Assignment_2_Solutions_COMM122_Jan2023

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Solutions to Assignment 2 19.4 To find the new stock price, we multiply the current stock price by the ratio of old shares to new shares, so: a. $78 × (3/5) = $46.80 b. $78 × (1/1.15) = $67.83 c. $78 × (1/1.425) = $54.74 d. $78 × (7/4) = $136.50. To find the new shares outstanding, we multiply the current shares outstanding times the ratio of new shares to old shares, so: a: 260,000 × (5/3) = 433,333 b: 260,000 × (1.15) = 299,000 c: 260,000 × (1.425) = 370,500 d: 260,000 × (4/7) = 148,571 19.5 The stock price is the total market value of equity divided by the shares outstanding, so: P 0 = $465,000 equity/12,000 shares = $38.75 per share Ignoring tax effects, the stock price will drop by the amount of the dividend, so: P X = $38.75 – $1.90 = $36.85 The total dividends paid will be: $1.90 per share × 12,000 shares = $22,800

The equity and cash accounts will both decline by $22,800. 19.13 a. If the company makes a dividend payment, we can calculate the wealth of a shareholder as: Dividend per share = $3,000/600 shares = $5.00 The stock price after the dividend payment will be: P X = $58 – $5 = $53 per share The shareholder will have a stock worth $53 and a $5 dividend for a total wealth of $58. If the company makes a repurchase, the company will repurchase shares worth $3,000: Shares repurchased = $3,000/$58 = 51.72 shares If the shareholder lets their shares be repurchased, they will have $58 in cash. If the shareholder keeps their shares, they’re still worth $58. b. If the company pays dividends, the current EPS is $1.50, and the P/E ratio is: P/E = $53/$1.50 = 35.33 If the company repurchases stock, the number of shares will decrease. The total net income is the EPS times the current number of shares outstanding. Dividing net income by the new number of shares outstanding, we find the EPS under the repurchase is: EPS = ($1.50 × 600)/(600 − 51.72) = $1.64 The stock price will remain at $58 per share, so the P/E ratio is: P/E = $58/$1.64 = 35.33 c. A share repurchase would seem to be the preferred course of action. Only those shareholders who wish to sell will do so, giving the shareholder a tax timing option that he or she doesn’t get with a dividend payment. 19.14 a. Since the firm has a 100 percent payout policy, the entire net income, $85,000 will be paid as a dividend. The current value of the firm is the discounted value one year from now, plus the current income, which is:

Value = $85,000 + $1,725,000/1.12 Value = $1,625,178.57 b. The current stock price is the value of the firm, divided by the shares outstanding, which is: Stock price = $1,625,178.57/25,000 Stock price = $65.01 Since the company has a 100 percent payout policy, the current dividend per share will be the company’s net income, divided by the shares outstanding, or: Current dividend = $85,000/25,000 Current dividend = $3.40 The stock price will fall by the value of the dividend to: Ex – dividend stock price = $65.01 – $3.40 Ex – dividend stock price = $61.61 c. i. According to MM, it cannot be true that the low dividend is depressing the price. Since dividend policy is irrelevant, the level of the dividend should not matter. Any funds not distributed as dividends add to the value of the firm, hence the stock price. These directors merely want to change the timing of the dividends (more now, less in the future). As the calculations below indicate, the value of the firm is unchanged by their proposal. Therefore, the share price will be unchanged. To show this, consider what would happen if the dividend were increased to $4.60. Since only the existing shareholders will get the dividend, the required dollar amount to pay the dividends is: Total dividends = $4.60 × 25,000 Total dividends = $115,000 To fund this dividend payment, the company must raise: Dollars raised = Required funds – Net income Dollars raised = $115,000 – $85,000 Dollars raised = $30,000

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This money can only be raised with the sale of new equity to maintain the all – equity financing. Since those new shareholders must also earn 12 percent, their share of the firm one year from now is: New shareholder value in one year = $30,000 × 1.12 New shareholder value in one year = $33,600 This means that the old shareholders' interest falls to: Old shareholder value in one year = $1,725,000 – $33,600 Old shareholder value in one year = $1,691,400 Under this scenario, the current value of the firm is: Value = $115,000 + $1,691,400/1.12 Value = $1,625,178.57 Since the firm value is the same as in part a, the change in dividend policy had no effect. ii. The new shareholders are not entitled to receive the current dividend. They will receive only the value of the equity one year hence. The present value of those flows is: Present value = $1,691,400/1.12 Present value = $1,510,178.57 And the current share price will be: Current share price = $1,510,178.57/25,000 Current share price = $60.41 So, the number of new shares the company must sell will be:

Shares sold = $30,000/$60.41 Shares sold = 496.63 shares 20.1 a. The new market value will be the number of current shares outstanding times the stock price plus the number of new shares offered times the offer price, so: New market value = 450,000 × $90 + 80,000 × $84 = $47,220,000 b. The number of rights associated with one new share is the number of current shares outstanding divided by the number of shares offered, so: Number of rights needed = 450,000 old shares/80,000 new shares = 5.63 rights per new share c. The ex – rights price of the stock will be the new market value of the company divided by the total number of shares outstanding after the rights offer, which will be: P X = $47,220,000/(450,000 + 80,000) = $89.09 d. The value of the right Value of a right = $90.00 – $89.09 = $0.91 e. A rights offering usually costs less, protects the proportionate interests of existing shareholders, and protects against underpricing. 20.8 The number of rights needed per new share is: Number of rights needed = 100,000 old shares/20,000 new shares = 5 rights per new share. Using P RO as the rights – on price, and P S as the subscription price, we can express the price per share of the stock ex – rights as: P X = (NP RO + P S )/(N + 1)

a. P X = (5 × $80 + $80)/6 = $80.00; No change. b. P X = (5 × $80 + $75)/6 = $79.17; Price drops by $0.83 per share. c. P X = (5 × $80 + $65)/6 = $77.50; Price drops by $2.50 per share. 20.10 a. Assume you hold three shares of the company’s stock. The value of your holdings before you exercise your rights is: Value of holdings = 3 × $68 Value of holdings = $204 When you exercise, you must remit the three rights you receive for owning three shares, and $11. You have increased your equity investment by $11. The value of your holdings after surrendering your rights is: New value of holdings = $204 + $11 New value of holdings = $215 After exercise, you own four shares of stock. Thus, the price per share of your stock is: Stock price = $215/4 Stock price = $53.75 b. The value of a right is the difference between the rights – on price of the stock and the ex – rights price of the stock: Value of rights = Rights – on price – Ex – rights price Value of rights = $68 – $53.75 Value of rights = $14.25 c. The price drop will occur on the ex – rights date, even though the ex – rights date is neither the expiration date nor the date on which the rights are first exercisable. If you

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purchase the stock before the ex – rights date, you will receive the rights. If you purchase the stock on or after the ex – rights date, you will not receive the rights. Since rights have value, the stockholder receiving the rights must pay for them. The stock price drop on the ex – rights day is similar to the stock price drop on an ex – dividend day. 21.10 If interest rates rise, the price of the bonds will fall. If the price of the bonds is low, the company will not call them. The firm would be foolish to pay the call price for something worth less than the call price. In this case, the bondholders will receive the coupon payment, C, plus the present value of the remaining payments. So, if interest rates rise, the price of the bonds in one year will be: P 1 = C + C/0.10 If interest rates fall, the assumption is that the bonds will be called. In this case, the bondholders will receive the call price, plus the coupon payment, C. So, the price of the bonds if interest rates fall will be: P 1 = $1,175 + C The selling price today of the bonds is the PV of the expected payoffs to the bondholders. To find the coupon rate, we can set the desired issue price equal to the present value of the expected value of end of year payoffs, and solve for C. Doing so, we find: P 0 = $1,000 = [0.60 × (C + C/0.10) + 0.40 × ($1,175 + C)]/1.09 C = $88.57 So the coupon rate necessary to sell the bonds at par value will be: Coupon rate = $88.57/$1,000 Coupon rate = 0.0886 or 8.86%

21.11 a. The price of the bond today is the present value of the expected price in one year. So, the price of the bond in one year if interest rates increase will be: P 1 = $70 + $70/0.09 P 1 = $847.78 If interest rates fall, the price if the bond in one year will be: P 1 = $70 + $70/0.05 P 1 = $1470 Now we can find the price of the bond today, which will be: P 0 = (0.40 × $847.78 + 0.60 × $1,470)/1.07 P 0 = $1,141.23 b. If interest rates rise, the price of the bonds will fall. If the price of the bonds is low, the company will not call them. The firm would be foolish to pay the call price for something worth less than the call price. In this case, the bondholders will receive the coupon payment, C, plus the present value of the remaining payments. So, if interest rates rise, the price of the bonds in one year will be: P 1 = C + C/0.09 If interest rates fall, the bonds will be called. In this case, the bondholders will receive the call price, plus the coupon payment, C. The call premium is the same as the coupon rate, so the price of the bonds if interest rates fall will be: P 1 = ($1,000 + C) + C P 1 = $1,000 + 2 × C

The selling price today of the bonds is the PV of the expected payoffs to the bondholders. To find the coupon rate, we can set the desired issue price equal to present value of the expected value of end of year payoffs, and solve for C. Doing so, we find: P 0c = $1,000 = [0.40 × (C + C/0.09) + 0.60 × ($1,000 + 2 × C)]/1.07 C = $77.76 So the coupon rate necessary to sell the bonds at par value will be: Coupon rate = $77.76/$1,000 Coupon rate = 0.07776 or 7.776% c. To the company, the value of the call provision is given by the difference between the value of an outstanding, non – callable bond and the value of the callable bond, with both bonds offering the same coupon rate. So, the value of a noncallable bond with the same coupon rate would be: Non – callable bond value = [0.4 × ($77.76 + $77.76/0.09) + 0.6 × ($77.76 + $77.76/0.05)]/1.07 = $1,267.74 Thus, the value of the call provision to the company is: Value = $1,267.74 – $1,000 Value = $267.74 Alternatively, the value of the call provision to the company can be determined using the PV of the expected savings from the provision: Value = {(0.4 × $0) + 0.6 × [($77.76 + $77.76/0.05) - ($1,000 + 2 × $77.76)]}/1.07 = $267.74

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23.6 Using put – call parity and solving for the put price, we get: $47 + P = $45 × e – (0.026)(3/12) + $3.80 P = $1.51 23.7 Using put – call parity and solving for the call price we get: $61 + $4.89 = $65 × e – (0.036)(.5) + C C = $2.05 23.8 Using put – call parity and solving for the stock price we get: S + $2.40 = $85 × e – (0.048)(3/12) + $5.09 S = $86.68 23.9 Using put – call parity, we can solve for the risk – free rate as follows: $57.30 + $2.65 = $55 × e – R(2/12) + $5.32 $54.63 = $55 × e – R(2/12) 0.9932 = e – R(2/12) ln(0.9932) = ln(e – R(2/12) ) – 0.0068 = – R × (2/12) R f = 4.05% 23.10 Using the Black – Scholes option pricing model to find the price of the call option, we find: d 1 = [ln($46/$50) + (0.06 + 0.54 2 /2)  (3/12)]/(0.54  12 / 3 ) = – 0.1183 d 2 = – 0.1183 – (0.54  12 / 3 ) = – 0.3883 N(d 1 ) = 0.4529 N(d 2 ) = 0.3489 Putting these values into the Black – Scholes model, we find the call price is: C = $46 × 0.4529 – ($50 × e – 0.06(0.25) × 0.3489) = $3.65

Using put – call parity, the put price is: Put = $50 × e – 0.06(0.25) + $3.65 – $46 = $6.90 23.11 Using the Black – Scholes option pricing model to find the price of the call option, we find: d 1 = [ln($93/$90) + (0.04 + 0.62 2 /2)  (5/12)]/(0.62  12 / 5 ) = 0.3237 d 2 = 0.3237 – (0.62  12 / 5 ) = – 0.0765 N(d 1 ) = 0.6269 N(d 2 ) = 0.4695 Putting these values into the Black – Scholes model, we find the call price is: C = $93 × 0.6269 – ($90 × e – 0.04(5/12) × 0.4695) = $16.75 Using put – call parity, the put price is: P = $90 × e – 0.04(5/12) + $16.75 – $93 = $12.26 23.18 If the exercise price is equal to zero, the call price will equal the stock price, which is $75. 23.19 If the standard deviation is zero, d 1 and d 2 go to +∞, so N(d 1 ) and N(d 2 ) go to 1. So: C = S N(d 1 ) – E N(d 2 ) e – rt C = $84 × (1) – $80 × (1) × e – 0.05(6/12) = $5.98

24.1 a. The inputs to the Black – Scholes model are the current price of the underlying asset (S), the strike price of the option (E), the time to expiration of the option in fractions of a year (t), the variance (  2 ) of the underlying asset, and the continuously – compounded risk – free interest rate (R). Since these options were granted at – the – money, the strike price of each option is equal to the current value of one share, or $50. We can use Black – Scholes to solve for the option price. Doing so, we find: d 1 = [ln(S/E) + (R +  2 /2)(t)]/(  2 t) 1/2 d 1 = [ln($50/$50) + (0.06 + 0.56 2 /2)  5]/(0.56  5 ) = 0.8657 d 2 =0.8657 – (0.56  5 ) = – 0.3865 Find N(d 1 ) and N(d 2 ), the area under the normal curve from negative infinity to d 1 and negative infinity to d 2 , respectively. Doing so: N(d 1 ) = N(0.8657) = 0.8067 N(d 2 ) = N( – 0.3865) = 0.3496 Now we can find the value of each option, which will be: C = S N(d 1 ) – E e – Rt N(d 2 ) C = $50 × 0.8067 – ($50 × e – 0.06(5) × 0.3496) C = $27.39 Since the option grant is for 30,000 options, the value of the grant is: Grant value = 30,000 × $27.39 Grant value = $821,700 b. Because he is risk – neutral, you should recommend the alternative with the highest net present value. Since the expected value of the stock option package is worth more than $750,000, he would prefer to be compensated with the options rather than with the immediate bonus. c. If he is risk – averse, he may or may not prefer the stock option package to the immediate bonus. Even though the stock option package has a higher net present value, he may not prefer it because it is undiversified. The fact that he cannot sell his options prematurely makes it much more risky than the immediate bonus. Therefore, we cannot say which alternative he would prefer.

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