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Chapter 5
Time Value of Money
Instructor’s Resources
Overview
This chapter introduces an important financial concept: the time value of money. The present value and future of a
sum, as well as the present and future values of an annuity, are explained. Special applications of the concepts
include intra-year compounding, mixed cash flow streams, mixed cash flows with an embedded annuity,
perpetuities, deposits to accumulate a future sum, and loan amortization. Numerous business and personal financial
applications are used as examples. The chapter drives home the need to understand time value of money at the
professional level because funding for new assets and programs must be justified using these techniques. Decisions
in a student’s personal life should also be acceptable on the basis of applying time-value-of-money techniques to
anticipated cash flows.
Suggested Answer to Opener-in-Review Question
The city of Cincinnati gave up the right to collect parking fees over a 30-year period in exchange for a lump
sum of $92 million plus a 30-year annuity of $3 million. Suppose that if the city had not entered into that
arrangement, it would have collected parking fees the following year of $6 million (net of operating costs),
and those fees would have grown at a steady 3% for the next 30 years. At an interest rate of 4%, what is the
present value of the parking revenue that the city could have collected? Using the same 4% to value the
payments that the city was set to receive in their privatization deal, do you think that the city made the right
decision? Why or why not?
Total PV of giving up the right to collect parking fees:
I = 4; N = 30; PMT = 3
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Chapter 5 Time Value of Money 76
Answers to Review Questions
1. Future value (FV), the value of a present amount at a future date, is calculated by applying compound interest
2. A single amount cash flow refers to an individual standalone value occurring at one point in time. An annuity
3. Compounding of interest occurs when an amount is deposited into a savings account and the interest paid
FVn PV(1 r)n
4. A decrease in the interest rate lowers the future amount of a deposit for a given holding period because the
5. Present value is the current dollar value of a future amount. It indicates how much money today would be
PV FVn (1 r)n
6. An increasing required rate of return would reduce the present value of a future amount because future dollars
7. Present value calculations are the exact inverse of compound interest calculations. Using compound interest,
8. Answers will vary for question because values are algorithmically generated in MyFinanceLab.
9. Answers will vary for question because values are algorithmically generated in MyFinanceLab.
10. An ordinary annuity is one for which payments occur at the end of each period. An annuity due is one for
11. The most efficient ways to calculate present value of an ordinary annuity are using an algebraic equation, a
12. You can calculate the future value of an annuity due by multiplying the value calculated for an ordinary
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Chapter 5 Time Value of Money 77
13. You can calculate the present value of an annuity due by multiplying the value calculated for an ordinary
14. A perpetuity is an infinite-lived annuity. By multiplying the PV by the required rate of return, i, the perpetual
15. Answers will vary for question because values are algorithmically generated in MyFinanceLab.
16. Answers will vary for question because values are algorithmically generated in MyFinanceLab.
17. Answers will vary for question because values are algorithmically generated in MyFinanceLab.
18. The future value of a mixed stream of cash flows is calculated by multiplying each year’s cash flow by
19. Answers will vary for question because values are algorithmically generated in MyFinanceLab.
20. As interest is compounded more frequently than once a year, both (a) the future value for a given holding
21. Continuous compounding assumes interest will be compounded an infinite number of times per year, at
22. The nominal annual rate is the contractual rate that is quoted to the borrower by the lender. The effective
23. Answers will vary for question because values are algorithmically generated in MyFinanceLab.
24. Answers will vary for question because values are algorithmically generated in MyFinanceLab.
25. Answers will vary for question because values are algorithmically generated in MyFinanceLab.
26. The size of the equal annual end-of-year deposits needed to accumulate a given amount over a certain time
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Chapter 5 Time Value of Money 78
27. Amortizing a loan into equal annual payments involves finding the future payments whose present value at
28. The best way to determine an unknown number of periods is through the use of a calculator or spreadsheet. In
Suggested Answer to Focus on Practice Box:
New Century Brings Trouble for Subprime Mortgages
As a reaction to problems in the subprime area, lenders tightened lending standards. What effect do you
think this change had on the housing market?
The tightening of lending standards following the subprime fiasco further depressed home prices, which in 2007
Suggested Answer to Focus on Ethics Box:
How Fair Is “Check into Cash”?
The 391% mentioned is an annual nominal rate [15% (365/14)]. Should the 2-week rate (15%) be
compounded to calculate the effective annual interest rate?
No, the rollover fee is a simple $15 per 2-week period. In other words, the 15% 2-week rate is only applied to the
Answers to Warm-Up Exercises
E5-1. Future value of a lump-sum investment
E5-2. Finding the future value
Answer: Because the interest is compounded monthly, the number of periods is 4 12 48 and the monthly
interest rate is 1/12th of the annual rate.
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Chapter 5 Time Value of Money 79
Column A Column B
Cell 1 Future value of a single amount
Cell B5 $2,420.99
E5-3. Comparing a lump sum with an annuity
Answer: This problem can be solved in either of two ways. Both alternatives can be compared as lump sums in
Method 1: Perform a lump sum comparison. Compare $1.3 million now with the present value of the
Method 2: Compare two annuities. Because the $100,000 per year is already an annuity, all that
E5-4. Comparing the present value of two alternatives
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Chapter 5 Time Value of Money 80
Answer: To solve this problem you must first find the present value of the expected savings over the
5-year life of the software.
Year Savings Estimate
Present Value
of Savings
1 $35,000 $32,110
E5-5. Compounding more frequently than annually
Answer: Partners’ Savings Bank:
1
2
1
2
1
1
$12,000 (1 0.03/2)
$12,000 (1 0.03/2) $12,000 1.030225 $12,362.70
m n
r
FV PV m
FV
FV
´
æ ö
= ´ +
ç ÷
è ø
= ´ +
= ´ + = ´ =
Selwyn’s:
rxn 0.0275 1
1
FV PV (e ) $12,000 (2.7183 )
$12,000 1.027882 $12,334.58
´
= ´ = ´
= ´ =
Joseph should choose the 3% rate with semiannual compounding.
E5-6. Determining deposits needed to accumulate a future sum
Answer: The financial calculator input is as follows:
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Chapter 5 Time Value of Money 81
Solutions to Problems
P5-1. Using a time line
LG 1; Basic
d. Financial managers rely more on present value than future value because they typically make
P5-2. Future value calculation
LG 2; Basic
Case
P5-3. Time to double
LG 1; Basic
Case A: Computer Inputs: I 12%, PV $100; FV $200
Case B: Computer Inputs: I 6%, PV $100; FV $200
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Chapter 5 Time Value of Money 82
P5-4. Future values
LG 2; Intermediate
Case Case
AN 20, I 5%, PV $200.BN 7, I/Y 8%; PV $4500.
P5-5. Personal finance: Time value
LG 2; Intermediate
a. (1) N 3, I 7%, PV $1,500 b. (1) Interest earned FV3 PV
$1,500.00
$337.56
(2) N 6, I 7%, PV $1,500 (2) Interest earned FV6 – FV3
–$1,837.56
$413.54
(3) N 9, I 7%, PV $1,500 (3) Interest earned FV9 FV6
–$2,251.10
$506.59
c. The fact that the longer the investment period is, the larger the total amount of interest collected will
P5-6. Personal finance: Time value
LG 2; Challenge
a. (1) N 5, I 2%, PV $14,000 (2) N 5, I 4%, PV $14,000
b. The car will cost $1,576.01 more with a 4% inflation rate than an inflation rate of 2%. This increase is
c. Future value at end of first 2 years:
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Chapter 5 Time Value of Money 83
P5-7. Personal finance: Time value
LG 2; Challenge
Deposit Now: Deposit in 10 Years:
P5-8. Personal finance: Time value
LG 2; Challenge
a. N 5, PV $10,200, FV 15,000 b. N 5, PV $8,150, FV $15,000
c. N 5, PV $7150, FV $15,000
P5-9. Personal finance: Single-payment loan repayment
LG 2; Intermediate
a. N 1, I 14%, PV $200 b. N 4, I 14%, PV $200
c. N 8, I 14%, PV $200
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Chapter 5 Time Value of Money 84
P5-10. Present value calculation:
1
PVIF (1 )
n
i
=+
LG 2; Basic
Case
AN 4, I 2, FV $1.00, Solve for PV $0.9238
P5-11. Present values
LG 2; Basic
Case PV
AN 4, I 12%, FV $7,000 $4,448.63
P5-12. Present value concept
LG 2; Intermediate
a. N 6, I 12%, FV $6,000 b. N 6, I 12%, FV $6,000
c. N 6, I 12%, FV $6,000
d. The answer to all three parts is the same. In each case, the same question is being asked but in a
P5-13. Personal finance: Time Value
LG 2; Basic
a. N 3, I 7%, FV $500
b. Jim should be willing to pay no more than $408.15 for this future sum given that his opportunity cost
P5-14. Time value: Present value of a lump sum
LG 2; Intermediate
© 2015 Pearson Education, Inc.
