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Fundamentals
of Corporate
Finance
Chapter 5
The Time Value of Money
Sixth Edition
Richard A. Brealey
Stewart C. Myers
Alan J. Marcus
Slides by
Matthew Will
McGraw
McGraw Hill/Irwin
Hill/Irwin
Copyright ©Copyright
2009 by The
McGraw-Hill
Companies, Inc.
All rights
reserved
© 2009
by The McGraw-Hill
Companies,
Inc.
All rights reserved
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Topics Covered
 Future Values and Compound Interest
 Present Values
 Multiple Cash Flows
 Level Cash Flows Perpetuities and Annuities
 Effective Annual Interest Rates
 Inflation & Time Value
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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.
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Future Values
Example - Simple Interest
Interest earned at a rate of 6% for five years on a
principal balance of $100.
Interest Earned Per Year = 100 x .06 = $ 6
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Future Values
Example - Simple Interest
Interest earned at a rate of 6% for five years on a
principal balance of $100.
Today
Interest Earned
Value
100
Future Years
1
2
3
4
6
6
6
6
106 112 118 124
Value at the end of Year 5 = $130
5
6
130
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Future Values
Example - Compound Interest
Interest earned at a rate of 6% for five years on the
previous year’s balance.
Interest Earned Per Year =Prior Year Balance x .06
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Future Values
Example - Compound Interest
Interest earned at a rate of 6% for five years on
the previous year’s balance.
Today
Interest Earned
Value
100
Future Years
1
2
3
4
5
6
6.36
6.74
7.15
7.57
106 112.36 119.10 126.25 133.82
Value at the end of Year 5 = $133.82
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Future Values
Future Value of $100 = FV
FV  $100  (1  r )
t
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Future Values
FV  $100  (1  r )
t
Example - FV
What is the future value of $100 if interest is
compounded annually at a rate of 6% for five years?
FV  $100  (1  .06 )  $133 .82
5
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Future Values with Compounding
FV of $100
1800
1600
0%
1400
5%
1200
1000
800
10%
15%
Interest Rates
600
400
200
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Number of Years
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Manhattan Island Sale
Peter Minuit bought Manhattan Island for $24 in 1626.
Was this a good deal?
To answer, determine $24 is worth in the year 2008,
compounded at 8%.
FV  $24 (1  .08)
 $140.63 trillion
382
FYI - The value of Manhattan Island land is
well below this figure.
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Present Values
Present Value
Discount Factor
Value today of a
future cash
flow.
Present value of
a $1 future
payment.
Discount Rate
Interest rate used
to compute
present values of
future cash flows.
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Present Values
Present Value = PV
PV =
Future Value after t periods
(1+r)
t
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Present Values
Example
You just bought a new computer for $3,000. The payment
terms are 2 years same as cash. If you can earn 8% on your
money, how much money should you set aside today in order
to make the payment when due in two years?
PV 
3000
2
(1.08 )
 $2,572
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Present Values
Discount Factor = DF = PV of $1
DF 
1
(1 r ) t
 Discount Factors can be used to compute the
present value of any cash flow.
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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.
PV  FV 
1
(1 r ) t
Present Values with Compounding
120
Interest Rates
100
PV of $100
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0%
5%
80
10%
15%
60
40
20
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Number of Years
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Time Value of Money
(applications)
 Value of Free Credit
 Implied Interest Rates
 Internal Rate of Return
 Time necessary to accumulate funds
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PV of 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 is
8%, which do you prefer?
Immediate payment 8,000.00
PV1 
4 , 000
(1.08 )1
 3,703.70
PV2 
4 , 000
(1.08 ) 2
 3,429.36
T otalPV
 $15,133.06
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Present Values
$8,000
$4,000
$ 4,000
Present Value
Year 0
Year
0
$8,000
4000/1.08
= $3,703.70
4000/1.082
= $3,429.36
Total
= $15,133.06
1
2
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Perpetuities & Annuities
Finding the present value of multiple cash flows by using a spreadsheet
Time until CF Cash flow Present value
0
8000
$8,000.00
1
4000
$3,703.70
2
4000
$3,429.36
SUM:
Discount rate:
Formula in Column C
=PV($B$11,A4,0,-B4)
=PV($B$11,A5,0,-B5)
=PV($B$11,A6,0,-B6)
$15,133.06 =SUM(C4:C6)
0.08
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PV of Multiple Cash Flows
 PVs can be added together to evaluate
multiple cash flows.
PV 
C1
(1 r )
 (1 r ) 2 ....
C2
1
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Perpetuities & Annuities
Perpetuity
A stream of level cash payments
that never ends.
Annuity
Equally spaced level stream of cash
flows for a limited period of time.
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Perpetuities & Annuities
PV of Perpetuity Formula
PV 
C = cash payment
r = interest rate
C
r
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Perpetuities & Annuities
Example - Perpetuity
In order to create an endowment, which pays
$100,000 per year, forever, how much money must
be set aside today in the rate of interest is 10%?
PV 
100 , 000
.10
 $1,000,000
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Perpetuities & Annuities
Example - continued
If the first perpetuity payment will not be received
until three years from today, how much money needs
to be set aside today?
PV 
1, 000 , 000
( 1 .10 ) 3
 $751,315
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Perpetuities & Annuities
PV of Annuity Formula
PV  C

1
r

1
r ( 1 r ) t

C = cash payment
r = interest rate
t = Number of years cash payment is received
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Perpetuities & Annuities
PV Annuity Factor (PVAF) - The present value
of $1 a year for each of t years.
PVAF 

1
r

1
r ( 1 r ) t

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Perpetuities & Annuities
Example - Annuity
You are purchasing a car. You are scheduled to
make 3 annual installments of $4,000 per year.
Given a rate of interest of 10%, what is the price you
are paying for the car (i.e. what is the PV)?

PV  4 ,000
1
.10

PV  $9,947.41
1
.10 ( 1 .10 ) 3

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Perpetuities & Annuities
Applications
 Value of payments
 Implied interest rate for an annuity
 Calculation of periodic payments
Mortgage payment
 Annual income from an investment payout
 Future Value of annual payments

FV   C  PVAF   (1  r )
t
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Perpetuities & Annuities
Example - Future Value of annual payments
You plan to save $4,000 every year for 20 years and
then retire. Given a 10% rate of interest, what will
be the FV of your retirement account?

FV  4 ,000
1
.10

FV  $229,100
1
.10 ( 1 .10 ) 20
  (1.10)
20
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Effective Interest Rates
Effective Annual Interest Rate - Interest rate
that is annualized using compound interest.
Annual Percentage Rate - Interest rate that is
annualized using simple interest.
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Effective Interest Rates
example
Given a monthly rate of 1%, what is the Effective
Annual Rate(EAR)? What is the Annual Percentage
Rate (APR)?
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Effective Interest Rates
example
Given a monthly rate of 1%, what is the Effective
Annual Rate(EAR)? What is the Annual Percentage
Rate (APR)?
12
EAR = (1 + .01) - 1 = r
EAR = (1 + .01)12 - 1 = .1268or 12.68%
APR = .01 x 12 = .12 or 12.00%
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Inflation
Inflation - Rate at which prices as a whole are
increasing.
Nominal Interest Rate - Rate at which money
invested grows.
Real Interest Rate - Rate at which the
purchasing power of an investment increases.
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Inflation
Annual Inflation, %
Annual U.S. Inflation Rates from 1900 - 2007
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Inflation
1+ nominal interest rate
1  real interest rate =
1+inflation rate
approximation formula
Real int. rate  nominal int. rate - inflation rate
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Inflation
Example
If the interest rate on one year govt. bonds is 6.0%
and the inflation rate is 2.0%, what is the real
interest rate?
1+.06
1  real interest rate = 1+.02
Savings
1  real interest rate = 1.039
Bond
real interest rate = .039or 3.9%
Approximation = .06 - .02 = .04 or 4.0%
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Inflation
 Remember: Current dollar cash flows must be
discounted by the nominal interest rate; real
cash flows must be discounted by the real
interest rate.
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