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Time Value of Money
Author's Overview
This is one of the most important chapters in the book as far as student comprehension is
concerned. The instructor should first determine how much prior knowledge of time value of
money the students have acquired from accounting or lower mathematics. While most
students are generally familiar with the concepts of future value and present value, they often
lack the ability to identify and categorize the nature of the problem before them.
The material in this chapter will serve as a springboard to the remaining chapters in this
section on valuation, cost of capital and capital budgeting related topics. A good background
in time value of money will ease the transition. The authors suggest a liberal use of
homework problems and a quiz to reinforce the importance of this material.
This chapter uses color-coordinated figures to explain the relationship between present value
and future value and present value to the present value of annuities, and future value to future
value of annuities. These color coded figures are helpful to those more visually oriented
students who may not understand the mathematical relationships between these time value
calculations.
For faculty who want to emphasize Excel spreadsheets and calculator keystrokes, this chapter
provides an excellent opportunity to develop both skills. Using spreadsheets with time-value
exercises can be especially instructive in understanding the concept of how higher discount
rates generate lower cash flows and visa versa.
Chapter Concepts
LO1. Money has a time value associated with it and therefore a dollar received today is worth
more than a dollar received in the future.
LO2. The future value is based on the number of periods over which the funds are to be
compounded at a given interest rate.
LO3. The present value is based on the current value of funds to be received.
LO4. Not only can future value and present value be computed, but other factors such as yield
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Education.
9-1
9
or rate of return can be determined as well.
LO5. Compounding or discounting may take place on a less than annual basis such as
semiannually or monthly.
Authors’ Note:
In previous editions of Foundations of Financial Management, the authors have emphasized
the use of time value tables in the text. This has presented several problems. With the advent
of computer generated problem sets, the tables were not precise and caused students who used
calculators and spreadsheets much frustration because of rounding errors created by
truncating the table values. This also caused difficulty for instructors because students could
get different answers using the tables than with calculators or spreadsheets. In this edition,
we have revamped this chapter to introduce calculator and Excel spreadsheets as the primary
way to calculate time value problems. Because many of our users like to use tables, we have
kept that pedagogy, but now it is found at the end of the chapter.
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Education.
9-2
Annotated Outline and Strategy
I. Relationship to the Capital Outlay Decision: Capital budgeting involves the analysis
of whether or not funds invested today will be more than offset by the value of the funds
received from the investment in the future.
II. Future Value -- Single Amount
A. In determining future value, we measure the value of an amount that is invested
now and allowed to grow at a given interest rate over a specified period of time.
B. The relationship may be expressed by the following formula:
The FVIF term is found using the calculator keystrokes. We use a $1,000 example
rather than $1 (also see Appendix A).
C. Using a financial calculator to derive the future value of a single payment, enter
the known values for the following:
N enter the number of interest periods
I/Y enter the annual interest rate
PV enter the present value amount
PMT enter zero
FV leave blank
Then press the CPT key and then the FV key
(The resulting value will appear as a negative number.)
Perspective 9-1: One or two numerical examples using this spreadsheet are helpful.
III. Present Value -- Single Amount
A. The present value of a future sum is the investment required today, at a given
interest rate that will equal the future sum at a specified point in time.
PPT Relationship of Present Value and Future Value (Figure 9-1)
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Education.
9-3
FV = PV(1 )n
i+
FV = PV FVIF
´
Perspective 9-2: Use Figure 9-1 to demonstrate the relationship between present and future
value.
B. The relationship may be expressed in the following formula:
C. The formula may be restated as:
The PVIF term is found using calculator keystrokes. We use $1,000 rather than
$1 in our example (also see Appendix B).
D. Using a financial calculator to derive the present value of a single future
payment, enter the known values for the following:
N enter the number of interest periods
I/Y enter the annual interest rate
PV leave blank
PMT enter zero
FV enter the future value amount
Then press the CPT key and then the PV key
(The resulting value will appear as a negative number.)
Perspective 9-3: We would suggest using the Excel spreadsheet example on page 261 to
show the relationship of the present value to the discount rate and time period.
IV. Interest Rate – Single Amount
A. When we know the future value, the present value, and the time period, we can
solve for the interest rate or the rate of return on an investment.
B. Using Formula 9-3 we find that:
C. We can use calculator keystrokes or the Excel spreadsheet to solve for the
interest rate.
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Education.
9-4
1
PV = FV (1 )n
i+
PV = FV PVIF
´
1
FV 1
PV
n
iæ ö
= -
ç ÷
è ø
V. Number of Periods – Single Amount
A. Following up on the interest rate example, we can use Formula 9-4 to solve for
the number of periods it takes to grow (compound) $1,000 to $1,464.10 if we
know the rate of return is 10%.
B. Once again, using the spreadsheet example on page 263 or the calculator
keystrokes, you can demonstrate how many years it takes to compound $1,000 to
$1,464.10.
VI. Future Value – Annuity
A. An annuity represents consecutive payments or receipts of equal amounts over
equal time intervals.
B. The annuity value is normally assumed to take place at the end of each period.
C. The future value of an annuity represents the sum of the future value of the
individual flows.
PPT Compounding Process for Annuity (Figure 9-2)
D. The formula for the future value for an annuity is given in Formula 9-5:
FVA = A FVIFA
The FVIFA term is found using the calculator keystrokes or the Excel spreadsheet
on page 265 (also see Appendix C).
E. Using a financial calculator to derive the future value of an annuity, enter the
known values for the following:
N enter the number of interest periods
I/Y enter the annual interest rate
PV enter zero
PMT enter the periodic payment
FV leave blank
Then press the CPT key and then the FV key
(The resulting value will appear as a negative number.)
Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill
Education.
9-5
Finance in Action: Powerball Jackpot Decisions
The odds of winning are now 1 in 292 million after Powerball changed the number of choices
for the white and red numbers and many students will be quite familiar with the $1.5 billion
Powerball prize won on January 13, 2016. However the purpose of the box is to demonstrate
the impact that present value has on an annuity prize taken as a lump sum payout. As a side bar
you might want to bring in what taxes do the lump sum payout.
VII. Present Value -- Annuity
A. The present value of an annuity represents the sum of the present value of the
individual cash flows.
B. The formula for the present value of an annuity is presented in Formula 9-6 (also
see Appendix D for PVIFA)
PVA = A PVIFA
C. Using a financial calculator to derive the present value of an annuity, enter the
known values for the following:
N enter the number of interest periods
I/Y enter the annual interest rate
PV leave blank
PMT enter the periodic payment
FV enter zero
Then press the CPT key and then the PV key
(The resulting value will appear as a negative number.)
D. The annuity value associated with a present value is often associated with
withdrawal of funds from an initial deposit or the repayment of a loan.
Perspective 9-4: An example that might ring true with the students is to have them use the
Excel spreadsheet on page 267 to calculate their annual or monthly student loan repayment
once they graduate from college and have a job. It might be an eye-opening exercise.
VIII. Graphical Presentation of Time Value Relationships
A. Use Figures 9-3 and 9-4 to show that present value and future value are inversely
related.
B. Use Figures 9-5 and 9-6 to demonstrate how annuities are the sums of single
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Education.
9-6
period values.
IX. Determining the Annuity Value (The amount of the periodic payment that would grow
to the requisite future value or be discounted to the requisite present value)
A. Annuity Equaling a Future Value
1. The formula for the future value for an annuity is found in Formula 9-7:
PPT Relationship of Present Value to Annuity (Table 9-1)
Perspective 9-3: Demonstrate how annuities work in everyday situations.
PPT Payoff Table for Loan (Amortization Table) (Table 9-2)
2. Using a financial calculator to derive the annuity value required to
achieve a future amount, enter the known values for the following:
N enter the number of interest periods
I/Y the periodic interest rate
PV enter zero
PMT leave blank
FV enter the future value amount
Then press the CPT key and then the PMT key
(The resulting annuity value will appear as a negative value.)
B. Annuity Equaling a Present Value
1. The formula for the present value of an annuity is presented in Formula 9-8:
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Education.
9-7
FV
(1 ) 1
A
n
Ai
i
=é ù
+ -
ê ú
ë û
2. Using a financial calculator to derive the annuity value required to achieve a
present amount, enter the known values for the following:
N enter the number of interest periods
I/Y the periodic interest rate
PV enter the present value amount
PMT leave blank
FV enter zero
Then press the CPT key and then the PMT key
(The resulting annuity value will appear as a negative value.)
3. Finding annuity payments using Excel’s PMT function: The PMT
function assumes the each payment is at the end of a period. Either the
PV or FV argument must be entered. See the example on page 276.
X. Finding Interest Rates and the Number of Payments
A. Finding Annuity Interest Rates Using Calculator Keystrokes or Excel: See
discussion on page 274.
B. Finding the Number of Annuity Payments Using Calculator Keystrokes or Excel:
See discussion on page 275.
XI. Compounding over Additional Periods
A. Compounding Semiannually requires us to cut the annual interest rate in half and
to multiply the number of periods by two. Consequently quarterly compounding
divides the interest rate by four and multiplies the periods by four. See case 1
and case 2 on page 275.
B. Patterns of Payment with a Deferred Annuity: Sometimes an annuity does not
start being received until several years in the future, and under these conditions,
we need to calculate the present value of each payment. There are several ways
to do the calculation. See examples on pages 276 and 277
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Education.
9-8
PV
1
1(1 )
A
n
A
i
i
=é ù
-
ê ú
+
ê ú
ê ú
ê ú
ë û
D. Annuities Due: Annuities that start at the beginning of the period rather than at
the end of the period. Use the Excel spreadsheet on page 278 to demonstrate both
present value and future values of annuities due.
XI. Alternative Calculations Using TVM Tables – APPENDICES 9A & B Page 288-294
A. Future Value -- Single Amount
1. In determining future value, we measure the value of an amount that is
invested now and allowed to grow at a given interest rate over a specified
period of time.
2. The relationship may be expressed by the following formula:
The FVIF term is found in Table 9-3 (also see Appendix A).
B. Present Value -- Single Amount
1. The present value of a future sum is the investment required today at a
given interest rate that will equal the future sum at a specified point in time.
2. The relationship may be expressed in the following formula:
1
PV = FV (1 )n
i+
The PVIF term is found in Table 9-4 (also see Appendix B).
C. Future Value -- Annuity
1. An annuity represents consecutive payments or receipts of equal amounts
over equal time intervals.
2. The annuity value is normally assumed to take place at the end of each
period.
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Education.
9-9
FV = PV(1 )n
i+
FV = PV FVIF
´
3. The future value of an annuity represents the sum of the future value of
the individual flows.
4. The formula for the future value for an annuity is:
FVA = A FVIFA
The FVIFA term is found in Table 9-5 (also see Appendix C).
D. Present Value -- Annuity
1. The present value of an annuity represents the sum of the present value of
the individual cash flows.
2. The formula for the present value of an annuity is:
PVA = A PVIFA
The PVIFA term is found in Table 9-6 (also see Appendix D).
Other Chapter Supplements
Cases for Use with Foundations of Financial Management
Case 10, Allison Boone, M.D. (time value of money)
Case 11, Billy Wilson, All American (time value of money)
Case 12, Sarah Gilbert, Retiree (time value of money)
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Education.
9-10
