Chapter 7
The Time
Value of
Money
1
Annuities - Future Sum
A. An annuity is a series of equal payments
or receipts that occur at evenly spaced
intervals. Leases and rental payments are
examples. The payments or receipts
occur at the end of each period for an
ordinary annuity while they occur at the
beginning of each period for an annuity
due.
2
FV Time Lines
Ordinary annuity
Year
Payment
0
1
2
3
-
$100
100
100
3
FV Time Lines
Annuity due
0
Year
Payment
$100
1
2
3
100
100
-
4
Future Value of an
Ordinary Annuity Illustrated
A. Future Value of an Ordinary Annuity
1. FVoa = PMT [((1 + i)n - 1) / i]
a) FVoa = Future Value of an Ordinary Annuity
b) PMT = Amount of each payment
c) i = Interest Rate per Period
d) n = Number of Periods
5
Future Value of an
Ordinary Annuity Illustrated
PV = 0
PMT = -100
I = 5
N = 3
FV = ?
= 315.25
6
Future Value of an
Annuity Due Illustrated
A. Future Value of an Annuity Due
1. FVad = FVoa (1+i)
a) FVad = Future Value of an Annuity Due
b) FVoa = Future Value of an Ordinary Annuity
c) i = Interest Rate per Period
7
Greater Terminal Values
Higher interest rates
Longer time periods
Result in greater terminal values
8
Greater Terminal Values
9
Present Value of an Annuity
The present value of an annuity
brings a series of payments in the
future back to the present.
10
Present Value of an Ordinary
Annuity Illustrated
1. PVoa = PMT [(1 - (1 / (1 + i)n)) / i]
a) PVoa = Present Value of an Ordinary
Annuity
b) PMT = Amount of each payment
c) i = Interest Rate per Period
d) n = Number of Periods
11
Present Value of an Ordinary
Annuity Illustrated
FV = 0
PMT = 100
I = 6
N = 3
PV = ?
= -267.30
12
Present Value of an
Annuity Due
C. Present Value of an Annuity Due
1. PVad = PVoa (1+i)
a) PV-ad = Present Value of an Annuity Due
b) PV-oa = Present Value of an Ordinary Annuity
c) i = Interest Rate per Period
13
Annuities - Present Value
Higher interest rates result in lower
present values.
But longer time periods increases
the present value (because more
payments are received).
14
Annuities - Present Value
15
Illustration 1
You deposit $1,000 in an account
at the end of each year for twenty
years. What is the total amount in
the account if you earn 6 percent
annually?
16
Future Value of an
Ordinary Annuity
The unknown: FV
The givens:
–PV = 0
–PMT = -1,000
–N = 20
–I = 6
The answer:
$36,786
17
Interpretation
For an annual cash payment of
$1,000, you will have $36,786 after
twenty years.
Of the $36,786
–$20,000 is the total cash outflow
–$16,786 is the earned interest.
18
Illustration 2
What is the present value of (or
required cash outflow to purchase)
an ordinary annuity of $1,000 for
twenty years, if the rate of interest
is 6 percent?
19
Present Value of an Annuity
The unknown: PV
The givens:
–FV = 0
–PMT = 1,000
–N = 20
–I = 6
The answer:
$11,470
20
Interpretation
For a present payment of $11,470,
the individual will annually receive
$1,000 for the next twenty years.
The $11,470 is an immediate cash
outflow.
The $1,000 annual payment to be
received is a cash inflow.
21
Illustration 6
If an investment pays $50 a year for
10 years and repays $1,000 after 10
years, what is this investment worth
today if you can earn 6 percent? (Like
a bond)
22
Determination of Present Value
The unknown: PV
The givens:
–FV = 1,000
The
answer:
–PMT = 50
$926
–I = 6
–N = 10
23
Interpretation
If you collect $50 a year for 10
years and receive $1,000 after 10
years, those cash inflows are
currently worth $926 at 6 percent.
24
Illustration 7
Time value is used to determine a
loan repayment schedule, such as
a mortgage.
25
Loan Repayment Schedule
Amount borrowed (PV) = $80,000
Interest rate (I) = 8%
Term of the loan (N) = 25 years
No future value since loan is repaid
Amount of the annual payment =
$7,494.30
26
Loan Repayment Schedule
Pmnt
Interest
Principal
Balance
Repayment
Owed
1
$6,400.00
$1,094.15
$78,905.85
2
6,312.47
1,181.68
77,724.17
.
.
.
.
.
.
.
.
.
.
.
.
25
555.13
6,939.17
.00
27
Illustration 8
You have $170,000 and spend
$36,000 a year. If you earn 8%
annually, how long will your funds
last?
28
Determination of
Number of Years
The unknown: N
The givens:
–PV = 170,000
–I = 8
–FV = 0
–PMT = -36,000
The answer:
6.2 years
29
Interpretation
If you have $170,000 and earn 8
percent annually, you can spend
$36,000 per year for approximately
6 years and 2 months.
30