2.0 pvs
TRANSCRIPT
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Topics Covered Today
Future Values and Present Values
Looking for ShortcutsPerpetuities and
Annuities
More ShortcutsGrowing Perpetuities and
Annuities
How Interest Is Paid and Quoted
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Present and Future Value
Present Value
Value today of a
future cash
flow.
Future Value
Amount to which an
investment will grow
after earning interest
How do we get from one to another?
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Future Values
FV r t
$100 ( )1
Example - FV
What is the future value of $100 if interest is
compounded annually at a rate of 7% for two years?
49.114$)07.1(100$
49.114)07.1()07.1(100$
2
FV
FV
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0
200
400
600
800
1000
1200
1400
1600
1800
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Number of Years
FVo
f$100
0%
5%
10%
15%
Future Values with Compounding
Interest Rates
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Present Value (PV)
Discount Factor = DF = PV of $1
Discount Factors can be used to compute the present value of any cash flow.
DF r t 1
1( )
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The PV formula has many applications. Given any
variables in the equation, you can solve for the
remaining variable. Also, you can reverse the prior
example.
10049.1142)07.1(1
22
PV
CDFPV
Present Value
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0
20
40
60
80
100
120
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Number of Years
PVo
f$1
00
0%
5%
10%
15%
Present Values with Compounding
Interest Rates
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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%
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Valuing an Office Building
Step 3: Discount future cash flows
Step 4: Go ahead if PV of payoff exceeds investment
000400051000420
1
1
,).(,
)(
r
C
PV
00030
000370000400
,
,,NPV
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Net Present Value
r
C
1
C=NPV
investmentrequired-PV=NPV
10
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Risk and Present Value
Higher risk projects require a higher rate ofreturnCENTRAL POINT OF FINANCE
Higher required rates of return cause lower
PVs
000,400.051
420,000PV
5%at$420,000CofPV 1
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Risk and Present Value
000,400.051
420,000PV
5%at$420,000CofPV 1
000,375.121
420,000PV
12%at$420,000CofPV 1
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Risk and Net Present Value
$5,000
370,000-75,0003=NPV
investmentrequired-PV=NPV
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Net Present Value Rule
Accept investments that have positivenet present value
ExampleUse the original example. Should we accept the
project given a 10% expected return?
000,30$1.05
420,000+-370,000=NPV
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Rate of Return Rule
Accept investments that offer rates of returnin excess of their opportunity cost of capital
Example
In the project listed below, the foregone investment
opportunity is 12%. Should we do the project?
13.5%or.135370,000
370,000420,000
investment
profitReturn
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Multiple Cash FlowsFor multiple periods we have the
Discounted Cash Flow (DCF)formula
tt
rC
rC
rCPV )1()1()1(0 ....2
21
1
T
tr
C
t
tCNPV1
)1(00
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Net Present Values
Present Value
Year 0
20,000/1.12
420,000/1.122
Total
= $17,900
= $334,800
= - $17,300
$20,000
- $370,000
Year0 1 2
$ 420,000
-$370,000
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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 calculationsquickly.
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Short Cuts
Perpetuity- Financial concept in which a cashflow is theoretically received forever.
PVCr
luepresent va
flowcashReturn
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Short Cuts
Perpetuity- Financial concept in which a cashflow is theoretically received forever.
r
CPV1
0
ratediscount
flowcashFlowCashofPV
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Present ValuesExample
What is the present value of $1 billion every year, for all eternity, if you
estimate the perpetual discount rate to be 10%??
billion10$10.0
bil$1 PV
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Present ValuesExample - continued
What if the investment does not start making money for 3 years?
billion51.7$31.101
10.0bil$1 PV
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Short Cuts
Annuity- An asset that pays a fixed sum each year for
a specified number of years.
r
CPerpetuity (firstpayment in year 1)
Perpetuity (first payment
in year t + 1)
Annuity from year
1 to year t
Asset Year of Payment
1 2..t t + 1
Present Value
t
rr
C
)1(
1
t
rr
C
r
C
)1(
1
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Example
Tiburon Autos offers you easy payments of $5,000 per year, at the endof each year for 5 years. If interest rates are 7%, per year, what is the
cost of the car?
Present Values
5,000
Year0 1 2 3 4 5
5,000 5,000 5,000 5,000
20,501NPVTotal
565,307.1/000,5
814,307.1/000,5
081,407.1/000,5
367,407.1/000,5
673,407.1/000,5
5
4
3
2
Present Valueat year 0
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Short Cuts
Annuity- An asset that pays a fixed sum eachyear for a specified number of years.
trrr
C1
11annuityofPV
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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?
trrr
C
1
11annuityofPV
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Annuity Short Cut
Example - continued
You agree to lease a car for 4 years at $300 per
month. You are not required to pay any money up
front or at the end of your agreement. If your
opportunity cost of capital is 0.5% per month,
what is the cost of the lease?
10.774,12$
005.1005.
1
005.
1300CostLease
48
Cost
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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 $11.828 million per year, at the end of each year.
If interest rates are 5.9% what is the true value of the lottery prize?
000,600,152$
059.1059.
1
059.
1828.11ValueLottery
25
Value
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FV Annuity Short Cut
Future Value of an Annuity
The future value of anasset that pays a fixed sum each year for a
specified number of years.
r
rC
t
11annuityofFV
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Annuity Short Cut
Example
What is the future value of $20,000 paid at the end of each of the following 5
years, assuming your investment returns 8% per year?
332,117$
08.
108.1000,20FV
5
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Constant Growth Perpetuity
gr
CPV
10
g = the annual growth
rate of the cash flow
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Constant Growth Perpetuity
gr
CPV
10
NOTE: This formula can be
used to value a perpetuity at
any point in time.
gr
CPV tt
1
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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%?
billion667.16$04.10.
10
PV
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Effective Interest Rates
Annual Percentage Rate - Interest rate that is
annualized using simple interest.
Effective Annual Interest Rate - Interest rate
that is annualized using compound
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)?
12.00%or.12=12x.01=APR
12.68%or.1268=1-.01)+(1=EAR
r=1-.01)+(1=EAR
12
12
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Web Resources
Click to access web sitesInternet connection required
www.smartmoney.com
http://finance.yahoo.comwww.in.gov/ifa/files/TollRoadFinancialAnalysis.pdf
www.mhhe.com/bma
http://www.smartmoney.com/http://finance.yahoo.com/http://www.in.gov/ifa/files/TollRoadFinancialAnalysis.pdfhttp://www.mhhe.com/bmahttp://www.mhhe.com/bmahttp://www.in.gov/ifa/files/TollRoadFinancialAnalysis.pdfhttp://finance.yahoo.com/http://www.smartmoney.com/