McGraw-Hill/Irwin Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

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Time Value of Money
Money has a time value. It can be expressed in
multiple ways:
A dollar today held in savings will grow.
A dollar received in a year is not worth as much as a dollar
received today.

FV = Initial investment (1 ) Compound ´ + r Simple ´ + r ´t
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Future Values
Future Value: Amount to which an investment will grow
after earning interest.
Let r = annual interest rate
Let t = # of years
Simple Interest Compound Interest
FV = Initial investment (1 )t

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Simple Interest: Example
Interest earned at a rate of 7% for five years on a
principal balance of $100.
Example - Simple Interest
Today Future Years
1 2 3 4 5
Interest Earned
Value 100
7
107
7
114
7
121
Value at the end of Year 5: $135
7
128
7
135

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Compound Interest: Example
Interest earned at a rate of 7% for five years on the
previous year’s balance.
Example - Compound Interest
Today Future Years
1 2 3 4 5
Interest Earned
Value 100
7
107
7.49
114.49
8.01
122.50
8.58
131.08
9.18
140.26
Value at the end of Year 5 = $140.26

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The Power of Compounding
Interest earned at a rate of 7% for the first forty years
on the $100 invested using simple and compound
interest.

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Present Value
What is it?
Why is it useful?

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Present Value
Discount Rate:
Discount Factor:
Present Value:
1
(1 r )t DF + =
1
(1 r )t PV FV + = ´
Recall: t = number of years
r

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Present Value: Example
Always ahead of the game, Tommy, at 8 years old, believes
he will need $100,000 to pay for college. If he can invest at a
rate of 7% per year, how much money should he ask his rich
Uncle GQ to give him?
FV =$100,000 t =10 yrs r =7%
PV = FV
´ 1 = $100,000 ´ 1
»
$50,835
(1 +
r
)t (1.07)
10
Note: Ignore inflation/taxes

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Time Value of Money
(applications)
The PV formula has many applications. Given
any variables in the equation, you can solve for
the remaining variable.
1
(1 r )t PV FV + = ´

Present Values: Changing Discount Rates
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The present value of $100 to be received in 1 to 20 years at varying discount rates:
120
100
80
60
40
20
0
Discount Rates
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Number of Years
PV of $100
0%
5%
10%
15%

Denote
C
C
C
:
The cash flow in year 1
The cash flow in year 2
The cash flow in year t (with any number of cash flows in between) t
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PV of Multiple Cash Flows
The present value of multiple cash flows can be calculated:
1 2
.... t
(1 )1 (1 )2 (1 ) t
C C C
r r r PV + + + = + + +
1
2
=
=
=
Recall: r = the discount rate

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Multiple Cash Flows: Example
Your auto dealer gives you the choice to pay $15,500 cash now or
make three payments: $8,000 now and $4,000 at the end of the
following two years. If your cost of money (discount rate) is 8%, which
do you prefer?
Initial Payment* 8,000.00
PV of C
PV of C
= =
= =
1 (1 .08)
1
4,000
2
+
4,000
2 (1 .08)
3,703.70
3,429.36
+
=
Total PV $15,133.06
* The initial payment occurs immediately and therefore would not be discounted.

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Perpetuities
Let C = Yearly Cash Payment
PV of Perpetuity:
Cr
PV =
What are they?
Recall: r = the discount rate

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Perpetuities: Example
In order to create an endowment, which pays $185,000 per year
forever, how much money must be set aside today if the rate of
interest is 8%?
PV = 185,000
= $2,312,500
.08 What if the first payment won’t be received until 3 years from
today?
PV = 2,312,500
=
$1,982,596 (1 + .08) 2

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Annuities
What are they?
Annuities are equally-spaced, level streams of cash flows lasting
for a limited period of time.
Why are they useful?

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Present Value of an Annuity
Let:
C = yearly cash payment
r = interest rate
t = number of years cash payment is received
= ´ é - ù ë û
1 1
r r (1 r)t PV C ´ +
The terms within the brackets are
collectively called the “annuity factor.”

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Annuities: Example
You are purchasing a home and are scheduled to make 30
annual installments of $10,000 per year. Given an interest
rate of 5%, what is the price you are paying for the house
(i.e. what is the present value)?
= é ù ë - 30
û
=
1 1
.05 .05(1 .05) $10,000
$153,724.51
PV
PV
+

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Future Value of Annuities:
Example
You plan to save $4,000 every year for 20 years and then retire.
Given a 10% rate of interest, how much will you have saved by
the time you retire?
= é - ù´ + ë û
=
$4,000 1 1 (1 .10)
20
.10 .10(1 .10) 20
$229,100
FV
FV
+

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Annuity Due
What is it?
How does it differ from an ordinary annuity?
(1 ) AnnuityDue Annuity PV = PV ´ + r
How does the future value differ from an ordinary annuity?
(1 ) AnnuityDue FV = FVAnnuity ´ + r
Recall: r = the discount rate

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Annuities Due: Example
FV FV (1 r) AD Annuity = ´ +
Example: Suppose you invest $429.59 annually at the
beginning of each year at 10% interest. After 50 years,
how much would your investment be worth?
FV = FV ´ +
r
AD Annuity
= ´
($500,000) (1.10)
$550,000
(1 )
=
AD
AD
FV
FV

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Interest Rates: EAR & APR
What is EAR?
What is APR?
How do they differ?

5-
EAR & APR Calculations
Effective Annual Interest Rate (EAR):
EAR = (1+MR)12 -1
Annual Percentage Rate (APR):
APR = MR´12
MR = monthly interest rate 23

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EAR and APR: Example
Given a monthly rate of 1%, what is the Effective Annual
Rate(EAR)? What is the Annual Percentage Rate (APR)?
(1.01)12 1 12.68%
= ´ =
= - =
(0.01) (12) 12.00%
EAR
APR

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Inflation
What is it?
What determines inflation rates?
What is deflation?

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Inflation and Real Interest
1+nominal interest rate
Exact calculation:
1 + real interest rate= 1+inflation rate
Approximation: Real interest rate » nominal interest rate - inflation rate

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Inflation: Example
If the nominal interest rate on your interest-bearing savings
account is 2.0% and the inflation rate is 3.0%, what is the
real interest rate?
1+.02
+
+
1 real interest rate= 1+.03
1 real interest rate= 0.9903
real interest rate = -.0097 or -.97%
Approximation = .02-.03 = - .01 = -
1%

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Appendix A: Inflation
Annual U.S. Inflation Rates from 1900 - 2010