FIN320 problem set #2

Fin 320: Green Practice Problems: II.B 3 4. In response to complaints about high prices, a grocery store chain runs the following advertising campaign: “If you pay your child 75 cents to go buy $25 dollars of groceries, then your child makes twice as much on the trip as we do”. You’ve collected the following information from the grocery chain’s financial statements: • Sales: $225 million • Net income: $3.375 million • Total Assets: $40 million • Total debt: $17 million Evaluate the grocery chain’s claim. What is the basis or the statement? Is this claim misleading? Why or why not?

Fin 320: Green Practice Problems: II.B 6 3. The financial statements for The English Leather Company are reproduced below. You are given the following information about the coming year. Prepare Pro Forma statements for the company for next year. • Sales are expected to increase by 10% and COGS will be 65% of sales • GA&S will remain at the same level and the cash level will remain the same. • Depreciation expense will be $91,000 and the company will add $170,000 of new assets and dispose of (sell) assets with book values of $30,000. The selling price of the sold assets will be equal to book value and therefore, there will be no tax effects. • The tax rate will be 40% • The company paid $500,000 in dividends in 2010 and will pay the same dividend next year. • AR will be reduced to 35 days’ worth of sales. Inventory management improvements will reduce inventories to 55 days of COGS. Accounts payable will remain at 30 days of COGS. The company uses a 360 day year in its calculations. • Accruals will be 1% of sales • The company is not allowed to borrow any more by way of long-term debt. Instead, any additional borrowing will come out of notes payable. The interest rate on long-term debt is 8%, while the interest rate on notes payable is 10%. Balance Sheets (in $000s) Income Statement Today Today Assets Cash 100 Sales 2200 Accounts receivable 275 Cost of goods sold 1452 Inventories 242 Gross margin 748 Total Current Assets 617 GA&S 470 Depreciation expense 83 Gross Fixed Assets 1200 Earnings before interest 195 Less: Acc. Depreciation 525 and taxes (EBIT) Net Fixed Assets 675 Interest expense 60 Earnings before taxes 135 Total Assets 1292 Taxes (40%) 54 Net income 81 Liabilities & Owners' Equity Accounts Payable 121 Notes Payable 100 Accruals 22 Total Current Liabilities Long-term Debt (8%) 625 Common Stock 120 Retained Earnings 304 Total Equity 424 Total Liabilities & Equity 1292

Fin 320: Green Practice Problems: III.A 9 III.A Time Value of Money From RWJ Chapter 5: Problems 9, 16, 17, 18, 20 Additional Problems 1. Bradbury Inc. is offering to sell you equipment with your choice of two payment schedules: $20,000 now plus $10,000 in each of the next two years and a final payment of $5,000 a year after that, or $30,000 now and $12,000 next year. If your borrowing rate is 7%, which deal should you take? If Bradbury is indifferent between these two deals (which they must be to offer them both), what does this say about their borrowing rate? 2. Find the amount to which $200 will grow under each of the following conditions. a) 12 % compounded annually for 5 years b) 12% compounded semiannually for 5 years c) 12% compounded quarterly for 5 years d) 12% compounded monthly for 1 year 3. First National City State Bank pays 9% interest, compounded annually, on deposits. Second National City State Bank pays 8% interest, compounded quarterly. In which bank would you prefer to deposit your money? 4. Your broker offers to sell you a note for $14,500 that will pay $2250 per year for 10 years. If you buy the note, what rate of interest will you be earning? 5. You have taken out a loan for $12,000 at an annual interest rate of 8%. If you repay the loan $1800 per year, how long will it take you to repay the loan? 6. How many years will it take to double an investment at the following interest rates? a) 7% b) 12% c) 100% 7. Find the present value of $200 due in five years at the following rates: a) 12% compounded semiannually b) 12% compounded quarterly c) 12% compounded monthly 8. You are planning a big party that will require $20,000. If you put $5,000 in your account today and you can earn an effective monthly rate of 1%, how many years before you can have your party? 9. If the nominal annual rate is 24% and your bank account has monthly compounding: a) What is the effective monthly rate? b) What is the effective quarterly rate? c) What is the effective 2-year rate?

Fin 320: Green Practice Problems: III.B 12 You can only afford to save $4,000 per year for the first ten years (first savings payment made one year from today). You expect that for the foreseeable future, your savings will earn 6 percent interest per year, compounded annually. a) Draw a time line indicating the cash flows (abbreviate where appropriate). b) Assuming you save the same amount each year in years 11 to 20, what must you save annually from years 11 to 20 to meet his objectives. 13. You have your choice of two investment accounts, Investment A is a seven year annuity that features end of month $100 payments and has an interest rate of 17 percent compounded monthly. Investment B is a 12 percent continuously compounded lump sum investment, also good for seven years. How much money would you need to invest in B today so it is worth as much as Investment A seven years from now? 14. This is a classic retirement problem. A time line will help in solving it. Your friend is celebrating her 35th birthday today and wants to start saving for her anticipated retirement at age 65. She wants to be able to withdraw $105,000 from her savings account on each birthday for 20 years following her retirement; the first withdrawal will be on her 66th birthday. Your friend intends to invest her money in the local credit union, which offers 7 percent interest per year. She wants to make equal annual payments on each birthday into the account established at the credit union for her retirement fund. a) If she starts making these deposits on her 36th birthday and continues to make deposits until she is 65 (the last deposit will be on her 65th birthday), what amount must she deposit annually to be able to make the desired withdrawals at retirement? b) Suppose your friend has just inherited a large sum of money. Rather than making equal annual payments, she has decided to make one lump sum payment on her 35th birthday to cover her retirement needs. What amount does she have to deposit? c) Suppose your friend’s employer will contribute $3,500 to the account every year as part of the company’s profit -sharing plan. In addition, your friend expects a $175,000 distribution from a family trust fund on her 55th birthday, which she will also put into the retirement account. What amount must she deposit annually now to be able to make the desired withdrawals at retirement?

Fin 320: Green Practice Problems: V 15 V: Security Valuation (Stocks) From RWJ Chapter 8: Problems 14, 16, 17, 18, 19, 20, 21, 22 Additional Problems 1. Assume the required return on Danish Designs Inc. stock is 16% and that the last dividend was $2.00. You expect 8% growth in dividends for the next 3 years after which you expect the dividends to grow constantly at 4%. What do you think is a fair price for Danish Design stock? 2. High Fly Corp. is expected to pay a dividend of $2.00 at the end of the first year. For the next three years after the first year, dividends will grow at 15%. The company then wishes to make certain capital investments, so the dividends will remain constant for two years. After the two years, dividends will again grow at a constant rate of 20% into the foreseeable future. The required rate for the stock is 22% pa. a) Draw a time line of these cash flows. b) What will the stock price be at the end of six years? c) What is the stock price today? d) What is the annually compounded growth rate of the stock price (not including dividends) per annum over the first six years? 3. The Feringi Corporation is expanding rapidly. Its dividend growth rate for the coming year is projected at 25%. This dividend growth rate will decline by 5 percentage points per year until it reaches the industry average of 5 percent. Once it reaches 5 percent, it will stay there indefinitely. The most recent dividend was $8.50 per share, and the market requires a return of 16 percent on investments such as this one. Calculate Feringi's share price. 4. The NYT Corporation will not pay dividends for the next 5 years. At the end of the 5 th year it will pay a dividend of $0.25. It will continue to pay the same dividend at the end of every year for another 3 years (i.e. the dividend at the end of the 8 th year will also be the same). After the 8 th year, dividends will grow at a constant rate of 8%. If the required rate of return on the stock is 14%, calculate the stock price today. 5. Remy Inc is a start-up pharmaceutical company that has 1 million shares outstanding. Since it is a start-up company, it is not expected to make any profit in the first two years of its life. At the end of the third year, however, the company expects net income to be $10 million. The chief financial office of the company has predicted that after the third year, net income will grow from this $10 million mark by 20% per annum for two years. The chief financial officer of the company refers to this 20% growth period as the “high growth phase.” Subsequent to this high growth phase, net income is expected to grow at a more modest rate of 6% per annum. The dividend payout ratio is 50% (and expected to remain at that level) and the required rate of return on the stock is 22%.

Fin 320: Green Practice Problems: VI 18 5. Consider the stock of Midterm Inc. It is expected to pay an annual per share dividend of $7 for the next 3 years, starting a year from today. After year three, the dividend will grow at a constant annual rate of 7% forever (i.e., four years from now the dividend is expected to be $7´1.07 = $7.49). Suppose the riskless rate of return is 7%, the  of Midterm stock is 0.8, and the market risk premium is 8.75%. a) What is Midterm's stock price today? b) What will be Midterm's stock price a year from today? c) What is Midterm's expected dividend yield over the first year? d) What is Midterm's expected capital gain yield over the first year? e) What is the total yield (dividend plus capital gain) over the first year? Why should this answer not surprise you? 6. Consider the stock of Adesh Inc. A dividend of $7 was just paid. Because of the weak economy, Mr. Adesh expects the dividend to remain at this level for the next two years. From that point on dividends are expected to grow at a constant rate of 7% per year for ever. The beta  of Adesh's rates of return is 0.5, the risk-free rate of return is 7%, and the market risk premium is 12%. a) What is Adesh stock price today? b) What will be Adesh stock price a year from today? c) What is the expected dividend yield for the first year? d) What is the expected capital gain yield for the first year? 7. You have a finicky Aunt who wants you to invest her money. However, she is not willing to allow you to choose a stock as you please, but rather, sets the condition that the annual return on the stock should be at least 10%. You have the following two options before you. • Stock X: The dividend this year for the stock is $3.00. Dividends are expected to grow at a constant rate of 5% every year, and the price of the stock today is $60. • Stock Y: The dividend next year will be $3.10. Dividends are expected to grow at a constant rate of 6% every year, and the price of the stock today is $90. a) Which of the above stocks, if any, satisfy your Aunt's criteria? b) Your Aunt attends a finance course at the local University, and learns of such terms as CAPM, Beta. She now inserts the additional criteria that any stock in her portfolio should not be overpriced according to the CAPM. Do X and Y satisfy this new criteria? Beta for X = 1.1 Beta for Y = 0.4 Risk free rate = 5%

Fin 320: Green Practice Problems: VIII 21 VIII. Capital Budgeting From RWJ Chapter 9: Problems 17, 25 Chapter 10: Problems 20 Chapter 11: Problem 25 (part b) Additional Problems 1. Copernicus Lines is considering the purchase of a new bulk carrier for $8 million. The forecast revenues are $5 million a year and operating costs are $4 million. A major refit costing $2 million will be required after both the fifth and tenth years. After 15 years, the ship is expected to be sold for scrap at $1.5 million. If the discount rate is 8 percent, what is the ship's NPV? Should you buy the ship? 2. Calculate the NPV and IRR for each of the following investments. The opportunity cost of capital is 20 percent for all four investments. ___________________________________________________________ Investment Initial Cashflow Cash Flow Next Year ___________________________________________________________ A -10,000 +20,000 B -5,000 +12,000 C -5,000 +5,500 D -2,000 +5,000 ___________________________________________________________ a) Which project(s) would you take? Which investment is most valuable? Which has the highest rate of return? b) Suppose the investments were mutually exclusive (you may only take one.) Which project would you take and why? 3. Consider the following projects: ___________________________________________________________________ Cashflows Project C 0 C 1 C 2 C 3 C 4 C 5 ___________________________________________________________________ A -1,000 +1,000 0 0 0 0 B -2,000 +1,000 +1,000 +1,000 +1,000 0 C -3,000 0 +1,000 +1,000 +2,000 +2,000 a) If the opportunity cost of capital is 10 percent, which projects have a positive NPV? b) Calculate the payback period for each project. c) Which projects would a firm using the payback rule accept if the cutoff period were 3 years?

Fin 320: Green Practice Problems: VIII 22 4. The president of Giant Enterprises has to make a choice between two possible investments: ___________________________________________ Cashflows Project C 0 C 1 C 2 __________________________________________ A -4,000 +2,410 +2,930 B -2,000 +1,130 +1,720 __________________________________________ a) Calculate the NPV and IRR or each project assuming the opportunity cost of capital is 9 percent. b) Mr. Clops is tempted to take B, which has the higher IRR. Explain in simple terms why this is not the correct procedure. 5. A factory costs $800,000. You expect it will generate cash inflows (net of operating costs) of $170,000 a year for 10 years. If the opportunity cost of capital is 14 percent, what is the net present value of the factory? If you were to sell the factory in 5 years, what would be a fair price? 6. The Financial Analysts Journal currently offers the following subscription options: 1 year, $48; 2 years, $85; 3 years, $108. If the interest rate is 10% and under each alternative you must pay your subscription at the beginning of the time period, what is your optimal strategy assuming you intend to be a permanent subscriber? 7. The XYZ company is trying to choose between two machines. Machine A lasts 3 years, costs $20,000, produces annual net revenues of $11,000 and requires a $2,000 payment each year for upkeep. Machine B lasts 7 years, costs $45,000, produces annual net revenues of $12,000, and requires a $1,500 per year upkeep payment. Revenues are received and upkeep payments are made at the end of the year in which they occur. Both machines are depreciated straight-line using the true life of the machine. The tax rate is 0.40 and the opportunity cost of capital is 8%. Which machine should be chosen? Assume that the tax law, prices, and upkeep costs will never change and that the company will produce forever. 8. Western Industries is a lumber company which is considering replacing its plywood gluing machine for two reasons. First, the current machinery would need some rather expensive new parts to be compatible with recent changes in the companies production methods and second, the new machine would allow the firm to produce certain new types of plywood and therefore increase sales and profits. The following information is available:

Fin 320: Green Practice Problems: VIII 23 Old Machine: • The new parts will cost $150,000 • The cost of the parts will be depreciated straight-line over the five year remaining life of the old machine. • The old machine (in its current condition) can be sold today for $20,000 but will have no salvage value in five years. • The old machine has been fully depreciated (i.e. the book value is zero). New Machine: • The new machine will cost $700,000 and last seven years. • The new machine will be depreciated straight-line over seven years. • The firm paid a consultant $10,000 (tax deductible) to determine exactly what specifications the new machinery has to meet in order to be compatible with the changes in the firm's production methods. This had to be done before the cost of the new machinery could be determined. • The new machine will increase net taxable income before depreciation to the firm by $140,000 a year over its seven year life. Neither machine is expected to have a salvage value at the end of its life. The expected income from manufacturing the company's existing products (which will be the same under each alternative) more than justifies choosing one alternative or the other. The required rate of return for investments of this type is 14% and the firm is in a 30% tax bracket. a) What is the net present value of the cashflows associated with buying the new parts for the existing machine? b) What is the net present value of the cashflows associated with replacing the existing machine? c) Which alternative should be chosen − you must address the fact that these projects have different lives. 9. The management of Fullerton Fresh Corp., a leading producer of fruit drinks, is considering the addition of a new product − fresh avocado-citron- kiwi juice (“Ack” Juice). A new juicing machine would be required to produce this new juice. The machine costs $300,000, has a depreciable life of 4 years and would be depreciated on a straight-line basis to a zero book value; Fullerton management believes that the machine could be sold for $25,000 at the end of the four year project. The new juicing machine would be placed in an unused portion of the existing plant; this portion of the plant was refurbished last year at a cost of $100,000. If the project is undertaken, inventories would have to be increased by $10,000 at the time of the initial investment. Sales of Ack Juice are expected to be 200,000 cartons per year at a price of $2.00 per carton, but $1.50 per carton would be needed to cover costs of the product. It’s estimated that sales of Ack Juice will result in a $10,000 per year reduction in sales of Fullerton’s “Limbaugh’s Sour Grape Juice”. Fullerton’s cost of

Fin 320: Green Practice Problems: VIII 24 raising capital has been estimated at 10%; the firm’s tax rate is 40%. Should Full erton undertake this new project? Show all calculations necessary to arrive at your decision and explain why Fullerton should or should not take the project. 10. As a hobby, you have developed and patented a new auto anti-theft device called the Wonderbar. If you were to undertake the manufacture of the Wonderbar, the initial costs would amount to $200,000, which your Uncle Ben has agreed to loan you provided that you repay him at the end of 20 years (he has generously refused to accept any interest on the loan). You estimate that the net benefit you would receive from the Wonderbar would be $20,000 per year for the first 20 years, and then $10,000 per year for the following 20 years (years 21-40), after which time the Wonderbar will made obsolete by other anti-theft technologies. At that time (the end of year 40), you wish to retire and buy a dream house for $1,000,000. During your golden years of retirement, you wish to receive $150,000 per year (the first payment at the end of year 41). You estimate that your “retirement period” will last 25 years. The nominal rate is 6.78% per year and interest is compounded monthly. Assume all cash flows occur at year-end. Will the Wonderbar allow you the wonderful retirement you desire? Show a timeline of the cash flows and all calculations used to arrive at your answer. 11. Jordan Enterprises needs someone to supply them with 70,000 cartons of machine screws to support their manufa cturing needs over the next three years, and you’ve decided to bid on the contract. It will cost you $360,000 to install the necessary equipment to start production; you’ll depreciate this cost straight -line to zero over the project’s life. You estimate th at in three years this equipment can be salvaged for $50,000. Your fixed production costs will be $100,000 per year, and your variable production costs will be $5.75 per carton. You also need an initial investment in net working capital of $40,000. If your tax rate is 35 percent and you require a 15 percent return on your investment, what bid price will you submit?

Fin 320: Green Practice Problems: IX 25 IX. The Cost of Capital From RWJ Chapter 14: Problems 15, 16, 30 Additional Problems 1. You have obtained the following information on Romeo Industries capital structure: Debt: Romeo has 10% annual bonds with a face value of $2,000,000 outstanding. The bonds mature in 6 years and are currently trading for $2,234,692. Preferred stock: 10,000 shares of $50 face value, 10% preferred stock trading at $55 per share. Common stock: 50,000 shares of stock which are trading at $75 per share. The next dividend is expected to be $4.50 and dividends are expected to grow at 6% from then on. Assume there are no retained earnings. Calculate the weighted average cost of capital for Romeo Industries assuming Romeo is in a 30% tax bracket. 2. The manager of Easymoney wants to invest in a project to manufacture exam crib sheets. He knows all the potential costs and revenues involved, but isn’t sure of how to discount the relevant cash flows. He approaches you and asks you to calculate the weighted average cost of capital. He gives you the following information for the calculations: • The firm has 60,000 shares of common stock. Last year, each share was trading at $25. The current price of the stock is $30. • The firm has 10,000 bonds outstanding, each of a face value $1000, a coupon rate of 10% and a maturity of 1 year. The bonds did not trade yesterday so you have no current price for these bonds. However, the firm’s investment bankers have assured the manager that if new bonds with similar characteristics were issued today, with a 1 year maturity, they would fetch $956.52 in the market. • The firm also borrowed $0.7 million from Citibank at a variable rate of 0.25% above the prime lending rate. However, Citibank recently sold the loan to Chase Manhattan for $0.5 million (so that Chase Manhattan is now the lender). • The firm has outstanding preferred stock worth $3 million. Each preferred stock is trading at $9, and is entitled to an annual dividend of $1.00. • The prime rate of return is 12%. The stock has a beta of 1.2, the risk free rate is 5%, and the historic risk premium is 6.8%. The firm has a marginal tax rate of 30%.

Fin 320: Green Solutions: II.A Solutions 26 II.A Financial Statements and Analysis RWJ 2.07 Change in NWC = ending NWC – beginning NWC = (3 ending CL) – (beginning CA – beginning CL) = ($5,360 – 2,970) – ($4,810 – 2,230) = = $2,390 - $2,580 = -$190 RWJ 2.11 To find the book value of current assets, we use: NWC = CA – CL. Rearranging to solve for current assets, we get: CA = NWC + CL = $235,000 + 895,000 = $1,130,000 Book value CA = $1,130,000 Book value NFA = $3,400,000 Book value assets = $4,530,000 The market value of NWC and fixed assets is given, so: Mkt value of NFA = $5,100,000 1 = $1,150,000 RWJ 2.12 To find the OCF, we first calculate net income. Income Statement 1A= $305,000 Costs 176,000 Other expenses 8,900 Depreciation 18,700 EBIT $101,400 Interest 12,900 Taxable income $88,500 Taxes 23,345 Net income $65,155 Dividends $19,500 Additions to RE $45,655 a. OCF = EBIT + Depreciation – Taxes = $101,400 + 18,700 – 23345, = $96,755 b. CFC = Interest – Net new LTD = $12,900 – ( – 4,900) = $17,800 Note that the net new long-term debt is negative because the company repaid part of its long- term debt. c. CFS = Dividends – Net new equity = $19,500 – 6,400 = $13,100 d. We know that CFA = CFC + CFS, so: CFA = $17,800 + 13,100 = $30,900 CFA is also equal to OCF – Net capital spending – Change in NWC. We already know OCF. Net capital spending is equal to:

Fin 320: Green Solutions: II.A Solutions 27 = Increase in NFA + Depreciation = $46,000 + 18,700 = $64,700 Now we can use: CFA = OCF – Net capital spending – Change in NWC $30,900 = $96,755 – $64,700 – Change in NWC Solving for the change in NWC gives $1,155, meaning the company increased its NWC by $1,155. RWJ 2.19 a. Total assets 2022 = $1,206 + 4,973 = $6,179 Total liabilities 2022 = $482 + 2,628 = $3,110 Owners’ e quity 2022 = $6,179 – 3,110 = $3,069 Total assets 2023 = $1,307 + 5,988 = $7,295 Total liabilities 2023 = $541 + 2,795 = $3,336 Owners’ equity 2023 = $7,295 – 3,336 = $3,959 b. NWC 2022 = CA17 – CL17 = $1206 – 482 = $724 NWC 2023 = CA18 – CL18 = $1307 – 541 = $766 Change in NWC = NWC18 – NWC17 = $766 – 724 = $42 c. We can calculate net capital spending as: Net capital spending = Net fixed assets 2023 – Net fixed assets 2022 + Depreciation Net capital spending = $5,988 – 4,973 + 1,363 = $2,378 So, the company had a net capital spending cash flow of $2,378. We also know that net capital spending is: Net capital spending = Fixed assets bought – Fixed assets sold $2,378 = $2,496 – Fixed assets sold Fixed assets sold = $2,496 – 2,378 = $120 To calculate the cash flow from assets, we must first calculate the operating cash flow. The income statement is: Income Statement Sales $ 15,301.00 Costs 7,135.00 Depreciation expense 1,363.00 EBIT $6,803.00 Interest expense 388.00 EBT $6,415.00 Taxes (21%) 1,347.15 Net income $5,067.85 So, the operating cash flow is: OCF = EBIT + Depreciation – Taxes = $6,803 + 1,363 – 1,347.15 = $6,818.85

Fin 320: Green Solutions: II.A Solutions 28 And the cash flow from assets is: Cash flow from assets = OCF – Change in NWC – Net capital spending. = $6,818.85 – $42 – $2,378= $4,477.85 d. Net new borrowing = LTD18 – LTD17 = $2,795 – 2,628 = $167 Cash flow to creditors = Interest – Net new LTD = $388 – 167 = $221 Net new borrowing = $167 = Debt issued – Debt retired Debt retired = 504 – 167 = $337 RWJ 3.18 Profit margin = net income / sales Total Asset Turnover = sales / total assets Equity Multiplier = 1 + Debt / Equity This is a multistep problem involving several ratios. The ratios given are all part of the DuPont Identity. The only DuPont Identity ration not given is the profit margin. If we know the profit margin we can find the net income since sales are given. So we begin with the DuPont Identity: ROE = PM x TAT x EM ROE = 0.11 = (PM)(TAT)(EM) = (PM)(S / TA)(1 + D/E) Solving the DuPont Identity for profit margin, we get: 0.11 = (PM)(S / TA)(1 + D/E) 0.11 = (PM)($6,183/ $2,974)(1 + .57) 0.11= (PM)(3.2641) PM = .0337 Now that we have the profit margin, we can use this number and the given sales figure to solve for net income: PM = .0337 = NI / S NI = .0337($6,183) = $208.37

Fin 320: Green Solutions: II.A Solutions 29 RWJ 3.23 This problem requires you to work backward through the income statement. First, recognize that Net income = (1- t ) EBT. Plugging in the numbers given and solving for EBT, we get: EBT = $16,481 / (1 – 0.21) = $20,862.93 Now, we can add interest to EBT to get EBIT as follows: EBIT = EBT + Interest paid = $20,862.03 + $3,681 = $24,543.03 To get EBITD (earnings before interest, taxes, and depreciation), the numerator in the cash coverage ratio, add depreciation to EBIT: EBITD = EBIT + Depreciation = $24,543.03+ $4,385= $28,928.03 Now, simply plug the numbers into the cash coverage ratio and calculate: Cash coverage ratio = EBITD / Interest = $28,928.03/ $4,385 = 6.60 times Additional Problems 1. Below are the financial statements for Lexon Industries. 2. a) b) The Du Pont formula is ROE = PM  Total asset turnover  Equity multiplier 530 Current ratio = = 2.41 220 290 + 40 Quick ratio = = 1.50 220 2,000 Inventory turnover = = 10 200 360 Days sales outstanding = = 41.76 2,500/290 2,500 Fixed asset turnover = = 2.38 1,050 2,500 Total assets turnover = = 1.58 1,580 1,120 Debt ratio = = .71 = 71% 1,580 128 Profit margin = = .0512 = 5.12% 2,500 128 Return on assets = = .081 = 8.1% 1,580 128 Return on equity = = .278 = 27.8% 300 +160 Sales 2,500 Cash 40 AP 220 COGS 2,000 Inv 200 CL 220 GM 500 AR 290 Ltd 900 S&A 100 CA 530 CS 300 Depr 150 PPE 1050 RE 160 EBIT 250 TA 1580 TL&OE 1580 Int 90 EBIT 160 Tax 32 NI 128 EPS 3.2

Fin 320: Green Solutions: II.A Solutions 30 For Lexon: 27.8% = 5.12%  1.58  3.43 For Bluffard 16.0% = 6.90%  1.20  1.93 These two firms have dramatically different returns on equity. However, the profit margin and asset turnovers are comparable. In fact, the return on assets (PM  Total asset turnover) is almost identical: 8.09% and 8.28%, respectively. The difference in ROE is primarily due to Lexon’s higher leverage. 3. (a) Ratios Selected Ratios 2020 2021 2022 Current Ratio 2.08 2.20 1.97 Quick Ratio 0.77 1.09 1.02 Average collection period 45.74 80.17 91.25 Fixed asset turnover 5.89 5.61 5.85 Total asset turnover 1.76 1.52 1.46 Times interest earned 4.48 2.96 1.71 ROE 22.9% 16.0% 6.6% (b) Liquidity − Essentially unchanged; current ratio has declined slightly, quick ratio has increased somewhat. Leverage − Has increased. Debt levels relative to assets and equity are up somewhat. The more troubling aspect of higher leverage is the substantial reduction in interest coverage, reflecting both more interest expense (due to more debt) and a significant decline in income. Profitability − Has plummeted. ROA and ROE have dropped dramatically. The problem appears to be a much lower profit margin (both COGS and GSA have risen relative to sales) combined with lower asset turnover. 4. Child: Profit Margin = $0.75 / $25=3% Store: Profit Margin = NI/S = $3.375 million/ $225 million = 1.5% The advertisement is referring (correctly) to the store’s profit margin, but a more appropriate earnings measure for the firm’s owners is the return on equity. ROE = NI/TE = NI/ (TA-TD) = $3.375 million/ ($40 million-$17 million) = 14.67%.

Fin 320: Green Solutions: II.B Solutions 31 II.B Financial Statements and Analysis: (Pro Formas) RWJ 4.19 We are given the profit margin. Remember that: ROA=PM(TAT) We can calculate ROA from the internal growth rate equation, and then use the ROA in this equation to find the total asset turnover. The retention ratio is: b = 1 – 0.25 b = 0.75 Using the internal growth rate equation and solving for ROA, we get: Internal growth rate = (ROA × b) / [1 – (ROA × b)] 0.071 = [ROA(0.75)] / [1 – ROA(0.75)] 0.071 - 0.05325ROA = 0.75ROA 0.071= 0.8033ROA ROA = 0.08839 or 8.83% Now we can plug ROA and PM into the equation we began with and solve for TAT, we get: ROA = (PM)(TAT) 0.0883 = (0.065)(TAT) TAT = (0.0883) / 0.065 TAT = 1.36 times RWJ 4.20 We should begin by calculating the D/E ratio. We calculate the D/E ratio as follows: Total debt ratio = 0.35 = TD / TA Inverting both sides we get: 1 / 0.35 = TA / TD Next, we need to recognize that TA / TD = 1 + TE / TD Substituting this into the previous equation, we get: 1 / 0.35 = 1 + TE /TD Subtract 1 (one) from both sides and inverting again, we get: D/E = 1 / [(1 / 0.35) – 1] D/E = 0.538 With the D/E ratio, we can calculate the EM and solve for ROE using the DuPont identity: ROE = (PM)(TAT)(EM) ROE = (0.063)(1.75)(1 + 0.538) ROE = 0.1696 or 16.96% Now we can calculate the retention ratio as: b = 1 – 0.30 b = 0.70 Finally, putting all the numbers we have calculated into the sustainable growth rate equation, we get: Sustainable growth rate = (ROE × b) / [1 – (ROE × b)] Sustainable growth rate = [0.1696(0.70)] / [1 – 0.1696(0.70)] Sustainable growth rate = 0.1347 or 13.47%

Fin 320: Green Solutions: II.B Solutions 32 RWJ 4.21 To calculate the sustainable growth rate, we first must calculate the retention ratio and ROE. The retention ratio is: b = 1 – $9,400 / $17,300 b = 1 - 0.5434 b = 0.45665 And the ROE is: ROE = $17,300 / $59,000 ROE = 0.2932 or 29.32% So, the sustainable growth rate is: Sustainable growth rate = (ROE × b) / [1 – (ROE × b)] Sustainable growth rate = [0.2932(0.4567)] / [1 – 0.2932(0.4567)] Sustainable growth rate = 0.15460 or 15.46% If the company grows at the sustainable growth rate, the new level of total assets is: New TA = 1.1546($77,000 + 59,000) = $157,025.44 To find the new level of debt in the company’s balance sheet, we take the percentage of debt in the capital structure times the new level of total assets. The additional borrowing will be the new level of debt minus the current level of debt. So: New TD = [D / (D + E)](TA) New TD = [$77,000 / ($77,000 + 59,000)]( $157,025.44) New TD = $88,904.22 And the additional borrowing will be: Additional borrowing = $88,904.11 – 77,000 Additional borrowing = $11,904.11 The growth rate that can be supported with no outside financing is the internal growth rate. To calculate the internal growth rate, we first need the ROA, which is: ROA = $17,300 / ($77,000 + 59,000) ROA = 0.1272 or 12.72% This means the internal growth rate is: Internal growth rate = (ROA × b) / [1 – (ROA × b)] Internal growth rate = [0.1272(0.45665)] / [1 – 0.1272(0.45665)] Internal growth rate = 0.0617 or 6.17%

Fin 320: Green Solutions: II.B Solutions 33 RWJ 4.24 Income Statement 2022 2023 Computations Sales $980,760 $1,176,912 980,760 x 1.2 (sales grow by 20%) Costs 792,960 951,552 792,960x 1.2 (grow with sales) Other Expenses 20,060 24,072 20,060 x 1.2 (grow with sales) EBIT 167,740 201,288 $1, 176,912 − 951,552 – 24,072 Interest 14,740 14,740 Assumed to be constant Taxable Inc. 153,000 186,548 201,288 – 14,740 Taxes (21%) 32,130 39,175.08 186,548 x .21 Net Inc. 120,870 147,372.92 186,548-39,175.08 Dividends 39,250 47,856.27 (39,250/ 120,870) x 147,372.92 Addition to R.E. 81,620 99,516.65 147,372.92 – 47,856.27 Step 1: Fill out the income statement (in this case we can do the whole income statement since interest expense is assumed to be constant and doesn’t depend on how much we borrow) Step 2: Fill out asset side of the balance sheet and determine total assets. Step 3: Fill out the liability and Equity side of the balance sheet and compare Balance Sheet 2022 2023 Computations Assets Current Assets Cash $27,920 $33,504 $27,920 x 1.2 (grows with sales) AR 42,630 55,156 42,630 x 1.2 (grows with sales) Inv. 95,910 115,092 95,910 x 1.2 (grows with sales) Current Total 166,460 199,752 $33,504+ 55,156+ 115,092 Fixed 455,980 547,176 455,980x 1.2 Total 622,440 746,928 199,752 + 547,176 Liabilities and O.E. Current AP 71,720 86,064 71,720 x 1.2 (grows with sales) NP 17,620 17,620 Constant Current Total 89,340 103,684 86,064+ 17,620 Long Term Debt 170,000 170,000 Constant Owner’s Equity Common Stock 140,000 140,000 Constant R.E. 223,100 322,616.65 223,100+ 99,516.65 Total Equity 363,100 462,616.65 140,000+ 322,616.65 Total L & O.E. 622,440 736,300.65 103,684 + 170,000 + 462,616.65 EFN = Total assets – Total liabilities and equity EFN = $746,928-736,300.65 EFN = $10,627.35

Fin 320: Green Solutions: II.B Solutions 34 Additional Problems 1. Step 1: Fill out the income statement up to EBIT Step 2: Fill out asset side of the balance sheet and determine total assets. Step 3: Forecast as many of the liabilities as possible and equity and retained earnings if possible. These steps are easy Projected Income Statement Projected Balance Sheet Sales 2250 Cash 75 Cost of Goods Sold 1125 Accounts Receivable 450 General/Admin. Expenses 375 Inventory 750 Depreciation Expense 300 Net Fixed Assets 750 EBIT 450 Total Assets 2025 Interest Expense (10%) EBT Accounts Payable Taxes (34%) Long-term debt Net Income Common Stock & Paid- in Capital Dividends Retained Earnings Addition to Retained Earnings Total Liabilities & Owners’ Equity 2025 Step 4: Substitute all known quantities into the balance sheet equation and solve for the plug. They want us to raise money from debt, but they have a cap on how much debt we can use. We have two ways to proceed. ( ii ) We can use the balance sheet identity to solve for debt (assuming common stock stays the same) and see if we go over the debt limit (40% of TA), or ( ii ) we can use the maximum amount of debt allowed and see if it’s too much or not enough (i.e. do we still need to raise additional equity). Let’s try ( i ) first Balance Sheet Identity (for next year’s projections): Total Assets = AP + LTDebt + Common Stock + Retained Earnings: 2025 = 375 + LTDebt + 500 + 200 + Addition to Retained Earnings Thus, 950 = LTDebt + Addition to Retained Earnings  Addition to Retained Earnings: = (2/3)  Net Income = (2/3)  [EBIT − (0.10)  LTDebt]  (1 − tax rate) = (2/3)  [450 − (0.10)  LTDebt]  (0.66) Thus, Addition to Retained Earnings = 198 − (0.044)  LTDebt  Plugging  into  you get: 950 = LTDebt + 198 − (0.044)  LTDebt 752 = 0.956  LTdebt LTDebt = 752 / 0.956 = $786.61 So, the total Long-term Debt required to fund the growth would be $786.81. The restriction on long-term debt is no more than 40% of total assets, 0.4 x 2025 = $810, so we are safely under our debt limit. No additional common stock needs to be issued, since all of the growth can be funded by the additional debt amount.

Fin 320: Green Solutions: II.B Solutions 35 If we chose option ( ii ) to solve the problem, we would have used $810 for the long-term debt and solved for common stock to balance the balance sheet (after finding interest expense and then addition to retained earnings). In this case, common stock would be he plug. Solving it this way, we could find the common stock is lower than last year, which would suggest we raised too much debt and as a result bought back some stock. This isn’t what our bosses had in mind, so we would have to redo the problem solving i t using approach ( i ) above. In real life you can iterate this way to do pro forma analysis (using a computer, of course). On a test I would be clear so you only have to solve it one way. Step 5: Substitute in the values for debt and equity into the balance sheet Projected Income Statement Projected Balance Sheet Sales 2250 Cash 75 Cost of Goods Sold 1125 Accounts Receivable 450 General/Admin. Expenses 375 Inventory 750 Depreciation Expense 300 Net Fixed Assets 750 EBIT 450 Total Assets 2025 Interest Expense (10%) 78.66 EBT 371.34 Accounts Payable 375 Taxes (34%) 126.25 Long-term debt 786.61 Net Income 245.08 Common Stock & Paid-in Capital 500 Dividends 81.69 Retained Earnings 363.39 Addition to Retained Earnings 163.39 Total Liabilities & Owners’ Equity 2025 2. All accounts on the LHS of the balance sheet will increase by 20% except land and accounts receivable. Because of restrictions on DSR (it must be equal to 30 days), the firm’s A/R will not increase by 20% but rather will be: 30 = A/R  (150,000 / 365) A/R = 12,329 Therefore, total current assets will be 10,800 (cash) + 12,329 (A/R) + 34,800 (inv) = 57,929 total fixed assets will be 20,000 (land) + 50,400 (net ppe) = 70,400 total assets will be 57,929 + 70,400 = 128,329 A/P increase by 20% and are 12,000. Apply the current ratio restriction: CR = CA / CL = 1.80  CL = 57,929 / 1.8  CL = 32,183 therefore, N/P = $20,183 Use the equation: TA = Liabilities + Equity 128,329 = CL + LTD + Common Stock + old Retained Earnings + Addition to retained earnings 128,329 = 32,183 + LTD + 31,000 +24,000 + Add to RE 128,329 = 32,183 + LTD + 31,000 +24,000 + {15,300 − [(20,183+LTD)* 0.1]} * (1 − 0.4)* 0.6 Solve for LTD  LTD = 37,723

Fin 320: Green Solutions: II.B Solutions 36 Projected Balance Sheet Cash $10,800 Accounts Payable $12,000 Accounts Receivable $12,328 Notes Payable $20,182 Inventories $34,800 Tot. Current Liabilities $32,182 Tot. Current Assets $57,928 Long-term Debt $37,723 Land $20,000 Net Plant, Property & Equipment $50,400 Common Stock $31,000 Total Fixed Assets $70,400 Retained Earnings $27,423 Total Assets $128,329 Tot. Liabilities & Equity $128,329 Income Statement Jan.1,2011 - Dec.31,2011 Sales $150,000 Cost of Goods Sold $90,000 Gross Margin $60,000 Selling Expense $25,200 Gen.& Adm. Expense $15,000 Depreciation Expense $4,500 EBIT $15,300 Interest Expense $5,790 EBT $9,509 Taxes $3,803 Net Income $5,705 Dividends $2,282 R / E $3,423

Fin 320: Green Solutions: II.B Solutions 37 3. Balance Sheets (in $000s) Income Statements Next Year Next Year Assets Cash 100 Sales 2,420 Accounts receivable 235.28 Cost of goods sold 1,573 Inventories 240.32 Gross margin 847 Total Current Assets 575.60 GA&S 470 Depreciation expense 91 Gross Fixed Assets 1,340 Earnings before interest 286 Less: Acc. Depreciation 616 and taxes (EBIT) Net Fixed Assets 724 Interest expense 98.26 Earnings before taxes 187.74 Total Assets 1,299.6 Taxes (40%) 75.09 Net income 112.64 Liabilities & Owners' Equity Accounts Payable 131.08 Notes Payable 482.68 Accruals 24.20 Total Current Liabilities 637.96 Long-term Debt (8%) 625 Common Stock 120 Retained Earnings -83.36 Total Equity 1,299.6 Total Liabilities & Equity Calculations: TA = AP + NP + Accruals + LT Debt + Common Stock + Old Retained Earnings + Add. Retained Earnings (2011) 1299.60 = 131.08 + NP + 24.20 + 625 + 120 + 304 + Add. RE (projected) 95.32 = NP + [ EBIT − (0.1) NP − (0.08)  625] [1 − 0.4] − 500 95.32 = NP + 141.6 − (0.06) NP − 500  453.72 = NP (1 − 0.06)  NP = 453.72 / 0.94 = $482.68 It’s strange that the company paid $500,000 in dividends last year when their net income was only $81,000. It’s possible the firm used the proceeds from raising debt to pay the dividend. This is what is assumed for next year. The solution shows the net income for next year is $112,640, which isn’t enough to cover the dividend payment (that’s why you have negative retained earnings). So a lot of the new NP you raise goes to help pay the dividend. This isn’t impossible, but it’s not very likely. 4. Forecast LHS of balance sheet and income statement up to EBIT. All accounts except AR grow with sales. AR = ( 1400 / 365 ) * 60 = 230.14 Therefore TA = 1770.137 TA = AP + long-term debt (LTdebt) + Common Stock + Retained Earnings 1770.137 = 280 + LTdebt + 500 + old RE + addition to R.E. or 1770.137 = 280 + LTdebt + 500 + 200 + addition to R.E.  790.137 = LTdebt + addition to R.E.

Fin 320: Green Solutions: II.B Solutions 38 We know that the projected addition to R.E. will be: Addition to R.E. = Net Income - dividend paid = [ ( EBIT - 0.10  LTdebt)(1-tax rate)] - [(EBIT − 0.10  LTdebt)(1 − tax rate) ]  (0.33) = [(448 - 0.10  LTdebt)(0.66)] - [(448 - 0.10  LTdebt)(0.66)]  0.33 add R.E. = 0.67  [(448 - 0.10  LTdebt)(0.66)] Substitute and solve for LTdebt: 790.137 = LTdebt + 0.67  [ ( 448 − 0.10  LTdebt )( 0.66) ] => LTdebt = 619.42 Projected Balance Sheet Assets Liabilities & Owners’ Equity Current Assets: Accounts Payable 280 Cash 140 Long-term Debt 619.42 Accounts Receivable 230.14 Owners’ Equity: Inventory 140 Common Stock & 500 Total Current Assets 510.14 Paid-in Capital Fixed Assets: Retained Earnings 370.71 Net PPE 1260 Total Owners’ Equity 870.71 Total Assets 1770.14 Total Liabilities & 1770.14 Owners’ Equity Projected Income Statement Net Sales 1400 Cost of Goods Sold 700 Depreciation Expense 70 General & Administrative Expenses 182 Earnings before interest and taxes (EBIT) 448 Interest Expense 61.94 Earnings before taxes (EBT) 386.06 Taxes (.34) 131.26 NET INCOME 254.80 Dividends 84.08 Addition to Retained Earnings 170.71 5. Forecast I/S up to EBIT

Fin 320: Green Solutions: II.B Solutions 39 Forecast LHS of B/S: Cash, prepaid expenses and deferred taxes grow with sales. Inventory remains at the same level. AR = ( 17307.40 / 365 )  60 = 2845.05 Net fixed assets = 8,988 + 2,000= 10,988 Forecast items on RHS of B/S that you can: AP = ( 5828.9 / 365 )  100 = 1596.96 Accrued liabilities and misc. liabilities stay the same. You do not need to sell equity because there are no restrictions on how much you can borrow. TA = TL + E 18,103.85 = NP + 1596.96 + 1859 + LTD + 2397 + 767 + 6355 + AddRE Since NP costs more than LTD, you should raise all the money that you need from LTD and make NP=0 Therefore 18,103.85 = 0 + 1596.96 + 1859 + LTD + 2397 + 767 + 6355 + AddRE The dividend payout ratio is 20%,  18,103.85 = LTD + AddRE + 12,974.96 18,103.85 = LTD + [ ( 3167.10 − 0.05 LTD )( 1 − 0.34 )( 1 − 0.2 ) ] + 12,974.96 LTD = 3550.39 From here you can fill out the pro forma Projected Income Statement: (in $ millions) Sales 17,307.40 Cost of Goods Sold 5,828.90 Gross Profit 11,478.50 General and Administrative Expenses 6,985.00 Research Expense 1,278.00 Other Expenses 48.40 EBIT 3,167.10 Interest Expense 177.52 EBT 2,989.58 Taxes (34%) 1,016.46 Net Income 1,973.12 Dividends 394.62 Addition to Retained Earnings 1,578.50 Projected Balance Sheet: Cash 774.40 Notes Payable 0 AR 2,845.05 AP 1,596.96 Inventory 2,161.00 Accrued Liabilities 1,859.00 Prepaid Exp. 695.20 Total Current Liabilities 3,455.96 Deferred Taxes 640.20 Total Current Assets 7,115.85 Long-term Debt 3,550.39 Net Fixed Assets 10,988.00 Misc Liabilities 2,397.00 Common Stock 767.00 Retained Earnings 7,933.50 Total Assets 18,103.85 Total Liabilities and Equity 18,103.85

Fin 320: Green Solutions: III.A Solutions 40 III.A Time Value of Money RWJ 5.9 To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is : FV = PV(1 + r ) t Solving for t , we get: t = ln(FV / PV) / ln(1 + r ) t = ln ($245,000 / $50,000) / ln 1.043 = 37.75 years RWJ 5.16 To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV=PV (1+ r) t Solving for r, we get : r = (FV / PV ) 1-t -1 a. PV = $100/ (1 + r ) 20 = $50 r = ($100/ $50) 1/20 – 1 = .03526 or 3.53% b. PV = FV / (1 + .1) 10 = $50 r = ($50)*(1.1) 10 = $129.69 c. PV = $259.37 / (1 + r ) 10 = $129.69 r = ($259.37 / $129.69) 1/10 – 1 = .07177 or 7.178% RWJ 5.17 To find the PV of a lump sum, we use: PV = FV / (1 + r) t PV = $245,000 / (1.112) 9 = $94,237.34 RWJ 5.18 To find the FV of a lump sum, we use: FV = PV(1 + r ) t FV = $5,500(1.10) 45 = $400,897.66 FV = $5,500(1.10) 35 = $154,563.40 The investment strategy is start saving early! RWJ 5.20 To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is : FV = PV(1 + r ) t

Fin 320: Green Solutions: III.A Solutions 41 Solving for t , we get: t = ln(FV / PV) / ln(1 + r ) t = ln($60,000 / $10,000) / ln(1.09) = 20.79 So, the money must be invested for 18.68 years. However, you will not receive the money for another two years. From now, you’ll wait: 2 years + 20.79 years = 22.79 years Additional Problems 1. Find the present value of each deal: PV = $20,000 + $10,000  + $5,000  PV = $42,162 PV = $30,000 + $12,000  = $41,215 You should take the second deal since its present value (of costs) is lower. Consider the effects of an increase in discount rates. Both deals will be worth less since they contain future cashflows. However, the first deal will be affected more since more of its cashflows occur in the future. There is some discount rate at which the two projects will be worth the same amount − and it must be a higher discount rate than 7% since lower rates would increase the value of the first deal relative to the second. 2. a) 12 % compounded annually for 5 years FV = (1+0.12) 5 = $352.47 b) 12% compounded semiannually for 5 years FV=[1+(0.12/2)] 10 = $358.17 c) 12% compounded quarterly for 5 years FV=[1+(0.12/4)] 20 =$361.22 d) 12% compounded monthly for 1 year FV=[1+(0.12/12)] 12 =$225.36 3. Deposit your money in First Bank since it has a higher EAR: First’s EAR=[1+(0.09/1)] 1 -1 = 9% (also its nominal rate) Second’s EAR=[1+(0.08/4)] 4 − 1=8.24% 4. 14,500=2250 Solving for r you get: 8.91% 5. 12,000=1800 Solving for t you get: 9.9  10 years 6. a) 7%: 2=1  (1.07) t , Solving for t => 10.24 years b) 12%: 2=1  (1.12) t , Solving for t => 6.12 years c) 100%: 2=1  (2.00) t , Solving for t => 1 year ( ) 2 1 1 1.07 .07         −           ( ) 3 1 1.07       ( ) 1 1 1.07       ( ) 10 1 1 1 r r         −     +       ( ) t 1 1 1.08 .08         −          

Fin 320: Green Solutions: III.A Solutions 42 7. a) 12% compounded semiannually PV=200/[1+(0.12/2)] 10 =$111.68 b) 12% compounded quarterly PV=200/[1+(0.12/4)] 20 =$110.74 c) 12% compounded monthly PV=200/[1+(0.12/12)] 60 =$110.09 8. FV = PV (1+r) t 20,000 = 5,000 (1.01) t 4= (1.01) t t = ln (4) / ln (1.101) = 139.32 months t=11.61 years 9. a) Eff. monthly rate = [1 + (0.24/12)] 1 − 1 = 2% b) Eff. quarterly rate = [1 + (0.24/12)] 3 − 1 = 6.12% c) Eff. 2-year rate = [1 + (0.24/12)] 24 − 1 = 60.84 %

Fin 320: Green Solutions: III.B Solutions 43 III.B Time Value of Money RWJ 6.35 Since we are looking to triple our money, the PV and FV are irrelevant as long as the FV is three times as large as the PV. The number of periods is four, the number of quarters per year. So: FV = $3 = $1(1 + r ) (12/3) r = .3161 or 31.61% RWJ 6.36 Here we need to compare two cash flows, so we will find the value today of both sets of cash flows. We need to make sure to use the monthly cash flows since the salary is paid monthly. Doing so, we find: EAR = [1 + (0.07 / 12)] 12 – 1 = 0.07229 or 7.229% PVA 1 = $85,000 {[1 – (1 / 1.07229) 2 ] / 0.07229} = $153,195.13 PVA 2 = $20,000 + $74,000{[1 – (1/1.07229) 2 ] /0.07229} = $153,369.87 You should choose the second option since it has a higher PV. RWJ 6.50 To find the value of the perpetuity at t = 7, we first need to use the PV of a perpetuity equation. Using this equation we find: PV = $6,400 / 0.053 = $120,754.72 Remember that the PV of a perpetuity (and annuity) equations give the PV one period before the first payment, so, this is the value of the perpetuity at t = 14. To find the value at t = 7, we find the PV of this lump sum as: PV = $120,754.72/ 1.053 7 = $84,121.21

Fin 320: Green Solutions: III.B Solutions 44 RWJ 6.53 a. If the payments are in the form of an ordinary annuity, the present value will be: PVA = C ({1 – [1/(1 + r ) t ]} / r ) ) PVA = $13,500[{1 – [1 / (1 + 0.068)] 5 }/ 0.06] PVA = $63,070.40 If the payments are an annuity due, the present value will be: PVA due = (1 + r ) PVA PVA due = (1 + 0.068)$63,070.40 PVA due = $67,359.18 b. We can find the future value of the ordinary annuity as: FVA = C {[(1 + r ) t – 1] / r } FVA = $13,500{[(1 + 0.068) 5 – 1] / 0.068} FVA = $77,325.75 If the payments are an annuity due, the future value will be: FVA due = (1 + r ) FVA FVA due = (1 + 0.068)$ 77,325.75 FVA due = $82,583.90 c. Assuming a positive interest rate, the present value of an annuity due will always be larger than the present value of an ordinary annuity. Each cash flow in an annuity due is received one period earlier, which means there is one period less to discount each cash flow. Assuming a positive interest rate, the future value of an ordinary due will always higher than the future value of an ordinary annuity. Since each cash flow is made one period sooner, each cash flow receives one extra period of compounding. RWJ 6.73 Here we need to find the interest rate that makes us indifferent between an annuity and a perpetuity. To solve this problem, we need to find the PV of the two options and set them equal to each other. The PV of the perpetuity is: PV = $35,000 / r And the PV of the annuity is: PVA = $47,000[{1 – [1 / (1 + r )] 15 } / r ] Setting them equal and solving for r , we get: $35,000 / r = $47,000[ {1 – [1 / (1 + r )] 15 } / r ] $35,000 / $47,000 = 1 – [1 / (1 + r )] 15 0.2553 1/15 = 1 / (1 + r ) 0.91300 = 1 / (1 + r ) 0.91300 r + 0.91300 = 1 1-0.91300 = 0.91300r r = 0.0953 or 9.53%

Fin 320: Green Solutions: III.B Solutions 45 Additional Problems 1. Find the annuity which gives a $20,000 present value: $20,000 = A  A = $2,653.90 2. Savings: FV 50 = $10,000  FV 50 = $572,749.99 FV 80 = $572,749.99(1.1) 30 = $9,994,144.98 Withdrawals: FV 79 = $15,000  FV 79 = $859,124.99 FV 80 = $859,124.99 (1.1) = $945,037.49 So the value of what you save minus what you spend in retirement (expressed at the date you die) will be your kids inheritance: $9,994,144.98 − $945,037.49 = $9,049,107.49 3. Find the annuity which provides a $20,000 future value: $20,000 = A  A = $3,275.95 This is an annuity due. You start saving earlier so you don’t have to save as much. FVA D = FVA O (1+r) $20,000 = A (1+r) $20,000/(1+r) = A  A = $2,978.14 4. Find the present value of each alternative: A: $100,000 B: PV = $180,000 / (1.12) 5  PV = $102,137 C: PV = ($11,400 / 0.12) = $95,000 D: PV = $19,000  PV = $107,354 E: PV = $6,500/(.12 − .05) = $92,857 Since D has the highest present value, it is the most valuable prize. 5. 2-year rate = [1 + (0.12/4) 8 ] − 1 = 26.67% at time = − 1.5, value of perpetuity = 250 / 0.2667 = 937.38 Therefore, at time = 0, value of perpetuity = 937.38  (1.2667) 0.75 = 1118.85 ( ) 12 1 1 1.08 .08         −           ( ) 20 1.1 1 .1   −     ( ) 20 1.1 1 .1   −     ( ) 5 1.1 1 .1   −     ( ) 5 1.1 1 .1   −     ( ) 5 1.1 1 .1   −     ( ) 10 1 1 1.12 .12         −          

Fin 320: Green Solutions: III.B Solutions 46 6. a) FVA = A   A = 1,500,000 / 442.59 = $3389.14 b) PV = = 1,500,000  0.0221 = $33,142.39 c) 1,500,000  0.06 = $90,000 per year (you live on the interest from the $1.5M and leave the investment intact for your kids) 7. 1st payment occurs at beginning of year 12, i.e. at end of year 11 Value of perpetuity at beginning of year 11 = 100/0.08 = $1250 We want value at the end of year 7, i.e. at the beginning of year 8 Therefore, discount three years: PV = 1250 / (1+0.08) 3 = 992.25 8. The present value of the investment (a perpetuity) as of year 2 will be: PV (year 2) = 1000/0.11 = $9090.91 The year 2 value must be discounted back to the present: PV today = 9090.91/(1.11) 2 =$7378.39 9. PV = (100/.05) = $2000 If interest rates in general were to double, and the appropriate discount rate rose to 10%, what would happen to the present value of the perpetuity? PV = (100/.10) = $1000 (the present value would diminish by 1/2) 10. PV=1000 =$3169.87 If the annuity were an “annuity due,” then what would be its present value? PV=1000 + 1000 =1000+2486.85=$3486.85 11. a) Loan Amount = 0.8  180,000 = 144,000 Nominal Annual Rate = 9% Monthly Rate = 9/12 = 0.75% 144,000 = A 144,000 = A  (124.281)  A = 1,158.65 b) EAR = [1 + (0.09/12) 12 ] − 1 = 9.38% ( ) 40 1.1 1 .1   −     ( ) 40 FV 1.1 ( ) 4 1 1 1.1 .1         −           ( ) 3 1 1 1.1 .1         −           ( ) 360 1 1 1.0075 .0075         −          

Fin 320: Green Solutions: III.B Solutions 47 12. a) Draw a time line indicating the cash flows (abbreviate where appropriate). 0 1 10 20 44 -|---|------------------|---------------------|-------------------------------------------|-- 4k ................ 4k-52k -50k ................................................... -50k b) Consolidate all cash flows at year 0 Retirement Money: PV 19 = 50,000  = $639,168 PV 0 = 639,168 / (1.06) 19 = 211,253 Purchase of Cabin: PV 0 = 52,000 / (1.06) 10 = 29,036 Savings from first 10 years: PV 0 = 4,000  = $29,440 Therefore, the present value at t=0 of necessary year 11 − 20 savings is: 211,253 + 29,036 − 29,440 = 210,849 Therefore annuity amount required in years 11-20 can be calculated as follows: 210,849 = / (1.06) 10 377,598 = A   A = 51,303 13. A: FVA = $100  = $15,951.16. B: FV = $15,951.16= PV e 0.12  7 ; PV = $15,951.16 e − 0.84 = $6,886.28. You need to invest $6,886.28 in B today. 14. a) Calculate PV of annuity at one time period before first cash flow. PV 22 = 1000  = 4967.63 Discount value of this annuity back to today. PV 20 = 4967.63 / (1.12) 2 = 3,960.16 b) Calculate PV of annuity with growth rate g (8.5%), valued at one time period before first cash flow. ( ) 25 1 1 1.06 .06         −           ( ) 10 1 1 1.06 .06         −           ( ) 10 1 1 1.06 A .06             −                    ( ) 10 1 1 1.06 .06         −           ( ) ( ) 84 1 .17 12 1 .17 12   + −       ( ) 8 1 1 1.12 .12         −          

Fin 320: Green Solutions: III.B Solutions 48 Calculate PV of this amount today. PV 20 = 6408.60 / (1.12) 2 = 5,108.89 15 Here we are solving a two-step time value of money problem. Each question asks for a different possible cash flow to fund the same retirement plan. Each savings possibility has the same FV, that is, the PV of the retirement spending when your friend is ready to retire. The amount needed when your friend is ready to retire is: PVA = $105,000{[1 – (1/1.07) 20 ] / .07} = $1,112,371.50 This amount is the same for all three parts of this question. a. If your friend makes equal annual deposits into the account, this is an annuity with the FVA equal to the amount needed in retirement. The required savings each year will be: FVA = $1,112,371.50= C [(1.07 30 – 1) / .07] C = $11,776.11 b. Here we need to find a lump sum savings amount. Using the FV for a lump sum equation, we get: FV = $1,112,371.50= PV(1.07) 30 PV = $146,129.04 8 22 1.085 1 1.12 PV 1000 6,408.60 0.12 0.085     −         = =   −      

Fin 320: Green Solutions: III.B Solutions 49 c. In this problem, we have a lump sum savings in addition to an annual deposit. Since we already know the value needed at retirement, we can subtract the value of the lump sum savings at retirement to find out how much your friend is short. Doing so gives us: FV of trust fund deposit = $175,000(1.07) 10 = $344,251.49 So, the amount your friend still needs at retirement is: FV = $1,112,371.50 – 344,251.49 = $768,120.01 Using the FVA equation, and solving for the payment, we get: $768,120.01= C [(1.07 30 – 1) / .07] C = $8,131.63 This is the total annual contribution, but your friend’s employer will contribute $ 3,500 per year, so your friend must contribute: Friend's contribution = $8,131.63 – 3,500 = $4631.63

Fin 320: Green Solutions: IV Solutions 50 IV. Security Valuation (Bonds) RWJ 7.20 Initially, at a YTM of 6 percent, the prices of the two bonds are: P J = $15(PVIFA3%,28) + $1,000(PVIFA3%,28) = $718.54 PK = $45(PVIFA3%,28) + $1,000(PVIF3%,28) = $1281.46 If the YTM rises from 6 percent to 8 percent: PJ = $15(PVIFA4%,28) + $1,000(PVIF4%,28) = $583.42 PK = $45(PVIFA4%,28) + $1,000(PVIF4%,28) = $1083.32 The percentage change in price is calculated as: Percentage change in price = (New price – Original price) / Original price  PJ% = ($583.42 – $718.54) / $718.54 = – 18.80%  PK% = ($1,083.32 – $1,281.46) / $1,281.46 = – 15.46% If the YTM declines from 6 percent to 4 percent: PJ = $15(PVIFA2%,28) + $1,000(PVIF2%,28) = $893.59 PK = $45(PVIFA2%,28) + $1,000(PVIF2%,28) = $1532.03  PJ% = ($893.59 – $718.54) / $718.54 = + 24.36%  PK% = ($1,532.03 – $1,281.46) / $1,281.46 = + 19.55% All else the same, the lower the coupon rate on a bond, the greater is its price sensitivity to changes in interest rates. RWJ 7.25 To find the number of years to maturity for the bond, we need to find the price of the bond. Since we already have the coupon rate, we can use the bond price equation, and solve for the number of years to maturity. We are given the current yield of the bond, so we can calculate the price as: Current yield = .0755 = $80/P0 P0= $80 / .0755 = 1059.60 Now that we have the price of the bond, the bond price equation is: P= $1059.60 = $80 + [(1-(1/1.072) t )/0.072] + $1000/1.072 t $1,059.60(1.072 t ) = $1,111.11 (1.072 t ) – 1,111.11 + 1,000 111.11 = =51.51 (1.072 t ) 2.1570=1.072 t t = ln (8/3.71) / ln(1.072) = 11.06 The bond has 11 years to maturity. RWJ 7.27

Fin 320: Green Solutions: IV Solutions 51 a .The bond price is the present value of the cash flows from a bond. The YTM is the interest rate used in valuing the cash flows from a bond. b .If the coupon rate is higher than the required return on a bond, the bond will sell at a premium, since it provides periodic income in the form of coupon payments in excess of that required by investors on other similar bonds. If the coupon rate is lower than the required return on a bond, the bond will sell at a discount since it provides insufficient coupon payments compared to that required by investors on other similar bonds. For premium bonds, the coupon rate exceeds the YTM; for discount bonds, the YTM exceeds the coupon rate, and for bonds selling at par, the YTM is equal to the coupon rate. c .Current yield is defined as the annual coupon payment divided by the current bond price. For premium bonds, the current yield exceeds the YTM, for discount bonds the current yield is less than the YTM, and for bonds selling at par value, the current yield is equal to the YTM. In all cases, the current yield plus the expected one-period capital gains yield of the bond must be equal to the required return. . RWJ 7.34 The price of any bond (or financial instrument) is the PV of the future cash flows. Even though Bond M makes different coupons payments, to find the price of the bond, we just find the PV of the cash flows. The PV of the cash flows for Bond M is: P M = $900(PVIFA 2.7%,16 )(PVIF 2.7%,12 ) + $1,300(PVIFA 2.7%,12 )(PVIF 2.7%,28 ) + $20,000(PVIF 2.7%,40 ) P M = $21,541.52 Notice that for the coupon payments of $1,300, we found the PVA for the coupon payments, and then discounted the lump sum back to today. Bond N is a zero coupon bond with a $20,000 par value, therefore, the price of the bond is the PV of the par, or: P N = $20,000(PVIF 2.7%,40 ) = $6,889.89 Additional problems: 1. You need to find the required rate of return for the second bond in order to price it. Since the two companies issuing the bonds are similar, their required rates will be the same. Thus, you first calculate the required rate on first bond. Start by finding the issue price (the present value): PV = $500,000(1+.064) = $532,000 Then solve the bond valuation for the required rate: PV = $532,000 = $50,000  + $500,000   r = 9.0026% Finally, find the value of the second bond: PV = $70,000  + $1,000,000  PV = $871,495 (or $871,647 using r=9%) The first sold at a premium since the coupon rate was higher than the discount rate while the reverse was true for the second. ( ) 10 1 1 1 r r         −     +       ( ) 10 1 1 r     +   ( ) 10 1 1 1.090026 .090026         −           ( ) 10 1 1.090026      

Fin 320: Green Solutions: IV Solutions 52 2. a) P = 105(5.95148) + 1000(0.36022) = 624.91 + 360.22 = 985.12 b) P = 105(5.54) + 1000(0.39092) = 972.31 3. a) Price (Slugworth) = = 319.42 + 680.58 = $1000 Price (Wonka) = = 785.45 + 214.55 = $1000 Both bonds are currently priced at par since their coupon rates equal the yield to maturity. b) Price (Slugworth) = = 356.15 +821.93 = $1178.08 Price (Wonka) = =1087.23 + 456.39 = $1543.62 c) Price (Slugworth) = = 288.38 +567.43 = $855.81 Price (Wonka) = = 597.56 + 103.67 = $701.23 d) ( ) ( ) 5 5 1 1 1.08 1000 80 .08 1.08         −          +         ( ) ( ) 20 20 1 1 1.08 1000 80 .08 1.08         −          +         ( ) ( ) 5 5 1 1 1.04 1000 80 .04 1.04         −          +         ( ) ( ) 20 20 1 1 1.04 1000 80 .04 1.04         −          +         ( ) ( ) 5 5 1 1 1.12 1000 80 .12 1.12         −          +         ( ) ( ) 20 20 1 1 1.12 1000 80 .12 1.12         −          +         400 600 800 1000 1200 1400 1600 0 0.04 0.08 0.12 bond value YTM Slugworth Wonka

Fin 320: Green Solutions: IV Solutions 53 e) Since Wonka has a longer time to maturity than Slugworth, and therefore the cash flows associated with the Wonka bond extend further into the future, the price of Wonka will fluctuate more with changes in market interest rates. They key here is that the longer time to maturity of the Wonka bond means that it is a riskier bond, primarily because the principal is paid very far in the future (Wonka is subject to more interest rate risk). 4. a) P 3 = = 937.951 b) 937.95  500,000 = $468.97 million c) 968.3013 = by trial and error or calculator, r = 10% 5. YTM at issuance: 1000 = 300 (1+r) 10 r = 12.79% Therefore current price = 1000 / (1.1379) 5 = 524.17 ( ) ( ) 4 4 1 1 1.11 1000 90 .11 1.11         −          +         ( ) ( ) 4 4 1 1 1 r 1000 90 r 1 r         −     +      +       +  

Fin 320: Green Solutions: V Solutions 54 V: Security Valuation (Stocks) RWJ 8.14 This stock has a constant growth rate of dividends, but the required return changes twice. To find the value of the stock today, we will begin by finding the price of the stock at Year 6, when both the dividend growth rate and the required return are stable forever. The price of the stock in Year 6 will be the dividend in Year 7, divided by the required return minus the growth rate in dividends. So: 𝑃 𝑖 = ? 𝑖+1 ? − 𝑔 => 𝑃 6 = $2.65 × 1.038 7 0.11 − 0.038 = $47.79 Now we can find the price of the stock in Year 3. We need to find the price here since the required return changes at that time. The price of the stock in Year 3 is the PV of the dividends in Years 4, 5, and 6, plus the PV of the stock price in Year 6. The price of the stock in Year 3 is: 𝑃 3 = 2.65(1.038) 4 1.13 + 2.65(1.038) 5 1.13 2 + 2.65(1.038) 6 1.13 3 + 47.79 1.13 3 = $40.61 Finally, we can find the price of the stock today. The price today will be the PV of the dividends in Years 1, 2, and 3, plus the PV of the stock in Year 3. The price of the stock today is: 𝑃 3 = 2.65(1.038) 1.15 + 2.65(1.038) 2 1.15 2 + 2.65(1.038) 3 1.15 3 + 40.61 1.15 3 = $33.20 RWJ 8.16 The price of a stock is the PV of the future dividends. This stock is paying four dividends, so the price of the stock is the PV of these dividends using the required return. The price of the stock is: 𝑃 0 = 7.25 1.11 + 11.75 1.11 2 + 16.25 1.11 3 + 20.75 1.11 4 + 25.25 1.11 5 = $56.60 RWJ 8.17 With supernormal dividends, we find the price of the stock when the dividends level off at a constant growth rate, and then find the PV of the future stock price, plus the PV of all dividends during the supernormal growth period. The stock begins constant growth in Year 4, so we can find the price of the stock in Year 4, at the beginning of the constant dividend growth, as: P 4 = D 4 (1 + g ) / ( R – g ) = $2.75(1.05) / (.1075 – .05) = $50.22 The price of the stock today is the PV of the first four dividends, plus the PV of the Year 3 stock price. So, the price of the stock today will be: P 0 = $13.00 / 1.1075 + $9.00 / 1.1075 2 + $6.00 / 1.1075 3 + $2.75 / 1.1075 4 + $50.22 / 1.1075 4 = $58.70 RWJ 8.18 With supernormal dividends, we find the price of the stock when the dividends level off at a constant growth rate, and then find the PV of the futures stock price, plus the PV of all dividends during the

Fin 320: Green Solutions: V Solutions 55 supernormal growth period. The stock begins constant growth in Year 4, so we can find the price of the stock in Year 3, one year before the constant dividend growth begins as: P3 = D3(1+g)/(R-g) = D0 (1+g1)^3 (1+g2)/(R-g) 𝑃 3 = 2.45(1.30) 3 (1.04) 0.11 − 0.04 = $79.97 The price of the stock today is the PV of the first three dividends, plus the PV of the Year 3 stock price. The price of the stock today will be: 𝑃 0 = 2.45(1.30) 1.11 + 2.45(1.30) 2 1.11 2 + 2.45(1.30) 3 1.11 3 + 79.97 1.11 3 = $68.64 RWJ 8.19 Here we need to find the divident next year for a stock experiencing supernormal growth. We know the stock price, the divident growth rates, and the required return, but not the divident. First, we need to realize that the divident in Year 3 is current divident times the FVIF. The divident in Year 3 will be: D 3 = D 0 (1.25) 3 And the dividend in Year 4 will be the dividend in Year 3 times one plus the growth rate, or: D 4 = D 0 (1.25) 3 (1.15) The stock begins constant growth in Year 4, so we can find the price of the stock in Year 4 as the dividend in Year 5, divided by the required return minus the growth rate. The equation for the price of the stock in Year 4 is: P 4 = D 4 (1 + g ) / ( R – g ) Now we can substitute the previous dividend in Year 4 into this equation as follows: P 4 = D 0 (1 + g 1 ) 3 (1 + g 2 ) (1 + g 3 ) / ( R – g ) P 4 = D 0 (1.25) 3 (1.15) (1.06) / (.10 – .06) = 59.52D 0 When we solve this equation, we find that the stock price in Year 4 is 59.52 times as large as the dividend today. Now we need to find the equation for the stock price today. The stock price today is the PV of the dividends in Years 1, 2, 3, and 4, plus the PV of the Year 4 price. So: P 0 = D 0 (1.25)/1.10 + D 0 (1.25) 2 /1.10 2 + D 0 (1.25) 3 /1.10 3 + D 0 (1.25) 3 (1.15)/1.10 4 + 59.52D 0 /1.10 4 We can factor out D 0 in the equation, and combine the last two terms. Doing so, we get: P 0 = $79 = D 0 {1.25/1.10 + 1.25 2 /1.10 2 + 1.25 3 /1.10 3 + [(1.25) 3 (1.15) + 59.52] / 1.10 4 } Reducing the equation even further by solving all of the terms in the braces, we get: $79 = $40.08D 0 D 0 = $79 / $40.08 D 0 = $1.97

Fin 320: Green Solutions: V Solutions 56 This is the dividend today, so the projected dividend for the next year will be: D 1 = $1.97(1.25) D 1 = $2.46 RWJ 8.20 The constant growth model can be applied even if the dividends are declining by a constant percentage, just make sure to recognize the negative growth. So, the price of the stock today will be: P0=D0(1+ g )/( R-g ) P0=$9.8(1-0.04)/[(0.095-(-0.04)] P0=$69.69 RWJ 8.21 We are given the stock price, the dividend growth rate, and the required return and are asked to find the dividend. Using the constant dividend growth model, we get: P 0 = $57 = D 0 (1 + g ) / ( R – g ) Solving this equation for the dividend gives us: D 0 = $57(0.11 – 0.0375) / (1.0375) D 0 = $3.98 RWJ 8.22 The price of a share of preferred stock is the dividend payment divided by the required return. We know the dividend payment in Year 20, so we can find the price of the stock in Year 19, one year before the first dividend payment. Doing so, we get: P 19 = $20.00 /0.0565 P 19 = $353.98 The price of the stock today is the PV of the stock price in the future, so the price today will be: P 0 = $353.98 / (1.0565) 19 P 0 = $124.58

Fin 320: Green Solutions: V Solutions 57 Additional Problems : 1. Calculate the next four dividends: D 1 = (1 + .08)  $2.00 = $2.16 D 2 = (1 + .08)  $2.16 = $2.33 D 3 = (1 + .08)  $2.33 = $2.52 D 4 = (1 + .04)  $2.52 = $2.62 Since every dividend from D 4 grows at a rate of 4.0%, we can apply the growth model to that dividend: PV 3 = $2.62 / (0.16 − 0.04) = $21.83 Now find the present value of each future cashflow where the value of the infinite dividend stream is one of those values: PV = [$2.16 / (1.16)] + [$2.33 / (1.16) 2 ] + [($2.52 + $21.83)/(1.16) 3 ] PV = $19.20 2. . a) 0 1 2 3 4 5 6 2.00--------------------------------->------------------------>--------> 15% growth constant div 20% growth b) P 6 = D 7 /(r-g) = 2(1.15) 3 (1.20)/(0.22-0.20) = $182.505 c) P 0 = D 1 /(1+r) + D 2 /(1+r) 2 + D 3 /(1+r) 3 + D 4 /(1+r) 4 + D 5 /(1+r) 5 + D 6 /(1+r) 6 + P 6 /(1+r) 6 = 2/1.22 + 2(1.15)/(1.22) 2 + 2(1.15) 2 /(1.22) 3 + 2(1.15) 3 /(1.22) 4 + 2(1.15) 3 /(1.22) 5 + 2(1.15) 3 /(1.22) 6 + P 6 /(1.22) 6 = $63.41 d) P 6 = P 0 (1+g) 6 From this, we get g = 19.26% 3. D 0 8.50 D 1 10.63 (growth at 25%) / 1.16 = 9.16 D 2 12.76 (growth at 20%) / (1.16) 2 = 9.48 D 3 14.67 (growth at 15%) / (1.16) 3 = 9.40 D 4 16.14 (growth at 10%) / (1.16) 4 = 8.91 D 5 16.94 (growth at 5%) P 4 = D 5 / (r − g)  P 4 = 16.94 / (0.16 - 0.05) = 153.95 Therefore, P 0 = 9.16 + 9.48 + 9.40 + 8.91 + [153.95/(1.16) 4 ] = 122.00 4. P0 = = 0.7284 / (1.14) 4 + 4.5 / (1.14) 8 ( ) ( ) ( ) 4 4 8 1 1 1.14 .25 1.08 .25 .14 .14 .08 1.14 1.14         −              −     +

Fin 320: Green Solutions: V Solutions 58 = 0.4313 + 1.5775 = 2.00 5. a) Div 3 = ($10mn / 1mn)  0.5 = $5.00 Div 4 = 5  (1 + 0.20) = $6.00 Div 5 = 5  (1+0.20) 2 = $7.20 Div 6 = 7.20  (1 + 0.06) = $7.63 P 5 = 7.63 / (0.22 - 0.06) = $47.7 P 3 = (6 / 1.22) + (7.2 / 1.22 2 ) + (47.7 / 1.22 2 ) = $41.80 b) P 0 = (5 / 1.22 3 ) + (41.80 / 1.22 3 ) = $25.77 6. P 0 = $92 = D 0 (1+g) /(r − g) D 0 = 92  (0.13 − 0.10) / (1.10) = $2.51. 7. a) P 11 = PV of all future cash flows P 11 =0.4(1.11) + = 4.81 b) P 0 = = 1.69 8. a) P 3 = D 4 / (r − g) = 2.589 / 0.16 − 0.07 = 28.77 b) r = = 0.5387 6 1.11 1 1.17 0.17 0.11     −           −       ( ) 6 6 0.4 1.11 (1.06) 1 ( ) 0.17 0.06 (1.17)     −     ( ) ( ) ( ) 7 4 11 1 1 1.17 0.4 4.81 .17 1.17 1.17         −              + (28.77 23) 2 2.2 2.42 23 − + + +

Fin 320: Green Solutions: VI Solutions 59 VI. The Risk-Return Relation and The Pricing of Risk RWJ 12.7 The average return is the sum of the returns, divided by the number of returns. The average return for each stock was: ? ̅ = 1 𝑁 (∑ ? 𝑖 𝑁 𝑖=1 ) = 0.12 + 0.28 + 0.09 − 0.07 + 0.10 5 = 0.104 𝑜? 10.4% ? ̅ = 1 𝑁 (∑ ? 𝑖 𝑁 𝑖=1 ) = 0.25 + 0.34 + 0.13 − 0.27 + 0.14 5 = 0.118 𝑜? 11.8% Remembering back to “statistics,” we calculate the variance of each stock as: 𝜎 2 = 1 (𝑁 − 1) [∑(? 𝑖 − ?̅) 2 𝑁 𝑖=1 ] 𝜎 ? 2 = 1 5 − 1 [(0.12 − 0.104) 2 + (0.28 − 0.104) 2 + (0.09 − 0.104) 2 + (−0.07 − 0.104) 2 + (0.10 − 0.104) 2 ] = 0.01543 𝜎 ? 2 = 1 5 − 1 [(0.25 − 0.118) 2 + (0.34 − 0.118) 2 + (0.13 − 0.118) 2 + (−0.27 − 0.118) 2 + (0.14 − 0.118) 2 ] = 0.05447 The standard deviation is the square root of the variance, so the standard deviation of each stock is:  X = (0.01543) 1/2 = .1242 or 12.42%  Y = (0.05447) 1/2 = .2334 or 23.34% RWJ 12.8 We will calculate the sum of the returns for each asset and the observed risk premium first. Doing so, we get: Year Large co. stock return T-bill return Risk premium 1970 3.94% 6.50% − 2.56% 1971 14.30 4.36 9.94 1972 18.99 4.23 14.76 1973 – 14.69 7.29 – 21.98 1974 – 26.47 7.99 – 34.46 1975 37.23 5.87 31.36 33.30 36.24 – 2.94 a . The average return for large company stocks over this period was: Large company stocks average return = 33.30% / 6 = 5.55% And the average return for T-bills over this period was: T-bills average return = 36.24% / 6 = 6.04%

Fin 320: Green Solutions: VI Solutions 60 b . Using the equation for variance, we find the variance for large company stocks over this period was: Variance = 1/5[(.0394 – .0555) 2 + (.1430 – .0555) 2 + (.1899 – .0555) 2 + ( – .1469 – .0555) 2 + ( – .2647 – .0555) 2 + (.3723 – .0555) 2 ] Variance = 0.053967 And the standard deviation for large company stocks over this period was: Standard deviation = (0.053967) 1/2 = 0.2323 or 23.23% Using the equation for variance, we find the variance for T-bills over this period was: Variance = 1/5[(.0650 – .0604) 2 + (.0436 – .0604) 2 + (.0423 – .0604) 2 + (.0729 – .0604) 2 + (.0799 – .0604) 2 + (.0587 – .0604) 2 ] Variance = 0.000234 And the standard deviation for T-bills over this period was: Standard deviation = (0.000234) 1/2 = 0.0153 or 1.53% c . The average observed risk premium over this period was: Average observed risk premium = – 2.94% / 6 = – 0.49% The variance of the observed risk premium was: Variance = 1/5[( – .0256 – ( – .0049)) 2 + (.0994 – ( – .0049)) 2 + (.1476 – ( – .0049))) 2 + ( – .2198 – ( – .0049)) 2 + ( – .3446 – ( – .0049)) 2 + (.3136 – ( – .0049)) 2 ] Variance = 0.059517 And the standard deviation of the observed risk premium was: Standard deviation = (0.059517) 1/2 = 0.2440 or 24.40% d. Before the fact, for most assets the risk premium will be positive; investors demand compensation over and above the risk-free return to invest their money in the risky asset. After the fact, the observed risk premium can be negative if the asset’s nominal return is unexpectedly low, the risk-free return is unexpectedly high, or if some combination of these two events occurs.

Fin 320: Green Solutions: VI Solutions 61 VI. RISK, RETURN, AND THE SECURITY MARKET LINE RWJ 13.9 a) To find the expected return of the portfolio, we need to find the return of the portfolio in each state of the economy. This portfolio is a special case since all three assets have the same weight. To find the expected return in an equally weighted portfolio, we can sum the returns of each asset and divide by the number of assets, so the expected return of the portfolio in each state of the economy is: Equally weighted, so 1/3 in each stock: 𝐵𝑜𝑜?: ?(𝑅 𝑝 ) = (0.08 + 0.17 + 0.24) 3 = 0.1633 𝑜? 16.33% 𝐵???: ?(𝑅 𝑝 ) = (0.11 − 0.05 − 0.08) 3 = −0.0067 𝑜? − 0.67% ?(𝑅 𝑝 ) = 0.75(0.1633) + 0.25(−0.0067) = 0.12083 𝑜? 12.083% b) This portfolio does not have an equal weight in each asset. We still need to find the return of the portfolio in each of the economy. To do this, we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy. Doing so, we get: 𝐵𝑜𝑜?: ?(𝑅 𝑝 ) = 0.2(0.08) + 0.2(0.17) + 0.6(0.24) = 0.194 𝑜? 19.4% 𝐵???: ?(𝑅 𝑝 ) = 0.2(0.11) + 0.2(−0.05) + 0.6(−0.08) = −0.036 𝑜? − 3.6% ?(𝑅 𝑝 ) = 0.75(0.194) + 0.25(−0.036) = 0.1365 𝑜? 13.65% To find variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, than add all of these up. The result is the variance. So, the variance and standard deviation of the portfolio is: 𝜎 𝑝 2 = 0.75(0.194 − 0.1365) 2 + 0.25(−0.036 − 0.1365) 2 = 0.00992 𝑜? 0.992% RWJ 13.13 CAPM states the relationship between the risk of an asset and its expected return. CAPM is: ?(𝑅 𝑝 ) = 0.031 + 1.15(0.103 − 0.031) = 0.1138 𝑜? 11.38% RWJ 13.20 a) Again we have a special case where the portfolio is equally weighted, so we can sum the returns of each asset and divide by the number of assets. The expected return of the portfolio is: ?(𝑅 𝑝 ) = 0.5(0.108) + 0.5(0.027) = 0.0675 𝑜? 6.75% b) We need to find the portfolio weights that result in a portfolio with a beta of 0.93. We know the beta of the risk-free asset is zero. We also know the weight of the risk-free asset is one minus the weight of the stock since the portfolio weights must sum to one, or 100 percent. So: 𝛽 𝑝 = 0.92

Fin 320: Green Solutions: VI Solutions 62 1.12? ? + 0(1 − ? ? ) = 0.92 => ? ? = 0.821 𝑜? 82.1% ? ?𝑓 = 1 − ? ? = 1 − 0.821 = 0.179 𝑜? 17.9% c) We need to find the portfolio weights that result in a portfolio with an expected return of 9 percent. We also know the weight of the risk-free asset is one minus the weight of the stock since the portfolio weights must sum to one, or 100 percent. So: ?(𝑅 𝑝 ) = 0.09 0.108? ? + 0.027(1 − ? ? ) = 0.09 => ? ? = 0.7778 𝑜? 77.78% ? ?𝑓 = 1 − ? ? = 1 − 0.7778 = 0.2222 𝑜? 22.22% 𝛽 𝑝 = 1.12(0.7778) + 0(0.2222) = 0.871 d) Solving for the beta of the portfolio as we did in part a , we find: 𝛽 𝑝 = 2.24 = 1.12? ? + 0(1 − ? ? ) => ? ? = 2 𝑜? 200% ? ?𝑓 = 1 − ? ? = 1 − 2 = −1 𝑜? − 100% The portfolio is invested 200% in the stock and – 100% in risk free asset. This represents borrowing at the risk free rate to the stock. RWJ 13.23 a) We need to find the return of the portfolio in each state of the economy. To do this, we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy. Doing so, we get: 𝐵𝑜𝑜?: ?(𝑅 𝑝 ) = 0.4(0.21) + 0.4(0.33) + 0.2(0.55) = 0.326 𝑁𝑜??𝑎?: ?(𝑅 𝑝 ) = 0.4(0.17) + 0.4(0.11) + 0.2(0.09) = 0.130 𝐵???: ?(𝑅 𝑝 ) = 0.4(0.00) + 0.4(−0.212) + 0.2(−0.45) = −0.174 ?(𝑅 𝑝 ) = 0.25(0.326) + 0.60(0.130) + 0.15(−0.174) = 0.133 𝑜? 13.3% To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We ten multiply each possible squared deviation by its probability, than add all of there up. The result is the variance. So, the variance and standard deviation of the portfolio is: 𝜎 𝑝 2 = 0.25(0.326 − 0.133) 2 + 0.6(0.130 − 0.133) 2 + 0.15(−0.174 − 0.133) 2 = 0.0235 𝜎 𝑝 = √𝜎 𝑝 2 = 0.1532 b) The risk premium is the return of a risky asset minus the risk-free rate. T-bills are often used as the risk- free rate, so: Risk Premium = ?(𝑅 𝑝 ) ? 𝑓 = 0.133 0.038 = 0.095 RWJ 13.27 Here we have the expected return and beta for two assets. We can express the returns of the two assets using CAPM. If the CAPM is true, then the security market line holds as well, which means all assets have the same risk premium. Setting the risk premiums of the assets equal to each other and solving for the risk- free rate, we find:

Fin 320: Green Solutions: VI Solutions 63 0.1323− ? 𝑓 1.25 = 0.0967− ? 𝑓 0.87 => ? 𝑓 = 0.0152 𝑜? 1.52% Now using CAPM to find the expected return on the market with both stocks, we find: 0.1323 = 0.0152 + 1.25(R M – 0.0152) 0.0967 = 0.0152 + 0.87(R M – 0.0152) R M = 0.1089 or 10.89% R M = 0.1089 or 10.89% Additional Problems: 1. a) When considering stocks held in isolation, we can compare expected return and total risk (standard deviation) combinations. Since A and B provide the same expected return, but the risk associated with holding Stock A alone is less than that of Stock B, you should choose Stock A. b) In this case, beta would be the appropriate risk measure to consider, since only systematic risk remains when stocks are held in a diversified portfolio. The beta for Stock A would be .003/.002 = 1.5, while Stock B’s beta would be .0024/.002 = 1.2, so you should choose Stock B to add to your portfolio. 2. a) (b) Using CAPM: E(R A ) =.08+ (.15 − .08)1.5 = .185 < .20 E(R B ) =.08+(.15 − .08)2.0 = .22 > .21 Since the expected return for stock A is less than that indicated by its current market price, Stock A appears to be “under priced.” Stock B’s expected return is greater than that indicated by its current price, so it appears to be “overpriced.” This can also be seen by the stocks’ positions relative to the SML. Stock A falls above the SML (indicating that it is offering more expected return for its risk level than the market), while stock B falls below the SML. You should add stock A to your portfolio. 3. Recall that = Cov(i,m) / 2 (m) = Corr(i,m)  i m / 2 (m) = Corr(i,m) i / m Therefore, firm risk depends on i and m as well as correlation. Hence, for the beta's (and consequently the same required rate of return) to be the same, B = F . Given the description of the companies however, it is more likely that F > B which means that F > B . 4. a) E(R p ) = (0.1)(0.5) + (0.18)(0.5) = 0.14 s p = [(0.12) 2 (0.5) 2 + (0.24) 2 (0.5) 2 + 0] 0.5 = 0.1342 b) E(R p ) = 0.14 s p = [(0.12) 2 (0.5) 2 + (0.24) 2 (0.5) 2 + 2 (0.5) (0.5) (0.12) (0.24) (0.4)] 0.5 = 0.1542 c) E(R p ) = 0.14 s p = [(0.12) 2 (0.5) 2 + (0.24) 2 (0.5) 2 + 2 (0.5) (0.5) (0.12) (0.24)] 0.5 = 0.18 0 0.05 0.1 0.15 0.2 0.25 0 0.5 1 1.5 2 2.5 E(R) beta SML market Stock A Stock B

Fin 320: Green Solutions: VI Solutions 64 5. First, we must calculate the required rate of return on Midterm Inc. stock: k = 7.00% + .80(8.75%) = 14.00% a) Since every dividend subsequent to D 3 grows at a rate of 7.0%, we can apply the growth model to that dividend: PV 2 = $7.00 / (.14 − .07) = $100.00 Now find the value of all future cashflows as of year 0: PV 0 = ($100.00 + $7.00) / (1 + .14) 2 + $7.00/(1 + .14) = $88.47 b) Find the value of all future cashflows as of year 1: PV 1 = ($100.00 + $7.00) / (1 + .14) = $93.86 c) The dividend yield is the dividend divided by the price: $7.00 / $88.47 = 7.91% d) The capital gain yield is the change in price divided by the initial price. We calculated the prices in parts (a) and (b), thus ($93.86 − $88.47) / $88.47 = 6.09% e) The total yield is the sum of dividend and capital gains yields and is equal to 7.91% + 6.09% = 14%. This should not come as any surprise since all the calculations we done such that the value of money over any given year is equal to 14%. 6. First, we must calculate the required rate of return on Adesh Inc. stock: k = 7.00% + .50(12%) = 13.00% a) Since every dividend subsequent to D 2 grows at a rate of 7.0%, we can apply the growth model to that dividend: PV 1 = $7.00 / (.13 − .07) = $116.67 Then the present value of cashflows is PV = ($116.67 + $7.00) / (1 + .13) = $109.44 b) This is just the value calculated in part (a) for the price at time 1: $116.67 c) The dividend yield is the dividend divided by the price: $7.00 / $109.44 = 6.4% d) The capital gain yield is the change in price divided by the initial price. We calculated the prices in parts (a) and (b), thus ($116.67 − $109.44) / $109.44 = 6.6% 7. a) Stock X: 60= 3(1.05)/(r − .05) r = 10.25%, which is greater than 10%. Thus, stock X satisfies the criteria. Stock Y: 90=3.10/(r − .06) r = 9.44%, which is less than 10%. Thus, stock Y doesn't satisfy the criteria. b) E(R x ) = 0.05 + 1.1(0.15 − 0.05) = 16%, which is greater than 10.25%, calculated in (a). Therefore the stock is overpriced. E(R y ) = 0.05 + 0.4(0.15 − 0.05) = 9%, which is less than the 9.44% calculated in (a). Therefore the stock is under priced. Only stock Y satisfies the second criteria. 8. If  =1, then does  =1? From equation above, this is true only when  E =  M Since we don’t know that the standard deviations are the same, the statement is uncertain. 1 0 D p r g = − E M E 2 2 M M M Cov(E, M)     = = =   

Fin 320: Green Solutions: VI Solutions 65 9. a) E(R p ) = 0.5(10) + 0.5(6) = 8% b) Covariance =  A  B = 0.8 (0.7) (0.5) = 0.28 Variance = (0.5) 2 (0.49) + (0.5) 2 (0.25) + 2 (0.5) (0.5) (0.28) = 0.325 Standard deviation = 0.57 c) W ITT =7/12 W HI = 5/12 Variance = (7/12) 2 (0.49) + (5/12) 2 (0.25) + 2 (7/12) (5/12) (0.28) = 0.3462 Standard deviation = 0.5884 10. Therefore,  = 0.8574 11. a) or 26.43% b) or 17.5 % 12. Portfolio A has more diversifiable risk because all the stocks in the portfolio are concentrated in one industry. i m 2 2 m m (0.8944)(0.5477) Cov 1.4 .3     = = = =   (2.95 2.8) 0.59 R 0.2643 2.8 − + = = (2.95 2.8) 0.59 0.25 R 0.175 2.8 − + − = =

Fin 320: Green Solutions: VII Solutions 66 VII. Market Efficiency and The Historical Record Additional Problems 1. False. Since the accident was unanticipated, news of the accident is new information and the markets would be expected to adjust to such information. Even if the market expected an accident with some probability less than certainty, the markets would adjust since news that the accident occurred would revise the prior probability. 2) a) The price drop when the firm announced the unexpected restrictions is consistent with semi-strong form efficiency. The price dropped because the information was new (unexpected). b) The low earnings will NOT "depress the stock price" for two years − as soon as the announcement is made, the information on the settlement is included and the price now reflects that information. The price will be lower, but is free to move as it normally would from then on (reflecting the risk of the stock and new information) and there is no reason for the stock price to rise at the end of the two years. There is NO reason to avoid buying the stock − it has already dropped to reflect all the bad news of the restrictions. 3. The market expected the growth rate in the coming year to be 2 percent, then there would be no change in security prices if this expectation had been fully anticipated and priced. However, if the market had been expecting a growth rate different than 2 percent and the expectation was incorporated into security prices, then the government’s announcement would most likely cause security prices in general to change; prices would drop if the anticipated growth rate had been more than 2 percent and prices would rise if the anticipated growth rate had been less than 2 percent. 4. E [r i ] = 0.05 + 0.123  1 , 0.20 > E[r m ] = 0.05 + 0.123 (1.2) = 0.1976; M plots above the SML and is undervalued E[r N ] = 0.05 + 0.123(0.9) = 0.1607 > 0.16; N plots below the SML and is overvalued. 5. a) Dearth Vader offer = 180/5 =$36 per share Death Star offer = $42 per share Take the Death Star offer. b) No, it merely implies some uncertainty about which takeover bid might succeed.

Fin 320: Green Solutions: VIII Solutions 67 VIII. Capital Budgeting RWJ 9.17 a . The payback period for each project is: A: 3 + ($182,000/$458,000) = 3.40 years B: 2 + ($5,000/$21,500) = 2.23 years The payback criterion implies accepting project B, because it pays back sooner than project A. b. The discounted payback for each project is: A: $46,000/1.11 + $68,000/1.11 2 + $68,000/1.11 3 = $146,352.78 $458,000/1.11 4 = $301,698.79 Discounted payback = 3 + ($364,000 – 146,352.78)/$301,698.79 = 3.72 years B: $25,000/1.11 + $22,000/1.11 2 = $40,378.22 $21,500/1.11 3 = $15,720.61 Discounted payback = 2 + ($52,000 – 40,378.22 )/$15,720.61=2.74 years The discounted payback criterion implies accepting project B because it pays back sooner than A. c .The NPV for each project is: A: NPV = – $364,000 + $46,000/1.11 + $68,000/1.11 2 + $68,000/1.11 3 + $458,000/1.11 4 NPV = $84,051.57 B: NPV = – $52,000 + $25,000/1.11 + $22,000/1.11 2 + $21,500/1.11 3 + $17,500/1.11 4 NPV = $15,626.62 NPV criterion implies we accept project A because project A has a higher NPV than project B. d. The IRR for each project is: A: $364,000 = $46,000/(1+IRR) + $68,000/(1+IRR) 2 + $68,000/(1+IRR) 3 + $458,000/(1+IRR) 4 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR =18.14% B: $52,000 = $25,000/(1+IRR) + $22,000/(1+IRR) 2 + $21,500/(1+IRR) 3 + $17,500/(1+IRR) 4 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR =25.29 % IRR decision rule implies we accept project B because IRR for B is greater than IRR for A. e. The profitability index for each project is: A: PI = ($46,000/1.11 + $68,000/1.11 2 + $68,000/1.11 3 + $458,000/1.11 4 ) / $364,000 = 1.23 B: PI = ($31,000/1.11 + $28,000/1.11 2 + $25,500/1.11 3 + $19,000/1.11 4 ) / $65,000 = 1.30 Profitability index criterion implies accept project B because its PI is greater than project A’s. f. In this instance, the NPV criteria implies that you should accept project A, while profitability index, payback period, discounted payback, and IRR imply that you should accept project B. The final decision should be based on the NPV since it does not have the ranking problem associated with the other capital budgeting techniques. Therefore, you should accept project A. RWJ 9.25 Here the cash inflows of the project go on forever, which is a perpetuity. Unlike ordinary perpetuity cash flows, the cash flows here grow at a constant rate forever, which is a growing perpetuity. If you remember back to the chapter on stock valuation, we presented a formula for valuing a stock with constant growth in dividends. This formula is actually the formula for a growing perpetuity, so we can use it here. The PV of the future cash flows from the project is: a) PV of cash inflows = C 1 / (r − g) = $145,000 /(.11 − .04) = $2,071,428.57 NPV is the PV of the inflows minus the PV of the outflows, so the NPV is: NPV of the project = − $1,900,000 + $2,071,428.57 = $171,428.571

Fin 320: Green Solutions: VIII Solutions 68 The NPV I positive, so we would accept the project. b) Here we want to know the minimum growth rate in cash flows necessary to accept the project. The minimum growth rate is the growth rate at which we would have a zero NPV. The equation for a zero NPV, using the equation for the PV of a growing perpetuity is: $145,000/(.11 − g) +-$1,900,000=0; g = 3.37% RWJ 10.20 To find the bid price, we need to calculate all other cash flows for the project, and then solve for the bid price. The aftertax salvage value of the equipment is: After-tax salvage value = $85,000(1 – 0.21) After-tax salvage value = $67,150 Now we can solve for the necessary OCF that will give the project a zero NPV. The equation for the NPV of the project is: NPV = 0 = – $910,000 – 90,000 + OCF(PVIFA 12%,5 ) + {($67,150 + 90,000) / 1.12 5 } Solving for the OCF, we find the OCF that makes the project NPV equal to zero is: OCF = $910,828.87 / PVIFA 12%,5 OCF = $252,672.79 The easiest way to calculate the bid price is the tax shield approach, so: OCF = $252,672.79= [(P – v)Q – FC ](1 – T) + TD $252,672.79= [(P – $17.35)(125,000) – $485,000](1 – 0.21) + 0.21($910,000/5) P = $23.40

Fin 320: Green Solutions: VIII Solutions 69 RWJ 11.25 (part b) The cash breakeven is: Q C = $500,000/($40,000 – 20,000) Q C = 25 And the accounting breakeven is: Q A = {$500,000 + [($700,000 – $700,000(0.21))/0.62]}/($40,000 – 20,000) Q A = 69.60 The financial breakeven is the point at which the NPV is zero, so: OCF F = $3,500,000/PVIFA 20%,5 OCF F = $1,170,328.96 Additional problems 1. Begin by calculating the cashflow for each period: C 0 = − $8.0 million C 1 − C 4 = 1.0 C 5 = − 1.0 C 6 − C 9 = 1.0 C 10 = − 1.0 C 11 - C 14 = 1.0 C 15 = 2.5 Then calculate the NPV at 8.0%: NPV = − $1,255,200 Since the NPV is negative, Copernicus Lines should not buy the new ship. 2. First calculate the NPV and IRR of each project: A: NPV = $20,000 / (1 + .20) − $10,000 = $6,667 IRR = ($20,000 − $10,000) / $10,000 = 100% B: NPV = $12,000 / (1 + .20) − $5,000 = $5,000 IRR = ($12,000 − $5,000) / $5,000 = 140% C: NPV = $5,500 / (1 + .20) − $5,000 = -$416 IRR = ($5,500 − $5,000) / $5,000 = 10% D: NPV = $5,000 / (1 + .20) − $2,000 = $2,167 IRR = ($5,000 − $2,000) / $2,000 = 150% a) Projects A, B and D would all be acceptable since all have positive NPVs (and also IRRs greater than 20%). The most valuable is project A since its NPV is highest. The award for the highest IRR goes to project D. b) You would take project A since it's NPV is highest − i.e. it would lead to the largest increase in the value of the firm. 3. a) The NPVs of the projects are: A: NPV = − $91 B: NPV = $1,170 C: NPV = $1,186 Projects B and C have positive NPVs.

Fin 320: Green Solutions: VIII Solutions 70 b) The payback period for each project is the length of time it takes for the project to generate cashflows equal to the firm's initial investment. In this case: 1 yr, 2 yrs, and 3.5 yrs, for projects A, B and C, respectively. c) With a three year cutoff, the firm would accept projects A and B. In other words, the firm would take one negative NPV project and forgo the most valuable of the three. 4. a) The NPVs and IRRs of each project are: A: NPV = $677 IRR = 20.85% B: NPV = $484 IRR = 25.19% b) While the return is higher with project B, wealth is increased by a larger amount with project A. Since Mr. Clops purchases goods and services with wealth, not with returns, project A is the better project. 5. Calculate the NPV at 14%: NPV@14% = $170,000  − $800,000 = $86,740 If you sell the factory, you would not sell it at a price that would given someone else a positive NPV (any positive NPV you would rather keep!!), so the sales price would be that which gives a zero NPV. This is just the present value of the future cashflows discounted at 14%: Price = PV 5 = $170,000  PVIFA(14%, 5) = $583,623 6. This is a comparison of choices with different lives. The easiest approach is to calculate the equivalent annual annuity (EAA). Note that the EAA is calculated assuming payments are made at the end of the year. A: $48 (1 + .10) = $52.80 B: $84 = A   A = $48.40 C: $108 = A   A = $43.43 Your optimal strategy is therefore to purchase 3 year subscriptions. 7. Find the NPV of each project and calculate the EAA: A: Find NPV: C 0 = $20,000 C 1-3 = ($11,000 − $2,000 − $6,667)(1 − .40) + 6,667 = $8,067 NPV@8% = $8,067  − $20,000 = $789 Find EAA: $789 = EAA   EAA = $306 B: Find NPV: C 0 = $45,000 C 1-3 = ($12,000 - $1,500 - $6,429)(1 − .40) + 6,429 ( ) 10 1 1 1.14 .14         −           ( ) 2 1 1 1.1 .1         −           ( ) 3 1 1 1.1 .1         −           ( ) 3 1 1 1.08 .08         −           ( ) 3 1 1 1.08 .08         −          