Chapter

4
TIME VALUE OF
MONEY
LEARNING OUTCOMES

1. Identify various types of cash flow

patterns (streams) seen in business.
2. Compute the future value of different
cash flow streams. Explain the results.
3. Compute the present value of different
cash flow streams. Explain the results.
4. Compute (a) the return (interest rate) on
an investment (loan) and (b) how long it
takes to reach a financial goal.

CFIN6 | CH4 2
LEARNING OUTCOMES (continued)

5. Explain the difference between the

Annual Percentage Rate (APR) and the
Effective Annual Rate (EAR). Explain when
it is appropriate to use each.
6. Describe an amortized loan. Compute (a)
amortized loan payments and (b) the
amount that must be paid on an amortized
loan at a specific date during the life of
the loan.

CFIN6 | CH4 3
Time Value of Money (TVM)

• The principles and computations used to revalue

cash payoffs from different times so they are
stated in dollars of the same time period.
• Dollar amounts from different time periods should
never be compared; rather, amounts should be
compared only when they are stated in dollars at the
same point in time.
• Dollars from different time periods have
opportunities to earn different amounts (numbers
of periods) of interest (return).

CFIN6 | CH4 4
Time Value of Money (TVM) (continued 1)

• At a 10 percent opportunity cost rate, which

is better, receiving $700 today or receiving
$935 in three years?

CFIN6 | CH4 5
 Time Value of Money (TVM) (continued 2)

• To answer the question, we must revalue the

cash payoffs so they are stated in dollars at
the same time period.

Year (n): 0 1 2 3
r = return = 10%
Cash flows:
Option A: VA= $700 Translate the current $700 into
? = FVA3
an FV amount by adding interest.

OR
Translate the future $935 into a $935 = FVB3
Option B: PVB= ?
PV amount by taking out interest.

CFIN6 | CH4 6
 Cash Flow Time Lines

Graphical representations used to show timing of cash flows

0 1 2 3
r = 10%
PVA = 700.00 ? = FV3

PV = Present Value—the beginning amount that can be

invested (current value of some future amount).
FV = Future Value—the value to which an amount invested
today will grow at the end of n periods at an opportunity cost
rate equal to r.

CFIN6 | CH4 7
Types of Cash Flow Patterns

• Lump Sum Amount—a single payment

(received or made) that occurs either
today or at some date in the future.
• Annuity—multiple payments of the same
amount over equal time periods.
• Uneven Cash Flows—multiple payments of
different amounts over a period of time.

CFIN6 | CH4 8
Future Value

• Compounding—to compute the future value

of an amount we push forward the current
amount by adding interest for each period in
which the money can earn interest in the
future.

CFIN6 | CH4 9
Ways to Solve TVM Problems

• Use a cash flow timeline

• Use an equation
• Use a financial calculator
• Use a spreadsheet

CFIN6 | CH4 10
 FVn Cash Flow Timeline Solution

The Future Value of $700 invested at 10% per

year for three (3) years
0 1 2 3
r = 10%
x 1.10 x 1.10 x 1.10
PVA = 700.00 770.00 847.00 931.70 = FV3

CFIN6 | CH4 11
FVn Equation Solution

The cash flow timeline solution can be written

in equation form as:
FV3 = $700(1.10)3
This relationship is generalized as:

FVn = PV(1 + r)n

= $700(1.10)3
= $700(1.33100)
= $931.70
CFIN6 | CH4 12
FVn Financial Calculator Solution

Inputs: 3 10 − 700 0 ?
N I/Y PV PMT FV
Outputs: = 931.70

CFIN6 | CH4 13
FVn Spreadsheet Solution—MS Excel

• Set up a table that contains the data used

to solve the problem.
• Click fx and choose the FV function.
• Click the cells containing the appropriate
data to enter the data into the FV function.
• Calculate the answer.

CFIN6 | CH4 14
FVn Spreadsheet Solution—MS Excel
(continued)

CFIN6 | CH4 15
Future Value of an Annuity—FVA

• Annuity—a series of equal amounts paid at

equal intervals.
• Ordinary (deferred) Annuity—an annuity
with payments that occur at the end of each
period.
• Annuity Due—an annuity with payments
that occur at the beginning of each period.

CFIN6 | CH4 16
What’s the Future Value of a Three-Year
Ordinary Annuity of $400 at 5%?

0 1 2 3 Value of Each
r = 5% Deposit
400 at the End of Year 3
400 400
x 400
x (1.05)1 (1.05)0
420
x (1.05)2
441
FVA3 = 1,
261.00

CFIN6 | CH4 17
FVAn Equation Solution

CFIN6 | CH4 18
FVAn Equation Solution (continued)

 (1 0 5 )3  1 
FVA 3
 400  
 0 .0 5 
 4 0 0 (3 .1 5 2 5 )

 1, 2 6 1 .0 0

CFIN6 | CH4 19
FVAn Financial Calculator Solution

Inputs: 3 5 0 − 400 ?
N I/Y PV PMT FV
Output: = 1,261.00

CFIN6 | CH4 20
 FV of an Annuity Due—FVA(DUE)n

Payments are made at the beginning of the year,

which means each payment earns one additional
year’s worth of 5 percent interest.

0 1 2 3 Value of Each Deposit

r = 5% at the End of Year 3
400 400 400
x (1.05)0 x (1.05)
420.00
x (1.05)1 x
(1.05) 441.00
x (1.05)2 x (1.05)
463.05
FVA(DUE)3 = 1,324.05

CFIN6 | CH4 21
 FVA(DUE)n Equation Solution

Include one additional year’s worth of 5 percent

interest in the computation.

CFIN6 | CH4 22
FVA(DUE)n Equation Solution (continued)

CFIN6 | CH4 23
FVA(DUE)n Financial Calculator Solution

BGN
Input 3 5 0 −400 ?
s: N I/Y PV PMT FV
Output: = 1,324.05

CFIN6 | CH4 24
Cash Flow Streams

• Payment = PMT = constant cash

flows—that is, an annuity stream.
• Cash flow = CF = cash flows in general,
both constant cash flows (i.e., annuities)
and uneven cash flows.

CFIN6 | CH4 25
Find the FV of an Uneven Cash Flow
Stream—FVCFn

0 1 2 3 Value of Each Deposit

r = 5% at the End of Year 3
400 300 250
x
250.00
x (1.05)1 (1.05)0
315.00
x (1.05)2
441.00
FVCF3 = 1,
006.00

CFIN6 | CH4 26
FVCFn Equation Solution

CFIN6 | CH4 27
FVCFn Equation Solution (continued)

CFIN6 | CH4 28
Present Value, PV

• 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
• Finding the present value (discounting) is
the reverse of finding the future value
(compounding); i.e., interest is taken out
of a future amount to determine its
present value.
CFIN6 | CH4 29
PV of a Lump-Sum Amount—PV

Discounting—to compute the present value of

an amount we bring back to the present a
future amount by taking out interest for
each period in which the money can earn
interest in the future.

0 1 2 3
r = 10%
1 x 1 x 1
x
1.10 1.10 1.10 935.00=FV3
PVB = 702.48 772.73 850.00

CFIN6 | CH4 30
PV of a Lump-Sum Amount—Equation Solution

CFIN6 | CH4 31
PV of a Lump-Sum Amount—Equation Solution
(continued)

What is the PV of $935 due in three (3) years

if r = 10%?

CFIN6 | CH4 32
PV of a Lump-Sum Amount—Financial
Calculator Solution

Inputs: 3 10 ? 0 935
N I/Y PV PMT FV
Outputs: = −702.48

CFIN6 | CH4 33
Present Value of an Annuity
(Ordinary)—PVAn

• PVAn = the present value of an annuity

with n payments, each made at the end of
the period.
• Each payment is discounted, and the sum
of the discounted payments is the present
value of the annuity.

CFIN6 | CH4 34
PVAn Cash Flow Timeline Solution

What is the PV of a three-year $400 ordinary

annuity if r = 5%?

Value of Each FV 0 1 2 3
Amount Today r = 5%
(Year 0) 1 x
400 400 400
380.95 1 x
(1.05)1 1 x
362.81 (1.05)2
345.54 (1.05)3
1,089.30 =
PVA3

CFIN6 | CH4 35
PVAn Equation Solution

CFIN6 | CH4 36
PVAn Equation Solution (continued)

CFIN6 | CH4 37
 PVAn Financial Calculator Solution

Inputs: 3 5 ? 400 0
N I/Y PV PMT FV
Outputs: = −1,089.30

CFIN6 | CH4 38
 Present Value of an Annuity Due—PVA(DUE)n

Payments are made at the beginning of the year,

which means one less year of 5 percent interest is
taken out of each payment.

PV of Each FV
Amount Today 0 1 2 3
(Year 0)
r = 5%
(1.05) x 1 x 400 400 400
400.00 (1.05) x 1 x
(1.05) 1
380.95 (1.05) x 1 x
(1.05) 1
362.81 (1.05)2
1,143.76 =
PVA(DUE)3

CFIN6 | CH4 39
PVA(DUE)n Equation Solution

Include one additional year’s worth of 5 percent interest.

CFIN6 | CH4 40
PVA(DUE)n Equation Solution (continued)

CFIN6 | CH4 41
PVA(DUE)n Financial Calculator Solution

BGN
Inputs: 3 5 ? −400 0
N I/Y PV PMT FV
Output: = 1,143.76

CFIN6 | CH4 42
Perpetuities—PVP

Streams of equal payments that are

expected to go on forever; perpetual
annuities

CFIN6 | CH4 43
Perpetuities—PVP (continued)

This example illustrates a fundamental principle

in finance: Everything else equal, the higher the
rate of return, the lower the value of an
investment.
CFIN6 | CH4 44
PV of an Uneven Cash Flow Stream—PVCFn

Value of Each FV 0 1 2 3
Amount Today r = 5%
(Year 0) 1 x
400 300 250
380.95 1 x
(1.05)1 1 x
272.11 (1.05)2
215.96 (1.05)3
869.02 = PVCF3

CFIN6 | CH4 45
PVCFn Equation Solution

CFIN6 | CH4 46
PVCFn Equation Solution (continued)

CFIN6 | CH4 47
PVCFn Financial Calculator Solution

• Input in “CF” register:

• CF0 = 0
• CF1 = 400
• CF2 = 300
• CF3 = 250
• Enter I = 5
• Press NPV button to get NPV = -869.02.

CFIN6 | CH4 48
Comparison of FV with PV

• FV contains interest, whereas PV does not.

• At an opportunity cost rate of 10 percent:
• a lump-sum payment of $700 today is the
same as a lump-sum payment of $931.70
in three years.
• The PV of $700 has no interest; the FV of
$931.70 contains three years of interest,
which equals $231.70.

CFIN6 | CH4 49
 Comparison of FV with PV (continued)

0 1 2 3r = 10%

PV = 700.00 FV1 = 770.00 FV2 = 847.00 FV3 = 931.70

PV at Year 0 = 700 = PV at Year 0 = 700 = PV at Year 0 = 700 = PV at Year 0 = 700

Interest from: Interest from: Interest from: Interest from:
Year 1 = Year 1 = Year 1 =
70.00 70.00
Year 2 = 70.00
Year 2 =
77.00 77.00
Year 3 =
Total =
< Total =
< Total =
< 84.70
Total =
70.00 70.00 147.00 231.70

The values given under the tick marks for each year
differ only because they contain different amounts of
interest.
CFIN6 | CH4 50
 Solving for Interest Rates (r)

Suppose you pay $78.35 for an investment

that promises to pay you $100 five years
from today. What annual rate of return will
you earn on your investment?

0 1 4 5
r= ? …
PV = -78.35 100.00 = FV5

CFIN6 | CH4 51
 Solving for r—Financial Calculator Solution

Inputs: 5 ? −78.35 0 100.00

N I/Y PV PMT FV
Output: = 5.00

CFIN6 | CH4 52
 Solving for Time (n)

A security that costs $68.30 will provide a

return of 10 percent per year. If you want
to keep the investment until it grows to a
value of $100, how long will you have to keep
it?
0 1 n-1 n = ?
r = 10% …
PV = -68.30 100.00 = FVn

CFIN6 | CH4 53
 Solving for n—Financial Calculator Solution

Inputs: ? 10 −68.3 0 100.00

N I/Y 0
PV PMT FV
Output: = 4.0

CFIN6 | CH4 54
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 earned (added) once per 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 per year.

CFIN6 | CH4 55
FV of a lump sum

• The FV of a lump sum be larger if interest is

compounded more often, holding the stated r
constant? Why?
• If compounding is more frequent than once
per year—for example, semi-annually,
quarterly, or daily—interest is earned on
interest—that is, compounded—more often.
• Compared to annual compounding, a greater
amount of interest is earned when interest
is compounded more than once per year.

CFIN6 | CH4 56
Distinguishing Between Different Interest
Rates

rSIMPLE = Simple (Quoted) Rate

used to compute the interest paid per period

APR = Annual Percentage Rate = rSIMPLE

rEAR = Effective Annual Rate

the annual rate of interest actually being
earned
CFIN6 | CH4 57
Comparison of Different Types of
Interest Rates

• rSIMPLE: Simple, or quoted rate; not

used in calculations.
• rPER: Periodic rat; rate per period (e.g.,
per year, per month, etc.); used in
calculations.
• rEAR: Effective annual rate; used to
compare returns on investments with
different interest compounding per year.

CFIN6 | CH4 58
Simple (Quoted) Rate

• rSIMPLE is stated in contracts

Interest periods per year (m) must be given

• Examples:
• 9%, compounded quarterly
• 9%, compounded daily (365 days)

CFIN6 | CH4 59
Periodic Rate, rPER

• Periodic rate = rPER = rSIMPLE/m, where m is

number of compounding periods per year
• Examples:
• 9%, compounded quarterly: m = 4 and rPER =
9%/4 = 2.25%
• 9%, compounded monthly: m = 12 and rPER =
9%/12 = 0.75%

CFIN6 | CH4 60
Effective Annual Rate, rEAR

The annual rate that causes PV to grow to

the same FV as it would with multi-period
compounding.

CFIN6 | CH4 61
Computing rEAR

• What is the effective annual return (EAR)

for an investment that pays 12 percent
interest, compounded monthly?

CFIN6 | CH4 62
Amortized Loans

• Amortized Loan—a loan that is repaid in equal

payments over its life.
• A portion of the payment represents interest
and the remainder represents repayment of the
amount that was borrowed.
• Amortization schedules show how much of each
payment represents principal repayment and
how much represents interest.

CFIN6 | CH4 63
Amortization Schedule

An amortization schedule for a $33,000, 6.5

percent loan that requires three equal
annual payments.

Year 0 1r2=3 6.5%

33,000 PMT = ? PMT = ? PMT = ?

CFIN6 | CH4 64
 PMT Financial Calculator Solution

Inputs: 3 6.5 33, ? 0

N I/Y 000
PV PMT FV
Output: = −12,460

CFIN6 | CH4 65
Amortization Schedule (continued)

Beginning Payment Interest Repayment Remaining

Year of Year (2) @ 6.5% of Principal Balance
Balance (1) (3) = (1) × (4) = (2) − (3) (5) = (1) −
0.065 (4)
1 $33,000.00 $12,460 $2,145 $10,315.00 $22,685.00
2 22,685.00 $12,460 1,474.53 10,985.48 11,699.53
3 11,699.53 $12,460 760.47 11,699.53 0.00

CFIN6 | CH4 66
Questions and Problems

If you invest $500 today in an account that pays 6

percent interest compounded monthly, how much
will be in your account after two years?

CFIN6 | CH4 67
Questions and Problems

To the closest year, how long will it take a $200

investment to double if it earns 7% interest? How
long will it take if the investment earns 1%?

CFIN6 | CH4 68
Questions and Problems

Martell Corporation’s sales were $12 million this

year. Sales were $6 million five years earlier. To
the nearest percentage point, at what annual
rate have sales grown?

CFIN6 | CH4 69
Questions and Problems

• Suppose you have been shopping for mortgages to

finance the house that you want to buy. The East Coast
Federal Credit Union (ECFCU) has offered a 30-year
fixed mortgage that requires you to pay 6 percent
interest compounded monthly. The purchase price of the
house is $260,000, and you plan to make a down payment
equal to $28,000. What would your monthly payments be
with the ECFCU mortgage?
• Suppose it is now 10 years later, such that you have lived
in the house for 10 years, and you are considering paying
off your mortgage. How much do you owe on the
mortgage if this month’s payment was made yesterday?
CFIN6 | CH4 70
Questions and Problems

CFIN6 | CH4 71
Questions and Problems

Allison wants to pay off her existing automobile

loan. Two years ago, Allison borrowed $35,600 with
terms that required her to make monthly
payments equal to $739 for a period of five years.
The interest rate on the loan is 9 percent. To the
nearest dollar, how much does Allison currently
owe on her automobile loan? The most recent
payment on the loan was made yesterday.

CFIN6 | CH4 72
Questions and Problems

CFIN6 | CH4 73
Questions and Problems

Jack just discovered that he holds the winning

ticket for the $87 million “mega” lottery in
Missouri. Now he must decide which alternative to
choose: (a) a $44 million lump-sum payment today
or (b) a payment of $2.9 million per year for 30
years. With the second option, the first payment
will be made today. If Jack’s opportunity cost is 5
percent, which alternative should he choose?

CFIN6 | CH4 74
Questions and Problems

CFIN6 | CH4 75
Questions and Problems

John has made the following annual deposits in his

bank account as shown below. How much money is
in his account (at year t=5) if the bank pays
monthly interests and the annual interest rate is
3%?
Years: 0 1 2 3 4 5
Deposits: $45 $75 $225$150$300?

CFIN6 | CH4 76