## 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 Cash Flow 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:

1. the future value of  240 monthly  payments of \$100  Plus
2. 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).