Chapter 5
Time Value of Money
Instructor’s Resources
Chapter Overview
This chapter introduces a key—indeed, perhaps the most important—concept in finance: the time
value of money. The present and future value of a sum and an annuity are explained. Special
applications include intra-year compounding, mixed cash-flow streams, mixed cash flows with an
embedded annuity, perpetuities, loan amortization, and deposits necessary to accumulate a future
sum. The discussion employs numerous business and personal examples to stress all applications as
variations on the same theme—sums received at different points in time are worth different amounts
to the recipient, with differences traceable to when the sums are received and how frequently interest
is compounded. The chapter drives home the need to understand the time value of money to analyze
project profitability as a finance professional and achieve personal-finance goals as an individual.
Answers to Review Questions
5-1 Future value (FV) is the sum an amount today will grow to equal by a future date with compound
interest. Present value (PV) is the dollar value today of an amount promised at a specific future date for
5-2 A single amount cash flow refers to a single payment or cash receipt, occurring at one point in
5-3 Compounding occurs when interest earned in the initial period is added to the original
principal and that sum grows by the interest rate in the second period, and so on. With
5-4 A fall in the interest rate reduces the future value of a deposit for a given holding period
because of the decline in the amount of interest on which additional interest is subsequently
Chapter 5 Time Value of Money 81
© 2019 Pearson Education, Inc.
()
n−1
⎡
⎧
⎫
5-14 A perpetuity is an infinite-lived annuity; the present value of an ordinary annuity is given by:
PV
0
=CF
1
r
⎛
⎝
⎜⎞
⎠
⎟×1−1
1+r
()
n
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
5-15 The future value (at retirement) of equal annual IRA contributions at the end of every year can
be determined using the formula for the future value of an ordinary annuity:
()
n
−1
⎡
⎧
⎫
5-16 Specific numerical answers will vary because values are algorithmically generated in
© 2019 Pearson Education, Inc.
E5-6 Determining deposits needed to accumulate a future sum (LG 6)
Ch
a
a
pter 5 Time
V
V
alue of Money
87
© 2019 Pearson Education, Inc.
P5-18 Calculating deposit needed (LG 2; Challenge)
Step 1: Determine future value of initial deposit at the end of the 7 years.
FVn = PV × (1 + r)n = $10,000 × (1 + 0.05)7 = $14,071
Step 2: Determine future value of second deposit
P5-19 Future value of an annuity (LG 3; Intermediate)
a. The future value of an ordinary annuity is given by:
()
n
−1
⎡
⎣
⎢⎤
⎦
⎥
⎧
⎪
⎪
⎫
⎪
⎪
spreadsheet program like Excel. In Excel, command format is FV(r, n, –CF1,0,0)
where the final “0” inside the parentheses indicates ordinary annuity (1 = annuity
due). For case A above, the specific Excel entry is: =FV(0.08,10,-2500,0,0)
The future value of an annuity due is given by:
()
n
−1
⎡
⎧
⎩
⎫
⎭
© 2019 Pearson Education, Inc.
C FV5 = $30,000 × {[(1 + 0.20)5 − 1] ÷ 0.20}= $267,897.60
D FV8 = $11,500 × {[(1 + 0.09)8 − 1] ÷ 0.09}= $138,241.92
E FV30 = $6,000 × {[(1 + 0.14)30 − 1] ÷ 0.14}= $2,440,422.03
Again, future value of an annuity due may be found with a financial calculator or
spreadsheet program like Excel. In Excel, command format is FV(r, n, –CF1,0,1) where
P5-20 Present value of an annuity (LG 3; Intermediate)
a. Using the formula for present value of an ordinary annuity:
PV
0
=CF
1
r
⎛
⎝
⎜⎞
⎠
⎟×1−1
1+r
()
n
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥