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 45–48)
• PowerPoint Alternative Example 4.4 (Slides 49–52)
4.3 Valuing a Stream of Cash Flows (Slides 53–54)
• Example 4.5: Present Value of a Stream of Cash Flows (Slides 55–59)
• PowerPoint Alternative Example 4.5 (Slides 60–61)
4.4 Calculating the Net Present Value (Slide 63)
• Example 4.6: Net Present Value of an Investment Opportunity (Slides 64–68)
• PowerPoint Alternative Example 4.6 (Slides 69–71)
• Using Excel: Calculating Present Values in Excel
4.5 Perpetuities and Annuities (Slides 72–73, 78)
• Perpetuities (Slides 72–73)
• Example 4.7: Endowing a Perpetuity (Slides 74–75)
• PowerPoint Alternative Example 4.7 (Slides 76–77)
• Common Mistake: Discounting One Too Many Times
• Annuities (Slide 78)
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– Present Value of an Annuity (Slides 79–81)
Example 4.8: Present Value of an Annuity Due (Slides 82–86)
– Future Value of an Annuity (Slide 87)
Example 4.9: Retirement Savings Plan Annuity (Slides 88–92)
Growing Cash Flows (Slide 93)
– Growing Perpetuity (Slide 93)
Example 4.10: Endowing a Growing Perpetuity (Slides 94–95)
PowerPoint Alternative Example 4.10 (Slides 96–97)
– Growing Annuity (Slide 98)
Example 4.11: Retirement Savings with a Growing Annuity (Slides 99–101)
PowerPoint Alternative Example 4.11 (Slides 102–103)
4.6 Using an Annuity Spreadsheet or Calculator (Slide 104)
• Example 4.12: Computing the Future Value in Excel (Slides 105–108)
• Example 4.13: Using the Annuity Spreadsheet (Slides 109–112)
4.7 Non-Annual Cash Flows (Slide 113)
• Example 4.14: Evaluating an Annuity with Monthly Cash Flows (Slides 114–116)
4.8 Solving for the Cash Payments (Slides 117–118)
• Example 4.15: Computing a Loan Payment (Slides 119–121)
• PowerPoint Alternative Example 4.15 (Slides 122–123)
4.9 The Internal Rate of Return (Slide 124)
• Example 4.16: Computing the IRR for a Perpetuity (Slides 125–126)
• Example 4.17: Computing the Internal Rate of Return for an Annuity (Slides 127–129)
• Using Excel: Excel’s IRR Function
Chapter 4 Appendix Solving for the Number of Periods (Slides 130–131)
• Example 4A.1 Solving for the Number of Periods in a Savings Plan (Slides 132–134)
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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Berk/DeMarzo • Corporate Finance, Fifth Edition
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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 that 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
𝑁
𝑃𝑉 = ∑𝑁
𝑛=0 𝑃𝑉(𝐶𝑛 ) = ∑𝑛=0
𝐶𝑛
(1+𝑟)𝑛
(4.4)
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Berk/DeMarzo • Corporate Finance, Fifth Edition
Future Value of a Cash Flow Stream with a Present Value of PV
𝐹𝑉𝑛 = 𝑃𝑉 × (1 + 𝑟)𝑛
(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.
𝑁𝑃𝑉 = 𝑃𝑉(𝑏𝑒𝑛𝑒𝑓𝑖𝑡𝑠) − 𝑃𝑉(𝑐𝑜𝑠𝑡𝑠)
(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
𝑃𝑉(𝐶 𝑖𝑛 𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦) =
𝐶
𝑟
(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
1
1
𝑃𝑉(annuity of 𝐶 for 𝑁 periods with interest rate 𝑟 = 𝐶 × 𝑟 (1 − (1+𝑟)𝑁)
(4.9)
Future Value of an Annuity
𝐹𝑉(𝑎𝑛𝑛𝑢𝑖𝑡𝑦) = 𝑃𝑉 × (1 + 𝑟)𝑁
=
𝐶
1
(1 − 1+𝑟𝑁) ×
𝑟
(1 + 𝑟)𝑁
1
= 𝐶 × 𝑟 (1 + 𝑟)𝑁 − 1
(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
𝑃𝑉(𝑔𝑟𝑜𝑤𝑖𝑛𝑔 𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦) =
𝐶
𝑟−𝑔
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(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
1+𝑔𝑁
1
𝑃𝑉 = 𝐶 × 𝑟−𝑔 (1 − ( 1+𝑟 ))
(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
𝑃
𝐶= 1
1
𝑟
(1−
)
(1+𝑟)𝑁
(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 are available as Excel Project problems in MyLab Finance
at www.pearson.com/mylab/finance: 12, 13, 24, 25, 31, 38, 44, and 45.
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