2
class #03
page 5
Multiple cash flows: present value
•
Suppose we expect to receive $1000 at the end of each
of the next 5 years. Interest rate is 6%. What is the
value today of this set of cash flows?
1.
PV
0
= 1,000 / (1.06)
5
+ 1,000 / (1.06)
4
+ 1,000 /
(1.06)
3
+ 1,000 / (1.06)
2
+ 1,000 / (1.06) =
$4212.37
2.
1,000 / (1.06) = 943.4; 1,943.40 / (1.06) = 1,833.40;
2,833.40 / (1.06) = 2,673.01; 3,673.01 / (1.06) =
3,465.11; 4465.11 / (1.06) =
$4,212.37
01
2
3
5
PV
0
= ?
+1,000
+1,000
+1,000
+1,000
+1,000
4
class #03
page 6
Multiple cash flows: present value
Present value
calculated by
discounting each
cash flow separately
0
1
2
3
4
5
$1,000
$1,000
$1,000
$1,000
$
943.40
890.00
839.62
792.09
747.26
$4,212.37
x 1/1.06
5
Total present value
Time
(years)
$1,000
r
= 6%
x 1/1.06
4
x 1/1.06
3
x 1/1.062
x 1/1.06
0
1
2
3
4
5
$4,212.37
0.00
$4,212.37
$3,465.11
1,000.00
$4,465.11
$2,673.01
1,000.00
$3,673.01
$1,833.40
1,000.00
$2,833.40
$
943.40
1,000.00
$1,943.40
$
0.00
1,000.00
$1,000.00
Present value
calculated by
discounting back one
period at a time
Time
(years)
Total present value = $4,212.37
r
= 6%
class #03
page 7
Multiple cash flows: future value
•
We invest $2,000 for 5 years at 10% per year. What is
the FV in five years?
Future value calculated by compounding forward one period at a time
Time
(years)
0
1
2
3
4
5
$0
0
$0
$
0
2,000
$2,000
$2,200
2,000
$4,200
$4,620
2,000
$6,620
$7,282
2,000
$9,282
$10,210.2
2,000
$12,210.2
x 1.1
x 1.1
x 1.1
x 1.1
x 1.1
Time
(years)
0
1
2
3
4
5
$2,000
$2,000
$2,000
$2,000
$2,000.0
2,200.0
2,420.0
2,662.0
2.928.2
$12,210.20
x 1.14
x 1.1
3
x 1.1
2
x 1.1
Total future value
Future value calculated by compounding each cash flow separately
class #03
page 8
Types of multiple cash flows
•
Annuities
–
Ordinary annuity
: a series of identical
cash flows
occurring at the end
of each period for some fixed
number of periods
•
Examples: consumer loans (e.g. car loans), home
mortgages, etc.