# SUMMARY AND CONCLUSIONS

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1 CHAPTER 6 Discounted Cash Flow Valuation 187 CONCEPT QUESTIONS 6.4a What is a pure discount loan? An interest-only loan? 6.4b What does it mean to amortize a loan? 6.4c What is a balloon payment? How do you determine its value? SUMMARY AND CONCLUSIONS This chapter rounds out your understanding of fundamental concepts related to the time value of money and discounted cash flow valuation. Several important topics were covered, including: 1. There are two ways of calculating present and future values when there are multiple cash flows. Both approaches are straightforward extensions of our earlier analysis of single cash flows. 2. Aseries of constant cash flows that arrive or are paid at the end of each period is called an ordinary annuity, and we described some useful shortcuts for determining the present and future values of annuities. 3. Interest rates can be quoted in a variety of ways. For financial decisions, it is important that any rates being compared be first converted to effective rates. The relationship between a quoted rate, such as an annual percentage rate (APR), and an effective annual rate (EAR) is given by: EAR [1 (Quoted rate/m)] m 1 where m is the number of times during the year the money is compounded or, equivalently, the number of payments during the year. 4. Many loans are annuities. The process of providing for a loan to be paid off gradually is called amortizing the loan, and we discussed how amortization schedules are prepared and interpreted. The principles developed in this chapter will figure prominently in the chapters to come. The reason for this is that most investments, whether they involve real assets or financial assets, can be analyzed using the discounted cash flow (DCF) approach. As a result, the DCF approach is broadly applicable and widely used in practice. For example, the next two chapters show how to value bonds and stocks using an extension of the techniques presented in this chapter. Before going on, therefore, you might want to do some of the problems that follow. 6.5 Chapter Review and Self-Test Problems 6.1 Present Values with Multiple Cash Flows Afirst-round draft choice quarterback has been signed to a three-year, \$25 million contract. The details provide for an immediate cash bonus of \$2 million. The player is to receive \$5 million in salary at the end of the first year, \$8 million the next, and \$10 million at the end of the last year. Assuming a 15 percent discount rate, is this package worth \$25 million? How much is it worth? 6.2 Future Value with Multiple Cash Flows You plan to make a series of deposits in an individual retirement account. You will deposit \$1,000 today, \$2,000 in two years, and \$2,000 in five years. If you withdraw \$1,500 in three years and

2 188 PART THREE Valuation of Future Cash Flows \$1,000 in seven years, assuming no withdrawal penalties, how much will you have after eight years if the interest rate is 7 percent? What is the present value of these cash flows? 6.3 Annuity Present Value You are looking into an investment that will pay you \$12,000 per year for the next 10 years. If you require a 15 percent return, what is the most you would pay for this investment? 6.4 APR versus EAR The going rate on student loans is quoted as 8 percent APR. The terms of the loans call for monthly payments. What is the effective annual rate (EAR) on such a student loan? 6.5 It s the Principal That Matters Suppose you borrow \$10,000. You are going to repay the loan by making equal annual payments for five years. The interest rate on the loan is 14 percent per year. Prepare an amortization schedule for the loan. How much interest will you pay over the life of the loan? 6.6 Just a Little Bit Each Month You ve recently finished your MBAat the Darnit School. Naturally, you must purchase a new BMW immediately. The car costs about \$21,000. The bank quotes an interest rate of 15 percent APR for a 72-month loan with a 10 percent down payment. You plan on trading the car in for a new one in two years. What will your monthly payment be? What is the effective interest rate on the loan? What will the loan balance be when you trade the car in? Answers to Chapter Review and Self-Test Problems 6.1 Obviously, the package is not worth \$25 million because the payments are spread out over three years. The bonus is paid today, so it s worth \$2 million. The present values for the three subsequent salary payments are: (\$5/1.15) (8/ ) (10/ ) (\$5/1.15) (8/1.32) (10/1.52) \$ million The package is worth a total of \$ million. 6.2 We will calculate the future values for each of the cash flows separately and then add them up. Notice that we treat the withdrawals as negative cash flows: \$1, \$1, \$ 1, \$2, \$2, , \$1, \$1, , \$2, \$2, , \$1, \$1, , Total future value \$ 3, This value includes a small rounding error. To calculate the present value, we could discount each cash flow back to the present or we could discount back a single year at a time. However, because we already know that the future value in eight years is \$3,995.91, the easy way to get the PV is just to discount this amount back eight years: Present value \$3,995.91/ \$3,995.91/ \$

3 CHAPTER 6 Discounted Cash Flow Valuation 189 We again ignore a small rounding error. For practice, you can verify that this is what you get if you discount each cash flow back separately. 6.3 The most you would be willing to pay is the present value of \$12,000 per year for 10 years at a 15 percent discount rate. The cash flows here are in ordinary annuity form, so the relevant present value factor is: Annuity present value factor (1 Present value factor)/r [1 (1/ )]/.15 (1.2472)/ The present value of the 10 cash flows is thus: Present value \$12, \$60,225 This is the most you would pay. 6.4 Arate of 8 percent APR with monthly payments is actually 8%/12.67% per month. The EAR is thus: EAR [1 (.08/12)] % 6.5 We first need to calculate the annual payment. With a present value of \$10,000, an interest rate of 14 percent, and a term of five years, the payment can be determined from: \$10,000 Payment {[1 (1/ )]/.14} Payment Therefore, the payment is \$10,000/ \$2, (actually, it s \$2, ; this will create some small rounding errors in the following schedule). We can now prepare the amortization schedule as follows: Beginning Total Interest Principal Ending Year Balance Payment Paid Paid Balance 1 \$10, \$2, \$1, \$1, \$8, , , , , , , , , , , , , , , , , Totals \$14, \$4, \$10, The cash flows on the car loan are in annuity form, so we only need to find the payment. The interest rate is 15%/ % per month, and there are 72 months. The first thing we need is the annuity factor for 72 periods at 1.25 percent per period: Annuity present value factor (1 Present value factor)/r [1 (1/ )]/.0125 [1 (1/2.4459)]/.0125 (1.4088)/

4 190 PART THREE Valuation of Future Cash Flows The present value is the amount we finance. With a 10 percent down payment, we will be borrowing 90 percent of \$21,000, or \$18,900. So, to find the payment, we need to solve for C in the following: \$18,900 C Annuity present value factor C Rearranging things a bit, we have: C \$18,900 (1/ ) \$18, \$ Your payment is just under \$400 per month. The actual interest rate on this loan is 1.25 percent per month. Based on our work in the chapter, we can calculate the effective annual rate as: EAR (1.0125) % The effective rate is about one point higher than the quoted rate. To determine the loan balance in two years, we could amortize the loan to see what the balance is at that time. This would be fairly tedious to do by hand. Using the information already determined in this problem, we can instead simply calculate the present value of the remaining payments. After two years, we have made 24 payments, so there are payments left. What is the present value of 48 monthly payments of \$ at 1.25 percent per month? The relevant annuity factor is: Annuity present value factor (1 Present value factor)/r [1 (1/ )]/.0125 [1 (1/1.8154)]/.0125 (1.5509)/ The present value is thus: Present value \$ \$14, You will owe about \$14,360 on the loan in two years. Concepts Review and Critical Thinking Questions 1. Annuity Factors There are four pieces to an annuity present value. What are they? 2. Annuity Period As you increase the length of time involved, what happens to the present value of an annuity? What happens to the future value? 3. Interest Rates What happens to the future value of an annuity if you increase the rate r? What happens to the present value? 4. Present Value What do you think about the Tri-State Megabucks lottery discussed in the chapter advertising a \$500,000 prize when the lump-sum option is \$250,000? Is it deceptive advertising? 5. Present Value If you were an athlete negotiating a contract, would you want a big signing bonus payable immediately and smaller payments in the future, or vice versa? How about looking at it from the team s perspective?

5 CHAPTER 6 Discounted Cash Flow Valuation Present Value Suppose two athletes sign 10-year contracts for \$80 million. In one case, we re told that the \$80 million will be paid in 10 equal installments. In the other case, we re told that the \$80 million will be paid in 10 installments, but the installments will increase by 5 percent per year. Who got the better deal? 7. APR and EAR Should lending laws be changed to require lenders to report EARs instead of APRs? Why or why not? 8. Time Value On subsidized Stafford loans, a common source of financial aid for college students, interest does not begin to accrue until repayment begins. Who receives a bigger subsidy, a freshman or a senior? Explain. 9. Time Value In words, how would you go about valuing the subsidy on a subsidized Stafford loan? 10. Time Value Eligibility for a subsidized Stafford loan is based on current financial need. However, both subsidized and unsubsidized Stafford loans are repaid out of future income. Given this, do you see a possible objection to having two types? 1. Present Value and Multiple Cash Flows Mercer Shaved Ice Co. has identified an investment project with the following cash flows. If the discount rate is 10 percent, what is the present value of these cash flows? What is the present value at 18 percent? At 24 percent? Questions and Problems Basic (Questions 1 28) Year Cash Flow 1 \$1, , Present Value and Multiple Cash Flows Investment X offers to pay you \$3,000 per year for eight years, whereas Investment Y offers to pay you \$5,000 per year for four years. Which of these cash flow streams has the higher present value if the discount rate is 5 percent? If the discount rate is 22 percent? 3. Future Value and Multiple Cash Flows Rasputin, Inc., has identified an investment project with the following cash flows. If the discount rate is 8 percent, what is the future value of these cash flows in Year 4? What is the future value at a discount rate of 11 percent? At 24 percent? Year Cash Flow 1 \$ , , , Calculating Annuity Present Value An investment offers \$4,100 per year for 15 years, with the first payment occurring one year from now. If the required return is 10 percent, what is the value of the investment? What would the value be if the payments occurred for 40 years? For 75 years? Forever? 5. Calculating Annuity Cash Flows If you put up \$20,000 today in exchange for a 8.25 percent, 12-year annuity, what will the annual cash flow be?

6 192 PART THREE Valuation of Future Cash Flows Basic 6. Calculating Annuity Values Your company will generate \$75,000 in annual revenue each year for the next eight years from a new information database. The computer system needed to set up the database costs \$380,000. If you can borrow the money to buy the computer system at 7.5 percent annual interest, can you afford the new system? 7. Calculating Annuity Values If you deposit \$1,500 at the end of each of the next 20 years into an account paying 9.5 percent interest, how much money will you have in the account in 20 years? How much will you have if you make deposits for 40 years? 8. Calculating Annuity Values You want to have \$50,000 in your savings account five 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 6.2 percent interest, what amount must you deposit each year? 9. Calculating Annuity Values Biktimirov Bank offers you a \$35,000, seven-year term loan at 10 percent annual interest. What will your annual loan payment be? 10. Calculating Perpetuity Values The Perpetual Life Insurance Co. is trying to sell you an investment policy that will pay you and your heirs \$5,000 per year forever. If the required return on this investment is 9 percent, how much will you pay for the policy? 11. Calculating Perpetuity Values In the previous problem, suppose the Perpetual Life Insurance Co. told you the policy costs \$58,000. At what interest rate would this be a fair deal? 12. Calculating EAR Find the EAR in each of the following cases: Stated Rate (APR) Number of Times Compounded Effective Rate (EAR) 12% Quarterly 8 Monthly 7 Daily 16 Infinite 13. Calculating APR Find the APR, or stated rate, in each of the following cases: Stated Rate (APR) Number of Times Compounded Effective Rate (EAR) Semiannually 7.2% Monthly 9.1 Weekly 18.5 Infinite Calculating EAR First National Bank charges 9.1 percent compounded monthly on its business loans. First United Bank charges 9.2 percent compounded semiannually. As a potential borrower, which bank would you go to for a new loan? 15. Calculating APR Cannone Credit Corp. wants to earn an effective annual return on its consumer loans of 14 percent per year. The bank uses daily compounding on its loans. What interest rate is the bank required by law to report to potential borrowers? Explain why this rate is misleading to an uninformed borrower. 16. Calculating Future Values What is the future value of \$600 in 20 years assuming an interest rate of 11 percent compounded semiannually?

8 194 PART THREE Valuation of Future Cash Flows Basic 28. Discounted Cash Flow Analysis If the appropriate discount rate for the following cash flows is 11.5 percent per year, what is the present value of the cash flows? Year Cash Flow 1 \$1, , Intermediate (Questions 29 59) 29. Simple Interest versus Compound Interest First Simple Bank pays 6 percent simple interest on its investment accounts. If First Complex Bank pays interest on its accounts compounded annually, what rate should the bank set if it wants to match First Simple Bank over an investment horizon of 10 years? 30. Calculating EAR You are looking at an investment that has an effective annual rate of 14 percent. What is the effective semiannual return? The effective quarterly return? The effective monthly return? 31. Calculating Interest Expense You receive a credit card application from Shady Banks Savings and Loan offering an introductory rate of 2.90 percent per year, compounded monthly for the first six months, increasing thereafter to 15 percent compounded monthly. Assuming you transfer the \$3,000 balance from your existing credit card and make no subsequent payments, how much interest will you owe at the end of the first year? 32. Calculating the Number of Periods You are saving to buy a \$150,000 house. There are two competing banks in your area, both offering certificates of deposit yielding 5 percent. How long will it take your initial \$95,000 investment to reach the desired level at First Bank, which pays simple interest? How long at Second Bank, which compounds interest monthly? 33. Calculating Future Values You have an investment that will pay you 1.72 percent per month. How much will you have per dollar invested in one year? In two years? 34. Calculating the Number of Periods You have \$1,100 today. You need \$2,000. If you earn 1 percent per month, how many months will you wait? 35. Calculating Rates of Return Suppose an investment offers to quadruple your money in 12 months (don t believe it). What rate of return per quarter are you being offered? 36. Comparing Cash Flow Streams You ve just joined the investment banking firm of Dewey, Cheatum, and Howe. They ve offered you two different salary arrangements. You can have \$75,000 per year for the next two years, or you can have \$55,000 per year for the next two years, along with a \$30,000 signing bonus today. If the interest rate is 10 percent compounded monthly, which do you prefer? 37. Calculating Present Value of Annuities Peter Piper wants to sell you an investment contract that pays equal \$10,000 amounts at the end of each of the next 20 years. If you require an effective annual return of 9.5 percent on this investment, how much will you pay for the contract today? 38. Calculating Rates of Return You re trying to choose between two different investments, both of which have up-front costs of \$30,000. Investment G returns \$55,000 in six years. Investment H returns \$90,000 in 11 years. Which of these investments has the higher return?

10 196 PART THREE Valuation of Future Cash Flows Intermediate 48. Calculating Present Value of Annuities Congratulations! You ve just won the \$15 million first prize in the Subscriptions R Us Sweepstakes. Unfortunately, the sweepstakes will actually give you the \$15 million in \$375,000 annual installments over the next 40 years, beginning next year. If your appropriate discount rate is 11 percent per year, how much money did you really win? 49. Present Value and Multiple Cash Flows What is the present value of \$1,000 per year, at a discount rate of 12 percent, if the first payment is received 8 years from now and the last payment is received 20 years from now? 50. Variable Interest Rates A10-year annuity pays \$1,500 per month, and payments are made at the end of each month. If the interest rate is 15 percent compounded monthly for the first four years, and 12 percent compounded monthly thereafter, what is the present value of the annuity? 51. Comparing Cash Flow Streams You have your choice of two investment accounts. Investment Ais a 10-year annuity that features end-of-month \$1,000 payments and has an interest rate of 11.5 percent compounded monthly. Investment B is an 8 percent continuously compounded lump-sum investment, also good for 10 years. How much money would you need to invest in B today for it to be worth as much as Investment A10 years from now? 52. Calculating Present Value of a Perpetuity Given an interest rate of 6.5 percent per year, what is the value at date t 7 of a perpetual stream of \$500 payments that begin at date t 13? 53. Calculating EAR Alocal finance company quotes a 13 percent interest rate on one-year loans. So, if you borrow \$20,000, the interest for the year will be \$2,600. Because you must repay a total of \$22,600 in one year, the finance company requires you to pay \$22,600/12, or \$1,883.33, per month over the next 12 months. Is this a 13 percent loan? What rate would legally have to be quoted? What is the effective annual rate? 54. Calculating Future Values If today is Year 0, what is the future value of the following cash flows five years from now? What is the future value 10 years from now? Assume a discount rate of 9 percent per year. Year Cash Flow 2 \$30, , , Calculating Present Values A5-year annuity of ten \$8,000 semiannual payments will begin 9 years from now, with the first payment coming 9.5 years from now. If the discount rate is 14 percent compounded monthly, what is the value of this annuity five years from now? What is the value three years from now? What is the current value of the annuity? 56. Calculating Annuities Due As discussed in the text, an ordinary annuity assumes equal payments at the end of each period over the life of the annuity. An annuity due is the same thing except the payments occur at the beginning of each period instead. Thus, a three-year annual annuity due would have periodic payment cash flows occurring at Years 0, 1, and 2, whereas a three-year annual ordinary annuity would have periodic payment cash flows occurring at Years 1, 2, and 3.

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