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Fundamentals of Corporate Finance
Third Edition
Chapter 3
The Time
Value of
Money
Brealey
Myers Marcus
slides by Matthew Will
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3- 2
Topics Covered
Future Values
Present Values
Multiple Cash Flows
Perpetuities and Annuities
Inflation & Time Value
Effective Annual Interest Rate
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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.
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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.
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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
Future Years
2
3
4
5
Interest Earned
Value
100
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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
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1
6
106
Future Years
2
3
4
5
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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
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1
6
106
Future Years
2
3
6
112
4
5
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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
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1
6
106
Future Years
2
3
6
6
112 118
4
5
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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
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1
6
106
Future Years
2
3
4
6
6
6
112 118 124
5
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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
1
6
106
Future Years
2
3
4
5
6
6
6
6
112 118 124 130
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.
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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
1
Future Years
2
3
4
5
Interest Earned
Value
100
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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
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1
6.00
106.00
Future Years
2
3
4
5
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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
Irwin/McGraw-Hill
Future Years
1
2
3
6.00 6.36
106.00 112.36
4
5
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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
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Future Years
1
2
3
6.00 6.36 6.74
106.00 112.36 119.10
4
5
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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
Irwin/McGraw-Hill
Future Years
1
2
3
4
6.00 6.36 6.74 7.15
106.00 112.36 119.10 126.25
5
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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
6.00 6.36 6.74 7.15 7.57
106.00 112.36 119.10 126.25 133.82
Value at the end of Year 5 = $133.82
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Future Values
Future Value of $100 = FV
FV  $100  (1  r )
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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?
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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
Interest Rates
70
0%
60
5%
FV of $1
50
10%
15%
40
30
20
10
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
0
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 2000,
compounded at 8%.
FV  $24  (1  .08)
 $75.979 trillion
374
FYI - The value of Manhattan Island land is
well below this figure.
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Present Values
Present Value
Value today of a
future cash
flow.
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Present Values
Present Value
Discount Factor
Value today of a
future cash
flow.
Present value of
a $1 future
payment.
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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 =
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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 
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3000
(1.08)2
 $2,572
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Present Values
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.
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Time Value of Money
3- 32
(applications)
The PV formula has many applications.
Given any variables in the equation, you
can solve for the remaining variable.
PV  FV 
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1
(1 r ) t
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Time Value of Money
3- 33
(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?
Immediatepayment 8,000.00
PV1 
4 , 000
(1.08)1
 3,703.70
PV2 
4 , 000
(1.08) 2
 3,429.36
Total PV
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 $15,133.06
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PV of Multiple Cash Flows
PVs can be added together to evaluate
multiple cash flows.
PV 
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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%?
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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 
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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?
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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 
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1, 000 , 000
(1 .10 ) 3
 $751,315
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Perpetuities & Annuities
PV of Annuity Formula
PV  C

1
r

1
t
r ( 1 r )

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 
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
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 )
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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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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
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 5.0%
and the inflation rate is 2.2%, what is the real
interest rate?
1 + real interest rate = 1+.050
1+.022
Savings
Bond
1 + real interest rate = 1.027
Real interest rate = .027 or 2.7%
Approximation = .050 - .022 or .028 or 2.8%
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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)?
EAR = (1+.01)
12
- 1 = r
EAR = (1+.01)
12
- 1 =.1268 or 12.68%
APR =.01 x 12 =.12 or 12.00%
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Web Resources
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