Fundamentals of Corporate Finance - Chapter 5 Solutions - Business Finance PDF

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Fundamentals of Corporate Finance Third European Edition

Solutions Manual Chapter 5 MAN2089 Teaching week 4 relevant questions: Q 1-3, 5, 6, 9 Teaching week 5 relevant questions: All other questions EXCEPT 8, 25, 32, 35, 37, 39, 40

BASIC 1.

Present Value and Multiple Cash Flows Investment X offers to pay you £6,000 per year for nine years, whereas Investment Y offers to pay you £8,000 per year for six years. Which of these cash flow streams has the higher present value if the discount rate is 5 per cent? If the discount rate is 15 per cent? Answer: To find the PVA, we use the equation: PVA = C({1 – [1/(1 + r)t] } / r ) At a 5 per cent interest rate: X@5%:

PVA = £6,000{[1 – (1/1.059) ] / 0.05 } = £42,646.93

Y@5%:

PVA = £8,000{[1 – (1/1.056) ] / 0.05 } = £40,605.54

And at a 15 per cent interest rate: X@15%: PVA = £6,000{[1 – (1/1.159) ] / 0.15 } = £28,629.50 Y@15%: PVA = £8,000{[1 – (1/1.156) ] / 0.15 } = £30,275.86 Notice that the PV of project X has a greater PV at a 5 per cent interest rate, but a lower PV at a 15 per cent interest rate. The reason is that X has greater total cash flows. At a lower interest rate, the total cash flow is more important since the cost of waiting (the interest rate) is not as great. At a higher interest rate, Y is more valuable since it has larger annual cash flows. At the higher interest rate, these bigger annual cash flows coming early are more important since the cost of waiting (the interest rate) is so much greater.

2.

Future Value and Multiple Cash Flows Herlige Narren AS has identified an investment project with the following cash flows. If the discount rate is 10 per cent, what is the future value of these cash flows at the end of year 5? What is the future value at a discount rate of 15 per cent? At 20 per cent? Assume all the cash flows have occurred at the beginning of the year.

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Year

Cash flow (DKK)

1

950

2

1,190

3

1,540

4 1,905 Answer: To solve this problem, we must find the FV of each cash flow and add them. To find the FV of a lump sum, we use: FV = PV(1 + r)t FV@10% = DKK950(1.10)4 + DKK1,190(1.10)3 + DKK1,540(1.10)2 + DKK1,905(1.10) = DKK6,933.69 FV@15% = DKK950(1.15)4 + DKK1,190(1.15)3 + DKK1,540(1.15)2 + DKK1,905(1.15) = DKK7,698.80 FV@20% = DKK950(1.10)4 + DKK1,190(1.10)3 + DKK1,540(1.10)2 + DKK1,905(1.10) = DKK8,529.84

3.

Calculating Annuity Present Value An investment offers €4,000 per year for 10 years, with the first payment occurring one year from now. If the required return is 9 per cent, what is the value of the investment? What would the value be if the payments occurred for 20 years? For 50 years? For ever? Answer: To find the PVA, we use the equation:

PV = C x ATr

The value if the payments occurred for 10 years PV = C x A100.09 = €4000 x 6.42 PV = €25,670.63 The value if the payments occurred for 20 years PV = C x A200.09 = €4000 x 9.1285 PV = €36,514.18 © McGraw-Hill Education 2017

Fundamentals of Corporate Finance Third European Edition

The value if the payments occurred for 50 years PV = C x A500.09 = €4000 x 10.9617 PV = €43,846.73 The value if the payments occurred for forever

€4000 PV = 0.09 PV = €44,444.44

4.

Calculating Annuity Values A 25-year fixed-rate mortgage has monthly payments of €717 per month and a mortgage interest rate of 6.14 per cent per year compounded monthly. If a buyer purchases a home with the cash proceeds of the mortgage loan plus an additional 20 per cent deposit, what is the purchase price of the home? Assume that the additional deposit amount is 20% of the mortgage loan amount.

Answer: PV = C x ATr r= 0.0614/12 = 0.005117 T= 25x12 = 300 PV = C x A3000.005117 = €717 x 153.1602 Mortgage loan= €109,815.89 Purchase Price = Mortgage loan + Deposit (20% of mortgage loan) = €109,815.89 + [€109,815.89 x 20%] = €131,779.07 5.

Calculating Annuity Values You want to have DKr450,000 in your savings account 10 years from now, and you’re prepared to make equal annual deposits into the account at the end of each year. If the account pays 7 per cent interest, what amount must you deposit each year?

Answer: Here we have the FVA, the length of the annuity, and the interest rate. We want to calculate the annuity payment. Using the FVA equation: FVA = C{[(1 + r)t – 1] / r} Dkr450,000 = C[(1.0710 – 1) / .07] We can now solve this equation for the annuity payment. Doing so, we get: C = Dkr450,000 / 13.816448 = Dkr32,569.88 © McGraw-Hill Education 2017

Fundamentals of Corporate Finance Third European Edition

Calculating Perpetuity Values An investor purchasing a British consol is entitled to receive annual payments from the British government forever. What is the price of a consol that pays £4 annually if the next payment occurs one year from today? The market interest rate is 4 per cent. Answer: This cash flow is a perpetuity. To find the PV of a perpetuity, we use the equation: 6.

PV = C / r PV = £4 / 0.04 = €100

7.

Calculating EAR Find the EAR in each of the following cases: Stated rate (APR)

Number of times compounded

4%

Quarterly

8%

Monthly

12%

Daily

16%

Infinite

Effective rate (EAR)

Answer: For discrete compounding, to find the EAR, we use the equation: EAR = [1 + (QR / m)]m – 1 EAR = [1 + (0.04 / 4)]4 – 1

= 0.0406 or 4.06%

EAR = [1 + (0.08 / 12)]12 – 1

= 0.0830 or 8.30%

EAR = [1 + (0.12 / 365)]365 – 1 = 0.1275 or 12.75% To find the EAR with continuous compounding, we use the equation: EAR = eq – 1 EAR = e.16 – 1 = 0.1735 or 17.35%

Q8 is not relevant to MAN2089 9.

Calculating Future Values If you invest €1,000 in a savings account that pays 4 per cent every year, how long would it take to triple your money?

Answer: PV = €1,000; r = 4%; FV = 3 x PV = €3,000; t = ? PV = FV /(1 + r)t 1,000 = 3,000/(1.04)t Solve for t. By trial and error,

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t = 28 years

10. Calculating Present Values An investment will pay you £58,000 in seven years. If the appropriate discount rate is 10 per cent compounded daily, what is the present value? Answer: For this problem, we simply need to find the PV of a lump sum using the equation: PV = FV / (1 + r)t It is important to note that compounding occurs daily. To account for this, we will divide the interest rate by 365 (the number of days in a year, ignoring leap year), and multiply the number of periods by 365. Doing so, we get: PV = £58,000 / [(1 + 0.10/365)7(365)] = £28,804.71

11. Calculating Cash Flows You are planning to save for retirement over the next 30 years. To do this, you will invest £500 a month in a share account and £500 a month in a bond account. The return of the share account is expected to be 7 per cent, and the bond account will pay 4 per cent. When you retire, you will combine your money into an account with a 6 per cent return. How much can you withdraw each month from your account, assuming a 25-year withdrawal period? Answer: Step 1: Find the monthly effective interest rate on the share investment:

Find the monthly effective interest rate on the bond investment:

Step 2: Calculate future value in 30 years of share investment

T=30 x 12 = 360 C = £500 r = 0.5654% FV=£584,706.30 Calculate future value in 30 years of bond investment

T=30 x 12 = 360 C = £500 r = 0.3274% FV=£342,654.30 Total Value of investment in 30 years is £584,706.30 + £342,654.30 = £927,360.60

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Step 3: Calculate monthly rate on retirement investment

Step 4: Calculate monthly payment T = 25 x 12 = 300 months PV = £927,360.60 r = 0.4868%

C = £5,885.53

12. Calculating EAR Friendly’s Quick Loans offers you ‘two for four or I knock on your door.’ This means you get £2 today and repay £4 when you get your wages in one week (or else). What’s the effective annual return Friendly’s earns on this lending business? If you were brave enough to ask, what quoted rate would Friendly’s say you were paying? Answer: Here we are trying to find the interest rate when we know the PV and FV. Using the FV equation: FV = PV(1 + r)t £4 = £2(1 + r) r = 4/2 – 1 = 100% per week The interest rate is 100% per week. To find the QR, we multiply this rate by the number of weeks in a year, so: QR = (52)100% = 5,200% And using the equation to find the EAR: EAR = [1 + (QR / m)]m – 1 EAR = [1 + 1]52 – 1 = 450,359,962,737,049,000%

13. Calculating Annuity Future Values You are planning to make monthly deposits of £400 into a retirement account that pays 7 per cent interest compounded monthly. If your first deposit will be made one month from now, how large will your retirement account be in 40 years? Answer: This problem requires us to find the FVA. The equation to find the FVA is: FVA = C{[(1 + r)t – 1] / r} FVA = £400[{[1 + (0.07/12) ]479 – 1} / (0.07/12)] = £1,043,438.63

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14. Calculating Annuity Present Values Beginning three months from now, you want to be able to withdraw €1,200 each quarter from your bank account to cover university expenses over the next four years. If the account pays 1 per cent interest per quarter, how much do you need to have in your bank account today to meet your expense needs over the next four years? Assume the money is withdrawn at the beginning of the quarter. Answer: The cash flows are simply an annuity with four payments per year for four years, or 16 payments. We can use the PVA equation: PVA = C({1 – [1/(1 + r)t] } / r ) PVA = €1,200{[1 – (1/1.01)16] / 0.01} = €17,661.45

15. Discounted Cash Flow Analysis If the appropriate discount rate for the following cash flows is 9 per cent per year, what is the present value of the cash flows? Assume cash flows are generated at the end of the year. Year

Cash flow (£)

1

2,650

2

0

3

6,200

4

3,430

Answer: Here the cash flows are annual and the given interest rate is annual, so we can use the interest rate given. We simply find the PV of each cash flow and add them together. PV = (£2,650 / 1.09) + (£6,200 / 1.093) + (£3,430 / 1.094) = £9,648.63

INTERMEDIATE 16. Effective Annual Rate You have been charged with putting together a redundancy package for a departing executive. The executive has asked for the following: (1) The present value of the next two years’ lost pay. The executive’s current annual salary is £120,000 and historical salary growth is 5 per cent per annum. (2) £100,000 for reputation management and a non-disclosure contract. (3) £20,000 non-competition agreement. If the effective annual interest rate is 6 per cent, what is the size of the settlement? If you were the departing employee, would you like to see a higher or lower interest rate? Answer: Here, we have cash flows that would occur in the future. We need to bring these cash flows to today. (1) t=1: £120,000*1.05 = £126,000 t=2: £126,000*1.05 = £132,300

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PV = (£126,000/1.06) + (£132,300/1.062) + (£100,000 + £20,000) = £356,614.45 The employee would prefer a lower interest rate.

17. Calculating Future Values You have an investment that will pay you 1.17 per cent per month. How much will you have per euro invested in one year? In two years? Answer: We need to find the FV of a lump sum in one year and two years. It is important that we use the So:

number of months in compounding since interest is compounded monthly in this case.

FV in one year

= €1(1.0117)12 = €1.15

FV in two years = €1(1.0117)24 = €1.32 There is also another common alternative solution. We could find the EAR, and use the number of years as our compounding periods. So we will find the EAR first: EAR = (1 + 0.0117)12 – 1 = 0.1498 or 14.98% Using the EAR and the number of years to find the FV, we get: FV in one year

= €1(1.1498)1 = €1.15

FV in two years = €1(1.1498)2 = €1.32 Either method is correct and acceptable. We have simply made sure that the interest compounding period is the same as the number of periods we use to calculate the FV.

18. Calculating Rates of Return Suppose an investment offers to triple your money in 12 months (don’t believe it). Assume the rate is compounded quarterly. What rate of return per quarter are you being offered? Answer: 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 = 0.3161 or 31.61%

19. Growing Perpetuities Oasis Telephony has been working on a new hands-free telephone that clips into your ear. The new gadget has now been cleared for manufacture and development. Oasis Telephony anticipates the first annual cash flow from the phone to be €200,000, received 2 years from today. Subsequent annual cash flows will grow at 5 per cent in perpetuity. What is the present value of the phone if the discount rate is 10 per cent?

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Fundamentals of Corporate Finance Third European Edition

Answer: c

1

PV =

×

=

r–g

(1+r)

200,000

1 =€ 3,636,363.64

× 0.1 – 0.05

(1+0.1)

20. Present Value and Interest Rates What is the relationship between the value of an annuity and the level of interest rates? Suppose you just bought a 20-year annuity of £19,000 per year at the current interest rate of 8 per cent per year. What happens to the value of your investment if interest rates suddenly drop to 3 per cent? What if interest rates suddenly rise to 13 per cent? Answer: The relationship between the PVA and the interest rate is: PVA falls as r increases, and PVA rises as r decreases FVA rises as r increases, and FVA falls as r decreases The present values of £19,000 per year for 20 years at the various interest rates given are: PVA@8% = £19,000{[1 – (1/1.08)20] / 0.08}

= £186,544.80

PVA@3% = £19,000{[1 – (1/1.03)20] / 0.03}

= £282,672.02

PVA@13% = £19,000{[1 – (1/1.13)20] / 0.13} = £133,470.28

Therefore if interest rates drop to 3 per cent the PVA will increase by: £282,672.02 - £186,544.80 = £96,127.22 If interest rates rise to 13 per cent the PVA will decrease by: £186,544.80 - £133,470.28 = £53,074.52

21. Calculating Annuity Present Values You want to borrow £100,000 from your local bank to buy a new yacht. You can afford to make monthly payments of £2,000, but no more. Assuming monthly compounding, what is the highest rate you can afford on a 60-month loan? Answer: Here we are given the PVA, number of periods, and the amount of the annuity. We need to solve for the interest rate. Using the PVA equation: © McGraw-Hill Education 2017

Fundamentals of Corporate Finance Third European Edition

PVA = £100,000 = £2,000[{1 – [1 / (1 + r)]60}/ r] To find the interest rate, we need to solve this equation on a financial calculator, using a spreadsheet, or by trial and error. If you use trial and error, remember that increasing the interest rate lowers the PVA, and decreasing the interest rate increases the PVA. Using a spreadsheet, we find: r = 0.618% per month The APR is: APR = (1 + 0.00618)12 - 1 = 7.68%

22. Balloon Payments Mario Guiglini has just sold his hotel and purchased a restaurant with the proceeds. The restaurant is on the Riccione seafront in northern Italy. The cost of the restaurant to Mario is €200,000 and the seller requires a 20 per cent up-front payment. Mario is able to pay the up-front payment from the proceeds of the hotel sale. He needs to take out a mortgage and has been able to arrange one with Unicredit Bank that charges a 12 per cent APR. Mario will make equal monthly payments over the next 20 years. His first payment will be due one month from now. However, the mortgage has a 10-year balloon payment option, meaning that the balance of the loan could be paid off at the end of year 10. There were no other transaction costs or finance charges. How much will Mario’s balloon payment be in 8 years? Answer: The amount borrowed is the value of the restaurant times one minus the down payment, or: Amount borrowed = € 200,000(1 – .20) = €160,000 The monthly payments with a balloon payment loan are calculated assuming a longer amortization schedule, in this case, 20 years. The monthly return rate is: r = 1.121/12 – 1 = 0.0094 The payments based on a 20-year repayment schedule would be: PVA = €160,000 = C({1 – [1 / (1 + .0094)]144} / .0094) C = €1,693.80 Now, at time = 8, we need to find the PV of the payments which have not been made. The balloon payment will be: PVA = €1,693.80 ({1 – [1 / (1 + .0094)]12(12)} / .0094) PVA = €132,687.24

23. Interest You work for a jewellers and have sourced a good goldsmith who is able to sell you 100 ounces of gold for £100,000. You approach your two main customers. Mr Noel © McGraw-Hill Education 2017

Fundamentals of Corporate Finance Third European Edition

says he will buy the gold from you in 6 months for £104,000, whereas Ms Biggs tells you that she will be able to buy the gold from you in 2 years’ time for £116,000. What is the annual percentage rate that Mr Noel and Ms Biggs are offering you? Answer: FV = PV (1+r)T Given FV = R £ 104,000

PV = £ 100,000

T= 6 months (1/2 year)

r=?

Mr. Noel 104,000 = 100,000 (1 + r)1/2 (1 + r)1/2 = 1.04 r = 1.042 = 1.0816 – 1 = 0.0816 = 8.16% Ms. Biggs £ 116,000 = £ 100,000 (1+r)2 1.16 = (1+r)2, 1.077 = 1+r r = 7.7% You should go for Mr. Noel

24. Comparing Cash Flow You have your choice of two investment accounts. Investment A is a 20-year annuity that features end-of-month NKr12,000 payments and has an interest rate of 6 per cent compounded monthly. Investment B is an 8 per cent continuously compounded lump sum investment, also good for 15 years. How much money would you need to invest in B today for it to be worth as much as investment A 20 years from now? Answer: Here we are trying to find the Norwegian kroner amount invested today that will equal the FVA with a known interest rate, and payments. First we need to determine how much we would have in the annuity account. Finding the FV of the annuity, we get: FVA = Nkr12,000 [{[ 1 + (0.06/12)]240 – 1} / (0.06/12)] = Nkr5,544,490.74 Now we need to find the PV of a lump sum that will give us the same FV. So, using the FV of a lump sum with continuous compounding, we get: FV = Nkr5,544,490.74= PVe.08(15) PV = Nkr5,544,490.74e–1.20 = Nkr1,669,968.52

Question 25 is not relevant for MAN2089 26. Annuities You have just read a life-enhancing book that tells you that if you believe things will happen, they will! You decide that you want to become a millionaire by the time you are 65. You have just turned 22 and you decide to play the stock market. Your fantastic corporate finance textbook leads you to believe that you can earn 11.8 per cent per annum from investing in equities. How much must you invest each year in order to reali...