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HOMEWORK SOLUTIONS 2.1 Assets Current assets Cash $ 4,000 Accounts receivable 8,000 Total current assets $ 12,000 Long-tern assets Machinery $34,000 Patents 82,000 Total long-term assets $116,000 Total assets $128,000 Liabilities and equity Current liabilities Accounts payable $ 6,000 Taxes payable 2,000 Total current liabilities $ 8,000 Long-term liabilities Bonds payable $7,000 Stockholders equity Preferred stock $19,000 Common stock 88,000 Retained earnings 6,000 Total stockholders equity $113,000 Total liabilities and equity $128,000 2.3 Income Statement Total operating revenues $500,000 Less: Cost of goods sold $200,000 Administrative expenses 100,000 300,000 Earnings before interest and taxes $200,000 Less: Interest expense 50,000 Earnings before taxes $150,000 Taxes 60,000 Net income $90,000 2.4 a. Income Statement The Flying Lion Corporation 2004 2005 Net sales $800,000 $500,000 Cost of goods sold (560,000) (320,000) Operating expenses (75,000) (56,000) Depreciation (300,000) (200,000) Earnings before taxes $(135,000) $(76,000) Taxes* 54,000 30,400 Net income $(81,000) $(45,600) * The problem states that Flying Lion has other profitable operations. Flying Lion can take advantage of tax losses by deducting the tax liabilities in the other operations that have taxable profits. If Flying Lion did not have other operations and tax losses could not be carried forward or backward, then taxes in each of these years would have been zero. b. Cash flow during 2002 = -$81,000 + $300,000 = $219,000 Cash flow during 2005 = -$45,600+ $200,000 = $154,400 2.5 The main difference between accounting profit and cash flow is that non-cash costs, such as depreciation expense, are included in accounting profits. Cash flows do not consider costs that do not represent actual cash expenditures. Cash flows deduct the entire cost of an investment at the time the cash flow occurs. 2.A1 a) No change Both inventory and cash are current assets. b) Increase or Decrease Both current assets and current liabilities would be reduced by the same amount. The net effect will depend on the relative effects. c) Increase or Decrease If the bank loan is a current liability then both the current assets and current liabilities will be reduced. The effect on the current ratio will depend on the relative values. However, if the bank loan is long-term debt then the current ratio would decrease because of the reduction in current assets. d) Decrease Current assets are reduced to pay the long-term debt. e) No change Accounts receivable and cash are current assets. f) No change Inventory, cash and accounts receivable are current assets. 4.1 $120,000 - ($150,000 - $100,000) (1.1) = $65,000 In order to consume $150,000 this year Jean will borrow $50,000 and pay $55,000 ($50,000 principal and $5,000 interest) next year, leaving him $65,000 potential consumption next year. 4.6 a. $90,000 / $80,000 - 1 = 0.125 = 12.5% b. He will invest $10,000 in financial assets and $30,000 in productive assets today. c. NPV = -$30,000 + $56,250 / 1.125 = $20,000 5.1 a. $1,000 1.0510 = $1,628.89 b. $1,000 1.0710 = $1,967.15 c. $1,000 1.0520 = $2,653.30 d. Interest compounds on the interest already earned. Therefore, the interest earned in part c, $2653.30, is more than double the amount earned in part a, $628.89. 5.2 a. $1,000 / 1.17 = $513.16 b. $2,000 / 1.1 = $1,818.18 c. $500 / 1.18 = $233.25 5.7 PV of Joneses’ offer = $150,000 / (1.1)3 = $ 112,697.22 Since the PV of Joneses’ offer is less than Smiths’ offer of $115,000, you should choose Smiths’ offer. 5.9 PV = $5,000,000 / 1.1210 = $1,609,866.18 5.11 NPV = -($340,000 + $10,000) + ($100,000 - $10,000) / 1.1 + $90,000 /1.12 + $90,000 / 1.13 + $90,000 / 1.14 + $100,000 / 1.15 = -$2,619.98 Since the NPV is negative, you should not buy it. If the relevant cost of capital is 9 percent, NPV = -$350,000 + $90,000 / 1.09 + $90,000 / 1.092 + $90,000 / 1.093 + $90,000 / 1.094 + $100,000 / 1.095 = $ 6,567.93 Since the NPV is positive, you should buy it. 5.12 a. Profit = PV of revenue - Cost = NPV NPV = $90,000 / 1.15 - $60,000 = -$4,117 No, the firm will not make a profit. b. Find r that makes zero NPV. $90,000 / (1+r)5 - $60,000 = $0 (1+r)5 = 1.5 r = 0.08447 = 8.447% 5.17 a. $1,000 e0.12 5 = $1,822.12 b. $1,000 e0.1 3 = $ 1,349.86 c. $1,000 e0.05 10 = $1,648.72 d. $1,000 e0.07 8 = $1,750.67 5.20 The price of the consol bond is the present value of the coupon payments. Apply the perpetuity formula to find the present value. PV = ₤120 / 0.15 = ₤800. 5.22 The value at t = 8 (beginning of year 9) is $120 / 0.1 = $1,200. Thus, the value at t = 5 (end of year 5) is $1,200 / 1.13 = $901.58. 5.24 Using the growing perpetuity formula, find the PV of dividends. The PV is the maximum you should be willing to pay for the share. P0 = 1.00 / (.10 - .04) = $16.67 5.29 Apply the NPV technique to answer this question. NPV = -6,200 + 1,200 A8.10 = -6,200 + 6,401.91 = 201.91, Since the result is positive, then yes, you should buy the asset. 5.32 The exact value could be obtained by solving the following equation for the interest rate (r). 1 (1 r ) T PVA PMT r 1 (1 r ) 10 12,800 2, 000 r r 0.09063 Financial calculators can compute the rate directly as 9.0626%. 6.2 The amount of the semi-annual interest payment is $40 (=$1,000 0.08 / 2). There are a total of 40 periods; i.e., two half years in each of the twenty years in the term to maturity. The annuity factor tables can be used to price these bonds. The appropriate discount rate to use is the semi-annual rate. That rate is simply the annual rate divided by two. Thus, for part a the rate to be used is 4.0%, for part b the rate to be used is 5% and for part c it is 3%. a. $40 0.04 + $1,000 / 1.0440 = $1,000 40 Notice that whenever the coupon rate and the market rate are the same, the bond is priced at par. b. $40 0.05 + $1,000 / 1.0540 = $828.41 40 Notice that whenever the coupon rate is below the market rate, the bond is priced below par. c. $40 0.03 + $1,000 / 1.0340 = $1,231.15 40 Notice that whenever the coupon rate is above the market rate, the bond is priced above par. $923.14 = C 0.05 + $1,000 / 1.0530 30 6.5 = (15.3725) C + $231.38 C = $45.00 The annual coupon rate = $45.00 2 / $1,000 = 0.09 = 9% 6.6 a. The semi-annual interest rate is $60 / $1,000 = 0.06. Thus, the effective annual rate is 1.062 - 1 = 0.1236 = 12.36%. b. Price = $30 12 + $1,000 / 1.0612 0.06 = $251.52 + $496.97 = $748.49 c. Price = $30 12 + $1,000 / 1.0412 0.04 = $281.55+ $624.60= $906.15 Note: In parts b and c we are implicitly assuming that the yield curve is flat. That is, the yield in year 5 applies for year 6 as well. 6.13 The price of a share of stock is the PV of its dividend payments. Since a dividend of $2 was paid yesterday, the next dividend payment, to be received one year from today, will be $2.16 (=$2 1.08). The dividend for each of the two successive years will also grow at eight percent. PV(Year 1 – 3) = Div1 / (1+r) + Div2 / (1+r)2 + Div3 / (1+r)3 = $2.16 / (1.12) + $2.33 / (1.12)2 + $2.52 / (1.12)3 = $5.58 The dividend at year 4 is $2.62 since the $2 dividend that occurred yesterday has grown three years at eight percent and one year at four percent [=$2 (1.08)3 1.04]. Applying the perpetuity formula to the dividends that begin in year 4 will generate the PV as of the end of year 3. Discount that value back three periods to find the PV as of today, year 0. PV(Year 4 - ) = [Div4 / (r – g)] / (1+r)3 = [$2.62 / (0.12 – 0.04)] / (1.12)3 = $23.31 The price of the bond is the sum of the PVs of the first three dividend payments and the PV of the dividend payments thereafter. P = Div1 / (1+r) + Div2 / (1+r)2 + Div3 / (1+r)3 + [Div4 / (r – g)] / (1+r)3 = $2.16 / (1.12) + $2.33 / (1.12)2 + $2.52 / (1.12)3 + [$2.62 / (0.12 – 0.04)] / (1.12)3 = $28.89 The price of the stock is $28.89. 6.15 a. P = $2 / (0.12 - 0.05) = $28.57 b. P10 = D11 / (r - g) = $2 (1.0510) / (0.12 - 0.05) = $46.54 6.16 Find the PV of the dividend payments. Since the dividend of $1.15 was just paid yesterday, the dividend payment in year 1 is $1.36 (=$1.15 1.18). Remember to adjust the dividend payment each year for the appropriate growth rate. Div1 = $1.15 1.18 = $1.36 Div2 = $1.15 1.182 = $1.60 Div3 = $1.15 1.182 1.15 = $1.84 Div4 = $1.15 1.182 1.15 1.06 = $1.95 Apply the growing perpetuity formula to find the PV of the dividend payments starting in year 4 and growing at six percent forever. Since the perpetuity formula yields the PV of the cash flows as of year 3, discount the perpetuity back three periods to find its value as of today. P = Div1 / (1+r) + Div2 / (1+r)2 + Div3 / (1+r)3 + [Div4 / (r – g)] / (1+r)3 = $1.36 / (1.12) + $1.60 / (1.12)2 + $1.84 / (1.12)3 + [$1.95 / (0.12 – 0.06)] / (1.12)3 = $26.93 The price of the stock is $26.93. 6A.1 Given a one-year bond and a two-year bond, one knows the spot rates for both. The forward rate is the rate of return implicit on a one-year bond purchased in the second year that would equate the terminal wealth of purchasing the one-year bond today and another in one year, with that of the two-year bond. 7.1 a. The payback period is the time that it takes for the cumulative undiscounted cash inflows to equal the initial investment. Project A: Cumulative Undiscounted Cash Flows Year 1 = $4,000 Cumulative Undiscounted Cash Flows Year 2 = $4,000 +$3,500 = $7,500 Payback period = 2 Project A has a payback period of two years. Project B: Cumulative Undiscounted Cash Flows Year 1 = $2,500 Cumulative Undiscounted Cash Flows Year 2 = $2,500+$1,200 = $3,700 Cumulative Undiscounted Cash Flows Year 3 = $2,500+$1,200+$3,000 = $6,700 Project B’s cumulative undiscounted cash flows exceed the initial investment of $5,000 by the end of year 3. Many companies analyze the payback period in whole years. The payback period for project B is 3 years. Project B has a payback period of three years. Companies can calculate a more precise value using fractional years. To calculate the fractional payback period, find the fraction of year 3’s cash flows that is needed for the company to have cumulative undiscounted cash flows of $5,000. Divide the difference between the initial investment and the cumulative undiscounted cash flows as of year 2 by the undiscounted cash flow of year 3. Payback period = 2 + ($5,000 - $3,700) / $3,000 = 2.43 Since project A has a shorter payback period than project B has, the company should choose project A. b. Discount each project’s cash flows at 15 percent. Choose the project with the highest NPV. Project A = -$7,500 + $4,000 / (1.15) + $3,500 / (1.15)2 + $1,500 / (1.15)3 = -$388.96 Project B = -$5,000 + $2,500 / (1.15) + $1,200 / (1.15)2 + $3,000 / (1.15)3 = $53.83 The firm should choose Project B since it has a higher NPV than Project A has. 7.3 a. The average accounting return is the average project earnings after taxes, divided by the average book value, or average net investment, of the machine during its life. The book value of the machine is the gross investment minus the accumulated depreciation. Average Book Value = (Book Value0 + Book Value1 + Book Value2 + Book Value3 + Book Value4 + Book Value5) / (Economic Life) = ($16,000 + $12,000 + $8,000 + $4,000 + $0) / (5 years) = $8,000 Average Project Earnings = $4,500 Divide the average project earnings by the average book value of the machine to calculate the average accounting return. Average Accounting Return = Average Project Earnings / Average Book Value = $4,500 / $8,000 = 0.5625 = 56.25% The average accounting return is 56.25%. b. 1. The average accounting return uses accounting data rather than net cash flows. 2. The average accounting return uses an arbitrary firm standard as the decision rule. The firm standard is arbitrary because it does not necessarily relate to a market rate of return. 3. The average accounting return does not consider the timing of cash flows. Hence, it does not consider the time value of money. 7.4 Determine the average book value of the investment. The book value is the gross investment minus accumulated depreciation. Purchase Year 1 Year 2 Year 3 Year 4 Year 5 Gross $2,000,000 $2,000,000 $2,000,000 $2,000,000 $2,000,000 $2,000,000 Investment Less: Accumulated 0 400,000 800,000 1,200,000 1,600,000 2,000,000 Depreciation Net $2,000,000 $1,600,000 $1,200,000 $800,000 $400,000 $0 Investment Average Book Value = ($2,000,000 + $1,600,000 + $1,200,000 + $800,000 + $400,000 + $0) / (6) = $1,000,000 Next, calculate average annual net income. Net Income Year 1 = $100,000 Net Income Year 2 = $100,000 (1.06) = $106,000 Net Income Year 3 = $100,000 (1.06)2 = $112,360 Net Income Year 4 = $100,000 (1.06)3 = $119,102 Net Income Year 5 = $100,000 (1.06)4 = $126,248 Average Net Income = ($100,000+$106,000+$112,360+$119,102+$126,248) /5 = $112,742 The average accounting return is the average net income divided by the average book value. Average Accounting Return = Average Net Income / Average Book Value = $112,742 / $1,000,000 = 0.1127 = 11.27% Since the machine’s average accounting return, 11.27%, is below the company’s cutoff of 20%, the machine should not be purchased. 7.7 a. The internal rate of return is the discount rate at which the NPV of the project’s cash flows equal zero. Set the project’s cash flows, discounted at the internal rate of return (IRR), equal to zero. Solve for the IRR. IRR = C0 + C1 / (1+IRR) + C2 / (1+IRR)2 + C3 / (1+IRR)3 0 = -$8,000 + $4,500 / (1+IRR) + $3,500 / (1+IRR)2 + $2,500 / (1+IRR)3 IRR = 0.1669 The IRR is 16.69%. b. Yes. An investing-type project is one with a negative initial cash outflow and positive future cash inflows. One accepts a project when the IRR is greater than the discount rate. Similarly, one rejects the project when the IRR is less than the discount rate. The project should be accepted because the IRR (16.69%) is greater than the discount rate (8%). 7.8 a. The profitability index, PI, is the ratio of the present value of the future expected cash flows after the initial investment to the amount of the initial investment. PI(A) = [C1 / (1+r) + C2 / (1+r)2 + C3 / (1+r)3] / (Initial Investment) = [$300 / (1.1) + $700 / (1.1)2 + $600 / (1.1)3] / ($500) = 2.6 The profitability index for project A is 2.6. PI(B) = [C1 / (1+r) + C2 / (1+r)2 + C3 / (1+r)3] / (Initial Investment) = [$300 / (1.1) + $1,800 / (1.1)2 + $1,700 / (1.1)3] / ($2,000) = 1.5 The profitability index for project B is 1.5. b. Greenplain should accept both projects A and B. The NPV of a project is positive whenever the profitability index (PI) is greater than one. 7.13a. The profitability index, PI, is the ratio of the present value of the future expected cash flows after the initial investment to the amount of the initial investment. PI(A) = [C1 / (1+r) + C2 / (1+r)2 + C3 / (1+r)3] / (Initial Investment) = [$300 / (1.1) + $700 / (1.1)2 + $600 / (1.1)3] / ($500) = 2.6 The profitability index for project A is 2.6. PI(B) = [C1 / (1+r) + C2 / (1+r)2 + C3 / (1+r)3] / (Initial Investment) = [$300 / (1.1) + $1,800 / (1.1)2 + $1,700 / (1.1)3] / ($2,000) = 1.5 The profitability index for project B is 1.5. c. Greenplain should accept both projects A and B. The NPV of a project is positive whenever the profitability index (PI) is greater than one. 7.18 a. Payback period for the New Sunday Early Edition: Use the payback period rule to calculate the number of years that it takes for the cumulative undiscounted cash inflows to equal the initial investment. Initial Investment = -$1,200 Year 1 = $600 = $600 Year 2 = $600 + $550 = $1,150 Year 3 = $600 + $550 + $450 = $1,600 The undiscounted cash flows exceed the initial investment of $1,200 by the end of year 3. Many companies analyze the payback period in whole years. The payback period for the project is 3 years. The New Sunday Early Edition has a payback period of three years. Companies can calculate a more precise value using fractional years. Calculate the fraction of year 3’s cash flow that is needed for the company to have cumulative undiscounted cash flows of $1,200. Find the difference between the initial investment and the cumulative undiscounted cash flows as of year 2, divided by the undiscounted cash flow of year 3. Payback Period = 2 + ($1,200 - $1,150) / $450 = 2.11 Payback period for the New Saturday Late Edition: Use the payback period rule to calculate the number of years that it takes for the cumulative undiscounted cash inflows to equal the initial investment. Initial Investment = -$2,100 Year 1 = $1,000 = $1,000 Year 2 = $1,000 + $900 = $1,900 Year 3 = $1,000 + $900 + $800 = $2,700 In year 3, the undiscounted cash flows exceed the initial investment of $2,100 by the end of year 3. Many companies analyze the payback period in whole years. The payback period for the project is 3 years. The payback period for the New Saturday Late Edition is three years. Companies can calculate a more precise value using fractional years. Calculate the fraction of year 3’s cash flows that is needed for the company to have cumulative undiscounted cash flows of $2,100. Find the difference between the initial investment and the cumulative undiscounted cash flows as of year 2, divided by the undiscounted cash flow of year 3. Payback Period = 2 + ($2,100 - $1,900) / $800 = 2.25 Using the whole number payback period, the projects are equally attractive. Using the fractional payback period calculation, the New Sunday Early Edition is more attractive because it has a shorter payback period than does the New Saturday Early Edition. b. New Sunday Early Edition IRR The internal rate of return is the discount rate at which the NPV of the project’s cash flows equals zero. Set the project’s cash flows, discounted at the internal rate of return (IRR), equal to zero. Solve for the IRR. IRR = C0 + C1 / (1+IRR) + C2 / (1+IRR)2 + C3 / (1+IRR)3 0 = -$1,200 + $600 / (1+IRR) + $550 / (1+IRR)2 + $450 / (1+IRR)3 IRR = 0.1676 New Saturday Late Edition IRR The internal rate of return is the discount rate at which the NPV of the project’s cash flows equals zero. Set the project’s cash flows, discounted at the internal rate of return (IRR), equal to zero. Solve for the IRR. IRR = C0 + C1 / (1+IRR) + C2 / (1+IRR)2 + C3 / (1+IRR)3 0 = -$2,100 + $1,000 / (1+IRR) + $900 / (1+IRR)2 + $800 / (1+IRR)3 IRR = 0.1429 The New Sunday Early Edition has a greater IRR than the New Saturday Late Edition. c. Find the IRR of the incremental cash flows. The incremental IRR is the IRR on the incremental investment from choosing the larger project instead of the smaller project. Incremental cash flows are defined as the New Saturday Late Edition’s Cash Flows minus the New Sunday Early Edition’s cash flows. Remember to subtract the cash flows of the project with the smaller initial investment from those of the project with the larger initial investment, so that the incremental initial investment is negative. Year 0 Year 1 Year 2 Year 3 Saturday Edition -$2,100 $1,000 $900 $800 Sunday Edition -1,200 600 550 450 Saturday – Sunday -$900 $400 $350 $350 IRR = C0 + C1 / (1+IRR) + C2 / (1+IRR)2 + C3 / (1+IRR)3 0 = -$900 + $400 / (1+IRR) + $350 / (1+IRR)2 + $350 / (1+IRR)3 IRR = 0.1102 For investing-type projects, accept the larger project when the incremental rate of return is greater than the discount rate. Since the discount rate of 12% is greater than the incremental IRR of 11.02%, choose the New Sunday Early Edition. d. Average Accounting Return for the New Sunday Early Edition: First, determine the average book value of the project. The book value is the gross investment minus accumulated depreciation. Annual Depreciation = $1,200 / 3 = $400 Year 0 Year 1 Year 2 Year 3 Gross Investment $1,200 $1,200 $1,200 $1,200 Accumulated $0 $400 $800 $1,200 Depreciation Book Value $1,200 $800 $400 $0 Average Investment = ($1,200 + $800 + $400 + $0) / (4) = $600 Calculate the average annual income of the project. Average Income = ($400 + $350 + $300) / 3 = $350 Divide the average project earnings by the average book value of the machine to calculate the average accounting return. Average Accounting Return = (Average Income) / (Average Investment) = $350 / $600 = 0.583 The average accounting return for the New Sunday Early Edition is 58.3%. Average Accounting Return for the New Saturday Late Edition: First, determine the average book value of the project. The book value is the gross investment minus accumulated depreciation. Annual Depreciation = $2,100 / 3 = $700 Year 0 Year 1 Year 2 Year 3 Gross Investment $2,100 $2,100 $2,100 $2,100 Accumulated $0 $700 $1,400 $2,100 Depreciation Book Value $2,100 $1,400 $700 $0 Average Investment = ($2,100 + $1,400 + $700 + $0) / (4) = $1,050 Calculate the average annual income of the project. Average Income = ($800 + $700 + $600) / 3 = $700 Divide the average project earnings by the average book value of the machine to calculate the average accounting return. Average Accounting Return = (Average Income) / (Average Investment) = $700 / $1,050 = 0.667 The average accounting return for the New Saturday Late Edition is 66.7%. 8.3 Tax Shield Approach Project A: t0 t1-14 t15 Revenues 300,500 300,500 Foregone rent (45,000) (45,000) Expenditures (170,000) (170,000) Restoration costs (14,750) EBT 85,500 70,750 Taxes at 40% (34,200) (28,300) Net operating cash flow 51,300 42,450 Building Modifications tax 2,560 2,560 shield Capital investments (472,000) Tax shield on equipment 77,793 After tax cash flows (394,207) 53,860 45,010 (1) Building Modifications: The tax shield on these is Straight-line and therefore can be included in the annual cash flow calculations. The calculation of the annual tax shield is: (96,000 / 15 ) x 0.40 = 2,560 (2) PV of CCA Tax Shield on the Equipment modifications: CDTc 1 0.5k x k d 1 k $376, 000 x0.2 x0.4 1 (0.5)(0.16) x $77, 793 0.16 0.2 1 0.16 NPV = -394,207 + 53,860 A14 0.16 + 45,010/(1.16)15 = -394,207 + 294,481 + 4,858 = (94,868) Project B t0 t1-14 t15 Revenues 373,600 373,600 Foregone rent (45,000) (45,000) Expenditures (212,000) (212,000) Restoration costs (112,550) EBT 116,600 4,050 Taxes at 40% 46,640 1,620 Net operating cash flow 69,960 2,430 Building Modifications tax 4,733 4,733 shield Capital investments (599,500) Tax shield on equipment 87,310 After tax cash flows (512,190) 74,693 7,163 (3) Building Modifications: The tax shield on these is Straight-line and therefore can be included in the annual cash flow calculations. The calculation of the annual tax shield is: (177,500 / 15 ) x 0.40 = 4,733 (4) PV of CCA Tax Shield on the Equipment modifications: CDTc 1 0.5k x k d 1 k $422, 000 x0.2 x0.4 1 (0.5)(0.16) x $87,310 0.16 0.2 1 0.16 NPV = -512,190 + 74,693 A14 0.16 + 7,163/(1.16)15 = -512,190 + 408,386 + 773 = (103,031) Since both projects have negative NPVs, Victoria should continue to rent the building. 8.4 The price will rise by the NPVGO per share EPS = $660,000 / 275,000 = $2.40 NPVGO = (-$1,100,000 + $1,650,000) / 275,000 = $2 Price = EPS / r + NPVGO = [$2.4 / 0.15] + $2 =$18.00 8.5 Real interest rate = (1.15 / 1.05) - 1 = 9.52381% NPVA = -$40,000+ $20,000 / 1.0952381 + $15,000 / 1.09523812 + $15,000 / 1.09523813 = $ -40,000 + 18,261 +12,505 +11,417 = 2,183 NPVB = -$50,000+ $10,000 / 1.15 + $20,000 / 1.152 + $40,000 / 1.153 = -50,000 + 8,696 + 15,123 + 26,301 = $120 Choose project A as it has the highest positive NPV. 8.12 Let I be the maximum price the Majestic Mining Company should be willing to pay for the equipment. Examine the incremental cash flows from purchasing the new equipment. 1 (1 0.10) 8 ( I 20, 000)(0.45)(0.20) 1.05 NPV 0 10, 000(1 0.45) 1.10 0.10 0.10 0.20 (5, 000 0)(.45)(0.20) 8 8 (1 0.10) (5, 000 0)(1 0.10) ( I 20, 000) 0.10 0.20 I $63, 404.28 ( w / o rounding ) The maximum price Majestic should be willing to pay for-the equipment is $ 63,404.28 8.14 Assume the tax rate is zero. t=0 t=1 t=2 t=3 t=4 t=5 t=6 ... $12,000 $6,000 $6,000 $6,000 $4,000 $12,000 $6,000 $6,000 ... The present value of one cycle is: PV = $12,000 + $6,000 3 + $4,000 / 1.064 0.06 = $12,000 + $16,038 + 3,168 = $31,206 The cycle is four years long, so use a four-year annuity factor to compute the equivalent annual cost (EAC). EAC = $31,206 / 0.06 4 = $31.206 / 3.4651 = $9,005.80 The present value of such a stream in perpetuity is $9,005.80 / 0.06 = $150,097 8.16 Mixer X Cost Savings NPV = -$400,000 + $120,000 5 0.11 = $43,507.64 EAC = $43,507.64 / 5 = $11,771.88 0.11 Mixer Y Cost Savings NPV = -$600,000 + $130,000 8 0.11 = $68,995.96 = $68,995.96 / 0.11 = $13,407.37 8 EAC Choose Mixer X. 10.1 a. Capital gains = $38 - $37 = $1 per share; Total capital gains is $500 ( $1 x 500) b. Total dollar returns = Dividends + Capital Gains = $1,000 + ($1*500) = $1,500 On a per share basis, this calculation is $2 + $1 = $3 per share c. On a per share basis, $3/$37 = 0.08108 = 8.108% On a total dollar basis, $1,500/(500*$37) = 0.08108 = 8.108% d. No, you do not need to sell the shares to include the capital gains in the computation of the returns. The capital gain is included whether or not you realize the gain. Since you could realize the gain if you choose, you should include it. (Note that for tax purposes, you must sell the stock.) 10.3 Apply the percentage return formula. Note that the stock price declined during the period. Since the stock price decline was greater than the dividend, your return was negative. Rt+1 = [Divt+1 + (Pt+1 – Pt)] / Pt = [$2.40 + ($31 - $42)] / $42 = -0.2048 The percentage return is –20.48%. 10.4 Apply the holding period return formula. The expected holding period return is equal to the total dollar return on the stock divided by the initial investment. Rt+2 = [Pt+2 – Pt] / Pt = [$54.75 - $52] / $52 = 0.0529 The expected holding period return is 5.29%. 0.035 0.013 0.619 0.061 0.213 0.40 0.259 10.8 a. R 0.2148 7 b. Year R R-E(R) (R-E(R))^2 -7 -0.035 -0.2498 0.0624 -6 -0.013 -0.2278 0.05189 -5 0.619 0.4042 0.16338 -4 0.061 -0.1538 0.023654 -3 0.213 -0.0018 0.00000324 -2 0.400 0.1852 0.034299 -1 0.259 0.0442 0.0.001954 E(R) 0.2148 Total 0.337580 2 0.337580/ 7 1 0.056263 0.056263 0.237198 23.7198% Note, because the data are historical data, the appropriate denominator in the calculation of the variance is T-1. 10.10 a. For Treasury Bills: In any state, the return on the Treasury Bills is 3.5%, so the expected return on T-Bills is 3.5% Return if Economic State Prob. (P) State Occurs PReturn Recession 0.25 -0.082 -0.0205 Moderate Growth 0.5 0.123 0.0615 Rapid Expansion 0.25 0.258 0.0645 Expected Return = 0 .1055 b. The expected risk premium is the difference between the expected market return and expected T-Bill return = 10.55% - 3.5% = 7.05% 10.13 The range with 95% probability is: Mean 2, Mean 2 17.5-2 8.5, 17.5+28.5 0.5%, 34.5% 11.1 a. Expected Return = (0.1)(-0.045) + (.2)(0.046) + (0.5)(0.125) + (0.2)(0.207) = 0.1086 = 10.86% The expected return on Q-mart’s stock is 10.86%. b. Variance (2) = (0.1)(-0.045 – 0.1086)2 + (0.2)(0.046 – 0.1086)2 + (0.5)(0.125 – 0.1086)2 + (0.2)(0.207 – 0.1086)2 = 0.005214 Standard Deviation () = (0.005214)1/2 = 0.0722 = 7.22% The standard deviation of Q-mart’s returns is 7.22%. 11.3 a. and b. Economic condition Probability Market return Trebli return Weighted Weighted average average Rapid expansion 0.12 0.0276 0.0144 Moderate expansion 0.40 0.072 0.036 No growth 0.25 0.0375 0.0125 Moderate contraction 0.15 0.0135 0.0015 Serious contraction 0.08 0.0024 -0.0016 Total 1.00 0.153 0.0628 The expected return on the market is 15.3% and on Trembli is 6.28% 11.5 The return on the T-Bills is 3.5% in very state, so the expected return is 3.5%. Economic condition Probability Market return Weighted average Recession 0.20 -1.64% Normal 0.60 7.38% Boom 0.20 5.16% Total 1.00 10.9% The expected return on the market is 10.9% and on T-Bills is 3.50%. The market risk premium is the difference of 7.4% ( 10.9% – 3.5%). 11.8 Value of Atlas stock in the portfolio = (120 shares)($50 per share) = $6,000 Value of Babcock stock in the portfolio = (150 shares)($20 per share) = $3,000 Total Value in the portfolio = $6,000 + $3000 = $9,000 Weight of Atlas stock = $6,000 / $9,000 = 2/3 The weight of Atlas stock in the portfolio is 2/3. Weight of Babcock stock = $3,000 / $9,000 = 1/3 The weight of Babcock stock in the portfolio is 1/3. 11.11 a. Textbook p. 323: Missing data: σA = 0.08 and σB = 0.20 should be added. Value of Macrosoft stock in the portfolio = (100 shares)($80 per share) = $8,000 Value of Intelligence stock in the portfolio = (300 shares)($40 per share) = $12,000 Total Value in the portfolio = $8,000 + $12,000 = $20,000 Weight of Macrosoft stock = $8,000 / $20,000 = 0.40 Weight of Intelligence stock = $12,000 / $20,000 = 0.60 The expected return on the portfolio equals: E(RP) = (WMAC)[E(RMAC)] + (WI)[E(RI)] where E(RP) = the expected return on the portfolio E(RMAC) = the expected return on Macrosoft stock E(RI) = the expected return on Intelligence Stock WMAC = the weight of Macrosoft stock in the portfolio WI = the weight of Intelligence stock in the portfolio E(RP) = (WMAC)[E(RMAC)] + (WI)[E(RM)] = (0.40)(0.15) + (0.60)(0.20) = 0.18 = 18% The expected return on her portfolio is 18%. The variance of the portfolio equals: 2P = (WMAC)2(MAC)2 + (WI)2(I)2 + (2)(WMAC)(WI)(MAC)(I)[Correlation(RMAC, RI)] where 2P = the variance of the portfolio WMAC = the weight of Macrosoft stock in the portfolio WI = the weight of Intelligence stock in the portfolio MAC = the standard deviation of Macrosoft stock I = the standard deviation of Intelligence stock RMAC = the return on Macrosoft stock RI = the return on Intelligence stock 2P = (WMAC)2(MAC)2 + (WI)2(I)2 + (2)(WMAC)(WI)(MAC)(I)[Correlation(RMAC, RI)] = (0.40)2(0.08)2 + (0.60)2(0.20)2 + (2)(0.40)(0.60)(0.08)(0.20)(0.38) = 0.018342 The standard deviation of the portfolio equals: P = (2P)1/2 where P = the standard deviation of the portfolio 2P = the variance of the portfolio P = (0.018342)1/2 = 0.1354 =13.54% The standard deviation of her portfolio is 13.54%. b. Janet started with 300 shares of Intelligence stock. After selling 200 shares, she has 100 shares left. Value of Macrosoft stock in the portfolio = (100 shares)($80 per share) = $8,000 Value of Intelligence stock in the portfolio = (100 shares)($40 per share) = $4,000 Total Value in the portfolio = $8,000 + $4,000 = $12,000 Weight of Macrosoft stock = $8,000 / $12,000 = 2/3 Weight of Intelligence stock = $4,000 / $12,000 = 1/3 E(RP) = (WMAC)[E(RMAC)] + (WI)[E(RI)] = (2/3)(0.15) + (1/3)(0.20) = 0.1667 = 16.67% The expected return on her portfolio is 16.67%. 2P = (WMAC)2(MAC)2 + (WI)2(I)2 + (2)(WMAC)(WI)(MAC)(I)[Correlation(RMAC, RI)] = (2/3)2(0.08)2 + (1/3)2(0.20)2 + (2)(2/3)(1/3)(0.08)(0.20)(0.38) = 0.009991 P = (0.009991)1/2 = 0.1000 =10.00% The standard deviation of her portfolio is 10.00%. 11.14 The expected return on the portfolio must be less than or equal to the expected return on the asset with the highest expected return. It cannot be greater than this asset’s expected return because all assets with lower expected returns will pull down the value of the weighted average expected return. Similarly, the expected return on any portfolio must be greater than or equal to the expected return on the asset with the lowest expected return. The portfolio’s expected return cannot be below the lowest expected return among all the assets in the portfolio because assets with higher expected returns will pull up the value of the weighted average expected return. 11.18 a. State Return on A Return on B Probability 1 15% 35% (0.40)(0.50) = 0.2 2 15% -5% (0.40)(0.50) = 0.2 3 10% 35% (0.60)(0.50) = 0.3 4 10% -5% (0.60)(0.50) = 0.3 b. E(RP) = (0.20)[(0.50)(0.15) + (0.50)(0.35)] + (0.20)[(0.50)(0.15) + (0.50)(-0.05)] + (0.30)[(0.50)(0.10) + (0.50)(0.35)] + (0.30)[(0.50)(0.10) + (0.50)(-0.05)] = 0.135 = 13.5% The expected return on the portfolio is 13.5%. 11.22 If we assume that the market has not stayed constant during the past three years, then the lack in movement of Southern Co.’s stock price only indicates that the stock either has a standard deviation or a beta that is very near to zero. The large amount of movement in Texas Instrument’ stock price does not imply that the firm’s beta is high. Total volatility (the price fluctuation) is a function of both systematic and unsystematic risk. The beta only reflects the systematic risk. Observing the standard deviation of price movements does not indicate whether the price changes were due to systematic factors or firm specific factors. Thus, if you observe large stock price movements like that of TI, you cannot claim that the beta of the stock is high. All you know is that the total risk of TI is high. 11.26 a. E(RP) = (1/3)(0.10) + (1/3)(0.14) + (1/3)(0.20) = 0.1467 = 14.67% The expected return on an equally weighted portfolio is 14.67%. b. The beta of a portfolio equals the weighted average of the betas of the individual securities within the portfolio. P = (1/3)(0.7) + (1/3)(1.2) + (1/3)(1.8) = 1.23 The beta of an equally weighted portfolio is 1.23. c. If the Capital Asset Pricing Model holds, the three securities should be located on a straight line (the Security Market Line). For this to be true, the slopes between each of the points must be equal. Slope between A and B = (0.14 – 0.10) / (1.2 – 0.7) = 0.08 Slope between A and C = (0.20 – 0.10) / (1.8 – 0.7) = 0.091 Slope between B and C = (0.20 – 0.14) / (1.8 – 1.2) = 0.10 Since the slopes between the three points are different, the securities are not correctly priced according to the Capital Asset Pricing Model. 11.28 According to the Capital Asset Pricing Model: E(r) = rf + (EMRP) where E(r) = the expected return on the stock rf = the risk-free rate the stock’s beta EMRP = the expected market risk premium In this problem: rf = 0.06 0.80 EMRP = 0.085 The expected return on Stock A equals: E(r) = rf + (EMRP) = 0.06 + 0.80(0.085) = 0.128 The expected return on Stock A is 12.8%. 11.37 a. According to the Capital Asset Pricing Model: E(r)= rf + (EMRP) where E(r) = the expected return on the stock rf = the risk-free rate the stock’s beta EMRP = the expected market risk premium In this problem: E(r) = 0.167 rf = 0.076 = 1.7 E(r) = rf + (EMRP) 0.167 = 0.076 + 1.7(EMRP) EMRP = (0.167 – 0.076) / 1.7 = 0.0535 The expected market risk premium is 5.35%. b. According to the Capital Asset Pricing Model: E(r) = rf + (EMRP) where E(r) = the expected return on the stock rf = the risk-free rate the stock’s beta EMRP = the expected market risk premium In this problem: rf = 0.076 = 0.8 EMRP = 0.0535 E(r) = rf + (EMRP) = 0.076 + 0.8(0.0535) = 0.1188 The expected return on Magnolia stock is 11.88%. c. The beta of a portfolio is the weighted average of the betas of the individual securities in the portfolio. The beta of Potpourri is 1.7, the beta of Magnolia is 0.8, and the beta of a portfolio consisting of both Potpourri and Magnolia is 1.07. Therefore: 1.07 = (WP)(1.7) + (WM)(0.8) where WP = the weight of Potpourri stock in the portfolio WM = the weight of Magnolia stock in the portfolio Because your total investment must equal 100%: WP = 1 - WM 1.07 = (1 – WM)(1.7) + (WM)(0.8) 1.07 = 1.7 – 1.7WM + 0.8WM -0.63 = -0.90WM WM = 0.70 WP = 1 - WM = 1 – 0.70 = 0.30 You have 70% of your portfolio ($7,000) invested in Magnolia stock and 30% of your portfolio ($3,000) invested in Potpourri stock. E(r) = (0.70)(0.1188) + (0.30)(0.167) = 0.1333 The expected return of the portfolio is 13.33%. 11.42 First, determine the beta of the portfolio. Total Amount Invested = $5,000 + $10,000 + $8,000 + $7,000 = $30,000 Weight of Stock A = $5,000 / $30,000 = 1/6 Weight of Stock B = $10,000 / $30,000 = 1/3 Weight of Stock C = $8,000 / $30,000 = 4/15 Weight of Stock D = $7,000 / $30,000 = 7/30 The beta of a portfolio is the weighted average of the betas of its individual securities. Portfolio = (1/6)(0.75) + (1/3)(1.1) + (4/15)(1.36) + (7/30)(1.88) = 1.293 Use the Capital Asset Pricing Model (CAPM) to find the expected return on the portfolio. According to the CAPM: E(r)= rf + [E(rm) – rf] where E(r) = the expected return on the portfolio rf = the risk-free rate the portfolio’s beta E(rm) = the expected return on the market portfolio In this problem: rf = 0.04 = 1.293 E(rm) = 0.15 E(r)= rf + [E(rm) – rf] = 0.04+ 1.293(0.15 – 0.04) = 0.1822 The expected return on the portfolio is 18.22%. 13.2 a. Calculate the average return for Douglas stock and the market. RD = (Sum of Yearly Returns) / (Number of Years) = (-0.05 + 0.05 + 0.08 + 0.15 + 0.10) / (5) = 0.066 RM = (-0.12 + 0.01 + 0.06 + 0.10 + 0.05) / (5) = 0.020 To calculate the beta of Douglas stock, calculate the variance of the market, (RM - R M) , and the covariance of Douglas stock’s return with the market’s return, (RD 2 - R D) (RM - R M). The beta of Douglas stock is equal to the covariance of Douglas stock’s return and the market’s return divided by the variance of the market. Remember to divide both the covariance of Douglas stock’s return and the market’s return and the variance of the market by 4. Because the data are historical, the appropriate denominator in the calculation of the variance is 4 (=T – 1). RD - R D RM - R M (RM - R M)2 (RD - R D) (RM - R M) -0.116 -0.14 0.0196 0.01624 -0.016 -0.01 0.0001 0.00016 0.014 0.04 0.0016 0.00056 0.084 0.08 0.0064 0.00672 0.034 0.03 0.0009 0.00102 0.0286 0.02470 D = [Cov (RD, RM) / (T-1)] / [Var (RM) / (T-1)] = (0.02470 / 4) / (0.0286 / 4) = 0.864 The beta of Douglas stock is 0.864. 13.12 a. Apply the CAPM to estimate Adobe Online’s cost of equity, RS. The following equation expresses the firm’s required return, RS, in terms of the firm’s beta, S, the risk-free rate, RF, and the market return, R M. RS = RF + ( R M – RF) = 0.07 + 1.29 (0.13 – 0.07) = 0.1474 Adobe Online’s cost of equity is 14.74%. b. Use the debt-to-equity ratio to determine the weighted average cost of capital. Calculate the equity to value ratio [S / (S+B)] and the debt to value ratio [B / (S+B)] by substitution. B/S = 1.0 B = S [S / (S+B)] = [S / (S + S)] = [S / (2 S)] =1/2 = 0.5 [B / (S+B] = [1.0 S / (S + 1.0 S)] = [(S) / (2 S)] =1/2 = 0.5 Remember that interest on debt is tax deductible at the corporate level, which will lower the firm’s overall cost of capital. Multiply the cost of debt, rB, by (1 – TC) to include the interest tax shield in the weighted average cost of capital. RWACC = [S / (S+B)] rS + [B / (S+B)] rB (1 – TC) = (0.5) (0.1474) + (0.5) (0.07) (1 – 0.35) = 0.09645 The weighted average cost of capital is 9.645%. 13.14 Use the market value of debt to calculate the weighted average cost of capital. To compute the market value of the debt, multiply the book value of First Data’s debt by 95 percent. B = $180,000,000 0.95 = $171,000,000 The market value of First Data Co.’s stock is equal to the number of shares outstanding multiplied by the market price per share. S = 20,000,000 $25 = $500,000,000 The value of the firm is the market value of First Data Co.’s debt plus the market value of the stock. V =B+S = $171,000,000 + $500,000,000 = $671,000,000 Calculate the debt-to-value and equity-to-value ratios. These ratios will be used to compute the weighted average cost of capital. B / (S + B) = $171,000,000 / $671,000,000 = 0.2548 S / (S + B) = $500,000,000 / $671,000,000 = 0.7452 Apply the weighted average cost of capital formula, using the cost of debt and cost of equity given in the problem, 10 percent and 20 percent, respectively. Remember that interest on debt is tax deductible at the corporate level, which will lower the firm’s overall cost of capital. Multiply the cost of debt, rB, by (1 - TC) to include the interest tax shield in the weighted average cost of capital. rWACC = [S / (S+B)] rS + [B / (S+B)] rB (1 – TC) = (0.7452) (0.20) + (0.2548) (0.10) (0.60) = 0.1643 The weighted average cost of capital for First Data Co. is 16.43%. 14.2 Weak form. Market prices reflect information contained in historical prices. Investors are unable to earn abnormal returns using historical prices to predict future price movements. Semi-strong form. In addition to historical data, market prices reflect all publicly- available information. Investors with insider, or private information, are able to earn abnormal returns. Strong form. Market prices reflect all information, public or private. Investors are unable to earn abnormal returns using insider information or historical prices to predict future price movements. 14.3 a. False. Market efficiency implies that prices reflect all available information, but it does not imply certain knowledge. Many pieces of information that are available and reflected in prices are fairly uncertain. Efficiency of markets does not eliminate that uncertainty and therefore does not imply perfect forecasting ability. b. True. Market efficiency exists when prices reflect all available information. To be efficient in the weak form, the market must incorporate all historical data into prices. Under the semi-strong form of the hypothesis, the market incorporates all publicly-available information in addition to the historical data. In strong form efficient markets, prices reflect all publicly and privately available information. c. False. Market efficiency implies that market participants are rational. Rational people will immediately act upon new information and will bid prices up or down to reflect that information. d. False. In efficient markets, prices reflect all available information. Thus, prices will fluctuate whenever new information becomes available. e. True. Competition among investors results in the rapid transmission of new market information. In efficient markets, prices immediately reflect new information as investors bid the stock price up or down. 14.5 False. The stock price would have adjusted before the founder’s death only if investors had perfect forecasting ability. The 12.5% increase in the stock price after the founder’s death indicates that either the market did not anticipate the death or that the market had anticipated it imperfectly. However, the market reacted immediately to the new information, implying efficiency. It is interesting that the stock price rose after the announcement of the founder’s death. This price behavior indicates that the market felt he was a liability to the firm. 14.8 Under the semi-strong form of market efficiency, the stock price should stay the same. The accounting system changes are publicly available information. Investors would identify no changes in either the firm’s current or its future cash flows. Thus, the stock price will not change after the announcement of increased earnings. 14.12 Stock prices should immediately and fully rise to reflect the announcement. Thus, one cannot expect abnormal returns following the announcement. 15.3 Corporate bonds yields are usually higher. Corporations may receive dividends on preferred stock tax free, so they are willing to accept a lower pretax yield. Other corporations and institutions are the big investors in preferred stock. 15.10 After-tax yield on the bonds is 3.99% = 7% x (1-.43). The investor seeks 5.49% in after-tax yield on preferreds (3.99 + 1.5). Using the information in Table 1A2, the before-tax yield on preferred shares should be: = 5.49/ [l –1.25(.43 - 0.1333 - 0.0513] = 5.49/0.69325 = 7.92% Note the gross up and dividend tax credit reduces the impact of the tax liability. 16.1 a. Since Alpha Corporation is an all-equity firm, its value is equal to the market value of its outstanding shares. Alpha has 5,000 shares of common stock outstanding, worth $20 per share. Therefore, the value of Alpha Corporation is $100,000 (= 5,000 shares * $20 per share). b. Modigliani-Miller Proposition I states that in the absence of taxes, the value of a levered firm equals the value of an otherwise identical unlevered firm. Since Beta Corporation is identical to Alpha Corporation in every way except its capital structure and neither firm pays taxes, the value of the two firms should be equal. Modigliani-Miller Proposition I (No Taxes): VL =VU Alpha Corporation, an unlevered firm, is worth $100,000 = VU. Therefore, the value of Beta Corporation (VL) is $100,000. c. The value of a levered firm equals the market value of its debt plus the market value of its equity. VL = B + S The value of Beta Corporation is $100,000 (VL), and the market value of the firm’s debt is $25,000 (B). The value of Beta’s equity is: S = VL – B = $100,000 - $25,000 = $75,000 Therefore, the market value of Beta Corporation’s equity (S) is $75,000. d. Since the market value of Alpha Corporation’s equity is $100,000, it will cost $20,000 (= 0.20 * $100,000) to purchase 20% of the firm’s equity. Since the market value of Beta Corporation’s equity is $75,000, it will cost $15,000 (= 0.20 * $75,000) to purchase 20% of the firm’s equity. e. Since Alpha Corporation expects to earn $350,000 this year and owes no interest payments, the dollar return to an investor who owns 20% of the firm’s equity is expected to be $70,000 (= 0.20 * $350,000) over the next year. While Beta Corporation also expects to earn $350,000 before interest this year, it must pay 12% interest on its debt. Since the market value of Beta’s debt at the beginning of the year is $25,000, Beta must pay $3,000 (= 0.12 * $25,000) in interest at the end of the year. Therefore, the amount of the firm’s earnings available to equity holders is $347,000 (= $350,000 - $3,000). The dollar return to an investor who owns 20% of the firm’s equity is $69,400 (= 0.20 * $347,000). f. The initial cost of purchasing 20% of Alpha Corporation’s equity is $20,000, but the cost to an investor of purchasing 20% of Beta Corporation’s equity is only $15,000 (see part d). In order to purchase $20,000 worth of Alpha’s equity using only $15,000 of his own money, the investor must borrow $5,000 to cover the difference. The investor must pay 12% interest on his borrowings at the end of the year. Since the investor now owns 20% of Alpha’s equity, the dollar return on his equity investment at the end of the year is $70,000 ( = 0.20 * $350,000). However, since he borrowed $5,000 at 12% per annum, he must pay $600 (= 0.12 * $5,000) at the end of the year. Therefore, the cash flow to the investor at the end of the year is $69,400 (= $70,000 - $600). Notice that this amount exactly matches the dollar return to an investor who purchases 20% of Beta’s equity. Strategy Summary: 1. Borrow $5,000 at 12%. 2. Purchase 20% of Alpha’s stock for a net cost of $15,000 (= $20,000 - $5,000 borrowed). g. The equity of Beta Corporation is riskier. Beta must pay off its debt holders before its equity holders receive any of the firm’s earnings. If the firm does not do particularly well, all of the firm’s earnings may be needed to repay its debt holders, and equity holders will receive nothing. 16.2 a. A firm’s debt-equity ratio is the market value of the firm’s debt divided by the market value of a firm’s equity. The market value of Acetate’s debt $10 million, and the market value of Acetate’s equity is $20 million. Debt-Equity Ratio = Market Value of Debt / Market Value of Equity = $10 million / $20 million =½ Therefore, Acetate’s Debt-Equity Ratio is ½. b. In the absence of taxes, a firm’s weighted average cost of capital (rwacc) is equal to: rwacc = {B / (B+S)} rB + {S / (B+S)}rS where B = the market value of the firm’s debt S = the market value of the firm’s equity rB = the pre-tax cost of a firm’s debt rS = the cost of a firm’s equity. In this problem: B = $10,000,000 S = $20,000,000 rB = 14% The Capital Asset Pricing Model (CAPM) must be used to calculate the cost of Acetate’s equity (rS) According to the CAPM: rS = rf + S{E(rm) – rf} where rf = the risk-free rate of interest E(rm) = the expected rate of return on the market portfolio S = the beta of a firm’s equity In this problem: rf = 4% E(rm) = 9% S = 0.9 Therefore, the cost of Acetate’s equity is: rS = rf + S{E(rm) – rf} = 0.04 + 0.9( 0.09 – 0.04) = 0.085 The cost of Acetate’s equity (rS) is 8.5%. Acetate’s weighted average cost of capital equals: rwacc = {B / (B+S)} rB + {S / (B+S)}rS = ($10 million / $30 million)(0.14) + ($20 million / $30 million)(0.085) = (1/3)(0.14) + (2/3)(0.085) = 0.1033 Therefore, Acetate’s weighted average cost of capital is 10.33%. c. According to Modigliani-Miller Proposition II (No Taxes): rS = r0 + (B/S)(r0 – rB) where r0 = the cost of capital for an all-equity firm rS = the cost of equity for a levered firm rB = the pre-tax cost of debt In this problem: rS = 0.085 rB = 0.14 B = $10,000,000 S = $20,000,000 Thus: 0.085 = r0 + (1/2)(r0 – 0.14) Solving for r0: r0 = 0.1033 Therefore, the cost of capital for an otherwise identical all-equity firm is 10.33%. This is consistent with Modigliani-Miller’s proposition that, in the absence of taxes, the cost of capital for an all-equity firm is equal to the weighted average cost of capital of an otherwise identical levered firm. 16.3 Since Unlevered is an all-equity firm, its value is equal to the market value of its outstanding shares. Unlevered has 10 million shares of common stock outstanding, worth $80 per share. Therefore, the value of Unlevered is $800 million (= 10 million shares * $80 per share). Modigliani-Miller Proposition I states that, in the absence of taxes, the value of a levered firm equals the value of an otherwise identical unlevered firm. Since Levered is identical to Unlevered in every way except its capital structure and neither firm pays taxes, the value of the two firms should be equal. Modigliani-Miller Proposition I (No Taxes): VL =VU Therefore, the market value of Levered, Inc. should be $800 million also. Since Levered has 4.5 million outstanding shares, worth $100 per share, the market value of Levered’s equity is $450 million. The market value of Levered’s debt is $275 million. The value of a levered firm equals the market value of its debt plus the market value of its equity. Therefore, the current market value of Levered, Inc. is: VL = B + S = $275 million + $450 million = $725 million The market value of Levered’s equity needs to be $525 million, $75 million higher than its current market value of $450 million, for MM Proposition I to hold. Since Levered’s market value is less than Unlevered’s market value, Levered is relatively underpriced and an investor should buy shares of the firm’s stock. 16.7 a. According to Modigliani-Miller the weighted average cost of capital (rwacc) for a levered firm is equal to the cost of equity for an unlevered firm in a world with no taxes. Since Rayburn pays no taxes, its weighted average cost of capital after the restructuring will equal the cost of the firm’s equity before the restructuring. Therefore, Rayburn’s weighted average cost of capital will be 18% after the restructuring. b. According to Modigliani-Miller Proposition II (No Taxes): rS = r0 + (B/S)(r0 – rB) where r0 = the cost of capital for an all-equity firm rS = the cost of equity for a levered firm rB = the pre-tax cost of debt In this problem: r0 = 0.18 rB = 0.10 B = $400,000 S = $1,600,000 The cost of Rayburn’s equity after the restructuring is: rS = r0 + (B/S)(r0 – rB) = 0.18 + ($400,000 / $1,600,000)(0.18 - 0.10) = 0.18 + (1/4)(0.18 – 0.10) = 0.20 Therefore, Rayburn’s cost of equity after the restructuring will be 20%. In accordance with Modigliani-Miller Proposition II (No Taxes), the cost of Rayburn’s equity will rise as the firm adds debt to its capital structure since the risk to equity holders increases with leverage. c. In the absence of taxes, a firm’s weighted average cost of capital (rwacc) is equal to: rwacc = {B / (B+S)} rB + {S / (B+S)}rS where B = the market value of the firm’s debt S = the market value of the firm’s equity rB = the pre-tax cost of the firm’s debt rS = the cost of the firm’s equity. In this problem: B = $400,000 S = $1,600,000 rB = 10% rS = 20% Rayburn’s weighted average cost of capital after the restructuring will be: rwacc = {B / (B+S)} rB + {S / (B+S)}rS = ( $400,000 / $2,000,000)(0.10) + ($1,600,000 / $2,000,000)(0.20) = (1/5)(0.10) + (4/5)(0.20) = 0.18 Consistent with part a, Rayburn’s weighted average cost of capital after the restructuring remains at 18%. 16.9 a. Without the power plant, Alberta Power expects to earn $27 million per year into perpetuity. Since Alberta is an all-equity firm and the required rate of return on the firm’s equity is 10%, the market value of Alberta’s assets is equal to the present value of a perpetuity of $27,000,000 per year, discounted at 10%. PV(Perpetuity) =C/r = $27,000,000 / 0.10 = $270,000,000 Therefore, the market value of Alberta’s assets before the firm announces that it will build a new power plant is $270,000,000. Since Alberta is an all-equity firm, the market value of Alberta’s equity is also $270,000,000. Alberta’s market-value balance sheet before the announcement of the buyout is made is: Alberta Power Assets = $ 270,000,000 Debt = $ - Equity = $ 270,000,000 Total Assets = $ 270,000,000 Total D + E = $ 270,000,000 Since the market value of Alberta’s equity is $270 million and the firm has 10 million shares outstanding, Alberta’s stock price before the announcement to build the new power plant is $27 per share (= $270 million / 10 million shares). b. i. According to the efficient-market hypothesis, the market value of Alberta’s equity will change immediately to reflect the net present value of the project. Since the new power plant will cost Alberta $20 million but will increase the firm’s annual earnings by $3 million in perpetuity, the NPV of the new power plant can be calculated as follows: NPVNEW POWER PLANT = -$20 million + ($3 million/ 0.10) = $10 million Remember that the required return on the firm’s equity is 10% per annum. Therefore, the market value of Alberta’s equity will increase to $280 million (= $270 million + $10 million) immediately after the announcement. Alberta’s market-value balance sheet after the announcement will be: Alberta Power Old Assets = $ 270,000,000 Debt = $ - NPVPOWER PLANT = $ 10,000,000 Equity = $ 280,000,000 Total Assets = $ 280,000,000 Total D + E = $ 280,000,000 Since Alberta has 10 million shares of common stock outstanding and the total market value of the firm’s equity is $280 million , Alberta’s new stock price will immediately rise to $28 per share (= $280 million / 10 million shares) after the firm’s announcement. ii. Alberta needs to issue $20 million worth of equity in order to fund the construction of the power plant. The market value of the firm’s stock will be $28 per share after the announcement. Therefore, Alberta will need to issue 714,285.71 shares (= $20 million / $28 per share) in order to fund the construction of the power plant. iii. Alberta will receive $20 million (= 714,285.71 shares * $28 per share) in cash after the equity issue. Since the firm now has 10,714,285.71 (= 10 million + 714,285.71) shares outstanding, where each share is worth $28, the market value of the firm’s equity increases to $300,000,000 (=10,714,285.71 shares * $28 per share). Alberta’s market-value balance sheet after the equity issue will be: Alberta Power Old Assets = $ 270,000,000 Debt = $ - Cash = $ 20,000,000 Equity = $ 300,000,000 NPVPOWER PLANT = $ 10,000,000 Total Assets = $ 300,000,000 Total D + E = $ 300,000,000 iv. Alberta will pay $20,000,000 in cash for the power plant. Since the plant will generate $3 million in annual earnings forever, its present value is equal to a perpetuity of $3 million per year, discounted at 10%. PVNEW POWER PLANT = $3 million / 0.10 = $30 million Alberta’s market-value balance sheet after the construction of the power plant will be: Alberta Power Old Assets = $ 270,000,000 Debt = $ - PVPOWER PLANT = $ 30,000,000 Equity = $ 300,000,000 Total Assets = $ 300,000,000 Total D + E = $ 300,000,000 v. Since Alberta is an all-equity firm, its value will equal the market value of its equity. Therefore, the value of Alberta Power will be $300 million if the firm issues equity to finance the construction of the power plant. c. i. Under the efficient-market hypothesis, the market value of the firm’s equity will immediately rise by $10 million following the announcement to reflect the NPV of the power plant. Therefore, the total market value of Alberta’s equity will be $280 million (= $270 million + $10 million) after the firm’s announcement. Alberta’s market-value balance sheet after the announcement will be: Alberta Power Old Assets = $ 270,000,000 Debt = $ - NPVPOWER PLANT = $ 10,000,000 Equity = $ 280,000,000 Total Assets = $ 280,000,000 Total D + E = $ 280,000,000 Since the firm has 10 million shares of common stock outstanding, Alberta’s new stock price will be $28 per share (= $280 million / 10 million shares). ii. Alberta will receive $20 million in cash after the debt issue. The market value of the firm’s debt will be $20 million. Alberta’s market-value balance sheet after the debt issue will be: Alberta Power Old Assets = $ 270,000,000 Debt = $ 20,000,000 Cash = $ 20,000,000 Equity = $ 280,000,000 NPVPOWER PLANT = $ 10,000,000 Total Assets = $ 300,000,000 Total D + E = $ 300,000,000 iii. Alberta will pay $20 million in cash for the power plant. Since the plant will generate $3 million of earnings forever, its present value is equal to a perpetuity of $3 million per year, discounted at 10%. PVPOWER PLANT = $3 million / 0.10 = $30 million Alberta’s market-value balance sheet after it builds the new power plant is: Alberta Power Old Assets = $ 270,000,000 Debt = $ 20,000,000 PVPOWER PLANT = $ 30,000,000 Equity = $ 280,000,000 Total Assets = $ 300,000,000 Total D + E = $ 300,000,000 iv. The value of a levered firm is the sum of the market values of the firm’s debt and equity. Since the market value of Alberta’s debt will be $20 million and the market value of Alberta’s equity will be $280 million, the value of Alberta Power will be $300 million if the firm decides to issue debt in order to fund the outlay for the power plant. Therefore, the value of Alberta Power will be $300 million regardless of whether the firm issues debt or equity to fund the construction of the new power plant. v. According to Modigliani-Miller Proposition II (No Taxes): rS = r0 + (B/S)(r0 – rB) where r0 = the required return on an unlevered firm’s equity rS = the required return on a firm’s equity rB = the required return on a firm’s debt In this problem: r0 = 0.10 rB = 0.08 B = $20 million S = $280 million The required return on Alberta’s levered equity is: rS = r0 + (B/S)(r0 – rB) = 0.10 + ($20 million / $280 million)(0.10 - 0.08) = 0.10 + (1/14)(0.10 – 0.08) = 10.14% Therefore, the required return on Alberta’s levered equity is 10.14%. vi. In the absence of taxes, a firm’s weighted average cost of capital (rwacc) is equal to: rwacc = {B / (B+S)} rB + {S / (B+S)}rS where B = the market value of the firm’s debt S = the market value of the firm’s equity rB = the required return on the firm’s debt rS = the required return on the firm’s equity. In this problem: B = $20 million S = $280 million rB = 8% rS = 10.14% Alberta’s weighted average cost of capital after the construction of the new power plant is: rwacc = {B / (B+S)} rB + {S / (B+S)}rS = ( $20 million / $300 million)(0.08) + ($280 million / $300 million)(0.1014) = (1/15)(0.08) + (14/15)(0.1014) = 0.10 Therefore, Alberta’s weighted average cost of capital will be 10% following either debt or equity financing. 16.10 a. False. A reduction in leverage will decrease both the risk of the stock and its expected return. Modigliani and Miller state that, in the absence of taxes, these two effects exactly cancel each other out and leave the price of the stock and the overall value of the firm unchanged. b. False. Modigliani-Miller Proposition II (No Taxes) states that the required return on a firm’s equity is positively related to the firm’s debt-equity ratio [rS = r0 + (B/S)(r0 – rB)]. Therefore, any increase in the amount of debt in a firm’s capital structure will increase the required return on the firm’s equity. 16.15 Modigliani-Miller Proposition I states that in a world with corporate taxes: V L = V U + T CB where VL = the value of a levered firm VU = the value of an unlevered firm TC = the corporate tax rate B = the value of debt in a firm’s capital structure Since the firm is an all-equity firm with 175,000 shares of common stock outstanding, currently worth $20 per share, the market value of this unlevered firm (VU) is $3,500,000 (= 175,000 shares * $20 per share). The firm plans to issue $1,000,000 debt and is subject to a corporate tax rate of 30%. In this problem: VU = $3,500,000 TC = 0.30 B = $1,000,000 The market value of a levered firm is: V L = V U + T CB = $3,500,000 + (0.30)($1,000,000) = $3,800,000 The value of a levered firm is equal to the sum of the market value of its debt and the market value of its equity. That is, the value of a levered firm is: VL = S + B Rearranging this equation, the market value of the firm’s levered equity, S, is: S = VL – B = $3,800,000 - $1,000,000 = $2,800,000 Therefore, the market value of the firm’s equity is $2,800,000 after the firm announces the stock repurchase plan. 16.19 a. Modigliani-Miller Proposition I states that in a world with corporate taxes: V L = V U + T CB where VL = the value of a levered firm VU = the value of an unlevered firm TC = the corporate tax rate B = the value of debt in a firm’s capital structure The value of an unlevered firm is the present value of its after-tax earnings: VU = [(EBIT)(1-TC)] / r0 where VU = the value of an unlevered firm EBIT = the firm’s expected annual earnings before interest and taxes TC = the corporate tax rate r0 = the after-tax required rate of return on an all-equity firm In this problem: EBIT = $4,000,000 TC = 0.35 r0 = 0.15 The value of Appalachian if it were unlevered is: VU = [(EBIT)(1-TC)] / r0 = [($4,000,000)(1 - 0.35)] / 0.15 = $17, 333, 333 The value of Appalachian if it were an all-equity firm is $17,333,333. Appalachian currently has $10,000,000 of debt in its capital structure and is subject to a corporate tax rate of 35%. Thus: VU = $17,333,333 TC = 0.35 B = $10,000,000 The value of Appalachian is: V L = V U + T CB = $17,333,333+ (0.35)($10,000,000) = $20,833,333 Therefore, the value of Appalachian is $20,833,333. b. According to Modigliani-Miller Proposition II with corporate taxes: rS = r0 + (B/S)(r0 – rB)(1 – TC) where r0 = the required return on the equity of an unlevered firm rS = the required return on the equity of a levered firm rB = the pre-tax cost of debt TC = the corporate tax rate B = the market value of the firm’s debt S = the market value of the firm’s equity In this problem: r0 = 0.15 rB = 0.10 TC = 0.35 B = $10,000,000 S = $10,833,333 (= $20,833,333 – $10,000,000) The required return on Appalachian’s levered equity is: rS = r0 + (B/S)(r0 – rB)(1 – TC) = 0.15 + ($10,000,000 / $10,833,333)(0.15 – 0.10)(1 – 0.35) = 0.15 + (0.9231)(0.15-0.10)(1 – 0.35) = 0.18 Therefore, the cost of Appalachian’s levered equity is 18%. c. In a world with corporate taxes, a firm’s weighted average cost of capital (rwacc) is equal to: rwacc = {B / (B+S)}(1 – TC) rB + {S / (B+S)}rS where B = the market value of the firm’s debt S = the market value of the firm’s equity rB = the required return on the firm’s debt rS = the required return on the firm’s equity. TC = the corporate tax rate The value of Appalachian’s debt is $10,000,000. Since the value of the firm ($20,833,333) is the sum of the value of the firm’s debt and the value of the firm’s equity, the market value of the firm’s equity is $10,833,333 (= $20,833,333 - $10,000,000). Thus: B = $10,000,000 S = $10,833,333 rB = 0.10 rS = 0.18 TC = 0.35 Appalachian’s weighted average cost of capital is: rwacc = {B / (B+S)}(1 – TC) rB + {S / (B+S)}rS = ($10,000,000 / $20,833,333)(1 – 0.35)(0.10) + ($10,833,333 / $20,833,333)(0.18) = (0.48)(1 – 0.35)(0.10) + (0.52)(0.18) = 0.1248 Therefore, Appalachian’s weighted average cost of capital is 12.48%. 16.20 a. In a world with corporate taxes, a firm’s weighted average cost of capital (rwacc) is equal to: rwacc = {B / (B+S)}(1 – TC) rB + {S / (B+S)}rS where B / (B+S) = the firm’s debt-to-value ratio S / (B+S) = the firm’s equity-to-value ratio rB = the pre-tax cost of debt rS = the cost of equity for a levered firm. TC = the corporate tax rate While the problem does not list Williamson’s debt-to-value ratio or Williamson’s equity-to-value ratio, it does say that the firm’s debt-to-equity ratio is 2.5. If Williamson’s debt-to-equity ratio is 2.5: B / S = 2.5 Solving for B: B = (2.5 * S) The above formula for rwacc uses the following ratio: B / (B+S) Since B = (2.5 * S): B/ (B+S) = (2.5 * S) / { (2.5 * S) + S} = (2.5 * S) / (3.5 * S) = (2.5 / 3.5) = 0.7143 Williamson’s debt-to-value ratio is 71.43% The above formula for rwacc also uses the following ratio: S / (B+S) Since B = (2.5 * S): Williamson’s equity-to-value ratio = S / {(2.5*S) + S} = S / (3.5 * S) = (1 / 3.5) = 0.2857 Williamson’s equity-to-value ratio is 28.57%. In order to solve for the cost of Williamson’s equity capital (rS), set up the following equation: rwacc = {B / (B+S)}(1 – TC) rB + {S / (B+S)}rS 0.15 = (0.7143)(1 – 0.35)(0.10) + (0.2857)(rS) rS = 0.3625 Therefore, the cost of Williamson’s equity capital is 36.25%. b. According to Modigliani-Miller Proposition II with corporate taxes: rS = r0 + (B/S)(r0 – rB)(1 – TC) where r0 = the cost of equity for an unlevered firm rS = the cost of equity for a levered firm rB = the pre-tax cost of debt TC = the corporate tax rate B/S = the firm’s debt-to-equity ratio In this problem: rS = 0.3625 rB = 0.10 TC = 0.35 B/S = 2.5 In order to solve for the cost of Williamson’s unlevered equity (r0), set up the following equation: rS = r0 + (B/S)(r0 – rB)(1 – TC) 0.3625 = r0 + (2.5)(r0 – 0.10)(1 – 0.35) r0 = 0.20 Therefore, Williamson’s unlevered cost of equity is 20%. c. If Williamson’s debt-to-equity ratio is 0.75, the cost of the firm’s equity capital (rS) will be: rS = r0 + (B/S)(r0 – rB)(1 – TC) = 0.20 + (0.75)(0.20 – 0.10)(1 – 0.35) = 0.2488 If Williamson’s debt-to-equity ratio is 0.75: B / S = 0.75 Solving for B: B = (0.75 * S) A firm’s debt-to-value ratio is: B / (B+S) Since B = (0.75 * S): Williamson’s debt-to-value ratio = (0.75 * S) / { (0.75 * S) + S} = (0.75 * S) / (1.75 * S) = (0.75 / 1.75) = 0.4286 Williamson’s debt-to-value ratio is 42.86% A firm’s equity-to-value ratio is: S / (B+S) Since B = (0.75 * S): Williamson’s equity-to-value ratio = S / {(0.75*S) + S} = S / (1.75 * S) = (1 / 1.75) = 0.5714 Williamson’s equity-to-value ratio is 57.14%. Williamson’s weighted average cost of capital (rwacc) is: rwacc = {B / (B+S)}(1 – TC) rB + {S / (B+S)}rS = (0.4286)(1 – 0.35)(0.10) + (0.5714)(0.2488) = 0.17 Therefore, Williamson’s weighted average cost of capital (rwacc) is 17% if the firm’s debt-to-equity ratio is 0.75. If Williamson’s debt-to-equity ratio is 1.5, then the cost of the firm’s equity capital (rS) will be: rS = r0 + (B/S)(r0 – rB)(1 – TC) = 0.20 + (1.5)(0.20 – 0.10)(1 – 0.35) = 0.2975 If Williamson’s debt-equity ratio is 1.5: B / S = 1.5 Solving for B: B = (1.5 * S) A firm’s debt-to-value ratio is: B / (B+S) Since B = (1.5 * S): Williamson’s debt-to-value ratio = (1.5 * S) / { (1.5 * S) + S} = (1.5 * S) / (2.5 * S) = (1.5 / 2.5) = 0.60 Williamson’s debt-to-value ratio is 60% A firm’s equity-to-value ratio is: S / (B+S) Since B = (1.5 * S): Williamson’s equity-to-value ratio = S / {(1.5*S) + S} = S / (2.5 * S) = (1 / 2.5) = 0.40 Williamson’s equity-to-value ratio is 40%. Williamson’s weighted average cost of capital (rwacc) is: rwacc = {B / (B+S)}(1 – TC) rB + {S / (B+S)}rS = (0.60)(1 – 0.35)(0.10) + (0.40)(0.2975) = 0.158 Therefore, Williamson’s weighted average cost of capital (rwacc) is 15.8% if the firm’s debt-to-equity ratio is 1.5. 17.2 a. The total value of a firm’s equity is the discounted expected cash flow to the firm’s stockholders. If the expansion continues, each firm will generate earnings before interest and taxes of $2 million. If there is a recession each firm will generate earnings before interest and taxes of only $800,000. Since Steinberg owes its bondholders $750,000 at the end of the year, its stockholders will receive $1.25 million (= $2 million - $750,000) if the expansion continues. If there is a recession, its stockholders will only receive $50,000 (= $800,000 - $750,000). The market value of Steinberg’s equity is: {(0.80)($1,250,000) + (0.20)($50,000)} / 1.15 = $878,261 The value of Steinberg’s equity is $878,261. Steinberg’s bondholders will receive $750,000 regardless of whether there is a recession or a continuation of the expansion. The market value of Steinberg’s debt is: {(0.80)($750,000) + (0.20)($750,000)} / 1.15 = $652,174 The value of Steinberg’s debt is $652,174. Since Dietrich owes its bondholders $1 million at the end of the year, its stockholders will receive $1 million (= $2 million - $1 million) if the expansion continues. If there is a recession, its stockholders will receive nothing since the firm’s bondholders have a more senior claim on all $800,000 of the firm’s earnings. The market value of Dietrich’s equity is: {(0.80)($1,000,000) + (0.20)($0)} / 1.15 = $695,652 The value of Dietrich’s equity is $695,652. Dietrich’s bondholders will receive $1 million if the expansion continues and $800,000 if there is a recession. The market value of Dietrich’s debt is: {(0.80)($1,000,000) + (0.20)($800,000)} / 1.15 = $834,783 The value of Dietrich’s debt is $834,783. b. The value of Steinberg is the sum of the value of the firm’s debt and equity. The value of Steinberg is: VL = B + S = $652,174 + $878,261 = $1,530,435 The value of Steinberg is $1,530,435. The value of Dietrich is the sum of the value of the firm’s debt and equity. The value of Dietrich is: VL = B + S = $834,783 + 695,652 = $1,530,435 The value of Dietrich is also $1,530,435. c. You should disagree with the CEO’s statement. The risk of bankruptcy per se does not affect a firm’s value. It is the actual costs of bankruptcy that decrease the value of a firm. Note that this problem assumes that there are no bankruptcy costs. 17.4 You should disagree with the statement. If a firm has debt, it might be advantageous to stockholders for the firm to undertake risky projects, even those with negative net present values. This incentive results from the fact that most of the risk of failure is borne by bondholders. Therefore, value is transferred from the bondholders to the shareholders by undertaking risky projects, even if the projects have negative NPVs. This incentive is even stronger when the probability and costs of bankruptcy are high. A numerical example illustrating this concept is included in the solution to question 17.3 under the heading “Incentive to take large risks”. 17.6 a. If the low-risk project is undertaken, the firm will be worth $500 if the economy is bad and $700 if the economy is good. Since each of these two scenarios is equally probable, the expected value of the firm is $600 {= (0.50)($500) + (0.50)($700)}. If the high-risk project is undertaken, the firm will be worth $100 if the economy is bad and $800 if the economy is good. Since each of these two scenarios is equally probable, the expected value of the firm is $450 {= (0.50)($100) + (0.50)($800)}. The low-risk project maximizes the expected value of the firm. b. If the low-risk project is undertaken, the firm’s equity will be worth $0 if the economy is bad and $200 if the economy is good. Since each of these two scenarios is equally probable, the expected value of the firm’s equity is $100 {= (0.50)($0) + (0.50)($100)}. If the high-risk project is undertaken, the firm’s equity will be worth $0 if the economy is bad and $300 if the economy is good. Since each of these two scenarios is equally probable, the expected value of the firm’s equity is $150 {= (0.50)($0) + (0.50)($300)}. c. Risk-neutral investors prefer the strategy with the highest expected value. Fountain’s stockholders prefer the high-risk project since it maximizes the expected value of the firm’s equity. d. In order to make stockholders indifferent between the low-risk project and the high-risk project, the bondholders will need to raise their required debt payment so that the expected value of equity if the high-risk project is undertaken is equal to the expected value of equity if the low risk project is undertaken. As shown in part a, the expected value of equity if the low-risk project is undertaken is $100. If the high-risk project is undertaken, the value of the firm will be $100 if the economy is bad and $800 if the economy is good. If the economy is bad, the entire $100 will go to the firm’s bondholders and Fountain’s stockholders will receive nothing. If the economy is good, Fountain’s stockholders will receive the difference between $800, the total value of the firm, and the required debt payment. Let X be the debt payment that bondholders will require if the high-risk project is undertaken: Expected Value of Equity = (0.50)($0) + (0.50)($800 – X) In order for stockholders to be indifferent between the two projects, the expected value of equity if the high-risk project is undertaken must be equal to $100. $100 = (0.50)($0) + (0.50)($800-X) X = $600 Therefore, the bondholders should promise to raise the required debt payment by $100 (= $600 - $500) if the high-risk project is undertaken in order to make Fountain’s stockholders indifferent between the two projects. 17.10 a. 1. If Fortune remains an all-equity firm, its value will equal VU, the value of Fortune as an unlevered firm. The general expression for the value of a levered firm in a world with both corporate and personal taxes is: VL = VU + [ 1 – {(1 – TC)(1 – TS) / (1 - TB)}] * B where VL = the value of a levered firm VU = the value of an unlevered firm B = the market value of the firm’s debt TC = the tax rate on corporate income TS = the personal tax rate on equity distributions TB = the personal tax rate on interest income In this problem: B = $13,500,000 TC = 0.40 TS = 0.30 TB = 0.30 The value of Fortune as a levered firm is: VL = VU + [ 1 – {(1 – TC)(1 – TS) / (1 - TB)}] * B = VU + [ 1 – {1 – 0.40)(1 – 0.30) / (1 – 0.30)}] * $13,500,000 = VU + (0.40)($13,500,000) = VU + $5,400,000 As was stated in Chapter 16, stockholders prefer a policy that maximizes the value of the firm. Equity holders will prefer Fortune to become a levered firm since the debt will increase the firm’s value by $5.4 million. 2. The CCRA will prefer the plan that generates the highest amount of tax revenue. The CCRA taxes corporate income at 40%, interest income at 30%, and equity distributions at 30%. Under the unlevered plan: The CCRA generates $1,200,000 (= 0.40 * $3,000,000) of corporate tax revenue on the firm’s earnings and $540,000 (= 0.30 * $1,800,000) of personal tax revenue on Fortune’s equity distributions. Since the firm has no debt, no interest payments are made, and the CCRA will not generate any tax revenue on interest. The CCRA generates $1,740,000 (= $1,200,000 + $540,000) of tax revenue under the unlevered plan. Under the levered plan: The CCRA generates $660,000 (= 0.40 * $1,650,000) of corporate tax revenue on the firm’s earnings, $297,000 (= 0.30 * $990,000) of personal tax revenue on Fortune’s equity distributions, and $405,000 (= 0.30 * $1,350,000) of personal tax revenue on the firm’s interest payments. The CCRA generates $1,362,000 (= $660,000 + $297,000 + $405,000) of tax revenue under the levered plan. Since the all-equity plan generates higher tax revenues, the CCRA will prefer an unlevered capital structure. 3. As an unlevered firm, Fortune would generate $1,800,000 of net income every year into perpetuity. Since the firm distributes all earnings to equity holders, this amount will be taxed at a rate of 30%, providing a $1,260,000 {= $1,800,000 * (1 – 0.30)} annual after-tax cash flow to the firm’s equity holders. Since stockholders demand a 20% return after taxes, the value of Fortune if it were an unlevered firm is equal to a perpetuity of $1,260,000 per year, discounted at 20%. VU = $1,260,000 / 0.20 = $6,300,000 The value of Fortune as an unlevered firm is $6.3 million. The general expression for the value of a levered firm in a world with both corporate and personal taxes is: VL = VU + [ 1 – {(1 – TC)(1 – TS) / (1 - TB)}] * B where VL = the value of a levered firm VU = the value of an unlevered firm B = the market value of the firm’s debt TC = the tax rate on corporate income TS = the personal tax rate on equity distributions TB = the personal tax rate on interest income In this problem: VU = $6,300,000 B = $13,500,000 TC = 0.40 TS = 0.30 TB = 0.30 The value of Fortune as a levered firm is: VL = VU + [ 1 – {(1 – TC)(1 – TS) / (1 - TB)}] * B = $6,300,000 + [ 1 – {1 – 0.40)(1 – 0.30) / (1 – 0.30)}] * $13,500,000 = $6,300,000 + (0.40)($13,500,000) = $11,700,000 The value of Fortune as a levered firm is $11.7 million. b. 1. Under the unlevered plan, the firm’s earnings available to equity holders is $1,800,000. Since equity distributions are taxed at a rate of 20%, the annual after-tax cash flow to Fortune’s equity holders is $1,440,000 {= $1,800,000 * (1 – 0.20)}. The annual after-tax cash flow to equity holders under the unlevered plan is $1,440,000. Under the levered plan, the firm’s earnings available to equity holders is $990,000. Since equity distributions are taxed at a rate of 20%, the annual after- tax cash flow to Fortune’s equity holders is $792,000 {= $990,000 * (1 – 0.20)}. The annual after-tax cash flow to equity holders under the levered plan is $792,000. 2. Under the unlevered plan, Fortune will have no debt. The annual after-tax cash flow to debt holders under the unlevered plan is $0. Under the levered plan, the firm will make annual interest payments of $1,350,000 to debt holders. Since interest income is taxed at a rate of 55%, the annual after-tax cash flow to Fortune’s debt holders is $607,500 {= $1,350,000 * (1 – 0.55)}. The annual after-tax cash flow to debt holder under the levered plan is $607,500 17.15 a. The general expression for the value of a levered firm in a world with both corporate and personal taxes is: VL = VU + [ 1 – {(1 – TC)(1 – TS) / (1 - TB)}] * B – C(B) where VL = the value of a levered firm VU = the value of an unlevered firm B = the market value of the firm’s debt TC = the tax rate on corporate income TB = the personal tax rate on interest income C(B) = the present value of the costs of financial distress and agency costs The value of an all-equity firm (VU) is the present value of the firm’s after-tax cash flows to equity holders. VU = {(EBIT)(1 - TC)(1 – TS)} / r0 where VU = the value of an unlevered firm EBIT = the firm’s earnings before interest and taxes TC = the tax rate on corporate income TS = the tax rate on equity distributions r0 = the required rate of return on the firm’s unlevered equity In this problem: EBIT = $800,000 TC = 0.35 TS = 0 r0 = 0.10 The value of Weinberg as an all-equity firm in a world with both corporate and personal taxes is: VU = {(EBIT)(1 - TC)(1 – TS)} / r0 = {($800,000)(1 – 0.35)(1 – 0)} / 0.10 = $5,200,000 Thus: VU = $5,200,000 B = $1,200,000 TC = 0.35 TS =0 TB = 0.15 C(B) = $60,000 (= 0.05 * $1,200,000) The value of Weinberg as a levered firm in a world with both corporate and personal taxes is: VL = VU + [ 1 – {(1 – TC)(1 – TS) / (1 - TB)}] * B – C(B) = $5,200,000 + [ 1 – {(1 – 0.35)(1 – 0) / (1 – 0.15)}]*$1,200,000 - $60,000 = $5,422,353 The value of Weinberg as a levered firm is $5,422,353 in a world with both corporate and personal taxes. b. Since the value of Weinberg is $5,200,000 as an unlevered firm and $5,422,353 as a levered firm, the added value of including debt in the firm’s capital structure is $222,353 (= $5,422,353 - $5,200,000). The added value of the firm’s debt is $222,353. 17.16 a. Since STC is an all-equity firm, its cost of capital is equal to the required return on its equity. Use the Capital Asset Pricing Model (CAPM) to determine the required return on STC’s unlevered equity. According to CAPM: rS = rf + S{E(rm) – rf} where rS = the required return on a firm’s equity rf = the expected return on a risk-free asset E(rm) = the expected rate of return on the market portfolio S = the beta of a firm’s equity The beta of a firm’s equity is equal to: S = Cov(x, m) / 2m where Cov(x, m) = the covariance between the return on the firm’s common stock and the return on the market portfolio m = the variance of returns on the market portfolio 2 In this problem: Cov(x, m) = 0.048 2m = 0.04 The beta of STC’s equity is: S = Cov(x, m) / 2m = 0.048 / 0.04 = 1.2 Thus: rf = 0.10 E(rm) = 0.20 S = 1.2 The required return on STC’s capital is: rS = rf + S{E(rm) – rf} = 0.10 + 1.2(0.20 – 0.10) = 0.22 STC’s overall cost of capital is 22%. b. STC should purchase the machine with the higher net present value (NPV). Remember that the firm will need a machine for the next four years. Since the economic life of the Heavy-Duty Model is four years, the firm will only need to purchase the machine once. NPVHEAVY DUTY = -Price + PV(Annual Cost Savings) + PV(CCA Tax Shield) The Heavy-Duty Model will generate $640 of cost savings every year for four years. This is equivalent to the firm generating $640 of additional earnings each year for four years. Because these earnings are subject to a corporate income tax of 40%, the annual after-tax cash flow created by the firm’s additional savings is $384.00 {= $640(1 – 0.4)}. Since the required return on STC’s equity is 22%, the firm’s after-tax annual cost savings can be valued as an annuity with four annual payments of $384, discounted at 22%. PV(Annual Cost Savings) = $384.00A40.22 = $957.56 InvestmentxTaxratexCCArate 1 0.5(cos tofcapital ) PVCCATS [ ][ ] CCArate cos tofcapital 1 cos tofcapital $1, 000 x0.40 x0.20 1 0.5(0.22) [ ][ ] $173.30 0.20 0.22 1 0.22 The NPV of the Heavy-Duty Model is: NPVHEAVY DUTY = -Price + PV(Annual Cost Savings) + PV(Depreciation Tax Shield) = -$1,000 + $957.56 + $ 173.30 = $130.86 The NPV of the Heavy-Duty Model is $130.86. Since the economic life of the Light-Weight Model is only 2 years, the firm will need to buy one machine now and one in two years. The cash flows associated with the second purchase must be discounted by two years. NPVLIGHT WEIGHT = -Price + PV(Annual Cost Savings) + PV(CCA Tax Shield) + {(- Price + PV(Annual Cost Savings) + PV(CCA Tax Shield)} / (1.22)2 The Light-Weight Model will generate annual cost savings of $616 for two years. This is equivalent to the firm generating $616 of additional earnings each year for two years. Since these earnings are subject to a corporate income tax of 40%, the annual after-tax cash flow created by the firm’s additional savings is $369.60 {= $616(1 – 0.40)}. Since the required return on STC’s equity is 22%, the firm’s after- tax annual cost savings can be valued as a two-year annuity with annual payments of $369.60, discounted at 22%. PV(Annual Cost Savings) = $369.60A20.22 = $551.27 InvestmentxTaxratexCCArate 1 0.5(cos tofcapital ) PVCCATS [ ][ ] CCArate cos tofcapital 1 cos tofcapital $500 x 0.40 x 0.20 1 0.5(0.22) [ ][ ] $82.71 0.20 0.22 1 0.22 82.71 should be 86.65 The NPV of the Light-Weight Model is: NPVLIGHT WEIGHT = -Price + PV(Annual Cost Savings) + PV(Depreciation Tax Shield) + {(- Price + PV(Annual Cost Savings) + PV(Depreciation Tax Shield)} / (1.22)2 = -$500 + $551.27 + $ 86.65 + {(-$500 + $551.27 + $ 86.65)} / (1.22)2 = $ 137.92 + $ 137.92/ (1.22)2 = $ 230.58 The NPV of the Light-Weight Model is $ 230.58. Since its NPV is higher, STC should purchase the Light-Weight Model. c. 1. Modigliani-Miller Proposition I states that in a world with corporate taxes: V L = V U + T CB where VL = the value of a levered firm VU = the value of an unlevered firm TC = the corporate tax rate B = the value of debt in a firm’s capital structure In this problem: VU = $10 million B = $2 million TC = 0.40 The new value of STC will be: V L = V U + T CB = $10,000,000 + (0.40)($2,000,000) = $10,800,000 The value of the STC will be $10,800,000 if the CFO’s plan adopted. 2. The value of a levered firm is the sum of the market value of the firm’s debt and the market value of the firm’s equity. VL = B + S In this problem: VL = $ 10,800,000 B = $2,000,000 Therefore, the value of STC’s levered equity must be: S = VL – B = $ 10,800,000 - $2,000,000 = $8,800,000 The value of STC’s levered equity is $8,800,000. d. Since the value of STC as a levered firm is $10,800,000 and costs of financial distress are 2% of this value, the STC’s costs of financial distress total $216,000 (= 0.02 * $10,800,000). This reduces the firm’s value to $10,584,000. Since this amount is greater than $10 million, the value of STC as an unlevered firm, adding debt to the firm’s capital structure increases the firm’s value. STC should not remain an unlevered firm. 18.2 a. The adjusted present value of a project equals the net present value of the project under all-equity financing plus the net present value of any financing side effects. In Peatco’s case, the NPV of financing side effects equals the after-tax present value of the cash flows resulting from the firm’s debt. APV = NPV(All-Equity) + NPV(Financing Side Effects) NPV(All-Equity) NPV = -Initial Investment + PV[(1-TC)(Earnings Before Taxes and Depreciation)] + PV(CCA Tax Shield) Assuming the company has other assets in this CCA pool, InvestmentxTaxRatexCCA 1 0.50( DiscountRate) PVCCATS [ ][ ] CCA DiscountRate 1 DiscountRate 2,100, 000 x0.40 x0.30 1 0.50(0.16) [ ][ ] $510, 045 0.30 0.16 1 0.16 NPV = -$2,100,000 + (1-0.40)($900,000)A30.16 + $510,045 = -$377,175 NPV(Financing Side Effects) The net present value of financing side effects equals the after-tax present value of cash flows resulting from the firm’s debt. NPV(Financing Side Effects) = -Flotation costs+ NPV(tax shield on flotation costs)+Amount borrowed - PV (Tax shield on interest) – Present Value of loan payments Given a known level of debt, debt cash flows should be discounted at the pre-tax cost of debt (rB), 7.5%. Since $21,000 in flotation costs will be amortized over the three-year life of the loan, $7,000 = ($21,000 / 3) of flotation costs will be expensed per year. NPV of Flotation costs = -21,000 + (0.40)($7,000)A30.075 = - 21,000 +7,281= -$13,719 NPV(Loan) = ($2,100,000 – (1 – 0.40)(0.075)($2,100,000)A30. 075 – [$2,100,000/(1.075)3] + = $2,100,000 - $245,750-$1,690,417 = $163,833 APV APV = NPV(All-Equity) + NPV(Flotation costs) + NPV(Loan) = -$377,175 - $13,719 +163,833 = -$227,061 Since the adjusted present value (APV) of the project is negative, Peatco should not undertake the project. b. NPV(Financing Side Effects) The only change is that the interest cost will be based on 6.5% but the cash flows will still be discounted at the company’s cost of debt which is 7.5%. The net present value of financing side effects equals the after-tax present value of cash flows resulting from the firm’s debt. NPV(Financing Side Effects) = -Flotation costs+ NPV(tax shield on flotation costs)+Amount borrowed - PV (Tax shield on interest) – Present Value of loan payments Given a known level of debt, debt cash flows should be discounted at the pre-tax cost of debt (rB), 7.5%. Since $21,000 in flotation costs will be amortized over the three-year life of the loan, $7,000 = ($21,000 / 3) of flotation costs will be expensed per year. NPV of Flotation costs = -21,000 + (0.40)($7,000)A30.075 = - 21,000 +7,281= -$13,719 NPV(Loan) = ($2,100,000 – (1 – 0.40)(0.065)($2,100,000)A30. 075 – [$2,100,000/(1.075)3] + = $2,100,000 - $212,983-$1,690,417 = $196,600 APV APV = NPV(All-Equity) + NPV(Flotation costs) + NPV(Loan) = -$377,175 - $13,719 +196,600 = -$194,294 Since the adjusted present value (APV) of the project is still negative, Peatco should still not undertake the project. Note: When a subsidy exists on a loan, you cannot use the PV(Tax Shield) method to compute the NPV of the loan. This is true because even with an interest rate subsidy, the appropriate rate by which to discount the after-tax interest payments is the market rate of interest. The PV(Tax Shield) method presumes the interest rate and the discount rate are the same. 18.4 The adjusted present value of a project equals the net present value of the project under all- equity financing plus the net present value of any financing side effects. In the joint venture’s case, the NPV of financing side effects equals the after-tax present value of cash flows resulting from the firms’ debt. APV = NPV(All-Equity) + NPV(Financing Side Effects) NPV(All-Equity) NPV = -Initial Investment + PV[(1 – TC)(Earnings Before Interest, Taxes, and Depreciation )] + PV(CCA Tax Shield) Assuming that the company has other assets in the class, InvestmentxTaxRatexCCA 1 0.50( DiscountRate) PVCCATS [ ][ ] CCA DiscountRate 1 DiscountRate 20, 000, 000 x0.25 x0.30 1 0.50(0.12) [ ][ ] $3,380,102 0.30 0.12 1 0.12 NPV = -$20,000,000 + [(1-0.25)($3,000,000)A200.12] + 3,380,102 = -$20,000,000 + $16,806,248 + 3,380,102 = $186,350 NPV(Financing Side Effects) The NPV of financing side effects equals the after-tax present value of cash flows resulting from the firms’ debt. Given a known level of debt, debt cash flows should be discounted at the pre-tax cost of debt (rB), 10%. NPV(Financing Side Effects) = Proceeds – After-tax PV(Interest Payments) – PV(Principal Repayments) = $10,000,000 – (1 – 0.25)(0.05)($10,000,000)A150.09 – [$10,000,000/((1.09)15] = $4,231,861 APV APV = NPV(All-Equity) + NPV(Financing Side Effects) = $186,350 + $4,231,861 = $4,418,211 The Adjusted Present Value (APV) of the project is $4,418,211. 18.7 a. Bolero has a capital structure with three parts: long-term debt, short-term debt, and equity. i. Book Value Weights: Type of Financing Book Value Weight Cost Long-term debt $5,000,000 25% 10% Short-term debt $5,000,000 25% 8% Common Stock $10,000,000 50% 15% Total $20,000,000 100% Since interest payments on both long-term and short-term debt are tax- deductible, multiply the pre-tax costs by (1-TC) to determine the after-tax costs to be used in the weighted average cost of capital calculation. rwacc = (WeightLTD)(CostLTD)(1-TC) + (WeightSTD)(CostSTD)(1-TC) + (WeightEquity)(CostEquity) = (0.25)(0.10)(1-0.4) + (0.25)(0.08)(1-0.4) + (0.50)(0.15) = 0.102 If Bolero uses book value weights, the firm’s weighted average cost of capital would be 10.2%. ii. Market Value Weights: Type of Market Financing Value Weight Cost Long-term debt $2,000,000 10% 10% Short-term debt $5,000,000 25% 8% Common Stock $13,000,000 65% 15% Total $20,000,000 100% Since interest payments on both long-term and short-term debt are tax- deductible, multiply the pre-tax costs by (1-TC) to determine the after-tax costs to be used in the weighted average cost of capital calculation. rwacc = (WeightLTD)(CostLTD)(1-TC) + (WeightSTD)(CostSTD)(1-TC) + (WeightEquity)(CostEquity) = (0.10)(0.10)(1-0.4) + (0.25)(0.08)(1-0.4) + (0.65)(0.15) = 0.1155 If Bolero uses market value weights, the firm’s weighted average cost of capital would be 11.55%. iii. Target Weights: If Bolero has a target debt-to-equity ratio of 100%, then both the target equity-to- value and target debt-to-value ratios must be 50%. Since the target values of long-term and short-term debt are equal, the 50% of the capital structure targeted for debt would be split evenly between long-term and short-term debt (25% Type of Financing Target Weight Cost Long-term debt 25% 10% Short-term debt 25% 8% Common Stock 50% 15% Total 100% each). Since interest payments on both long-term and short-term debt are tax- deductible, multiply the pre-tax costs by (1-TC) to determine the after-tax costs to be used in the weighted average cost of capital calculation. rwacc = (WeightLTD)(CostLTD)(1-TC) + (WeightSTD)(CostSTD)(1-TC) + (WeightEquity)(CostEquity) = (0.25)(0.10)(1-0.4) + (0.25)(0.08)(1-0.4) + (0.50)(0.15) = 0.102 If Bolero uses target weights, the firm’s weighted average cost of capital would be 10.2%. b. The differences in the WACCs are due to the different weighting schemes. The firm’s WACC will most closely resemble the WACC calculated using target weights since future projects will be financed at the target ratio. Therefore, the WACC computed with target weights should be used for project evaluation. 18.9 a. In a world with corporate taxes, a firm’s weighted average cost of capital (rwacc) equals: rwacc = {B / (B+S)}(1 – TC) rB + {S / (B+S)}rS where B / (B+S) = the firm’s debt-to-value ratio S / (B+S) = the firm’s equity-to-value ratio rB = the pre-tax cost of debt rS = the cost of equity TC = the corporate tax rate Since the firm’s target debt-to-equity ratio is 200%, the firm’s target debt-to- value ratio is 2/3, and the firm’s target equity-to-value ratio is 1/3. The inputs to the WACC calculation in this problem are: B / (B+S) = 2/3 S / (B+S) = 1/3 rB = 0.10 rS = 0.20 TC = 0.34 rwacc = {B / (B+S)}(1 – TC) rB + {S / (B+S)}rS = (2/3)(1 – 0.34)(0.10) + (1/3)(0.20) = 0.1107 NEC’s weighted average cost of capital is 11.07%. Use the weighted average cost of capital to discount NEC’s unlevered cash flows. NPV = -$20,000,000 + $8,000,000 / 0.1107 = $52,267,389 Since the NPV of the project is positive, NEC should proceed with the expansion. 19.2 Based on Miller and Modigliani reasoning, the stock will sell for $8.65. This is the same price you paid for the stock, and you are selling before the ex-dividend date. When the stock goes ex-dividend, the price is expected to fall $0.75 a share. 19.3 a. If the dividend is declared, the price of the stock will drop on the ex-dividend date by the value of the dividend, $5. It will then trade for $95. b. If it is not declared, the price will remain at $100. c. Mann’s outflows for investments are $2,000,000. These outflows occur immediately. One year from now, the firm will realize $1,000,000 in net income and it will pay $500,000 in dividends, but the need for financing is immediate. Mann must finance $2,000,000 through the sale of shares worth $100. It must sell $2,000,000 / $100 = 20,000 shares. d. The MM model is not realistic since it does not account for taxes, brokerage fees, uncertainty over future cash flows, investors’ preferences, signaling effects, and agency costs. 19.6 a. The price is the PV of the dividends, and there are only 2 more cash flows associated with this stock: D1 $2 and D2 $17.5375 . Find the present value of this cash flow series: $2 $17.5375 PV 1.14 1.142 $15.25 b. The current value of your shares is ($15.25)(500) = $7,625. Since you want equal payments, you want an annuity, which solves: X X $7,625 1.14 1.142 Solving for X, the cash flows are $4,630.5841 each year, However, you will receive $1,000 in dividends in the first year, so you must sell shares to make up the difference, At the end of the first year, you must sell just enough shares to generate $3,630.5841. In order to do that, first you must determine the stock price. At that time, the price will be the PV of the liquidating dividend: $17.5375 $15.38 1.14 and $3,630.5841 236 shares $15.38 So you must sell 236shares. At the end of the 2nd year, the remaining shares will each earn the liquidating dividend. To check your work, note that you will receive $ 4,630 [(500 - 236) x $17.5375]. (Rounding causes the discrepancy). 19.9 For either alternative, assume the $2,000,000 cash is after corporate tax. Alternative 1: Firm invests cash, either in T-bills or in preferred stock, and then pays out as special dividend in 3 years If the firm invests in T-Bills: after corporate tax yield 7(1- .35) 4.55 FV =2,000,000 1 .0455 3 2, 285,609.89 After personal tax cash flow to the stock holders: ATCF 2, 285,609.89 1 .31 1,577,070.82 If the firm invests in preferred stock (assume dividends will be re-invested in the same preferred stock): after corporate tax dividend: preferred dividends: 11% ( 2,000,000) = $220,000 Since 70% of preferred dvds are excluded from tax: Taxable dvds = 220,000 x .3 = 66,000 Tax = 66,000 x .35 = 23,100 after corporate tax dividend = 220,000 - 23,100 = 196,900 Therefore, 19.9 (continued) 196,900 after corp tax yield on pref stock .09845 2,000,000 FV 2,000,000 1.09845 3 2,650,762.85 After personal tax cash flow to the stock holders: ATCF 2,650,762.82 1 .31 1,829,026.37 Alternative 2: Firm pays out dividend now, and individuals invest in T-bills: After personal tax cash to be invested = 2,000,000(1-.31) = 1,380,000 After personal tax yield on T-bills = .07 (1-.31) = .0483 After personal tax cash flow to the stock holders: ATCF FV 1,380,000 1.0483 3 1,589,775.66 The after-tax cash flow for the shareholders is maximized when the firm invests the cash in the preferred stocks. 19.15 To minimize her tax burden, your aunt should divest herself of high dividend yield stocks and invest in low dividend yield stock. Or, if possible, she should keep her high dividend stocks, borrow an equivalent amount of money and invest that money in a tax deferred account. 23.1 a. An option is a contract giving its owner the right to buy or sell an asset at a fixed price on or before a given date. b. Exercise is the act of buying or selling the underlying asset under the terms of the option contract. c. The strike price is the fixed price in the option contract at which the holder can buy or sell the underlying asset. The strike price is also called the exercise price. d. The expiration date is the maturity date of the option. It is the last date on which an American option can be exercised and the only date on which a European option can be exercised. e. A call option gives the owner the right to buy an asset at a fixed price during a particular time period. f. A put option gives the owner the right to sell an asset at a fixed price during a particular time period. 23.5 a. The payoff to the owner of a call option at expiration is the maximum of zero and the current stock price minus the strike price. The payoff to the owner of a call option on Stock A on December 21 is: max[0, ST - K] = max[0, 55-50] = $5 where ST = the price of the underlying asset at expiration K = the strike price b. The payoff to the seller of a call option at expiration is the minimum of zero and the strike price minus the current stock price. The payoff to the seller of a call option on Stock A on December 21 is: min[0, K- ST] = min[0, 50-55] = -$5 In other words, the seller must pay $5. c. The payoff to the owner of a call option at expiration is the maximum of zero and the current stock price minus the strike price. The payoff to the owner of a call option on Stock A on December 21 is: max[0, ST - K] = max[0, 45-50] = $0 d. The payoff to the seller of a call option at expiration is the minimum of zero and the strike price minus the current stock price. The payoff to the seller of a call option on Stock A on December 21 is: min[0, K- ST] = min[0, 50-45] = $0 e. 25 20 Payoff to Owner 15 10 5 0 30 35 40 45 50 55 60 65 70 Stock Price at Expiration f. 0 30 35 40 45 50 55 60 65 70 Payoff to Seller -5 -10 -15 -20 -25 Stock Price at Expiration g. The seller of a call option receives a premium, the price of the option, at the time of sale. At expiration, if the buyer chooses not to exercise, the premium becomes pure profit for the seller. Therefore, an individual will write (sell) a call option if he does not believe the stock price will rise above the strike price before expiration. 23.10 a. Yes, there is an arbitrage opportunity. You should buy the American call option for $8, exercise the option (buy the underlying stock for the option’s strike price of $50), and sell the stock at the market price of $60. This strategy yields a riskless arbitrage profit of $2 (= $60 - $50 - $8). b. Arbitrage opportunities such as this imply that the lower bound on the price of an American call option is the value of immediate exercise, which is equal to the current stock price minus the strike price of the option (S – K). c. An upper bound on the price of an American call option is the current price of the underlying asset. A call option, which gives its owner the right to buy an underlying asset, cannot cost more than the underlying asset. If it did, selling the option, using the proceeds to purchase the underlying asset, and pocketing the difference would yield an arbitrage profit. Suppose Stock A is trading for $8 and a call option on Stock A with a strike price of $6 is selling for $10. Consider the following investment strategy designed to take advantage of the mispricing: Strategy Cash Flow 1. Sell call option +$10.00 2. Buy stock -$8.00 Arbitrage Profit +$2.00 If the buyer of the call option decides to exercise, the seller will exchange the stock for the option’s strike price of $6. Since the seller already owns the stock (and therefore does not need to purchase it), this results in an additional cash inflow of $6 for the seller, regardless of the price of the stock at the time of exercise. If the buyer decides not to exercise, the seller keeps both the stock and the $2 arbitrage profit. In either case, it is impossible for the seller of the call option to lose money. 23.13 According to Put-Call Parity, for two options with the same strike price and time to expiration, the cost of a call must equal the cost of a put plus the cost of the stock minus the present value of the strike price: According to Put-Call Parity: C = P + S – PV(K) where C = the cost of a call option P = the cost of a put option S = the current price of the underlying asset PV(K) = the present value of the strike price Solving for the stock price, this equation shows that the stock price must equal the cost of a call minus the cost of a put plus the present value of the strike price: Put-Call Parity: S = C – P + PV(K) The cost of a call with a strike of $40 written on General Eclectic Stock is $8. The cost of a put with a strike of $40 written on General Eclectic Stock is $2. The present value of the strike price is $ 38.10 (= $40 / 1.05). S = C – P + PV(K) = $8 - $2 + $38.10 = $44.10 The price of General Eclectic stock must be $44.10 per share in order to prevent arbitrage. 23.15 a. In order to solve a problem using the two-state option model, first draw a stock price tree containing both the current stock price and the stock’s possible values at the time of the option’s expiration. Next, draw a similar tree for the option, designating what its value will be at expiration given either of the 2 possible stock price movements. Eastjet’s stock price today is $100. It will either increase to $120 or decrease to $80 in one year. If the stock price rises to $120, Ken will exercise his call option for $110 and receive a payoff of $10 at expiration. If the stock price falls to $80, Ken will not exercise his call option, and he will receive no payoff at expiration. Eastjet's Stock Price Ken's European Call Option with a Strike of 110 Today 1 Year Today 1 Year 120 10 = max(0, 120 -110) 100 ? 80 0 = max(0, 80 -110) If Eastjet’s stock price rises, its return over the period is 20% [= (120/100) – 1]. If Eastjet’s stock price falls, its return over the period is –20% [= (80/100) –1]. Use the following expression to determine the risk-neutral probability of a rise in the price of Eastjet’s stock: Risk-Free Rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall) = (ProbabilityRise)(ReturnRise) + (1 - ProbabilityRise)(ReturnFall) 0.025 = (ProbabilityRise)(0.20) + (1 – ProbabilityRise)(-0.20) ProbabilityRise = 0.5625 ProbabilityFall = 1 - ProbabilityRise = 1 – 0.5625 = 0.4375 The risk-neutral probability of a rise in Eastjet stock is 50%, and the risk-neutral probability of a fall in Eastjet stock is 50%. Using these risk-neutral probabilities, determine the expected payoff to Ken’s call option at expiration. Expected Payoff at Expiration = (.5625)($10) + (.4375)($0) = $5.625 Since this payoff occurs 1 year from now, it must be discounted at the risk-free rate of 2.5% in order to find its present value: PV(Expected Payoff at Expiration) = ($5.625 / 1.025) = $5.49 Therefore, given the information Ken has about Eastjet’s stock price movements over the next year, a European call option with a strike price of $110 and one year until expiration is worth $5.49 today. b. Yes, there is a way for Ken to create a synthetic call option with identical payoffs to the call option described above. In order to do this, Ken will need to buy shares of Eastjet’s stock and borrow at the risk-free rate. The number of shares that Ken should buy is based on the delta of the option, where delta is defined as: Delta = (Swing of option) / (Swing of stock) Since the call option will be worth $10 if Eastjet’s stock price rises and $0 if it falls, the swing of the call option is 10 (= 10 – 0). Since the stock price will either be $120 or $80 at the time of the option’s expiration, the swing of the stock is 40 (= 120 - 80). Given this information: Delta = (Swing of option) / (Swing of stock) = (10 / 40) = 1/4 Therefore, Ken’s first step in creating a synthetic call option is to buy 1/4 of a share of Eastjet’s stock. Since Eastjet’s stock is currently trading at $100 per share, this will cost him $25.00 [= (1/4)($100)]. In order to determine the amount that Ken should borrow, compare the payoff of the actual call option to the payoff of delta shares at expiration. Call Option If the stock price rises to $120: payoff = $10 If the stock price falls to $80: payoff = $0 Delta Shares If the stock price rises to $120: payoff = (1/4)($120) = $30.00 If the stock price falls to $80: payoff = (1/4)($80) = $20.00 Ken would like the payoff of his synthetic call position to be identical to the payoff of an actual call option. However, owning 1/4 of a share leaves him exactly $20.00 above the payoff at expiration, regardless of whether the stock price rises or falls. In order to reduce his payoff at expiration by $20.00, Ken should borrow the present value of $20.00 now. In one year, his obligation to pay $20.00 will reduce his payoffs so that they exactly match those of an actual call option. Ken should purchase 1/4 of a share of Eastjet’s stock and borrow $19.51 (= $20.00 / 1.025) in order to create a synthetic call option with a strike price of $110 and 1 year until expiration. c. Since Ken pays $25.00 to purchase 1/4 of a share and borrows $19.51, the total cost of the synthetic call option is $5.49 (= $25.00 - $19.51). This is exactly the same price that Ken would pay for an actual call option. Since an actual call option and a synthetic call option provide Ken with identical payoff structures, he should not expect to pay more for one than the other. 23.17 a. Maverick would be interested in purchasing a call option on the price of gold with a strike price of $375 per ounce and 3 months until expiration. This option will compensate Maverick for any increases in the price of gold above the strike price and places a cap on the amount the firm must pay for gold at $375 per ounce. b. In order to solve a problem using the two-state option model, first draw a price tree containing both the current price of the underlying asset and the underlying asset’s possible values at the time of the option’s expiration. Next, draw a similar tree for the option, designating what its value will be at expiration given either of the 2 possible stock price movements. The price of gold is $350 per ounce today. If the price rises to $400, Maverick will exercise its call option for $375 and receive a payoff of $25 at expiration. If the price of gold falls to $325, Maverick will not exercise its call option, and the firm will receive no payoff at expiration. Price of Gold (per ounce) Maverick's Call Option with a Strike of 375 Today 3 months Today 3 months 400 25 = max(0, 400-375) 350 ? 325 0 = max(0, 325-375) If the price of gold rises, its return over the period is 14.29% [= (400/350) – 1]. If the price of gold falls, its return over the period is -7.14% [= (325/350) –1]. Use the following expression to determine the risk-neutral probability of a rise in the price of gold: Risk-Free Rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall) = (ProbabilityRise)(ReturnRise) + (1 - ProbabilityRise)(ReturnFall) The risk-free rate over the next three months must be used in the order to match the timing of the expected price change. Since the risk-free rate per annum is 16.99%, the risk-free rate over the next three months is 4% [= (1.1699)1/4 –1]. 0.04 = (ProbabilityRise)(0.1429) + (1 – ProbabilityRise)(-0.0714) ProbabilityRise = 0.5198 ProbabilityFall = 1 - ProbabilityRise = 1 – 0.5198 = 0.4802 The risk-neutral probability of a rise in the price of gold is 51.98%, and the risk- neutral probability of a fall in the price of gold is 48.02%. Using these risk-neutral probabilities, determine the expected payoff to Maverick’s call option at expiration. Expected Payoff at Expiration = (.5198)($25) + (.4802)($0) = $13.00 Since this payoff occurs 3 months from now, it must be discounted at the risk-free rate of 16.99% per annum in order to find its present value: PV(Expected Payoff at Expiration) = [$13.00 / (1.1699)1/4 ] = $12.50 Therefore, given the information Maverick has about gold’s price movements over the next three months, a European call option with a strike price of $375 and three months until expiration is worth $12.50 today. b. Yes, there is a way for Maverick to create a synthetic call option with identical payoffs to the call option described above. In order to do this, Maverick will need to buy gold and borrow at the risk-free rate. The amount of gold that Maverick should buy is based on the delta of the option, where delta is defined as: Delta = (Swing of option) / (Swing of price of gold) Since the call option will be worth $25 if the price of gold rises and $0 if it falls, the swing of the call option is 25 (= 25 – 0). Since the price of gold will either be $400 or $325 at the time of the option’s expiration, the swing of the price of gold is 75 (= 400 - 325). Given this information: Delta = (Swing of option) / (Swing of price of gold) = (25 / 75) = 1/3 Therefore, Maverick’s first step in creating a synthetic call option is to buy 1/3 of an ounce of gold. Since gold currently sells for $350 per ounce, Maverick must pay $116.67 (= 1/3 * $350) to purchase 1/3 of an ounce of gold. In order to determine the amount that Maverick should borrow, compare the payoff of the actual call option to the payoff of delta shares at expiration. Call Option If the price of gold rises to $400: payoff = $25 If the price of gold falls to $325: payoff = $0 Delta Shares If the price of gold rises to $400: payoff = (1/3)($400) = $133.33 If the price of gold falls to $325: payoff = (1/3)($325) = $108.33 Maverick would like the payoff of his synthetic call position to be identical to the payoff of an actual call option. However, buying 1/3 of a share leaves him exactly $108.33 above the payoff at expiration, regardless of whether the price of gold rises or falls. In order to decrease the firm’s payoff at expiration by $108.33, Maverick should borrow the present value of $108.33 now. In three months, the firm must pay $108.33, which will decrease its payoffs so that they exactly match those of an actual call option. Maverick should buy 1/3 of an ounce of gold and borrow $104.17 [= $108.33 / (1.1699)1/4] in order to create a synthetic call option with a strike price of $375 and 3 months until expiration. d. Since Maverick pays $116.67 in order to purchase gold and borrows $104.17, the total cost of the synthetic call option is $12.50 (= $116.67 – $104.17). This is exactly the same price that Maverick would pay for an actual call option. Since an actual call option and a synthetic call option provide Maverick with identical payoff structures, the firm should not expect to pay more for one than the other. 23.22 a. The inputs to the Black-Scholes model are the current price of the underlying asset (S), the strike price of the option (K), the time to expiration of the option in fractions of a year (t), the variance of the underlying asset (2), and the continuously- compounded risk-free interest rate (r). In this problem, the inputs are: S = $60 2 = 0.36 K =$30 r = 0.03 t = 0.25 After identifying the inputs, solve for d1 and d2: d1 = [ln(S/K) + (r + ½2)(t) ] / (2t)1/2 = [ln(60/30) + {0.03 + ½(0.36)}(0.25) ] / (0.36*0.25)1/2 = 2.4855 d2 = d1 - (2t)1/2 = 2.4855- (0.36*.25)1/2 = 2.1855 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(2.4855) = 0.9935 N(d2) = N(2.1855) = 0.9856 According 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-rtN(d2) = (60)(0.9935) – (30)e-(0.03)(0.25) (0.9856) = $30.26 The Black-Scholes Price of the call option is $30.26. b. Put-Call Parity implies that the cost of a European call option (C) must equal the cost of a European put option with the same strike price and time to expiration (P) plus the current stock price (S) minus the present value of the strike price [PV(K)]. In this problem: C = $30.26 S = $60 PV(K) = $30 / e(.03*0.25) = $29.78 Rearranging the Put-Call Parity formula: P = C – S + PV(K) = $30.26 - $60 + $29.78 = $0.04 Therefore, Put-Call Parity implies that the Black-Scholes price of a European put option with a strike price of $30 and 3 months until expiration should be $0.04. 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 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 $8, the current price of General Modem’s common stock. One would never pay more than $8 to receive the right to purchase a share of the company’s stock if the firm’s stock were only worth $8. b. If General Modem’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 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 Modem’s stock when he could purchase a share outright for $12. 25.6 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 firm’s 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 firm’s 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 Ricketti’s 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 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 value of each warrant ($2 = $17 - $15), the stockholders gain and the warrant holders lose. 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 Hannon’s common stock, the conversion ratio of the convertible bonds is 28. 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 Hannon’s 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 Hannon’s 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 Hannon’s 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 Hannon’s 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 bond’s 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 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. b. 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 bond’s 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. 25.17 a. The straight value of a convertible bond is the bond’s 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 bond’s 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 A’s 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: ($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.

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