6-1
Chapter 6
The Time Value
of Money
Future Value
Present Value
Rates of Return
Amortization
Copyright © 2000 by Harcourt, Inc.
All rights reserved. Requests for permission
to make copies of any part of the work should
be mailed to the following address:
Permissions Department, Harcourt, Inc., 6277
Sea Harbor Drive, Orlando, Florida 328876777.
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-2
Cash Flow Time Lines
Graphical representations used to
show the timing of cash flows.
0
1
2
CF1
CF2
3
i%
CF0
CF3
Tick marks at ends of periods, so t=0 is today;
t=1 is the end of Period 1; or the beginning of
Period 2.
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-3
Time line for a
$100 lump sum due
at the end of Year 2.
0
1
2
Year
i%
100
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-4
Time line for an ordinary
annuity of $100 for 3 years.
0
1
2
100
100
3
i%
100
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-5
Time line for uneven CFs
-$50 at t=0 and $100, $75, and $50
at the end of Years 1 through 3.
0
1
2
100
75
3
i%
-50
50
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-6
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.
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?
FVn = FV1 = PV + INT
= PV + (PV x i)
= PV (1 + i)
= $100(1+0.05) = $100(1.05) = $105
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-7
What’s the FV of an initial $100
after 3 years if i = 10%?
0
1
2
3
i=10%
100
FV = ?
Finding FVs is Compounding.
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-8
Future Value
After 1 year:
FV1 =
PV + i = PV + PV (i)
=
PV(1 + i)
=
$100 (1.10)
=
$110.00.
After 2 years:
FV2 =
PV(1 + i)2
=
$100 (1.10)2
=
$121.00.
After 3 years:
FV3 =
PV(1 + i)3
=
100 (1.10)3
=
$133.10.
In general, FVn = PV (1 + i)n
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-9
Three ways to find
Future Values:
 Solve the Equation with a
Regular Calculator
 Use Tables
 Use a Financial Calculator
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-10
Financial Calculator Solution:
Financial calculators solve this equation:
FV n PV 1  i .
n
There are 4 variables. If 3 are known,
the calculator will solve for the 4th.
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-11
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
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-12
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 cash flow or
series of cash flows, the reverse of
compounding.
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-13
What is the PV of $100 due
in 3 years if i = 10%?
0
1
2
3
10%
PV = ?
100
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-14
What is the PV of $100 due
in 3 years if i = 10%?
Solve FVn = PV (1 + i )n for PV:
n
FV
1 

n
PV =
= FV 

n
 1+ i 
1 + in
3
1



PV = $100
= $100PVIF 
1.10
i,n 
= $1000.7513 = $75.13.
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-15
Financial Calculator Solution:
INPUTS
OUTPUT
3
N
10
I/YR
PV
-75.13
0
PMT
100
FV
Either PV or FV must be negative. Here
PV = -75.13. Put in $75.13 today, take
out $100 after 3 years.
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-16
If sales grow at 20% per year,
how long before sales double?
Solve for n:
FVn = 1(1 + i)n;
2 = 1(1.20)n .
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-17
Financial Calculator Solution:
INPUTS
OUTPUT
Graphical
Illustration: 2
20
I/YR
N
3.8
-1
PV
0
PMT
2
FV
FV
3.8
1
Year
0
1
2
3
4
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-18
Future Value of an Annuity
 Annuity: A series of payments of an
equal amount 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.
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-19
Ordinary Annuity
Versus
Annuity Due
0
i%
0
1
2
PMT
PMT
1
2
PMT
PMT
3
PMT
3
i%
PMT
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-20
What’s the FV of a 3-year
Ordinary Annuity of $100 at 10%?
0
1
2
100
100
3
10%
100
110
121
FV
= 331
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-21
Financial Calculator Solution:
INPUTS
OUTPUT
3
10
0
-100
N
I/YR
PV
PMT
FV
331.00
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-22
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
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-23
What is the PV of this
Ordinary Annuity?
0
1
2
100
100
3
10%
100
90.91
82.64
75.13
248.69 = PV
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-24
Financial Calculator Solution:
INPUTS
3
N
OUTPUT
10
I/YR
100
PV
PMT
0
FV
-248.69
Have payments but no lump sum FV,
so enter 0 for future value.
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-25
Find the FV and PV if the
Annuity were an Annuity Due.
0
1
2
100
100
3
10%
100
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-26
Financial Calculator Solution:
Switch from “End” to “Begin”.
Then enter variables to find PVA3 = $273.55.
INPUTS
OUTPUT
3
10
N
I/YR
100
PV
PMT
0
FV
-273.55
Then enter PV = 0 and press FV to find
FV = $364.10.
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-27
Solving for Interest Rates
with Annuities
You pay $864.80 for an investment that promises to
pay you $250 per year for the next four years, with
payments made at the end of the year. What interest
rate will you earn on this investment?
0
1
2
3
4
250
250
250
i=?
- 864.80
250
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-28
Financial Calculator Solution:
PVAn
= PMT(PVIFAi,n)
$846.80
= $250(PVIFA i = ?,4)
INPUTS
OUTPUT
4
?
N
I/YR
- 846.80 250
PV
PMT
0
FV
=7.0
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-29
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
 Cash flow (CF) designates cash flows
in general, including uneven cash
flows
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-30
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
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-31
Financial Calculator Solution:
 Input in “Cash Flow” register:
CF0 =
0
CF1 =
100
CF2 =
300
CF3 =
300
CF4 =
-50
 Enter I = 10%, then press NPV button to
get NPV = 530.09. (Here NPV = PV.)
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-32
What interest rate would cause
$100 to grow to $125.97 in 3 years?
$100 (1 + i )3 = $125.97.
INPUTS
3
N
OUTPUT
-100
I/YR
PV
0
PMT
125.97
FV
8%
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-33
Semiannual and Other
Compounding Periods
 Annual compounding is the arithmetic
process of determining the final value of
a cash flow or series of cash flows when
interest is added once a year.
 Semiannual compounding is the
arithmetic process of determining the
final value of a cash flow or series of
cash flows when interest is added twice a
year.
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-34
Will the FV of a lump sum
be larger or smaller if we
compound more often,
holding the state i% constant?
Why?
LARGER! If compounding is more
frequent than once a year--for example,
semi-annually, quarterly, or daily-interest is earned on interest more often.
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-35
0
1
2
3
10%
100
133.10
Annually: FV3 = 100(1.10)3 = 133.10.
0
1
0
1
2
2
3
4
3
5
6
5%
100
134.01
Semi-annually: FV6/2 = 100(1.05)6 = 134.01.
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-36
Distinguishing Between
Different Interest Rates
iSIMPLE = Simple (Quoted) Rate
used to compute the interest paid per period
EAR = Effective Annual Rate
the annual rate of interest actually being earned
APR = Annual Percentage Rate
periodic rate X the number of periods per year
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-37
How do we find EAR for a
simple rate of 10%,
compounded semi-annually?
m
i

EAR = 1 + SIMPLE - 1

m 
2
0.10

 - 1.0
=  1+
2 
2
= 1.05 - 1.0
= 0.1025 = 10.25%.
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-38
FV of $100 after 3 years
under 10% semi-annual
compounding? Quarterly?
iSIMPLE 

FVn = PV  1 +

m 

FV3s
mxn
0.10

= $100 1 +


2 
2x3
= $100(1.05)6 = $134.01
FV3Q = $100(1.025)12 = $134.49
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-39
Fractional Time Periods
Example: $100 deposited in a bank
at 10% interest for 0.75 of the year
0
0.25
0.50
0.75
1.00
8%
- 100
INPUTS
FV = ?
0.75
N
OUTPUT
10
I/YR
- 100
PV
0
PMT
?
FV
=107.41
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-40
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, etc.
They are very important!
 Financial calculators (and spreadsheets)
are great for setting up amortization
tables.
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-41
Construct an amortization schedule
for a $1,000, 10% annual rate loan
with 3 equal payments.
Step 1: Find the required payments
0
1
2
PMT
PMT
3
10%
-1000
INPUTS
OUTPUT
3
10
-1000
N
I/YR
PV
PMT
0
PMT
FV
402.11
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-42
Step 2: Find interest charge for Year 1
INTt = Beg balt (i)
INT1 = 1000(0.10) = $100.
Step 3: Find repayment of principal in Year 1
Repmt. = PMT - INT
= 402.11 - 100
= $302.11.
Step 4: Find ending balance after Year 1
End bal = Beg bal - Repmt
= 1000 - 302.11 = $697.89.
Repeat these steps for Years 2 and 3
to complete the amortization table.
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-43
Loan Amortization Table
10 Percent Interest Rate
YR
1
Beg Bal
$1000
PMT
$402
2
698
402
70
332
366
3
366
402
37
366
0
1,206.34
206.34
1,000
Total
INT Prin PMT End Bal
$100
$302
$698
Interest declines. Tax Implications.
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-44
$
402.11
Interest
302.11
Principal Payments
0
1
2
3
Level payments. Interest declines because
outstanding balance declines. Lender earns
10% on loan outstanding, which is falling.
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-45
Comparison of Different
Types of Interest Rates
 iSIMPLE:
Written into contracts, quoted by banks
and brokers. Not used in calculations or shown on
time lines.
 iper:
Used in calculations, shown on time lines.
If iSIMPLE has annual compounding,
then iper = iSIMPLE/1 = iSIMPLE.
 EAR : Used to compare returns on investments
with different payments per year.
(Used for calculations if and only if dealing with annuities where
payments don’t match interest compounding periods.)
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-46
Simple (Quoted) Rate
 iSIMPLE is stated in contracts.
Periods per year (m) must also be given.
 Examples:
8%; Quarterly
8%, Daily interest (365 days)
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-47
Periodic Rate
 Periodic rate = iper = iSIMPLE/m, where m is
periods per year. m = 4 for quarterly, 12
for monthly, and 360 or 365 for daily
compounding.
 Examples:
8% quarterly: iper = 8/4 = 2%
8% daily (365): iper = 8/365 = 0.021918%
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-48
Effective Annual Rate
Effective Annual Rate:
The annual rate which cause PV to grow to the same
FV as under multiperiod compounding.
Example: EAR for 10%, semiannual:
FV
EAR
=
=
=
(1 + iSIMPLE/m)m
(1.05)2 = 1.1025.
10.25% because
(1.1025)1 = 1.1025.
Any PV would grow to same FV at 10.25% annually or
10% semiannually.
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-49
End of Chapter 6
The Time Value
of Money
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.