Time Value of Money
Multiple Cash Flows
Multiple Cash Flows
• We have learned the future and
present value of single cash flow
• An investment is a claim to a stream of
multiple cash flows
• How to value multiple cash flows?
• Rule:
Only values at the same point in time
can be compared or combined
Future Value
If you deposit $100 in one year, $200 in two
years and $300 in three years.
• How much will you have in three years at 7%
interest?
• How much in five years if you don’t add
additional amounts?
TIMELINE
0
1
2
3
$100.00
$200.00
$300.00
7%
4
5
Future Value
TIMELINE
0
1
2
3
4
$100.00
$200.00
$300.00
5
7%
100*(1.07) =
$107.00
$307.00
307*(1.07) =
$328.49
$628.49
628.49* (1.07)^2 =
$719.56
Future Value
TIMELINE
0
1
2
3
4
$100.00
$200.00
$300.00
5
7%
$300.00
200*(1.07) =
$214.00
100*(1.07)^2 =
$114.49
$628.49
* (1.07)^2 =
$719.56
Present Value
You are borrowing from bank now and will pay back
 $200 in a year
 $400 in 2 years
 $600 at the end of the 3rd year
If the bank’s lending rate is 12%, at most how much
can you borrow now?
TIMELINE
0
1
2
3
$200.00
$400.00
$600.00
12%
Present Value
TIMELINE
0
1
2
3
$200.00
$400.00
$600.00
12%
$178.57
$318.88
$427.07
$924.52
= 1/1.12*
= 1/(1.12)^2*
=1/(1.12)^3*
Present Value
TIMELINE
0
1
2
3
$200.00
$400.00
$600.00
12%
$535.71
$935.71
$835.46
$1,035.46
$924.52
= *1/1.12
= *1/1.12
= *1/1.12
Buying a Car
Two years ago, you put $10,000 in a savings account
earning an annual interest rate of 8%, compounded
semiannually. At that time, you thought that these
savings would grow enough for you to buy a new car
five years later (i.e. three years from now). However,
you just re-estimated the price that you will have to pay
for the new car in three years at $18,000.
• How much more money do you need to put in your
savings account now for it to grow to this new
estimate in three years?
Buying a Car
• The compounding rate is
8%
2
= 4%
• You put $10,000 in account 2 years ago, so the number of
compounding periods is 2 × 2 = 4. Now, you have in
account:
10,000 × 1 + 4% 4 = $11,698.59
• You need $18,000 in 3 years to buy a car, so the number
of periods for discounting is 2 × 3 =6. The account should
have amount:
18,000
= $14,225.66
6
1 + 4%
• So you need to put the difference in the account now:
14,225.66 − 11,698.59 = $2,527.07
Buying a Car
Now suppose that you know that the car company will
offer you to pay for the car over some time. In
particular, you will have the opportunity to make a
down payment of $6,000 at the time you get the car
(three years from now) and to make additional
payments of $6,500 at the end of each of the following
two years. With this new offer, how much money do
you need to add to your account now?
Buying a Car
• The car company’s offer asks you to pay $6,000 in
the 3rd year and $6,500 in the 4th and 5th year.
The present value of all three payments is
6000
6000
6500
+
+
= $13,882.54
2×3
2×4
2×5
1 + 4%
1 + 4%
1 + 4%
• So you need to put the difference in the account
now,
13,882.54 − 11,698.59 = $2,183.54