Chapter 3
The Time Value
of Money
© 2005 Thomson/South-Western
Time Value of Money
 The most important concept in finance
 Used in nearly every financial decision
Business decisions
Personal finance decisions
2
Cash Flow Time Lines
Graphical representations used to
show timing of cash flows
0
CF0
k%
1
2
3
CF1
CF2
CF3
Time 0 is today
Time 1 is the end of Period 1 or the beginning
of Period 2.
3
Time line for a $100 lump
sum due at the end of Year 2
0
1
2
Year
k%
100
4
Time line for an ordinary
annuity of $100 for 3 years
0
k%
1
2
100
100
3
100
5
Time line for uneven CFs
- $50 at t = 0 and $100, $75,
and $50 at the end of Years 1
through 3
0
-50
k%
1
2
100
75
3
50
6
Future Value
The amount to which a cash flow or
series of cash flows will grow over a
period of time when compounded at
a given interest rate.
7
Future Value
Calculating FV is compounding!
Question: How much would you have at the end of one
year if you deposited $100 in a bank account that pays 5
percent interest each year?
Translation: What is the FV of an initial $100 after 3 years
if k = 10%?
Key Formula: FVn = PV (1 + k)n
8
Three Ways to Solve Time
Value of Money Problems
Use Equations
Use Financial Calculator
Use Electronic Spreadsheet
9
Numerical (Equation) Solution
Solve this equation by plugging in the
appropriate values:
FVn  PV(1  k)
n
PV = $100, k = 10%, and n =3
3
FVn  $100(1.10)
 $100(1.3310)  $133.10
10
Financial Calculator Solution
Financial calculators solve this equation:
FVn  PV(1  k)
n
There are 4 variables (FV, PV, k, n).
If 3 are known, the calculator will solve
for the 4th.
11
Financial Calculator Solution
First: set calculator to show 4 digits to the
right of the decimal place
To enter “Format” register:
Type: 2nd, period
See: DEC (decimal) = (varies)
Type: 4, enter
See: DEC = 4
Exit Format register: hit CE/C
See 0.0000
12
Financial Calculator Solution
Here’s the setup to find FV:
INPUTS
3
N
10 -100
I/YR PV
OUTPUT
0
PMT
?
FV
133.10
Clearing automatically sets everything to 0, but for
safety enter PMT = 0.
Set:
P/YR = 1, END
13
Spreadsheet Solution
Set up Problem
Click on Function Wizard and
choose Financial/FV
14
Spreadsheet Solution
Reference cells:
Rate = interest
rate, k
Nper = number of
periods interest is
earned
Pmt = periodic
payment
PV = present value
of the amount
15
Present Value
Present value is the value today of a future
cash flow or series of cash flows.
Discounting is the process of finding the
present value of a future cash flow or series
of future cash flows; it is the reverse of
compounding.
16
What is the PV of $100 due in
3 years if k = 10%?
0
PV = ?
10%
1
2
3
100
17
Financial Calculator Solution
INPUTS
OUTPUT
3
10
?
0
N
I/YR
PV
PMT
100
FV
-75.13
VITAL: Either PV or FV must be negative.
Here PV = -75.13. Put in $75.13 today, take
out $100 after 3 years.
18
If sales grow at 20% per year,
how long before sales double?
19
Financial Calculator Solution
INPUTS
OUTPUT
?
20
N
I/YR
-1
0
PV
PMT
2
FV
3.8
Graphical
FV
Illustration: 2
3.8
1
0
1
2
3
4
Year
20
Future Value of an Annuity
 Annuity: A series of payments of equal
amounts at fixed intervals for a specified
number of periods.
 Ordinary (deferred) Annuity: An annuity
whose payments occur at the end of each
period.
 Annuity Due: An annuity whose payments
occur at the beginning of each period.
21
Ordinary Annuity Versus
Annuity Due
Ordinary Annuity
0
k%
1
2
PMT
PMT
1
2
PMT
PMT
3
PMT
Annuity Due
0
3
k%
PMT
22
What’s the FV of a 3-year
Ordinary Annuity of $100 at 10%?
0
10%
1
2
3
100
100
100
110
121
FV
= 331
23
Financial Calculator Solution
INPUTS
OUTPUT
3
10
0
-100
?
N
I/YR
PV
PMT
FV
331.00
24
Present Value of an Annuity
 PVAn = the present value of an annuity
with n payments.
 Each payment is discounted, and the
sum of the discounted payments is the
present value of the annuity.
25
0
What is the PV of this
Ordinary Annuity?
10%
1
2
100
100
3
100
90.91
82.64
75.13
248.69 = PV
26
Financial Calculator Solution
INPUTS
3
N
OUTPUT
10
?
I/YR
PV
100
PMT
0
FV
-248.69
We know the payments but no lump sum FV,
so enter 0 for future value.
27
Find the FV and PV if the
Annuity were an Annuity Due.
0
100
10%
1
2
100
100
3
28
Financial Calculator Solution
ANNUITY Due: Switch from “End” to “Begin”
Method: (2nd BGN, 2nd Enter)
Then enter variables to find PVA3 = $273.55.
INPUTS
3
N
OUTPUT
10
?
I/YR
PV
100
PMT
0
FV
-273.55
Then enter PV = 0 and press FV to find
FV = $364.10.
29
What is the PV of a $100
perpetuity if k = 10%?
You MUST know the formula for a perpetuity:
PV = PMT
k
So, here: PV = 100/.1 = $1000
30
Solving for Interest Rates
with Annuities
You pay $846.80 for an investment that promises
to pay you $250 per year for the next four years,
with payments made at the end of each year.
What interest rate will you earn on this
investment?
0
k=?
- 846.80
1
250
2
3
4
250
250
250
31
Financial Calculator Solution
INPUTS
OUTPUT
4
?
-846.80 250
N
I/YR
PV
PMT
0
FV
7.0
32
What interest rate would
cause $100 to grow to
$125.97 in 3 years?
33
What interest rate would
cause $100 to grow to
$125.97 in 3 years?
INPUTS
OUTPUT
3
?
N
I/YR
-100
PV
0
PMT
125.97
FV
8%
34
Uneven Cash Flow Streams
A series of cash flows in which the amount
varies from one period to the next:
Payment (PMT) designates constant cash
flows—that is, an annuity stream.
Cash flow (CF) designates cash flows in
general, both constant cash flows and
uneven cash flows.
35
What is the PV of this
Uneven Cash Flow Stream?
0
1
2
3
4
100
300
300
-50
10%
90.91
247.93
225.39
-34.15
530.08 = PV
36
Financial Calculator Solution
 In “CF” register, input the following:





CF0
C01
C02
C03
C04
=
=
=
=
=
0
100
300
300
-50
F01
F01
F01
F01
=
=
=
=
1
1
1
1
 In “NPV” Register:




Enter I = 10%
Hit down arrow to see “NPV = 0”
Hit CPT for compute
See “NPV = 530.09” (Here NPV = PV.)
37
Semiannual and Other
Compounding Periods
 Annual compounding is the process of
determining the future value of a cash flow
or series of cash flows when interest is
added once a year.
 Semiannual compounding is the process
of determining the future value of a cash
flow or series of cash flows when interest is
added twice a year.
38
Will the FV of a lump sum be larger or
smaller if we compound more often,
holding the stated k constant? Why?
39
Will the FV of a lump sum be larger or
smaller if we compound more often,
holding the stated k constant? Why?
LARGER!
If compounding is more frequent than once a
year—for example, semi-annually, quarterly,
or daily—interest is earned on interest—that
is, compounded—more often.
40
0
Compounding
Annually1 vs. Semi-Annually
2
3
10%
100
133.10
Annually: FV3 = 100(1.10)3 = 133.10.
0
1
2
0
100
5%
1
2
3
4
5
3
6
134.01
41
Semi-annually: FV6/2 = 100(1.05)6 = 134.01.
Distinguishing Between
Different Interest Rates
kSIMPLE = Simple (Quoted) Rate
kPER = Periodic Rate
EAR = Effective Annual Rate
APR = Annual Percentage Rate
42
kSIMPLE
kSIMPLE = Simple (Quoted) Rate
*used to compute the interest paid per period
*stated in contracts, quoted by banks & brokers
*number of periods per year must also be given
*Not used in calculations or shown on time lines
Examples:
8%, compounded quarterly
8%, compounded daily (365 days)
43
Periodic Rate = kPer
 kPER: Used in calculations, shown on time lines.
 If kSIMPLE has annual compounding, then kPER = kSIMPLE
 kPER = kSIMPLE/m, where m is number of compounding periods
per year.
 Determining m:
 m = 4 for quarterly
 m = 12 for monthly
 m = 360 or 365 for daily compounding
 Examples:
 8% quarterly: kPER = 8/4 = 2%
 8% daily (365): kPER = 8/365 = 0.021918%
44
APR = ksimple
APR = Annual Percentage Rate
= kSIMPLE periodic rate X
the number of periods per year
45
EAR
EAR = Effective Annual Rate
* the annual rate of interest actually being earned
* The annual rate that causes PV to grow to the same
FV as under multi-period compounding.
* Use to compare returns on investments with
different payments per year.
* Use for calculations when dealing with annuities
where payments don’t match interest compounding
periods .
46
How to find EAR for a simple rate of
10%, compounded semi-annually
 Hit 2nd then 2 to enter “ICONV” register:
 NOM = simple interest rate
 Type: 10, enter, down arrow twice
 C/Y = compounding periods per year
 Type: 2, enter, up arrow (or down arrow twice)
 EFF = effective annual rate = EAR
 Type: CPT (for compute)
 See: 10.25
POINT: Any PV would grow to same FV at 10.25%
annually or 10% semiannually.
47
Continuous Compounding
 The formula is FV = PV(e kt)
 k = the interest rate (expressed as a decimal)
 t = number of years
 Calculator “workaround”
 Store 9999999999 (as many nines as possible) in your
calculator under STO + 9
 Then, for N: N = # of years times “RCL 9”
 Then, for I: I = simple interest divided by “RCL 9”
48
Continuous Compounding
 Question: What is the value of a $1,000 deposit
invested for 5 years at an interest rate of 10%,
compounded continuously?
5* 10/
INPUTS RCL9 RCL9 -1000
N
OUTPUT
I/YR
PV
0
?
PMT
FV
1648.72
49
Fractional Time Periods
What is the value of $100 deposited in a
bank at EAR = 10% for 0.75 of the year?
0
- 100
10%
0.25
0.50
0.75
1.00
FV = ?
50
Fractional Time Periods
What is the value of $100 deposited in a
bank at EAR = 10% for 0.75 of the year?
0
10%
0.25
0.50
0.75
- 100
INPUTS
FV = ?
0.75
N
OUTPUT
1.00
10
-100
I/YR
PV
0
PMT
?
FV
107.41
51
Amortized Loans
 Amortized Loan: A loan that is repaid in equal
payments over its life.
 Amortization tables are widely used for home
mortgages, auto loans, business loans,
retirement plans, and so forth to determine how
much of each payment represents principal
repayment and how much represents interest.
They are very important, especially to homeowners!
 Financial calculators (and spreadsheets) are
great for setting up amortization tables.
52
Task: Construct an amortization
schedule for a $1,000, 10 percent
loan that requires three equal annual
payments.
0
-1,000
10%
1
2
PMT
PMT
3
PMT
53
Step 1: Determine the
required payments
0
10%
-1000
INPUTS
3
N
OUTPUT
1
2
PMT
PMT
10
I/YR
-1000
PV
3
PMT
?
PMT
0
FV
402.11
54
Enter “Amort” Register
Hit 2nd, PV to enter “Amort Register”
For 1st principal payment, 1st interest payment, and 1st year
remaining balance, enter:
P1 = 1
P2 = 1
Down arrow
See:
Bal = -697.88, down arrow
PRN = 302.11
INT = 100
55
Step 2: Create Loan
Amortization Table
YR Beg Bal
PMT INT
Prin PMT End Bal
1
$1000.00 $402.11 $100.00 $302.11 $697.88
2
697.88
402.11
* Rounding difference
56
Interest declines, which has tax implications.
Enter “Amort” Register
Hit 2nd, PV to enter “Amort Register”
For 2nd principal payment, 2nd interest payment, and 2nd
year remaining balance, enter:
P1 = 2
P2 = 2
Down arrow
See:
Bal = -365.56, down arrow
PRN = 332.32
INT = 69.79
57
Step 2: Create Loan
Amortization Table
YR Beg Bal
PMT
INT Prin PMT End Bal
1
$1000.00 $402.11 $100.00 $302.11 $697.89
2
697.89
402.11
3
365.57
402.11
69.79
332.32
365.57
Total
* Rounding difference
58
Interest declines, which has tax implications.
Enter “Amort” Register
Hit 2nd, PV to enter “Amort Register”
For 3rd principal payment, 3rd interest payment, and 3rd
year remaining balance, enter:
P1 = 3
P2 = 3
Down arrow
See:
Bal = 0, down arrow
PRN = 365.56
INT = 36.65
59
Step 2: Create Loan
Amortization Table
YR Beg Bal
PMT
INT Prin PMT End Bal
1
$1000.00 $402.11 $100.00 $302.11 $697.89
2
697.89
402.11
69.79
332.32
365.57
3
365.57
402.11
36.55
365.56
.01*
1206.33
206.34
1000.00
Total
* Rounding difference
60
Interest declines, which has tax implications.
Before Next Class
1.
2.
3.
3.
Review Chapter 3 materials
Do Chapter 3 homework
Prepare for Chapter 3 quiz
Read Chapter 4
61