CORPORATE FINANCE 1 FEBRUARY 2008 ANSWERS TO EXERCISES & PROBLEMS

I. REVIEW: TIME VALUE OF MONEY & DCF Time Value of Money 1. You can buy a 1-year T-bill for $955 now. The bill will pay $1,000 one year from now. What is your rate of return on this investment? 2. What is the PV of €100 received in 3 years at 5%, 10% and 20%. 3. What interest rate would turn 110 into 121 in 1 year? 4. Neon Danders, a professional football player will be paid $15,000,000 by his NFL franchise. He will be paid $5,000,000 now and $10,000,000 1 year from now. Using a discount rate of 12 percent, calculate the PV of Danders’ pay. 5. Calculate the discount factors for the following discount rates (for one year): (a) 20 percent, (b) 30 percent, and (c) 100 percent. 6. How much would €1,000 be worth in 5 years, invested at 5%, 10% and 20%. Why is the sum invested at 20% not double that invested at 10%? 7. At 7%, would you rather have €100 today or €131.1 in 4 years’ time? Why?

SOLUTIONS: Time Value of Money 1.

($1,000-$955)/$955 = 4.71 percent

2.

PV of € 100 received in 3 years

@ 5%: @ 10%: @ 20%:

PV = 100/(1.05)3 = € 86.38 PV = 100/(1.10)3 = € 75.13 PV =100/(1.20)3 = € 57.87

3.

FV = PV × (1 + INT) INT = (FV – PV)/PV = (121 – 110)/110 = 10%

4.

$5,000,000 + PV of $10,000,000 received 1 year from now = $5,000,000 + $10,000,000/1.12 = $13,928,571.14

5.

(a) 1/1.2 = 0.833---- (b) 1/1.3 = 0.769---- (c) 1/2 = 0.5

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6. Answers: €1,276 ; €1,611 ; €2,488. The sum invested at 20% is not the double of that invested at 10%, because the principal (€1,000) remains the same and interest more than doubles as a result of the process of compound interest. 7. Answer: It makes no difference, because €100 invested at 7% a year would be worth €131.1 in 4 years.

Discounted Cash Flows 1. John Miser has €20,000 to invest. He is considering two projects. Project A requires an investment of €10,000 and will pay €11,000 after one year. Project B also requires €10,000 but will pay only €10,700 after one year. What should Mr. Miser do if his opportunity cost of capital is: (a) 6 percent, (b) 9 percent, or (c) 12 percent? 2. ABC Corp. has opportunities to invest in three projects with potential payoffs next year linked to the state of the economy. Each project requires an investment of €3 million. The payoffs for different states of the economy are given below:

Projects

State of the Economy and Project Payoffs

Discount Rate

Recession

Normal

Boom

A

€2 million

€4 million

€6 million

20 percent

B

€3 million

€3.5 million

€4 million

10 percent

C

€1.5 million

€4 million

€6.5 million

35 percent

a. Assume in your answer that each state of the economy has the same probability to occur. Calculate the expected cash flows next year, the rate of return, and the NPV for each project. Which project(s) should be accepted by the company? b. Same question if the probabilities associated with the recession, normal and boom economic scenarios were 20%, 50% and 30% respectively. 3. A factory costs €400,000. It will produce an inflow after operating costs of €100,000 in year 1, €200,000 in year 2, and €300,000 in year 3. The opportunity cost of capital is 12%. Calculate the NPV. 4. You are considering a renewal of your subscription to Guitar Player Magazine. You have to pay in advance for any subscription renewals. A three-year subscription would be like three annual subscriptions but the payments would have to be at the beginning of the year. You assume that the annual subscription rate will remain at €14. Also, you currently have a savings account that is paying you a 6% annual rate. The magazine is offering the following subscription alternatives: 1-year renewal: €14.00 2-year renewal: €26.00 3-year renewal: €32.00

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Assuming that you plan to continue reading the magazine for at least 3 years and that the annual subscription rate in future years will remain at €14, which alternative is the most financially sound? A. Subscribe for now and then subscribe again at the beginning of each of the next two years. B. Subscribe for 2 years now and then subscribe again for the third year in two years. C. Subscribe for all 3 years now.

SOLUTIONS: Discounted Cash Flows 1.

Calculate the NPVs at each cost of capital for both projects. a.

NPV of project A at 6% NPV of project B at 6% Accept both projects.

= (€11,000/1.06) - €10,000 = €377.36 = (€10,700/1.06) - €10,000 = €94.34

b.

NPV of project A at 9% NPV of project B at 9% Accept A, reject B.

= (€11,000/1.09) - €10,000 = €91.74 = (€10,700/1.09) - €10,000 = -€183.49

c.

NPV of project A at 12% NPV of project B at 12% Reject both projects.

= (€11,000/1.12) - €10,000 = -€178.57 = (€10,700/1.12) - €10,000 = -€446.43

Alternative: calculate rate of return on each project. Project A: 10%, project B 7%. Leads to the same answers.

2.

a. Same probability for each scenario Project A: Expected CF next year = (€2m + €4m + €6m)/3 = €4m PV of cash flow = €4m/1.20 = €3.333m; NPV = €0.333m Project B:

Expected CF next year = (€3m + €3.5m + €4m)/3 = € 3.5m PV of cash flow = €3.5m/1.10 = €3.182m; NPV = € 0.182m

Project C:

Expected cash flow next year = (€1.5m + €4m + €6.5m)/3 = €4m PV of cash flow = €4m/1.35 = €2.963m; NPV = - €0.037m

Accept projects A and B. Reject C.

b. probabilities of 20%, 50% and 30% respectively for recession, normal and boom

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Project A: Project B: Project C:

Expected CF next year = (€2m x 0.2 + €4m x 0.3 + €6m x 0.5)= €4.2m PV of cash flow = €4.2m/1.20 = €3.5m; NPV = €0.5m Expected CF next year = (€3m x 0.2 + €3.5m x 0.5 + €4m x 0.3) = €3.55m PV of cash flow = €3.55m/1.10 = €3.227m; NPV = €0.227m Expected CF next year = (€1.5m x 0.2 + €4m x 0.5 + €6.5m x 0.3) = €4.25m PV of cash flow = €4.25m/1.35 = €3.148m; NPV = €0.148 m

3. − 400,000 +

100,000 200 ,000 300,000 + + = 62.26 1.12 1.12 2 1.12 3

4. Correct answer: C PV of the €14 payments at the beginning of each period = € 39.67 PV of €26 payment now + €14 in two years = € 38.46 Subscribe for all three years now = € 32 After C favor B over A

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II. REVIEW: ANNUITIES & PERPUITIES 1. I will receive from my late uncle’s estate €40 in 1 year’s time and annually thereafter in perpetuity. What is the value of this perpetuity at an interest rate of (a) 8 percent and (b) 10 percent? 2. How much is the previous perpetuity worth if it begins in 5 years’ time instead of 1? 3. I now discover that my uncle’s will provides that I receive €40 in 1 year’s time and that this amount is to be increased annually at a rate of 6 percent. What is the present value of this growing stream of income at an interest rate of (a) 8 percent and (b) 10 percent? 4. What is the present value at 10% of €100 paid annually for 3 years? 5. You have the opportunity to buy the right to park in a given parking place for 75 years, at a price of €30,000. You could also rent a parking place for €2,000 a year. If the opportunity cost is 5%, what would you choose? 6. You have found your dream house and you have the choice between renting it with a lease in perpetuity for €12,000 per year (paid at the beginning of each year) or buying it. At what purchase price would you be better off renting, if the loan you needed to buy the house costs you 7%, and the rent increases by 3% per year? 7. I took a car loan of $20,000 from my bank. The loan carries an annual interest rate of 12 percent and has to be repaid in 30 equal monthly payments. Calculate the monthly payment. 8. You can buy a color television set for either €320 cash or with the following credit terms to a price of only: only €20 down now and 18 monthly payments of €20. (a) Is this an attractive proposition if I can borrow at 1 percent per month? (b) What monthly interest rate is being charged? (c) What annual rate is being charged? 9. What annually compounded rate is equivalent to an interest rate of 12 percent compounded: (a) semi-annually, (b) quarterly, (c) monthly, (d) weekly (or 52 times a year), (e) daily (365 times a year)?

SOLUTIONS: Annuities & Perpetuities 1. a. PV of perpetuity = C/r = $40/0.08 = $500 b. PV of perpetuity = $40/0.10 = $400 2. Using Table 1, PV factor for 5 years, 8 percent = 0.681 Present value of the perpetuity = 0.681 x $500 = $340.50 3. a. This is a growing perpetuity. PC = C/(r-g) = $40/(0.08 - 0.06) = $2,000 b. If r = 10 percent, PV = $40/(0.1 - 0.06) = $1,000 4. PV = €100/1.10 + €100/(1.10)2 + €100/(1.10)3 = € 248.69 5. €2,000 over 75 years growing at 2% would be worth €38,969.94. So, it would be better to buy. PV (rent) = €2,000 × [(1 / (1+5%)) - (1 / (5% × (1+5)75)) = €38,969.94 PV (purchase) = €30,000 EDC 4YFIN – Extra Session – Answers to Exercises & Problems (VF) Feb 2008

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6. PV = €300,000 = €12,000 / (7% - 3%). 7. Calculator solution: N = 30, I = 1, PV = $20,000, FV = 0, PMT = solve = -$774.96. You can also do this using the Appendix Table 3 interpreting the 12 percent annual interest rate as 1 percent/month and finding the factor for 30 "years". Annuity PV factor from Table 3 for 30 periods and 1 percent = 25.81. Payment = $20,000/25.81 = $774.89

8.

a) You are taking a loan of $320-$20 = $300. If your interest rate is 1 percent per month, your payment will be $300 divided by the annuity factor for 18 periods, 1 percent (Table 3). Annuity factor for 18 periods, 1 percent = 16.40; Monthly payment = $300/16.40 = $18.29. Since you are required to pay more than this amount, this is not an attractive proposition. b) For payment of $20/month, the annuity factor has to be = $300/$20 = 15. This is very close to the factor for 18 periods and 2 percent (14.99). So the effective interest being charged is 2 percent/month. c) Effective annual rate = (1.02)12 – 1 = 26.82 percent.

9. The equivalent annually compounded rate or the effective annual rate (EAR) is given by the formula (1+i/m)m – 1, where i is the nominal annual rate or the annual percentage rate (APR) and m is the frequency of compounding. The EAR for the different compounding frequencies is: (a) m = 2, EAR = 12.36%; (b) m = 4, EAR = 12.5509%; (c) m = 12, EAR = 12.6825%; (d) m = 52, EAR = 12.7341%; (e) m = 365, EAR = 12.7475%

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III. VALUING SECURITIES: BONDS & STOCKS (1) 1. Calculate the value of a bond which pays an 8 percent coupon, has a face value of €100, ten years until maturity, and for which the investors require a rate of return of 10 percent. 2. If the above bond is selling at a price of $107, what is the yield to maturity (use Excel)? 3. A 20-year German Government bond has a coupon of 6.5 percent and a par value of 100. The bond is currently yielding 4 percent. Calculate its price using the tables. 4. A 20-year Brazilian Government bond has a coupon of 8.75 percent and a face value of 100. It is currently trading at a price of 120. Using Excel, calculate its yield to maturity. 5. Based on the class discussion, can you think of one factor that could explain some of the difference between the yield of the 20-year German and Brazilian government bonds in the two preceding problems? 6. You just paid €20 for a share and expect to sell it in one year for €25. The share will pay a dividend of €2 in one year. a. What is your expected return on this investment? b. We are now one year later, and the company has just published results lower than expected. You receive the €2 dividend, but only manage to sell the share for €20. What actual return did you make on that investment? 7. A share is expected to pay dividends of €2, €2.5 and €3 for the next 3 years respectively. At the end of the three years, you anticipate that you will be able to sell the share for €50. What maximum price should you be willing to pay for that share, given your 13% required rate of return? 8. You can buy a share for €50. You expect that you can sell it for €55 in one year. Your required rate of return is 20%. What is the minimum dividend payment that will make this investment acceptable for you? 9. A U.S. Treasury bond of 15 years maturity, 9 percent coupon and $1000 face-value is quoted with a yield to maturity of 8 percent. (a) Calculate its correct price, given that it makes semiannual coupon payments and the quoted yield is semi-annually compounded. (b) Calculate the (incorrect) price that would have been obtained by assuming annual payments and compounding. 10. An investment promises four annual payments of €52 over the next four years. You require an 8% return. How much would you be prepared to pay for this asset? The share is currently trading at €165. Would you be prepared to buy or sell? Why? If you buy at that price, how much will you have gained? Will the rate of return on your investment be greater or less than 8%? Why? If you buy at €172, what will your return on investment be? Why?

SOLUTIONS: Bonds & Stocks (1) 1. DF (10, 10) = 0.3855; AF (10, 10) = 6.1446 P = 8 x 6.1446 + 100 x 0.3855 = 87.7068 EDC 4YFIN – Extra Session – Answers to Exercises & Problems (VF) Feb 2008

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2. 7% 3. DF (4, 20) = 0.4564; AF (4, 20) = 13.5903 P = 6.5 x 13.5903 + 100 x 0.4564 = 133.977 4. .YTM = 6.88 5. Among other things, based on the class discussion: Ratings Brazil S&P: Local currency LT Debt. BB+ / Foreign currency LT Debt. BB MOODY'S: Local currency LT Debt. Ba2 / Foreign currency LT Debt. Ba2 Other answers include: difference in currency, difference in economic cycles/outlooks… 6. . a.

(5+2)/20 = 35%

b. 2/20 = 10% 7. €40.46 8. €5 9. (a) Calculator solution: N = 2x15=30, I = 4, PMT = $45, FV = $1,000, PV = solve = -$1,086.46; (b) Incorrect price: N = 15, I = 8, PMT = $90, FV = $1,000, PV = solve = -$1,085.59 Using the Appendix tables: (a) Price = $45 x 17.29 + 1,000 x 0.308 = $1,086.05 (b) Price = $90 x 8.559 + $1,000 x 0.315 = $1,085.31 10. Date Cash Flows Required return PV of Cash flows

1 52

2 52

3 52

4 52

= 8% = 52/(1+0.08) + 52/(1+0.08)2 + 52/(1+0.08)3 + 52/(1+0.08)4 = 172.23

Market price = 165 Buy because NPV = 172.23 – 165 = 7.23 > 0 Return is greater than 8% (9.95%) If you buy at 172 : return = 8% (approximately)

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IV. VALUING SECURITIES: BONDS & STOCKS (2) 1. JBH Inc. is expected to pay dividends of $2, $3, and $5 for the next three years. Thereafter, the dividends are expected to grow at a constant rate of 8 percent. If the required rate of return is 16 percent, what will be the current stock price? What will be the stock price next year and at the end of three years? 2. M Corp. is expected to pay a dividend of $6 a share next year. The dividends are expected to grow at the rate of 6 percent annually. If the current stock price is $60, what is the implied market capitalization rate? 3. You forecast that ITT will pay a dividend of $2.40 next year and that dividends will grow at a rate of 9 percent a year. What price would you expect to see for ITT stock if the market capitalization rate is 15 percent? 4. If instead the price of ITT is $30, what market capitalization rate is implied by your forecasts of problem 3? 5. Bear Corp. shares are expected to pay a dividend of $4 next year. The dividends are expected to decrease (because the company’s sales are declining on account of an industry wide decline) at the rate of 15 percent annually. If the market capitalization rate is 15 percent, what will be the current stock price? 6. The current earnings of M & M Corp. are $5 a share, and it has just paid an annual dividend of $2. The company is expected to continue to retain 60 percent of its earnings for the next 3 years and that both earnings and dividends will grow at 20 percent a year over that period. From year 4 on, the payout ratio is expected to increase to 70 percent and the growth rate to fall to 10 percent. If the capitalization rate for this stock is 15 percent, calculate (a) its price, (b) its price-earnings ratio. 7. Big Bull Corp. is expected to pay dividends of $3, $5, and $7 for the next three years. Thereafter, the dividend is expected to grow at the rate 8 percent. If the market capitalization rate is 16 percent, calculate the current stock price and the stock price for each of the next three years. 8. Mac & Burgers Corp. has the following estimated earnings and net investments. Dollar millions Years Assets Earnings Investments

1 24.00 4.80 16

2 40.00 8.00 12.00

3 52.00 10.40 8.00

4 60.00 10.80 8.00

5 68.00 10.88 5.44

6 73.44 11.75 5.88

The company will continue a payout ratio of 50 percent beyond year 6 and earn 16 percent on equity. If the market capitalization rate is 15 percent, estimate the value of the business as the present value of free cash flows.

SOLUTIONS: Bonds & Stocks (2) 1. P0 = DIV1/(1+r) + DIV2/(1+r)2 + DIV3/(1+r)3 + P3/(1+r)3 ; where P3 = DIV4/(r-g) = 5 x 1.08/(0.16 - 0.08) = $67.50; P0 = (2/1.16) + (3/1.162) + (5 + 67.5)/1.163 = 50.40; P1 = (3/1.16) + (5 + 67.5)/1.162 = $56.47 EDC 4YFIN – Extra Session – Answers to Exercises & Problems (VF) Feb 2008

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2. r = DIV1/P + g = $6/$60 + 0.06 = 0.16 = 16% 3. P0 = DIV1/(r-g) = $2.4/(0.15 - 0.09) = $40 4. r = DIV1/P + g = $2.40/$30 + 0.09 = 0.17 = 17% 5. P0 = $4/[0.15-(-0.15)] = $13.33 6. DIV0 = $2, DIV1 = $2 x 1.2 = $2.40, DIV2 = $2.40 x 1.2 = $2.88, DIV3 = $2.88 x 1.2 = 3.46 EPS4 = $5 x 1.23 x 1.1 = $9.50, DIV4 = $9.50 x 0.7 = $6.65, P3 = $6.65/(0.15 - 0.1) = $133 P0 = (2.4/1.15) + ($2.88/1.153) + [($3.46 + $133)/1.153] = $93.98, P/E = $93.98/2 = $46.99 PVGO = $93.98 – (EPS1/r) = $93.98 - ($6/0.15) = $53.98 7. DIV4 = $7 x 1.08 = $7.56, P3 = $7.56/(0.16 - 0.08) = $94.50, P0 = $3/1.16 + ($5/1.162) + [($7 + $94.50)/1.163] = $71.33, P1 = $5/1.16 + ($101.5/1.162) = $79.74, P2 = $101.5/1.16 = $87.50 8. Mac & Burgers Corp. has the following estimated earnings and net investments. Dollar millions Years Assets Earnings Investments

1 24.00 4.80 16

2 40.00 8.00 12.00

3 52.00 10.40 8.00

4 60.00 10.80 8.00

5 68.00 10.88 5.44

6 73.44 11.75 5.88

The company will continue a payout ratio of 50 percent beyond year 6 and earn 16 percent on the assets. If the market capitalization rate is 15 percent, estimate the value of the business as the present value of free cash flows. The earnings, investments, and free cash flows (in $ millions) for the first 5 years are given in the table below. From year six, the free cash flows are growing at 8 percent. We can calculate the horizon value at the end of year 5 and then discount all cash flows and the horizon value to the present at the market capitalization rate of 15 percent. Years Earnings Investments Free cash flow

1 4.80 16.00 -11.20

2 8.00 12.00 -4.00

3 10.40 8.00 2.40

4 10.80 8.00 2.80

5 10.88 5.44 5.44

Value of cash flows from year 6 onwards = Free cash flow6)/(r-g) = (5.44x1.08)/(0.15-0.08) = 5.88/0.07 = $84 million Value of the business = PV(Free cash flows) − 11.20 −4 2.40 2.80 5.44 + 84 + 2 + 3 + 4 + 115 . (115 . ) (115 . ) (115 . ) (115 . )5 = $34.88 million =

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