Transcript Review on present value

Present value, annuity,
perpetuity
Financial Economics
2012 höst
How to Calculate Present Values
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LEARNING OBJECTIVES
Time value of money
how to calculate present value of future cash flows.
To calculate the present value of perpetuities, growing
perpetuities, annuities and growing annuities.
Compound interest and simple interest
Nominal and effective interest rates.
To understand value additive property and the concept
of arbitrage.
the net present value rule and the rate of return rule.
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Time value of money
• The value of 1 € today is not the same as 1 € in a
year’s time
• Suppose the interest rate on savings is 2% per
year.
• At the end of year 1: 1 € × (1+0,02)=1,02
• At the end of year 2: 1,02€ × (1+0,02) =1,0404
year 3: 1,0404 × (1+0,02)=1,0612
i.e. FV = 1€× (1 + 0,02)3
year
FV
1
1
2
1,02
3
1,0404
4
1,061208
5
1,082432
6
1,104081
7
1,126162
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Topics Covered
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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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Calculating future value of 1$
Value of 1 $ investment in t year’s time
compound interest rate at 2%
2.5
2
1.5
1
FV
0.5
Year
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37
Note that the 1 $ doubled in about
37 year’s time, given interest rate
2%.
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future value and present value
• FV = PV× (1 + r)t
where
FV = Future value
PV = Present value
r = interest rate
t = number of years (Periods)
It is readily seen, PV = FV/ (1 + r)t
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Manhattan Island Sale
The Power of Compounding!
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
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FYI - The value of Manhattan Island land is well below this
figure.
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Future Values
Compound Interest - Interest earned on interest.
Simple Interest - Interest earned only on the
original investment.
Future Value - Amount to which an investment
will grow after earning interest.
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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 × .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
1
Interest Earned
Value
100
6
106
Future Years
2
3
6
112
6
118
4
5
6
124
6
130
Value at the end of Year 5 = $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.
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
10%
15%
Interest Rates
800
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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Present Values
Present Value
Discount Factor
Value today of a
future cash
flow.
Equals to
Present value of
a $1 future
payment.
r
Discount Rate
Interest rate used
to compute
present values of
future cash flows.
DF 
1
(1 r ) t
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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 5.2
You just bought a new computer for $3,000. The payment due
in 2 years. That means you need to pay 3000$ after 2 years. 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
Note that discount factor is the same as PV of $1
in t year´s time
DF 
1
t
(1 r )
• Discount Factors can be used to compute the present
value of any cash flow.
• PV=DF*FV
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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
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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
1
2
$8,000
4000/1.08
= $3,703.70
4000/1.082
= $3,429.36
Total
= $15,133.06
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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
r
C = cash payment
r = interest rate
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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

1
.10 ( 1 .10 ) 3

PV  $9,947.41
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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

1
.10 ( 1 .10 ) 20
  (1.10)
20
FV  $229,100
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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)?
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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 interest rate  nominal interest 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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