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Chapter 4
Time Value of Money
y
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Money Has “Time
“Time Value”
Value”


Money today is preferred to
money next month, next year
or in 10 years.
The further in the future that
money is to be received, the
less valuable it appears now.
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
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Of all of the concepts of finance,
none is more important than
the time
time--value of money (TVM
TVM).
).
The core to understanding TVM is
compounding..
compounding
• Even Albert Einstein, the most brilliant
and famous scientist of the 20th
Century, called it “one of the most
important forces in the world.”
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A Dollar Available Today:
Today:

Is safer than the same dollar
received in the future.

Won’t lose value to inflation.

Can be spent and enjoyed now.

Can be invested to earn income or
increase in value, so that more
will be available tomorrow.
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(Or) A Dollar Available Later:
Later:

Is riskier than the same dollar
received now.

Will lose value to inflation.

Cannot be spent and enjoyed now.

Cannot be invested to earn income
or increase in value.
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Interest Rate

The inducement to encourage
someone to save and invest
rather than spend today.
today
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Time Value of Money

Future Value & Compounding

Present Value & Discounting
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Future Value (FV)
&
Compounding
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Compounding

The growth over time of a
beginning amount when each
period’s increase is added to
the base for the next period.
period.
• Ex: $1000 is invested for 5 years w/
all of the P+I to be left untouched
until maturity.
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If earnings are withdrawn, it
is NOT compounding!!!
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Growth Over Time (1)



Inflation averages 2.70% growth
per year-year--compounded
compounded..
Y
Your
wages grow by
b 6.25%/year6.25%/year
6 25%/
-compounded
compounded..
The world population continues to
grow by 1.5% annually-annually-compounded..
compounded
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Growth Over Time (2)

Cost of buying a car

Cost of buying a house

Value of investments for
retirement
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Cash Flows
Lump sum:
sum: a single cash flow (or
series of different cash flows)

Annuity: a series of CFs of the
Annuity:
same size

• Ex: A salary of $5000/month
• Ex: Monthly car payments of $525
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Future Value


What is the value in the future
of a cash flow today ?
How
H
d
do we combine
bi
multiple
lti l
cash flows over several periods
into a single future value?
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Future Value

The ending value

A single lump sum

The amount to which a beginning
lump sum (or series of cash flows)
will be compounded
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Future Value


The higher the compounding rate
(i) the greater will be the future
value.
The higher the number of
compounding periods (n) the
greater will be the future value.
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Time Lines


A graph showing all relevant
information in a TVM problem
Especially helps to visualize time
periods
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Time Line of Cash Flows
• Tick marks at end of periods
• Time 0 is today;
• Time 1 is the end of Period 1
0
1
2
3
CF1
CF2
CF3
r%
CF0
Future Value of Two Cash Flows
Future Value of Three Cash
Flows
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FVn= PV * (1 + i)n = P0 (1 + i)n

FVn : FV of the principal (i.e., the
PV) at the end of n periods

PV:: The
PV
Th original
i i l principal
i i l (P0)

i: The interest rate per period

n: number of periods
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



1st period: P + P*i = P(1+i)1
2d period: P(1+i)(1+i)
= P(1+i)2
3d period:
i d P(1+i)(1+i)(1+i)
= P(1 + i)3
nth period: P(1+i)(1+i)(1+i)(1+i)..
= P(1 + i)n = FV
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Compounding More Than
Once Per Year

Number of periods: n*m

Interest rate: i÷m

Also: Pmt÷
Pmt÷m
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n
i
n*m
i÷m
PV
Pmt
FV
Pmt
Pmt÷
÷m
nc
nc
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Present Value (PV)
&
Discounting
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Present Value

The beginning value

A single lump sum


The
h amount that
h iis to b
be
compounded to a future value
The amount to which a future
value is to be discounted
discounted..
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Present Value


What is the value of future cash
flows today?
How
H
d
do we combine
bi
multiple
lti l
future cash flows into a single
value today?
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Present Value


The higher the discounting rate
(i) the lesser will be the
present value.
The higher the number of
discounting periods (n) the
lesser will be the present value.
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Present Value of Three Cash
Flows
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FV vs. PV

Alternative choices can be
compared on either a PV or a
FV basis.
• However, all alternatives must be
evaluated in the same timeframe—
timeframe—
either PV or FV.
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FV vs. PV


Compounding is the process of
converting PV(s) to a single FV.
Discounting is the process of
converting FV(s) to a single PV.
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

With compounding, i is called
the compound rate.
rate.
With discounting,
di
ti
i is
i called
ll d the
th
discount rate.
rate.
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FVn= PV * (1 + i)n
FV
= PV * (1 + i)n
(1 + i)n
(1 + i)n
PV = FVn / (1 + i)n
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Financial Calculator
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Financial Calculator: TVM
n

i
PV
Pmt
FV
Knowing any 4 of these variables,
we can calculate the 5th
• Either PV, Pmt or FV is often not
given & can be assumed to = 0.
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Check Your Calculator!!!



Always clear the memory before
beginning a calculation.
1 payment per period
Ordinary annuity (“end
(“end”)
”)—
—not
“beg” (beginning)
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Watch Your Sign!!!

Either PV or FV must be negative.
negative.
• If you enter a negative value for PV,
the FV answer will be positive.
positive
• If you enter a positive value for PV,
the FV answer will be negative.
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

If you make an investment or if
the price is given, PV is negative
and FV will be positive.
positive.
If you borrow money, PV is
positive and FV will be negative.
negative.
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Financial Calculator: TVM
n
i
PV
Pmt
FV
n:
Number of periods (years,
quarters, months, etc.)
i:
Periodic interest rate
PV:: LumpPV
Lump-sum beginning value
Pmt:: Periodic payment (year, month)
Pmt
FV:: LumpFV
Lump-sum ending value
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“Rule of 72”
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“Rule of 72”
72”

Used to determine the
compounding rate (i) or the
number of periods (n) that will
be necessary
y to approximately
pp
y
double a beginning value.
value.
• If you know n: 72 ÷ n ~ i
• If you know i: 72 ÷ i ~ n
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