The idea that money available at the present time is worth more than the same amount in the future due to its potential earning capacity. This core principle of finance holds that, provided money can earn interest, any amount of money is worth more the sooner it is received.

1. Corporate Finance
Time Value of Money
Session One

2. Decision Dilemma—Take a Lump Sum
or Annual Installments
 A suburban Chicago couple won the
Lottery.
 They had to choose between a single
lump sum $104 million, or $198 million
paid out over 25 years (or $7.92
million per year).
 The winning couple opted for the lump
sum.
 Did they make the right choice? What
basis do we make such an economic
comparison?
 To make such comparisons (the lottery
decision problem), we must be able to
compare the value of money at
different point in time.

3. Time Value of Money
“The idea that money available
at the present time is worth
more than the same amount in
the future due to its potential
earning capacity.”

4. Future Values and Compound Interest
Future Value – Amount to which an investment
will grow after earning interest.
Compound Interest – Interest earned on interest.
Simple Interest – Interest earned only on original
investment; no interest is earned on interest.

5. Future Values
You have $100 invested in a bank account. Suppose banks are
currently paying an interest rate of 6 percent per year on
deposits. So after a year, your account will earn interest of $6:
FV = PV (1 + r)
ͭ
FV = 100 (1+06)¹
Value of investment after 1 year = $106

6. 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
Value at the end of Year 5 = $130
Future Years
2
3
6
112
6
118
4
6
124
5
6
130

7. 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

8. Future Values
Example - Compound Interest
Interest earned at a rate of 6% for five years on the
previous year’s balance.
Today
1
Interest Earned
Value
100
6
106
Future Years
2
3
6.36
112.36
Value at the end of Year 5 = $133.82
6.74
119.10
4
7.15
126.25
5
7.57
133.82

9. Practice Problem: Warren Buffett’s Berkshire
Hathaway
• Went public in 1965: $18 per share
• Worth today (August 22, 2003):
$76,200
• Annual compound growth: 24.58%
• Current market value: $100.36
Billion
• If he lives till 100 (current age: 73
years as of 2003), his company’s
total market value will be ?

10. Market Value
• Assume that the company’s stock will continue to
appreciate at an annual rate of 24.58% for the next 27
years.
F
$100.36(1
0.2458)
$37.902 trillions
27

11. Manhattan Island Sale
Peter Minuit bought Manhattan Island for $24 in 1629.
Was this a good deal? Suppose the interest rate is 8%.

12. Manhattan Island Sale
Peter Minuit bought Manhattan Island for $24 in 1629.
Was this a good deal?
To answer, determine $24 is worth in the year 2003,
compounded at 8%.
FV
$ 24
(1
. 08 )
374
$ 75 . 979 trillion
FYI - The value of Manhattan Island land is
well below this figure.

13. Present Values
P resen t V a lu e = P V
PV =
F u tu re V alu e after t p erio d s
(1 + r)
t

14. 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
3000
( 1 . 08 )
2
$ 2 ,572

15. 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?
Im ediate
m
PV 1
PV 2
Total PV
paym
ent
4 , 000
(1
1
3 , 703 . 70
2
3 , 429 . 36
. 08 )
4 , 000
(1
. 08 )
8,000.00
$15,133.06

16. Present Values
$8,000
$4,000
$ 4,000
Present Value
Year 0
0
$8,000
4000/1.08
= $3,703.70
4000/1.082
= $3,429.36
Total
= $15,133.06
1
2
Year

17. PV of Multiple Cash Flows
• PVs can be added together to evaluate
multiple cash flows.
PV
C1
(1 r )
C2
1
(1 r )
2
....

18. 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.

19. Perpetuities
PV of Perpetuity Formula
PV
C = cash payment
r = interest rate
C
r

20. Perpetuities
Example - Perpetuity
In order to create an endowment, which pays
$100,000 per year, forever, how much money must
be set aside today if the rate of interest is 10%?
PV
100 ,000
.1 0
$ 1, 0 0 0 , 0 0 0

21. Perpetuities
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 .1 0 )
3
$ 7 5 1, 3 1 5

22. 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

23. 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

24. 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)?
1
.1 0
PV
4 ,0 0 0
PV
$ 9 , 9 4 7 .4 1
1
.1 0 ( 1 .1 0 )
3

25. 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
t
FV
C PVAF
(1 r )

26. 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?
1
.1 0
FV
4 ,0 0 0
FV
$ 2 2 9 ,1 0 0
1
.1 0 ( 1 .1 0 )
20
( 1 .1 0 )
20

27. 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.

28. 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%

29. 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.

30. Inflation
1
1 + n o m in a l in te re st ra te
re a l in te re st ra te =
1 + in fla tio n ra te
approximation formula
R eal in t. rate
n o m in al in t. rate - in flatio n rate

31. 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
1+.06
real int erestrat e =
1+.02
1
real int erestrat e = 1.039
real int erestrat e= .039or 3.9%
Approximat
ion = .06 - .02 = .04 or 4.0%

32. The End