Topic 5: Time Value of Money-Series of
Payments
Learning Objectives Satisfied:
1. Introduction to Financial Management
Also cover the major foundations of Finance such as
3 Time Value of Money
3 Cash Flow and Taxes and their implications for financial managers
2. Financial Markets and Interests Rates
Objectives: Understand the following topics
3 Inflation and interest rates and their relationship
3. Mathematics of Finance
Objectives: Understand the following concepts
3 Present and future value of perpetuities, annuities, annuities due
3 Loan amortization
Present value of a series of cash flows:
PV = C 0 +
C1
+
(1 +R)1
C2
+
...
+
(1 +R)2
Cn
(1 +R)n
Example 1 (for further practice, see Set 2, #1)
PV = 0
+
110
(1.10) 1
+
121
(1.10) 2
+
133.10
(1.10) 3
PV = 100 + 100 + 100 = 300
Internal Rate of Return:
C0 =
Example1
C1
(1 +R)1
+
...
+
(1 +R)2
Cn
(1 +R)n
(for further practice, see Set 2, #4)
110
300 =
C2
+
(1+R) 1
121
+
+
(1+R) 2
133.10
(1+R) 3
R = 10%
Future value of a series of cash flows:
FV = C 0 (1 +R)n + C 1 (1 +R)n-1 + ... + C n (1 +R)
Example 2 (for further practice, see Set 2, #4)
FV = 0 + 110(1.10)2 + 121(1.10)1 + 133.10
FV = 0 + 133.10
+ 133.10
+ 133.10 = $399.30
FV = 300 (1.10)3 = 339.30
n-n
How to incorporate inflation into a series:
Example (cash flows adjusted for inflation):
PV = 0
+ 100
(1.03)
1
+ 100
(1.03)
2
+ 100 3 = 282.86
(1.03)
Example (nominal cash flows):
PV = 0
+ 110
1
(1.133)
+ 121
2
(1.133)
+ 133.103 = 282.86
(1.133)
r = 3%, i = 10%, therefore R = 13.3%
Annuities are a special kind of cash flow series
• All payments are equal
• The payments are equally spaced through time
Time Value of Ordinary Annuities: (Example 4)
R=10%
0
PV
1
2
3
100
100
100
90.91
82.64
75.13
248.69
FV
0
1
2
3
100
100
100
110
121
331
For further practice, see Set 2, #5-11
Time Value of Ordinary Annuities
-n
1 - (1+R)
PV = PMT
R
[
]
n
FV = PMT
[
(1+R) - 1
R
]
More examples: Let payment = $100, n = 5, and R = 10%.
Then PV=$379.08 and FV=$610.51
Perpetuities
• Perpetuities are ordinary annuities in which n
approaches infinity
PV = PMT/R
FV approaches infinity, so is not calculated
Perpetuities
• Perpetuities are ordinary annuities in which n
approaches infinity
PV = PMT/R
FV approaches infinity, so is not calculated
Examples:
PMT =$100, R = 10%; Then PV = $100/.1 =
$1000
Perpetuities
• Perpetuities are ordinary annuities in which n
approaches infinity
PV = PMT/R
FV approaches infinity, so is not calculated
Examples:
PMT =$100, R = 10%; Then PV = $100/.1 =
$1000
PMT =$40, R = 8%; Then PV = $40/.08 = $500
Time Value of Annuities Due: (Example 5)
R=10%
PV
0
1
2
100
100
100
3
90.91
82.64
273.55
FV
0
1
2
100
100
100
3
110
121
133.10
364.10
More examples: Let payment = $100, n = 5, and R = 10%.
Then PV=$416.99 and FV=$671.56
Side-by-Side
R=10%
PV
0
1
2
3
100
100
100
R=10%
90.91
90.91
82.64
82.64
75.13
0
1
100
2
100
3
273.55
248.69
FV
0
100
PV
1
2
3
100
100
100
110
121
FV
0
1
2
100
100
100
3
110
121
133.10
331
364.10
End Mode
Begin Mode
Ordinary Annuity with Balloon Payment (Example 6)
R=10%
PV
0
1
2
3
100
100
1100
90.91
82.64
826.45
751.31 +75.13
1,000.00
For further practice, see Set 2, #18-19
Annuity Due with Balloon Payment (Example 7)
R=10%
0
1
2
100
100
100
3
1000
90.91
82.64
751.31
1,024.87
Insight into Calculator Software
-n
1 - (1+R)
x = PMT
R
{ [
•
•
•
•
]
k
(1+R) +
Balloon
n
(1+R)
k = 0, j = 0; then x is PV of ordinary annuity
k = 0, j = 1; then x is FV of ordinary annuity
k = 1, j = 0; then x is PV of annuity due
k = 1, j = 1; then x is FV of annuity due
}
jn
(1+R)
Amortizing Annuities: (Example 8)
•
Consider an annuity of $100 per year for 3 years, with interest of
10% compounded annually. Then the present value would be
$248.69, and the amorization table would be as follows:
Payment
number
1
2
3
Interest
Principal
Balance
$24.87
$17.36
$9.09
$75. 13
$82. 64
$90. 91
$173. 55
90. 91
0
Grouped Cash Flows: (Example 9)
R=10%
0
1
2
50 50
3
4
5
50 100 100
45.45
41.32
37.57
68.30
62.09
254.74
For further practice, see Set 2, #23
Deferred Annuities: (Example 10)
R=10%
0
1
2
3
0
0
0
4
5
100 100
68.30
62.09
130. 39
For further practice, see Set 2, #24
Annuities with Missed Payments:
R=10%
0
1
2
3
4
5
100 100 100 100 100
-100
100 100 0
90.91
82.64
0
68.30
62.09
303.95
100 100
Continuous Payment Streams
Rt
FV = Sum of all payments ×
e
-1
Rt
-Rt
PV = Sum of all payments ×
1-e
Rt
Example 12:
• Future value of $10,000 per year for 5 years,
discounted at 10% compounded continuously, with
payments spread continuously over the life of the
annuity, would be the following:
.5
FV = 50,000 ×
e
-1
= $64,872.13
.5
For further practice, see Set 2, #25
Example:
• Present value of $10,000 per year for 5 years,
discounted at 10% compounded continuously, with
payments spread continuously over the life of the
annuity, would be the following:
-.5
PV = 50,000 ×
1- e
= $39,346.93
.5
For further practice, see Set 2, #26