 2t)1/2= 0.4562 - (0.5625*0.50)1/2= -0.0742Find 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(0.4562) = 0.6759N(d2) = N(-0.0742) = 0.4704According to the Black-Scholes formula, the price of a European call option (C) on a non-dividend paying common stock is:C = SN(d1) – Ke-rt N(d2)= (100)(0.6759) – (95)e-(0.10)(0.50) (0.4704)= $25.08The Black-Scholes price of the call option is $25.08.Inserting this value into the equation above:$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)]*($25.08)#W = 507,634Therefore, in order to pay off $10 million worth of debt in 6 months, Omega should issue 507,634warrants today.24.11Since a convertible bond gives its holder the right to a fixed payment plus the right to convert, itmust 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 A, the market value of the convertible bond is $1,000. Since this amount is greater than theconvertible’s straight value ($900), Scenario A is feasible.In Scenario B, the market value of the convertible bond is $900. Since this amount is less than theconvertible’s straight value ($950), Scenario B is not feasible.Scenario A is more likely.24.12 a. The conversion price indicates that for each $25 of face value of the bond, the convertible bondholdercan receive 1 share. Since the $25 conversion price divides into the $1,000 face value of the bond 40times (= $1,000 / $25), each convertible bond can be exchanged for 40 shares of Sportime’s stock.Since each share is currently trading for $24, the value of immediate conversion of a single convertiblebond 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. Themarket price of a convertible bond includes a premium over the value of immediate conversion thataccounts for the possibility of increases in the price of the firm’s stock before the maturity of the bond.If the stock price rises, a convertible bondholder will convert and receive valuable shares of equity. Ifthe stock price decreases, the convertible bondholder holds the bond and retains his right to a fixedinterest and principal payments.24.13 a. Rob Stevens currently owns 500,000 of Isner’s 4,000,000 shares. Therefore, he owns 12.5%(= 500,000 / 4,000,000) of the firm’s common stock.b.The conversion price indicates that for every $20 of face value of convertible bonds outstanding, Isnerwill be obligated to issue a new share upon conversion. Since there is currently $20 million worth ofconvertible bonds (face value) outstanding, Isner will issue 1,000,000 (= $20,000,000 / $20) newshares when it calls the convertible bonds and forces conversion. This increases the number of Isner’soutstanding shares to 5,000,000 (= 4,000,000 + 1,000,000). After conversion, Rob Stevens will onlyown 10% (= 500,000 / 5,000,000) of the firm’s common stock.24.14 a. The conversion ratio is defined as the number of shares that will be issued upon conversion. Sinceeach bond is convertible into 28 shares of Hannon’s common stock, the conversion ratio of th econvertible bonds is 28.b.The conversion price is defined as the face amount of a convertible bond that the holder must surrenderin order to receive a single share. Since the conversion ratio indicates that each bond is convertible into28 shares and each convertible bond has a face value of $1,000, one must surrender $35.71 (= $1,000face value per bond / 28 shares per bond) in order to receive one share of Hannon’s common stock.c.The conversion premium is defined as the percentage difference between the conversion price of theconvertible bonds and the current stock price. Since Hannon’s common stock is trading for $31.25 pershare and the conversion price of each of its convertible bonds is $35.71, the conversion premium is14.27% [= ($35.71 / $31.25) – 1].d.The conversion value is defined as the amount that each convertible bond would be worth if it wereimmediately converted into common stock. Since each convertible bond gives its owner the right to 28shares of Hannon’s common stock, currently wor th $31.25 per share, the conversion value of the eachbond is $875 (= 28 shares * $31.25 per share).e.If Hannon’s common stock price increases by $2, the new conversion value of the bonds will be $931(= 28 shares * 33.25 per share).24.15 a. The straigh t value of a convertible bond is the bond’s value if it were not convertible into commonstock. Since the bond will pay $1,000 in 10 years and the appropriate discount rate is 10%, the presentvalue of $1,000, discounted at 10% per annum, equals the straight value of this convertible bond.Straight Value = $1,000 / (1.10)10= $385.54Therefore, 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 wereimmediately converted into common stock. Since the convertible bond gives its owner the right to 25shares of MGH’s 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 thebond and the maximum of its straight value and conversion value. In this problem, the bond’s marketvalue 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= $14.46Therefore, the option value of the convertible bond is $14.46.24.16 The conversion price is defined as the face amount of a convertible bond that the holder must surrenderin order to receive a single share of stock. In this problem, the conversion price is $180. Since the bond hasa face value of $1,000, it is convertible into 5.56 (= $1,000 / $180) shares.The conversion value is defined as the amount that the convertible bond would be worth if it wereimmediately converted into common stock. Since the convertible bond gives its owner the right to 5.56shares of common stock, currently worth $60 per share, the conversion value of the bond is $333.33 (= 5.56 shares * $60 per share).Therefore, the conversion value of this convertible bond is $333.33.24.17 a. The straight value of a con vertible bond is the bond’s value if it were not convertible into commonstock. The bond makes annual coupon payments of $60 (= 0.06 * $1,000) at the end of each year for30 years. In addition, the owner will receive the bond’s face value of $1,000 w hen the bond matures in30 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 valueof the coupon payments can be found by calculating the present value of an annuity that makespayments of $60 for 30 years, discounted at 12%.PV(Coupon Payments) = $60A300.12 = $483.31Since the repayment of principal occurs in 30 years, the present value of the principal payment can befound by discounting the $1,000 face value of the bond by 12% for 30 years.PV(Principal Payment) = $1,000 / (1.12)30 = $33.38Straight Value = PV(Coupon Payments) + PV(Principal Payment)= $483.31 + $33.38= $516.69Therefore, 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 surrenderin order to receive a single share. In this problem, the conversion price is $125. Since the bond has aface 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 wereimmediately converted into common stock. Since the convertible bond gives its owner the right to 8shares 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 A’s stock price were g rowing by 15% per year forever, each share of its stock would be worthapproximately $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:($35*8)(1.15)t = $1,100t = 9.79Therefore, it will take 9.79 years for the conversion value of the convertible bond to exceed $1,100.。