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Principles
of
Corporate
Finance
Chapter 3
How To Calculate
Present Values
Ninth Edition
Slides by
Matthew Will
McGraw Hill/Irwin
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
3- 2
Topics Covered
Valuing Long-Lived Assets
Looking for Shortcuts – Perpetuities and
Annuities
More Shortcuts – Growing Perpetuities and
Annuities
Compound Interest & Present Values
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Present Values
C1
PV  DF  C1 
1  r1
DF 
1
(1 r ) t
Discount Factors can be used to compute
the present value of any cash flow.
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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 
McGraw Hill/Irwin
3000
(1.08)2
 $2,572
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Present Values
Example
You have the opportunity to purchase the baseball hit by
Barry Bonds to break Hank Arron’s home run record
(home run # 756). You estimate this baseball will be worth
$2,000,000 when you retire at the end of twenty years. If
you expect a 12% return on your investment, how much
will you pay for the baseball ?
PV 
McGraw Hill/Irwin
2 , 000, 000
(1.12) 20
 $207,334
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Present Values
Ct
PV  DF  Ct 
t
(1  r )
Replacing “1” with “t” allows the formula
to be used for cash flows that exist at any
point in time
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Present Values
Example
You will receive $200 risk free in two years. If the annual
rate of interest on a two year treasury note is 7.7%, what is
the present value of the $200?
PV  (1.200
2  $172.42
077)
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Present Values
PVs can be added together to evaluate
multiple cash flows.
PV 
McGraw Hill/Irwin
C1
(1 r )
 (1r )2 ....
C2
1
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Present Values
PVs can be added together to evaluate
multiple cash flows.
PV 
McGraw Hill/Irwin
100
(1.07)1
 (1200
2  265.88
077)
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Present Values
$200
$100
Present Value
Year 0
Year
0
1
100/1.07
= $93.46
200/1.0772
= $172.42
Total
= $265.88
McGraw Hill/Irwin
2
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Present Values
 Given two dollars, one received a year from now
and the other two years from now, the value of
each is commonly called the Discount Factor.
Assume r1 = 20% and r2 = 7%.
McGraw Hill/Irwin
DF1 
1.00
(1.20)1
 .83
DF2 
1.00
(1.07 ) 2
 .87
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
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Present Values
Example
Assume that the cash flows
from the construction and sale
of an office building is as
follows. Given a 5% required
rate of return, create a present
value worksheet and show the
net present value.
Year 0
Year 1
Year 2
 170,000  100,000  320,000
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Present Values
Example - continued
Assume that the cash flows from the construction and sale of an office
building is as follows. Given a 5% required rate of return, create a
present value worksheet and show the net present value.
Period
Discount
0
Factor
1.0
1
1
1.05
2
1
1.052
McGraw Hill/Irwin
 .952
 .907
Cash
Present
Flow
 170,000
Value
 170,000
 100,000
 95,238
 320,000
 290,249
NPV  Total 
$25,011
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Present Values
Example - continued
Assume that the cash flows from the construction and sale of an office
building is as follows. Given a 5% required rate of return, create a
present value worksheet and show the net present value.
+$320,000
-$100,000
-$170,000
Present Value
Year 0
-170,000
Year
0
1
2
= -$170,000
-100,000/1.05 = $95,238
320,000/1.052 = $290,249
Total = NPV = $25,011
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Short Cuts
Sometimes there are shortcuts that make it
very easy to calculate the present value of
an asset that pays off in different periods.
These tools allow us to cut through the
calculations quickly.
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Short Cuts
Perpetuity - Financial concept in which a cash
flow is theoretically received forever.
cash flow
Return 
present va lue
C
r
PV
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Short Cuts
Perpetuity - Financial concept in which a cash
flow is theoretically received forever.
cash flow
PV of Cash Flow 
discount rate
C1
PV0 
r
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Present Values
Example
What is the present value of $1 billion every year, for all
eternity, if you estimate the perpetual discount rate to be
10%??
PV 
McGraw Hill/Irwin
$1 bil
0.10
 $10 billion
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Short Cuts
Annuity - An asset that pays a fixed sum each year for
a specified number of years.
Asset
Perpetuity (first
payment in year 1)
Year of Payment
1
2…..t
t+1
Present Value
C
r
Perpetuity (first payment
in year t + 1)
C  1
 
t
 r  (1  r )
Annuity from year
1 to year t
 C   C  1 

    
t 
 r   r  (1  r ) 
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Present Values
Example
Tiburon Autos offers you “easy payments” of $5,000 per year, at the end
of each year for 5 years. If interest rates are 7%, per year, what is the
cost of the car?
5,000
Present Value at
0
year 0
5,000
5,000
5,000
5,000
Year
1
2
3
4
5
5,000 / 1.07  4,673
5,000 / 1.07   4,367
2
5,000 / 1.07   4,081
3
5,000 / 1.07   3,814
4
5,000 / 1.07   3,565
5
Total NPV  20,501
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Short Cuts
Annuity - An asset that pays a fixed sum each
year for a specified number of years.
1
1 
PV of annuity  C   
t
 r r 1  r  
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Annuity Short Cut
Example
You agree to lease a car for 4 years at $300 per month.
You are not required to pay any money up front or at the
end of your agreement. If your opportunity cost of capital
is 0.5% per month, what is the cost of the lease?
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Annuity Short Cut
Example - continued
You agree to lease a car for 4 years at $300 per
month. You are not required to pay any money up
front or at the end of your agreement. If your
opportunity cost of capital is 0.5% per month,
what is the cost of the lease?
 1

1
Lease Cost  300  

48 
 .005 .0051  .005 
Cost  $12,774.10
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Annuity Short Cut
Example
The state lottery advertises a jackpot prize of $295.7
million, paid in 25 installments over 25 years of $11.828
million per year, at the end of each year. If interest rates
are 5.9% what is the true value of the lottery prize?
 1

1
Lottery Value  11.828  

25 
.
059
.0591  .059 

Value  $152,600,000
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FV Annuity Short Cut
Future Value of an Annuity – The future value of
an asset that pays a fixed sum each year for a
specified number of years.
 1  r   1
FV of annuity  C  

r


t
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Annuity Short Cut
Example
What is the future value of $20,000 paid at the end of each
of the following 5 years, assuming your investment returns
8% per year?
 1  .085  1
FV  20,000  

.08


 $117,332
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Constant Growth Perpetuity
C1
PV0 
rg
g = the annual growth rate of the
cash flow
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Constant Growth Perpetuity
NOTE: This formula can be used
to value a perpetuity at any point
in time.
C1
PV0 
rg
McGraw Hill/Irwin
C t 1
PVt 
rg
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Constant Growth Perpetuity
Example
What is the present value of $1 billion paid at the end of
every year in perpetuity, assuming a rate of return of 10%
and a constant growth rate of 4%?
1
PV0 
.10  .04
 $16.667 billion
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Perpetuities
A three-year stream of cash flows that grows at the rate g is
equal to the difference between two growing perpetuities.
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Compound Interest
i
ii
Periods Interest
per
per
year
period
iii
APR
(i x ii)
iv
Value
after
one year
v
Annually
compounded
interest rate
1
6%
6%
1.06
2
3
6
1.032
= 1.0609
6.090
4
1.5
6
1.0154 = 1.06136
6.136
12
.5
6
1.00512 = 1.06168
6.168
52
.1154
6
1.00115452 = 1.06180
6.180
365
.0164
6
1.000164365 = 1.06183
6.183
McGraw Hill/Irwin
6.000%
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Simple and Compound Interest
The value of a $100 investment earning 10% annually.
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Compound Interest
Compound interest versus simple interest. The top two ascending lines show the
growth of $100 invested at simple and compound interest. The longer the funds
are invested, the greater the advantage with compound interest. The bottom line
shows that $38.55 must be invested now to obtain $100 after 10 periods.
Conversely, the present value of $100 to be received after 10 years is $38.55.
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Compound Interest
The same story as the previous chart, except that the vertical scale is logarithmic.
A constant compound rate of growth means a straight ascending line. This graph
makes clear that the growth rate of funds invested at simple interest actually
declines as time passes.
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18
16
14
12
10
8
6
4
2
0
10% Simple
30
27
24
21
18
15
12
9
6
10% Compound
3
0
FV of $1
Compound Interest
Number of Years
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Compound Interest
Example
Suppose you are offered an automobile loan at an APR of
6% per year. What does that mean, and what is the true
rate of interest, given monthly payments?
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Compound Interest
Example - continued
Suppose you are offered an
automobile loan at an APR of 6% per
year. What does that mean, and what
is the true rate of interest, given
monthly payments? Assume $10,000
loan amount.
Loan Pmt  10,000  (1.005)
12
 10,616.78
APR  6.1678%
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Web Resources
Click to access web sites
Internet connection required
www.bankrate.com
www.money.cnn.com
www.quicken.com
www.smartmoney.com
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