# DUKE UNIVERSITY Fuqua School of Business. FINANCE CORPORATE FINANCE Problem Set #1 Prof. Simon Gervais Fall 2011 Term 2.

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1 DUKE UNIVERSITY Fuqua School of Business FINANCE CORPORATE FINANCE Problem Set #1 Prof. Simon Gervais Fall 2011 Term 2 Questions 1. Two years ago, you put \$20,000 dollars in a savings account earning an annual interest rate of 8%. At the time, you thought that these savings would grow enough for you to buy a new car five years later (i.e., in three years from now). However, you just re-estimated the price that you will have to pay for the new car in three years at \$36,000. (a) How much more money do you need to put in your savings account now for it to grow to this new estimate in three years? (b) Now suppose that you know that the car company will offer you to pay for the car over some time. In particular, you will have the opportunity to make a down-payment of \$12,000 at the time you get the car (three years from now) and to make additional payments of \$13,000 at the end of each of the following two years. With this offer, how much money do you need to add to your account now? 2. A rich entrepreneur would like to set up a foundation that will pay a scholarship to one deserving student every year. The first such scholarship will pay \$5,000 (in nominal terms) and is to be awarded in three years from now. A scholarship will be awarded in perpetuity every year after that (even after the entrepreneur s death), but will be indexed at 2% a year to account for inflation. How much money should the entrepreneur put in the foundation s account, if that account earns 10% a year? 3. Your parents make you the following offer. They will give you \$10,000 at the end of every year for the next five years if you agree to pay them back \$10,000 at the end of every year for the following ten years. Should you accept this offer if your opportunity cost (discount rate) is 12% a year? 4. You have a firm that starts out with \$60,000 in cash in the bank. You have three investment opportunities: (1) You can invest \$30,000 today and get back a gross payoff of \$45,000 next year; (2) you can invest \$20,000 today and get back a payoff of \$24,000 next year; (3) you can invest \$10,000 today and get back a payoff of \$20,000 next year. You can undertake any or all of these investment opportunities. (a) Suppose the interest rate is 30%. What investment policy do you undertake, and what is the value of your firm today after you announce your investment policy? (b) Suppose you want to consume \$30,000 today and the rest next year. How much money do you need to borrow or lend? How much can you consume tomorrow? (c) Suppose a fourth investment opportunity (4) is added to the problem: invest \$15,000 today and get back \$50,000 next year. In which projects should you invest? 1

2 5. In the following figure, the sloping straight line represents the opportunities for investment in the capital market, and the solid curved line represents the opportunities for investment in plant and machinery (real assets). The company s only asset at present is \$2.6 million in cash. \$ tomorrow Owner's preferred consumption patttern \$ today (a) What is the interest rate? (b) How much should the company invest in plant and machinery? (c) How much will this investment be worth next year? (d) What is the average rate of return on the investment? (e) What is the marginal rate of return? (f) What is the present value of this investment? (g) What is the net present value of this investment? (h) What is the total present value of the company? (i) How much will the individual consume today? (j) How much will he consume tomorrow? (k) Is he a borrower or a lender? 6. Draw a figure (like the ones we drew for Mr. Rossi in the course s first lecture) to represent the following situation. A firm starts out with \$10 million in cash. The rate of interest r is 10%. To maximize net present value, the firm invests today \$6 million in real assets. This leaves \$4 million which can be paid out to shareholders. The net present value of the investment is \$2 million. When you have finished, answer the following questions. (a) How much cash is the firm going to receive in year 1 from its investment? (b) What is the marginal return from the firm s investment? 2

3 (c) What is shareholders wealth (today) after the firm has announced its investment plan? (d) What is the present value of the projects cash flows? (e) Suppose shareholders want to spend \$6 million today. How can they do this? (f) How much will they then have to spend next year? 7. Sonos is in the process of assessing the attractiveness of a new project. The project has an estimated life of three years. The project s revenue estimates are as follows: Sales = 50,000 units/year. Per unit price: \$100 in year 1, \$110 in year 2, \$130 in year 3. The project s cost estimates are as follows: Up-front for new equipment = \$2,400,000. Annual overhead = \$1,300,000. Per unit cost = \$50. The expected life of the new equipment is 4 years, and it will be depreciated straight-line over that time. Because the project will be over by the end of year 3 however, Sonos plans to resell this equipment for \$800,000 at the end of year 3. In terms of net working capital, the following assumptions have been agreed upon: The project will have no cash or inventory requirement (i.e., manufactured products are shipped directly to customers). Payables are expected to be 10% of annual COGS (as Sonos may purchase some of the products necessary for production on credit). Receivables are expected to be 10% of annual sales (as customers may take some time to pay for the goods, and so not all cash flows are received by year end). Net working capital will fall back to zero in year 4 (i.e., all outstanding payments from customers are received, and all outstanding bills are paid that year). What is the net present value of this project if the corporate tax rate is 35% and the cost of capital (i.e., discount rate) is 16%? 8. The firm Nalyd is considering an investment in equipment to produce a new product. The cost of the equipment is \$150,000. This equipment falls into the 5-year asset class and thus would have to be capitalized and depreciated over 6 years at rates 20%, 32%, 19.2%, 11.52%, 11.52%, and 5.76%. Nalyd expects to use the equipment for three years and then to sell it for \$60,000. For the three years of operation, the equipment will generate revenues of \$40,000 per year and will have operating costs of \$3,000 per year. If the opportunity cost of capital for Nalyd is 12% and its tax rate is 35%, should Nalyd purchase this equipment? For simplicity, assume that net working capital stays at zero throughout the project s life. Also, assume that Nalyd s other projects are and will remain profitable, so that any negative tax on this one project can be considered a positive cash flow. 3

4 (Difficult) 9. Conocococonut is considering the purchase of a new harvester. They are currently involved in deliberations with the manufacturer and as of yet the parties have not come to a settlement regarding the final purchase price. The management of Conocococonut has hired you, a highpriced consultant, to establish the maximum price it should be willing to pay for the harvester (i.e., the break-even price such that the NPV of the project would be zero). This will be of obvious use to Conocococonut in their haggling with the capital equipment manufacturer. You are given the following facts: The new harvester would replace an existing one that has a current market value of \$20,000. The new harvester is not expected to affect revenues, but before-tax operating costs will be reduced by \$10,000 per year for ten years. The old harvester is now five years old. It is expected to last for another ten years and to have a resale value of \$1,000 at the end of those ten years. The old harvester was purchased for \$50,000 and is being depreciated to zero on a straight-line basis over ten years. The new harvester will be depreciated straight-line over its 10 year life to a zero book value. Conocococonut expects to be able to sell the harvester for \$5,000 at the end of the ten years. The corporate tax rate is 34% and the firm s cost of capital is 15%. For simplicity, assume that net working capital stays at zero throughout. Also, if you solve this problem using a spreadsheet feel free to assume that annual revenues are \$50,000 (for both harvesters) and that annual costs are \$20,000 (\$10,000) for the old (new) harvester. (Optional) 10. (Mini-case, taken from Ross, Westerfield and Jaffe, 6th edition) After extensive research and development, Goodweek Tires, Inc., has recently developed a new tire, the SuperTread, and must decide whether to make the investment necessary to produce and market the SuperTread. Thetirewouldbeideal fordriversdoingalarge amount ofwet weather andoff-road driving in addition to its normal freeway usage. The research and development costs so far total about \$10 million. The SuperTread would be put on the market beginning this year and Goodweek expects it to stay on the market for a total of four years. Test marketing costing \$5 million shows that there is a significant market for a SuperTread-type tire. As a financial analyst at Goodweek Tires, you are asked by your CFO, Mr. Adam Smith, to evaluate the SuperTread project and provide a recommendation on whether to go ahead with the investment. You are informed that all previous investments in the SuperTread are sunk costs and only future cash flows should be considered. Except for the initial investment (in production equipment and working capital) which will occur immediately, assume all cash flows will occur at year-end. Goodweek must initially invest \$120 million in production equipment to make the SuperTread. The equipment is expected to have a seven-year useful life. This equipment can be sold for \$51,428,571 at the end of four years. Goodweek intends to sell the SuperTread to two distinct markets: 4

5 (a) The Original Equipment Manufacturer (OEM) Market. The OEM market consists primarily of the large automobile companies (e.g., General Motors) who buy tires for new cars. In the OEM market, the SuperTread is expected to sell for \$36 per tire. The variable cost to produce each tire is \$18. (b) The Replacement Market. The replacement market consists of all tires purchased after the automobile has left the factory. This market allows higher margins and Goodweek expects to sell the SuperTread for \$59 per tire there. Variable costs are the same as in the OEM market. Goodweek Tires intends to raise prices at 1 percent above the inflation rate. 1 Variable costs will also increase 1 percent above the inflation rate. In addition, the SuperTread project will incur \$25 million in marketing and general administration costs the first year (this figure is expected to increase at the inflation rate in the subsequent years). Goodweek s corporate tax rate is 40 percent. Annual inflation is expected to remain constant at 3.25 percent. The company uses a 15.9 percent discount rate to evaluate new product decisions. Automotive industry analysts expect automobile manufacturers to produce 2 million new cars this year and production to grow at 2.5 percent per year thereafter. Each new car needs four tires (the spare tires are undersized and are in a different category). Goodweek Tires expects the SuperTread to capture 11 percent of the OEM market. Industry analysts estimate that the replacement tire market size will be 14 million tires this year and that it will grow at 2 percent annually. Goodweek expects the SuperTread to capture an 8 percent market share. You decide to use the MACRS depreciation schedule (seven-year property class), which is as follows. Year Depreciation % 14.29% 24.49% 17.49% 12.49% 8.93% 8.93% 8.93% 4.45% You also decide to consider net working capital (NWC) requirements in this scenario. The immediate initial working capital requirement is \$11 million, and thereafter the net working capital requirements will be 15 percent of sales. What will be the NPV and IRR on this project? Note: You should use a spreadsheet to solve this problem. 1 Feel free to use g = i + 1% or g = (1 + i)(1.01) 1, where i is the inflation rate. I will use the latter in my solution. 5

6 Solutions 1. (a) Let us first figure out how much money (FV) is now in the account ,000 (1.08) 2 FV FV = 20,000(1.08) 2 = 23, Now, the account should have an amount PV in it for it to grow to \$36,000 in three years PV PV = 36,000 (1.08) 3 = 28, (1.08) 3 36,000 So, you need to put \$28, \$23, = \$5, in the account. (b) The present value (at time 0) of these three payments is PV = 12,000 (1.08) ,000 (1.08) ,000 (1.08) 5 = 27, So, you need to add \$27, \$23, = \$4, to the account. 2. First, let us calculate how much money will need to be in the account at the end of the second year; let us denote that amount by PV ,000 5,000(1.02) 5,000(1.02) 2 PV = 5, For the account to be worth this much in two years, the amount that the entrepreneur needs to contribute initially is PV = PV 3. The present value of what you get is given by ( 1 PV + = 10, (1.10) 2 = 5, (1.10) 2 = 51, (1.12) 5 ) = 36, The present value of what you will have to pay back is given by PV = 10,000 ( ) (1.12) 10 (1.12) 5 = 32, Since the present value of the money you will get is larger than that you will have to pay back (PV + > PV ), you should accept the offer. 6

7 4. (a) Compute the rate of returns on the three projects: r 1 = 45,000 30,000 30,000 r 2 = 24,000 20,000 20,000 r 3 = 20,000 10,000 10,000 = 50% > 30%, = 20% < 30%, = 100% > 30%. You should therefore invest in (1) and (3) and have \$20,000 for today s consumption. The projects NPVs are NPV 1 = 30, , = 4,615, NPV 2 = 20, , = 1,538, NPV 3 = 10, , = 5,385, After your announcement of the selection of projects (1) and (3) (before consumption), the firm should be worth 60,000+NPV 1 +NPV 3 = 70,000. In other words, this is how much you can sell the firm for at the time. (b) From the answer to part (a), you need to borrow an extra 30,000 20,000 = 10,000 in order to achieve the consumption goal of \$30,000 today. (You should not forego any of the positive NPV projects, instead you can make use of the capital market to adjust your consumption.) Tomorrow s consumption level is 45,000+20,000 (10, ) = 52,000. (c) Also invest in project (4), in addition to projects 1 and 3: r 4 = 50,000 15,000 15,000 = %. 5. (a) = 25%. (b) \$2.6 million \$1.6 million = \$1.0 million. (c) \$3 million. (d) = 200%. (e) 25%. (f) \$4 million \$1.6 million = \$2.4 million. (g) \$2.4 million \$1 million = \$4 million \$2.6 million = \$1.4 million. (h) \$4 million. (i) \$1 million. 7

8 (j) \$3.75 million. (k) He is a lender (since he consumes less than \$1.6 million today). 6. The following figure shows the solution to the first part of the question. \$million period Real investment opportunities Real investment NPV \$million period 0 Here is how the numbers in that figure are calculated: (a) (1.1) = \$8.8 millions. 4 = 10 6; 11 = 10(1+r) = 10(1.1); 12 = 10+NPV = 10+2; 13.2 = 12(1+r) = 12(1.1); 8.8 = (12 4)(1+r) = 8(1.1). (b) The firm will invest (in real assets) until its marginal rate of return (from these assets) is equal to the interest rate, i.e. the marginal return is 10%. (c) 10+NPV = \$12 millions. (d) The net present value is NPV = PV(future cash flow from projects) I. We know that I = 6, and we know that NPV = 2. This implies that PV(future cash flow from projects) = NPV +I = 2+6 = 8. (e) Let the firm invest \$6 millions in real assets; this leaves the shareholders with \$4 millions today. Then borrow \$2 millions at 10% from period 1 (next year) consumption using the capital markets; this provides shareholders with an additional \$2 millions today, for a total of \$6 millions. 8

9 (f) Next year, the firm s real investments will have generated \$8.8 millions, but the shareholders owe 2(1.1) = \$2.2 millions. Their consumption (spending) in period 1 will therefore be = \$6.6 millions. 7. We can forecast the unlevered net income for this project as follows (all numbers in 000): End of year Sales 5,000 5,500 6,500 Cost of Goods Sold -2,500-2,500-2,500 Gross Profit 2,500 3,000 4,000 General & Administrative -1,300-1,300-1,300 Depreciation EBIT 600 1,100 2,100 Income Tax (35%) Unlevered Net Income ,365 To calculate the project s free cash flows, we need to add back depreciation, subtract capital expenditures, and subtract annual changes in net working capital. The initial capital expenditure is \$2,400,000. In year 3, the equipment sale will generate a positive cash flow of \$800,000. The excess over the book value of the machine (2.4M 0.6M 0.6M 0.6M = 600,000) is considered a taxable gain. The tax is 35% (800, ,000) = 70,000. Therefore, the net capital inflow from the sale of the machine in year 3 is 800,000 70,000 = 730,000. The following table shows how the annual changes in net working capital are calculated: End of year Cash Requirements Plus: Inventory Plus: Receivables Minus: Payables Net Working Capital Increase in NWC

10 The project s free cash flows are therefore as follows: End of year Unlevered Net Income ,365 Plus: Depreciation Minus: CapEx -2, Minus: Increase in NWC Free Cash Flow -2, ,265 2, The project s net present value is NPV = 2, ,265 (1.16) 2 + 2,595 (1.16) (1.16) 4 = 1, The solution spreadsheet is available on the Downloads section of the course s website. As shown in this spreadsheet, the project s free flows are as follows: End of year Revenues 40,000 40,000 40,000 Operating Costs -3,000-3,000-3,000 Depreciation -30,000-48,000-28,000 EBIT 7,000-11,000 8,200 Income Tax (35%) -2,450 3,850-2,870 Unlevered Net Income 4,550-7,150 5,330 Depreciation 30,000 48,000 28,800 Capital Expenditures -150,000 54,120 Free Cash Flow -150,000 34,550 40,850 88,250 The project s net present value is NPV = 150, , ,850 (1.12) ,250 (1.12) 3 = 23,772. Alternatively, one could calculate the project s NPV as NPV = cost of equipment+pv(after-tax net operating profits) +PV(depreciation tax shield)+pv(equipment sale) PV(tax on equipment sale). 10

11 These PV s are calculated as follows: [ ( ) ] 40,000 3, PV(after-tax net operating profits) = (1 0.35) 1 = 57,764.04; [ 0.2 P V(depreciation tax shield) = 0.35(150,000) (1.12) ] (1.12) 3 = 29,942.60; PV(equipment sale) = 60,000 (1.12) 3 = 42,706.81; PV(tax on equipment sale) = 0.35[ 60,000 ( )150,000 ] (1.12) 3 = 4, Again, this gives us NPV = 23, < 0. So Nalyd should not purchase this equipment. 9. The solution spreadsheet is available on the Downloads section of the course s website. To find the break-even price (of 60,019) using Excel, we make use of Goal Seek : Without Excel, the calculations are as follows. The current book value of the old harvester is ( ) 50,000 50,000 5 = 25, The incremental cash flow at t = 0 if the new harvester is purchased is therefore C 0 = 20, (25,000 20,000) P = 21,700 P where we have have taken into account the tax effect of the book loss on the sale of the old harvester. The incremental cash flow between years 1 and 5 equals the after-tax cost savings plus the tax effect of the incremental depreciation: ( ) P C t = (1 0.34)10, ,000 = 4, P (t = 1,...,5) After year 5, the old harvester would have been completely depreciated, so that the incremental cash flow between years 6 and 9 is: C t = (1 0.34)10, P 10 = 6, P (t = 6,...,9) Finally, the incremental cash flow in year 10 reflects the incremental salvage value of the new harvester and the consequent tax increase: C 10 = (1 0.34)10, P +(5,000 1,000) 0.34(5,000 1,000) 10 = 9, P 11

12 The NPV of the project is then: NPV = 10 t=0 C t 1.15 t = 21,700 P + 4, P 0.15 ( 1 + 6, P 0.15 (1.15) 5 = 49, P. ( 1 ) 1 (1.15) 5 ) 1 (1.15) 4 + 9, P (1.15) 10 Setting NP V = 0 and solving for P gives P = 60,019. This is the maximum price that Conocococonut would be willing to pay for the new harvester. 10. See solution spreadsheet posted on the Downloads section of the course s website. 12

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