TVM: More on Future Value
Helpful Tool: Cash Flow Diagrams
A cash flow diagram is a picture of a financial problem that shows all cash inflows and
outflows plotted along a horizontal time line. It can help you to visualize a
financial problem and to determine if it can be solved using TVM methods.
Constructing a CashFlow Diagram
The time line is a horizontal line divided into equal periods such as days, months, or
years. Each cash flow, such as a payment or receipt, is plotted along this line at
the beginning or end of the period in which it occurs. Funds that you pay
out such as savings deposits or lease payments are negative cash flows
that are represented by arrows which extend downward from the time line with their
bases at the appropriate positions along the line. Funds that you receive
such as proceeds from a mortgage or withdrawals from a saving account are positive
cash flows represented by arrows extending upward from the line.
Example: You are 40 years old and have accumulated $50,000 in your
savings account. You can add $100 at the end of each month to your account which
pays an annual interest rate of 6% compounded monthly. Will you be able to retire in
20 years?
The time line is divided into 240 monthly
periods (20 years times
12 payments per year) since the
payments are made monthly and the
interest is also
compounded monthly. The $50,000 that
you have now (present value) is a negative cash outflow since
you will treat it as though you were just now depositing it into the account.
It is represented with a downward pointing arrow with its base at the beginning of the
first period. The 240 monthly $100 deposits are also negative outflows
represented with downward pointing arrows placed at the end of each period. Finally
you will withdraw some unknown amount (the
future value) after 20
years. Represent this positive inflow with an upward pointing arrow with its
base at the very end of the last period.
This diagram was drawn from your point of view. From the bank's point of view,
the present value and the series of deposits are positive cash inflows, and the final
withdrawal of the future value will be a negative outflow.
Future Value of Annuities
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.
Future Value of an Ordinary Annuity
The Future Value of an Ordinary Annuity (FVoa) is the value that
a stream of expected or promised future payments will grow to after a given number of
periods at a specific compounded interest.
The Future Value of an Ordinary Annuity could be solved by calculating
the future value of each individual payment in the series using the future value formula
and then summing the results. A more direct formula is:
|
FVoa = PMT [((1 + i)n - 1) / i] |
Where:
- FVoa = Future Value of an Ordinary Annuity
- PMT = Amount of each payment
- i = Interest Rate Per Period
- n = Number of Periods
Example: What amount will accumulate if we deposit $5,000
at the end of each year for the next 5 years? Assume an interest of
6% compounded annually.
PV = 5,000
i = .06
n = 5
|
FVoa = 5,000 [ (1.3382255776 - 1) /.06 ] =
5,000 (5.637092) = 28,185.46 |
|
Year |
1 |
2 |
3 |
4 |
5 |
|
Begin |
0 |
5,000.00 |
10,300.00 |
15,918.00 |
21,873.08 |
|
Interest |
0 |
300.00 |
618.00 |
955.08 |
1,312.38 |
|
Deposit |
5,000.00 |
5,000.00 |
5,000.00 |
5,000.00 |
5,000.00 |
|
End |
5,000.00 |
10,300.00 |
15,918.00 |
21,873.08 |
28,185.46 |
Exercise: Draw a Cash Flow
Diagram for the problem above.
Example 2: In practical problems, you may need to calculate both
the future value of an annuity (a stream of future periodic payments) and the
future value of a single amount that you have today:
For example, you are 40 years old and have accumulated $50,000 in your savings
account. You can add $100 at the end of each month to your account
which pays an interest rate of 6% per year, compounded monthly.
How much will this contribute to your retirement in 20 years? (See Cash
Flow Diagram above).
You can treat this as the sum of two separate calculations:
- the future value of 240 monthly payments of $100 Plus
- the future value of the $50,000 now in your account.
PMT = $100 per period
i = .06 /12 = .005 Interest per period
(6% annual rate / 12
payments per year)
n = 240 periods
FVoa = 100 [ (3.3102 - 1) /.005 ] = 46,204
+
PV = 50,000 Present value (the amount you have today)
i = .005 Interest per period
n = 240 Number of periods
FV = PV (1+i)240 = 50,000 (1.005)240
= 165,510.22
After 20 years you will have accumulated $211,714.22
(46,204.00 + 165,510.22).
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