The future value of an ordinary annuity formula can be used to calculate the future value of a stream of payments over time, the formula is as follows:
Where:
PMT = payment
r = rate
n = periods
Assume you were to invest $1,000 per year, in an investment that would grow at a 7% rate of return over ten years. What would the future value be at the end of the 10th year?
We can solver for the future value by plugging in the variables as follows:
You can calculate the value above with an HP12C using the following keystrokes:
[1000][PMT]
[7][i]
[10][n][FV]
Using Excel, we can model the growth of the investment at different PMTs and rates of growth over time.
Assuming the same variables above we can construct a table of values for each of the periods:
| Period | PV | rate | FV | PMT |
| 1 | $ 1,000.00 | |||
| 2 | $ 1,000.00 | 7.00% | $ 1,070.00 | $ 2,070.00 |
| 3 | $ 2,070.00 | 7.00% | $ 2,214.90 | $ 3,214.90 |
| 4 | $ 3,214.90 | 7.00% | $ 3,439.94 | $ 4,439.94 |
| 5 | $ 4,439.94 | 7.00% | $ 4,750.74 | $ 5,750.74 |
| 6 | $ 5,750.74 | 7.00% | $ 6,153.29 | $ 7,153.29 |
| 7 | $ 7,153.29 | 7.00% | $ 7,654.02 | $ 8,654.02 |
| 8 | $ 8,654.02 | 7.00% | $ 9,259.80 | $ 10,259.80 |
| 9 | $ 10,259.80 | 7.00% | $ 10,977.99 | $ 11,977.99 |
| 10 | $ 11,977.99 | 7.00% | $ 12,816.45 | $ 13,816.45 |
How would these values look if we reduced the return from 7% to 4%:
| Period | PV | rate | FV | PMT |
| 1 | $ 1,000.00 | |||
| 2 | $ 1,000.00 | 4.00% | $ 1,040.00 | $ 2,040.00 |
| 3 | $ 2,040.00 | 4.00% | $ 2,121.60 | $ 3,121.60 |
| 4 | $ 3,121.60 | 4.00% | $ 3,246.46 | $ 4,246.46 |
| 5 | $ 4,246.46 | 4.00% | $ 4,416.32 | $ 5,416.32 |
| 6 | $ 5,416.32 | 4.00% | $ 5,632.98 | $ 6,632.98 |
| 7 | $ 6,632.98 | 4.00% | $ 6,898.29 | $ 7,898.29 |
| 8 | $ 7,898.29 | 4.00% | $ 8,214.23 | $ 9,214.23 |
| 9 | $ 9,214.23 | 4.00% | $ 9,582.80 | $ 10,582.80 |
| 10 | $ 10,582.80 | 4.00% | $ 11,006.11 | $ 12,006.11 |
We can illustrate the two tables graphically as well:
Generally speaking, over longer periods of time, the higher the rate of return, or the larger the annual contributions, the larger the difference between the two ending values will become.
In the world of financial planning, this formula can be applied to determine the approximate amount of money you will have at retirement on a pre-tax basis.
The variables will be defined by the amount of money you are contributing into your employer sponsored 401(k) plan (a type of tax-deferred account), Roth IRA or Traditional IRA, on an annual basis, any company matching contributions you may receive, and the number of years until you reach your retirement age.
A copy of the Excel model used to calculate the future value of an annuity can be found here:
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