Time

Time Value
Value of
of
Money
Money

1
 Time Value of Money
 refers to the observation that it is
better to receive money sooner
than later. Money that you have
in hand today can be invested to
earn a positive rate of return,
producing more money in the
future.
2
 The
The Time
Time Value
Value of
of Money
Money
 The Interest Rate
 Simple Interest
 Compound Interest
 Amortizing a Loan
 Compounding More Than
Once per Year
3
 Future Values
Future Value - Amount to which an
investment will grow after earning interest.

Compound Interest - Interest earned on

interest.

Simple Interest - Interest earned only on the

original investment.

4
 Why
Why TIME?
TIME?

Why is TIME such an important

element in your decision?

TIME allows you the opportunity to

postpone consumption and earn
INTEREST.
INTEREST

5
 Types
Types of
of Interest
Interest
 Simple Interest
Interest paid (earned) on only the original
amount, or principal, borrowed (lent).
 Compound Interest
Interest paid (earned) on any previous
interest earned, as well as on the
principal borrowed (lent).

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 Simple
Simple Interest
Interest Formula
Formula

Formula SI = P0(i)(n)
SI: Simple Interest
P0 : Deposit today (t=0)
i: Interest Rate per Period
n: Number of Time Periods
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 Simple
Simple Interest
Interest Example
Example
 Assume that you deposit $1,000 in an
account earning 7% simple interest for
2 years. What is the accumulated
interest at the end of the 2nd year?
 SI = P0(i)(n)
= $1,000(.07)(2)
= $140
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 Simple
Simple Interest
Interest (FV)
(FV)
 What is the Future Value (FV)
FV of the
deposit?
FV = P0 + SI
= $1,000 + $140
= $1,140
 Future Value is the value at some future
time of a present amount of money, or a
series of payments, evaluated at a given
interest rate.
9
 Simple
Simple Interest
Interest (PV)
(PV)
 What is the Present Value (PV)
PV of the
previous problem?
The Present Value is simply the
$1,000 you originally deposited.

 Present Value is the current value of a

future amount of money, or a series of
payments, evaluated at a given interest
10
rate.
 Why
Why Compound
Compound Interest?
Interest?
Future Value of a Single $1,000 Deposit
Future Value (U.S. Dollars)

20000
10% Simple
15000 Interest
10000 7% Compound
Interest
5000 10% Compound
Interest
0
1st Year 10th 20th 30th
Year Year Year
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 Present Value
Today's value of a lump sum received at a
future point in time:

FVn  PV  1  i 
n

FVn
PV 
(1  i ) n

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 Future Value of Single Cash
Flow

FV  PV  (1  i) n

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 Future
Future Value
Value
Single
Single Deposit
Deposit (Graphic)
(Graphic)
Assume that you deposit $1,000 at
a compound interest rate of 7% for
2 years.
years
0 1 2
7%
$1,000
FV2
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 Future
Future Value
Value
Single
Single Deposit
Deposit (Formula)
(Formula)
FV1 = P0 (1+i)1 = $1,000 (1.07)
= $1,070
Compound Interest
You earned $70 interest on your $1,000
deposit over the first year.
This is the same amount of interest you
would earn under simple interest.
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 General
General Future
Future
Value
Value Formula
Formula
FV1 = P0(1+i)1
FV2 = P0(1+i)2
etc.

General Future Value Formula:

FVn = P0 (1+i)n
or FVn = P0 (FVIFi,n) -- See Table I
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 Valuation
Valuation Using
Using Table
Table II

Period 6% 7% 8%
1 1.060 1.070 1.080
2 1.124 1.145 1.166
3 1.191 1.225 1.260
4 1.262 1.311 1.360
5 1.338 1.403 1.469

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 Using
Using Future
Future Value
Value Tables
Tables
FV2 = $1,000 (FVIF7%,2)
= $1,000 (1.145)
= $1,145 [Due to Rounding]
Period 6% 7% 8%
1 1.060 1.070 1.080
2 1.124 1.145 1.166
3 1.191 1.225 1.260
4 1.262 1.311 1.360
5 1.338 1.403 1.469
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 Story
Story Problem
Problem Example
Example
Julie Miller wants to know how large her deposit
of $10,000 today will become at a compound
annual interest rate of 10% for 5 years.
years

0 1 2 3 4 5
10%
$10,000
FV5
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 Story
Story Problem
Problem Solution
Solution
 Calculation based on general formula:
FVn = P0 (1+i)n
FV5 = $10,000 (1+ 0.10)5
= $16,105.10
 Calculation based on Table I: FV5 =
$10,000 (FVIF10%, 5)
= $10,000 (1.611)
= $16,110 [Due to Rounding]
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 Present
Present Value
Value
Single
Single Deposit
Deposit (Graphic)
(Graphic)
Assume that you need $1,000 in 2 years.
Let’s examine the process to determine
how much you need to deposit today at a
discount rate of 7% compounded annually.
0 1 2
7%
$1,000
PV0 PV1
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 Present
Present Value
Value
Single
Single Deposit
Deposit (Formula)
(Formula)
PV0 = FV2 / (1+i)2 = $1,000 / (1.07)2
= FV2 / (1+i)2 = $873.44

0 1 2
7%
$1,000
PV0
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 General
General Present
Present
Value
Value Formula
Formula
PV0 = FV1 / (1+i)1
PV0 = FV2 / (1+i)2
etc.

General Present Value Formula:

PV0 = FVn / (1+i)n
or PV0 = FVn (PVIFi,n) -- See Table II
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 Valuation
Valuation Using
Using Table
Table IIII
PVIFi,n is found on Table II
at the end of the book.
Period 6% 7% 8%
1 .943 .935 .926
2 .890 .873 .857
3 .840 .816 .794
4 .792 .763 .735
5 .747 .713 .681
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 Using
Using Present
Present Value
Value Tables
Tables
PV2 = $1,000 (PVIF7%,2)
= $1,000 (.873)
= $873 [Due to Rounding]

Period 6% 7% 8%
1 .943 .935 .926
2 .890 .873 .857
3 .840 .816 .794
4 .792 .763 .735
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5 .747 .713 .681
 Story
Story Problem
Problem Example
Example
Julie Miller wants to know how large of a
deposit to make so that the money will grow
to $10,000 in 5 years at a discount rate of
10%.
0 1 2 3 4 5
10%
$10,000
PV0
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 Story
Story Problem
Problem Solution
Solution
 Calculation based on general formula:
PV0 = FVn / (1+i)n
PV0 = $10,000 / (1+ 0.10)5 =
$6,209.21
 Calculation based on Table I: PV0 =
$10,000 (PVIF10%, 5) = $10,000 (.621) =
$6,210.00 [Due to Rounding]

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 Future Value of a Single Amount

Jane places P800 in a savings

account paying 6% interest
compounded annually. She wants to
know how much money will be in the
account at the end of 5 years.

FV= P 1,070.58
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 Present Value of a Single Amount

Paul has an opportunity to receive P300

one year from now. If he can earn 6%
on his investments in the normal
course of events, what is the most he
should pay now for this opportunity?

PV= P 283.02

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 Pam wishes to find the present
value of P1,700 that she will
receive 8 years from now. Pam's
opportunity cost is 8%.

PV = P918.46

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 DEPOSIT
SINGLE CASH
INTEREST RATE PERIOD
FLOW
(YEARS)
A P 200 5% 20
B 4,500 8% 7
C 10,000 9% 10
D 25,000 10% 12
E 37,000 11% 5
F 40,000 12% 9
G 56,000 7% 5
H 125,000 17% 20
I 1,500 18% 12
J 15,000 15% 15
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32
 Types
Types of
of Annuities
Annuities
 An Annuity represents a series of equal
payments (or receipts) occurring over a
specified number of equidistant periods.
 Ordinary Annuity:
Annuity Payments or receipts
occur at the end of each period.
 Annuity Due:
Due Payments or receipts
occur at the beginning of each period.

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 Parts
Parts of
of an
an Annuity
Annuity
(Ordinary Annuity)
End of End of End of
Period 1 Period 2 Period 3

0 1 2 3

$100 $100 $100

Today Equal Cash Flows

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Each 1 Period Apart
 Parts
Parts of
of an
an Annuity
Annuity
(Annuity Due)
Beginning of Beginning of Beginning of
Period 1 Period 2 Period 3

0 1 2 3

$100 $100 $100

Today Equal Cash Flows

Each 1 Period Apart
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 Fran Abrams is evaluating two
annuities. Both are 5-year, $1,000
annuities; annuity A is an ordinary
annuity and annuity B is an annuity
due. The amount of each annuity
totals $5,000.

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37
 Although the cash flows of both
annuities total $5,000, the annuity
due would have a higher FV than the
ordinary annuity because each of its
five annual cash flows can earn
interest for 1 year more than each of
the ordinary annuity's cash flows.

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 Finding the Future Value
of an Ordinary Annuity

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  Fran Abrams wishes to determine how
much money she will have at the end
of 5 years if she chooses annuity A,
the ordinary annuity. She will deposit
$1,000 annually, at the end of each of
the next 5 years, into a savings
account paying 7% annual interest.

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41
 Finding the Present Value
of an Ordinary Annuity

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  Braden Company, a small producer of
plastic toys, wants to determine the
most it should pay to purchase a
particular ordinary annuity. The annuity
consists of cash flows of $700 at the
end of each year for 5 years. The firm
requires the annuity to provide a
minimum return of 8%.
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44
45
 Finding the Future Value
of an Annuity Due

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  Fran Abrams wishes to determine how
much money she will have at the end
of 5 years if she chooses annuity A,
the ordinary annuity. She will deposit
$1,000 annually, at the end of each of
the next 5 years, into a savings
account paying 7% annual interest.

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  Ordinary annuity = $5,750.74

versus

 Annuity due = $6,153.29

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  The future value of an annuity due is
always greater than the future value of an
otherwise identical ordinary annuity.
 Because the cash flow of the annuity due
occurs at the beginning of the period rather
than at the end (that is, each payment comes
one year sooner in the annuity due), its future
value is greater.

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 Finding the Present
Value of an Annuity Due

50
  Braden Company, a small producer of
plastic toys, wants to determine the
most it should pay to purchase a
particular ordinary annuity. The annuity
consists of cash flows of $700 at the
end of each year for 5 years. The firm
requires the annuity to provide a
minimum return of 8%.
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 Ordinary annuity = $2,794.90

versus

Annuity due = $3,018.49

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  The present value of an annuity due is
always greater than the present value
of an otherwise identical ordinary
annuity.
 Because the cash flow of the annuity due
occurs at the beginning of the period
rather than at the end, its present value is
greater.

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 Future value of annuity
Amount Deposit
Interest
Case of period
Rate
Annuity (years)
A $2,500 8% 10

B 500 12% 6

C 30,000 20% 5
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 Present Value of Annuity
Amount of Interest Period
Case
Annuity Rate (years)

A $12,000 7% 3

B 55,000 12% 15

C 700 20% 9

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  A perpetuity is an annuity with an infinite life
—in other words, an annuity that never stops
providing its holder with a cash flow at the
end of each year (for example, the right to
receive $500 at the end of each year forever).
 If a perpetuity pays an annual cash flow of
CF, starting one year from now, the present
value of the cash flow stream is

56
  Ross Clark wishes to endow a chair in
finance at his alma mater. The university
indicated that it requires $200,000 per year to
support the chair, and the endowment would
earn 10% per year. To deter- mine the
amount Ross must give the university to fund
the chair, we must determine the present
value of a $200,000 perpetuity discounted at
10%.

57
 PV = $200,000 , 0.10 = $2,000,000

In other words, to generate $200,000 every

year for an indefinite period requires $2,000,000
today if Ross Clark’s alma mater can earn 10%
on its investments. If the university earns 10%
interest annually on the $2,000,000, it can
withdraw $200,000 per year indefinitely.

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 Mixed Streams
 Mixed stream is a stream of
unequal periodic cash flows that
reflect no particular pattern.
 Financial managers frequently need
to evaluate opportunities that are
expected to provide mixed streams
of cash flows.
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  Shrell Industries, a cabinet manufacturer,
expects to receive the following mixed
stream of cash flows over the next 5 years
from one of its small customers.
 If Shrell expects to earn 8% on its
investments, how much will it accumulate by
the end of year 5 if it immediately invests
these cash flows when they are received?

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 Future Value of a Mixed
Stream

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  Frey Company, a shoe manufacturer, has
been offered an opportunity to receive the
following mixed stream of cash flows over the
next 5 years.
 If the firm must earn at least 9% on its
investments, what is the most it should pay for
this opportunity?

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64
 Compounding Interest More
frequently than Annually
 Interest is often compounded more
frequently than once a year. Savings
institu- tions compound interest
semiannually, quarterly, monthly,
weekly, daily, or even continuously.

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 Semiannual Compounding

 Semiannual compounding of interest

involves two compounding periods
within the year. Instead of the stated
interest rate being paid once a year,
one-half of the stated interest rate is
paid twice a year.

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 Fred Moreno has decided to invest $100 in a
savings account paying 8% interest
compounded semiannually. If he leaves his
money in the account for 24 months (2 years),
he will be paid 4% interest com- pounded over
four periods, each of which is 6 months long.

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 Quarterly Compounding
 Quarterly compounding of interest
involves four compounding periods
within the year. One-fourth of the
stated interest rate is paid four
times a year.

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 Fred Moreno has found an institution that will pay
him 8% interest compounded quarterly. If he
leaves his money in this account for 24 months
(2 years), he will be paid 2% interest
compounded over eight periods, each of which is
3 months long.

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70
 A General Equation for
Compounding more frequently
than annually

The future value formula can be rewritten for use when

compounding takes place more frequently. “m” equals
the number of times per year interest is compounded.

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  Fred Moreno would have at the end of 2
years if he deposited $100 at 8% interest
compounded semiannually and
compounded quarterly.
 For semi- annual compounding, m
would equal 2 in Equation 5.15; for
quarterly com- pounding, m would equal
4.
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