Chapter 2
Principles of
Corporate Finance
Tenth Edition
How to Calculate
Present Values
Slides by
Matthew Will
McGraw-Hill/Irwin
Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.
Topics Covered
Future Values and Present Values
Looking for Shortcuts—Perpetuities and
Annuities
More Shortcuts—Growing Perpetuities and
Annuities
How Interest Is Paid and Quoted
2-2
Present and Future Value
Future Value
Amount to which an
investment will grow
after earning interest
Present Value
Value today of a
future cash
flow.
2-3
2-4
Future Values
Future Value of $100 = FV
FV  $100  (1  r )
t
Present Value
PresentValue = PV
PV = discount factor C1
2-5
Present Value
Discount Factor = DF = PV of $1
DF 
1
t
(1 r )
Discount Factors can be used to compute the present value of
any cash flow.
2-6
Valuing an Office Building
Step 1: Forecast cash flows
Cost of building = C0 = 370,000
Sale price in Year 1 = C1 = 420,000
Step 2: Estimate opportunity cost of capital
If equally risky investments in the capital market
offer a return of 5%, then
Cost of capital = r = 5%
2-7
Valuing an Office Building
Step 3: Discount future cash flows
PV 
C1
(1r )

420 , 000
(1 .05 )
 400,000
Step 4: Go ahead if PV of payoff exceeds investment
NPV  400,000 370,000
 30,000
2-8
Net Present Value
NP V = P V - required investment
C1
NP V = C0 
1 r
2-9
Risk and Present Value
Higher risk projects require a higher rate of
return
Higher required rates of return cause lower
PVs
P V of C1  $420,000at 5%
420,000
PV 
 400,000
1  .05
2-10
Risk and Present Value
2-11
PV of C1  $420,000at 12%
420,000
PV 
 375,000
1  .12
P V of C1  $420,000at 5%
420,000
PV 
 400,000
1  .05
Net Present Value Rule
Accept investments that have positive net
present value
Example
Use the original example. Should we accept the
project given a 10% expected return?
420,000
NPV = -370,000 +
 $30,000
1.05
2-12
Rate of Return Rule
Accept investments that offer rates of return
in excess of their opportunity cost of capital
Example
In the project listed below, the foregone investment
opportunity is 12%. Should we do the project?
profit
420,000  370,000
Return 

 .135 or 13.5%
investment
370,000
2-13
2-14
Multiple Cash Flows
For multiple periods we have the
Discounted Cash Flow (DCF) formula
PV0  (1 r )1  (1 r )2  ....  (1 r )t
C1
Ct
C2
T
NPV0  C0   (1 r )t
t 1
Ct
2-15
Net Present Values
- $370,000
$20,000
$ 420,000
Present Value
Year 0
Year
0
-$370,000
20,000/1.12 = $17,900
420,000/1.122 = $334,800
Total = - $17,300
1
2
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.
2-16
Short Cuts
Perpetuity - Financial concept in which a cash
flow is theoretically received forever.
cash flow
Return 
present value
C
r
PV
2-17
Short Cuts
Perpetuity - Financial concept in which a cash
flow is theoretically received forever.
cash flow
P V of Cash Flow 
discount rat e
C1
PV0 
r
2-18
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 
$1 bil
0.10
 $10 billion
2-19
Present Values
Example - continued
What if the investment does not start making money for 3
years?
PV 
$1 bil
0.10

   $7.51 billion
1
1.103
2-20
2-21
Short Cuts
Annuity - An asset that pays a fixed sum each year for
a specified number of years.
Asset
Year of Payment
1
Perpetuity (first
payment in year 1)
2…..t
Present Value
t+1
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 ) 
2-22
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 year 0
5,000
5,000
5,000
Year
0
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
T ot alNP V  20,501
5
5,000
1
2
3
4
5
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  
2-23
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?
2-24
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
2-25
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
LotteryValue  11.828 

25 
.
059
.0591  .059 

Value  $152,600,000
2-26
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
2-27
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
2-28
Constant Growth Perpetuity
C1
PV0 
rg
g = the annual growth
rate of the cash flow
2-29
Constant Growth Perpetuity
NOTE: This formula can be
used to value a perpetuity at
any point in time.
C1
PV0 
rg
C t 1
PVt 
rg
2-30
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
2-31
Perpetuities
A three-year stream of cash flows that grows at
the rate g is equal to the difference between two
growing perpetuities.
2-32
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.
2-33
Effective Interest Rates
example
Given a monthly rate of 1%, what is the Effective
Annual Rate(EAR)? What is the Annual
Percentage Rate (APR)?
2-34
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%
2-35
Web Resources
Click to access web sites
Internet connection required
www.smartmoney.com
http://finance.yahoo.com
www.in.gov/ifa/files/TollRoadFinancialAnalysis.pdf
www.mhhe.com/bma
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