BD IM 4e c04 - BD IM 4e c04 PDF

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Chapter 4 The Time Value of Money I. Chapter Outline The following chapter outline is correlated to the PowerPoint Lecture Slides. The PowerPoint slides are referenced in bold. Alternative Examples to selected textbook examples are also available in the PowerPoint Lecture Slides and are also referenced in bold. 4.1 The Timeline (Slides 8–11) • Example 4.1 Constructing a Timeline (Slides 12–13) • PowerPoint Alternative Example 4.1 (Slides 14–15) 4.2 The Three Rules of Time Travel (Slide 16) • Rule 1: Comparing and Combining Values (Slide 17) • Rule 2: Moving Cash Flows Forward in Time (Slides 18-19) • Figure 4.1 The Composition of Interest over Time (Slide 25) • Example 4.2 The Power of Compounding (Slides 26–28) • PowerPoint Alternative Example 4.2 (Slides 29–31) • Rule 3: Moving Cash Flows Back in Time (Slide 32) • Example 4.3 Present Value of a Single Future Cash Flow (Slides 33–35) • PowerPoint Alternative Example 4.3 (Slides 36-38) • Applying the Rules of Time Travel (Slides 39–43) • Table 4.1 The Three Rules of Time Travel (Slide 44) • Example 4.4 Computing the Future Value (Slides 45–47) • PowerPoint Alternative Example 4.4 (Slides 48–51) 4.3 Valuing a Stream of Cash Flows (Slides 52–53) • Example 4.5 Present Value of a Stream of Cash Flows (Slides 54–56) • PowerPoint Alternative Example 4.5 (Slides 57–58) 4.4 Calculating the Net Present Value (Slide 60) • Example 4.6 Net Present Value of an Investment Opportunity (Slides 61–63) • PowerPoint Alternative Example 4.6 (Slides 64–66) 4.5 Perpetuities and Annuities (Slides 67–68) • Perpetuities (Slides 67–68) • Example 4.7 Endowing a Perpetuity (Slides 69–70) • PowerPoint Alternative Example 4.7 (Slides 71–72) • Common Mistake: Discounting One Too Many Times • Annuities (Slide 73) – Present Value of an Annuity (Slides 74–76)

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• Example 4.8 Present Value of an Annuity Due (Slides 77–80) – Future Value of an Annuity (Slide 81) • Example 4.9 Retirement Savings Plan Annuity (Slides 82–85) • Growing Cash Flows (Slide 86) – Growing Perpetuity (Slide 86) • Example 4.10 Endowing a Growing Perpetuity (Slides 87–88) • PowerPoint Alternative Example 4.10 (Slides 89–90) – Growing Annuity (Slide 91) • Example 4.11 Retirement Savings with a Growing Annuity (Slides 92–93) • PowerPoint Alternative Example 4.11 (Slides 94–95) 4.6 Using an Annuity Spreadsheet or Calculator (Slide 96) • Example 4.12 Computing the Future Value in Excel (Slides 97–98) • Example 4.13 Using the Annuity Spreadsheet (Slides 99–100) 4.7 Non-Annual Cash Flows (Slide 101) • Example 4.14 Evaluating an Annuity with Monthly Cash Flows (Slides 102–103) 4.8 Solving for the Cash Payments (Slides 104–105) • Example 4.15 Computing a Loan Payment (Slides 106–107) • PowerPoint Alternative Example 4.15 (Slides 108–109) 4.9 The Internal Rate of Return (Slide 110) • Example 4.16 Computing the IRR for a Perpetuity (Slides 111–112) • Example 4.17 Computing the Internal Rate of Return for an Annuity (Slides 113–114) Chapter 4 Appendix Solving for the Number of Periods (Slide 116) • Example 4A.1 Solving for the Number of Periods in a Savings Plan (Slides 117–118)

II. Learning Objectives 4-1 Draw a timeline illustrating a given set of cash flows. 4-2 List and describe the three rules of time travel. 4-3 Calculate the future value of a. A single sum b. An uneven stream of cash flows, starting either now or sometime in the future c. An annuity, starting either now or sometime in the future d. Several cash flows occurring at regular intervals, which grow at a constant rate each period 4-4 Calculate the present value of a. A single sum b. An uneven stream of cash flows, starting either now or sometime in the future c. An infinite stream of identical cash flows d. An annuity, starting either now or sometime in the future

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e. An infinite stream of cash flows that grow at a constant rate each period f. Several cash flows occurring at regular intervals, which grow at a constant rate each period 4-5 Given four out of the following five inputs for an annuity, compute the fifth: (a) present value, (b) future value, (c) number of periods, (d) periodic interest rate, and (e) periodic payment. 4-6 Given three out of the following four inputs for a single sum, compute the fourth: (a) present value, (b) future value, (c) number of periods, and (d) periodic interest rate. 4-7 Given cash flows and present or future value, compute the internal rate of return for a series of cash flows.

III. Chapter Overview The introduction to Chapter 4 draws on the discussion of NPV in Chapter 3. The text puts a great deal of emphasis on drawing a timeline when approaching every problem, so that students may better visualize the problems they are solving. Another distinguishing feature of this text is that this chapter puts virtually everything in the context of present value. Hence all future value problems are illustrated as they relate to present value. The chapter covers present and future values of a single sum, an uneven stream of cash flows, and an annuity; present value of a perpetuity; present value of a growing perpetuity; and present and future value of a growing annuity. 4.1 The Timeline Drawing a timeline as the first step in solving a problem will often clarify any confusion about amount or timing of cash flows, making problems much clearer. The timeline is a very important part of calculating the time value of money. 4.2 The Three Rules of Time Travel Following are the three rules of time travel: 1. It is only possible to compare or combine values at the same point in time. 2. To move a cash flow forward in time, you must compound it. 3. To move a cash flow backward in time, you must discount it. To compound cash flows, multiply the amount by (1 + r)n (Equation 4.1), where r is the periodic interest rate and n is the number of compounding periods. To discount cash flows, divide the amount by (1 + r)n (Equation 4.2), where r and n are as defined previously. Figure 4.1 shows the composition of interest over time. The rules of time travel allow us to compare and combine cash flows that occur at different points in time. In order to compare or combine cash flows, they must be either discounted or compounded. 4.3 Valuing a Stream of Cash Flows This section shows how to calculate the present value or future value of an uneven cash flow stream. Note that FV can be solved by calculating the present value (discounting each cash flow), then compounding that total, n periods into the future. This information makes FV of an uneven cash flow stream solvable in a financial calculator. Present Value of a Cash Flow Stream (4.4)

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Future Value of a Cash Flow Stream with a Present Value of PV (4.5) 4.4 Calculating the Net Present Value The logic developed in Chapter 3 is reexamined here in the context of the time value of money. (4.6) 4.5 Perpetuities and Annuities Several alternatives to the formulas developed earlier are addressed at this point in the chapter. In every case, the formula is derived using algebra: 1. Present value of a perpetuity (4.7) 2. If we have several cash flows of the same size occurring at regular intervals, starting either now or sometime in the future, this is a present value or future value of an annuity. Annuities due are treated as ordinary annuities with one extra period of interest. Present Value of an Annuity (4.9) Future Value of an Annuity

(4.10)

3. The authors introduce the concept of valuing an infinite stream of cash flows that grow at a constant rate each period, using the formula for present value of a growing perpetuity. This development is particularly useful in later chapters. Present Value of a Growing Perpetuity (4.11)

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4. Sometimes several cash flows occur at regular intervals, which grow at a constant rate each period. Present value or future value of a growing annuity is used. Present Value of a Growing Annuity (4.12) 4.6 Using an Annuity Spreadsheet or Calculator Time value of money problems lend themselves well to spreadsheet solutions. The following functions are mentioned in the chapter: NPV, FV, PV, PMT, IRR, and NPER. Example 4.12 uses the annuity spreadsheet available on the textbook’s Web site to solve for future value, and Example 4.13 uses the annuity spreadsheet to solve for the amount of money left in an account after periodic withdrawals are made. 4.7 Non-Annual Cash Flows This section shows how to adjust the formulas for non-annual cash flows by simply adjusting the interest rate to be a monthly rate and adjusting the number of periods to be number of months. 4.8 Solving for Cash Payments This section explains how to solve for the annuity payment. Loan or Annuity Payment (4.14)

4.9 The Internal Rate of Return This section shows how to solve for the internal rate of return using algebra for perpetuities and trial and error method and spreadsheet method for annuities. Appendix: Solving for the Number of Periods This appendix shows how to solve for the number of periods using both spreadsheets and logarithms.

IV. Spreadsheet Solutions in Excel The following Problems for Chapter 4 have spreadsheet versions of the problems available: 12, 13, 14, 15, 23, 24, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 40, 41, 42, 43, 45, 46, and A-2. These spreadsheets can be downloaded from the Instructor’s Resource Center at: www.pearsonglobaleditions.com/berk. If you do not have a login and password for this Web site, contact your Pearson sales representative.

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