Compared to an ordinary annuity, the present value of an annuity due can be calculate by modifying the formula above with the addition of the quantity (1 + r) as follows:
present value of annuity due formula
Where: PMT = payment r = rate n = periods
Assume an individual won the lottery and the prize was to be a series of $1,000 payments received at the beginning of each year, over a ten year period. The winner has the option of choosing between the stream of payments or a lump sum discounted at a required rate of 7%, we can calculate what the present value of the stream of payments is as follows:
where; PMT = $1,000, r = 0.07, n = 10
Using an HP12C calculator, we can solve the equation above using the following keystrokes:
Compared to an ordinary annuity, payments for an annuity due are received at the beginning of each period. Due to this extra time that the payments have to compound, we can modify the ordinary annuity formula with the addition of the quantity (1 + r) to calculate the future value of an annuity due as follows:
future value of an annuity due formula
Where: PMT = payment r = rate n = periods
Assume that an individual was to invest $1,000 over a period of 10 years, in a security with a 7% rate of return, and the investment was made at the beginning of each year. We can calculate the future value of the investment as follows:
where; PMT = $1,000, r = 0.07, n = 10
Using an HP12C calculator, we can calculate the future value of an annuity due using the variables above as follows:
The present value of an ordinary annuity formula can be used to calculate the present value of a stream of income payments, the formula is as follows:
present value of an ordinary annuity formula
Where: PMT = payment r = rate n = periods
Assume you won the lottery and the prize is a $1,000 series of payments to be received over the next ten years, at the end of each year. As the winner you could choose either the $1,000 stream of payments or a lump sum discounted at a required rate of 7%. We can calculate the lump sum as follows:
where; PMT = $1,000, r = 0.07, n = 10
Give the result above, as long as the lump sum is exactly $7,023.58, and assuming you could realize a 7% return over ten years, you should be indifferent to receiving a lump sum or the stream of payments. If the lump sum offered is less than $7,023.58, you should chose the income; however, if the lump sum offered is greater than $7,023.58 you should choose the lump sum over the payment stream.
Using an HP12C calculator, you can calculate the present value of an ordinary annuity with the variables above using the following keystrokes:
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:
future value of ordinary annuity formula
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:
where; PMT = $1,000, r = 0.07, n = 10
You can calculate the value above with an HP12C using the following keystrokes:
hp12c
[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
FV of an annuity table; PMT = $1,000, r = 0.07
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
FV of an annuity table; PMT = $1,000, rate = 0.04
We can illustrate the two tables graphically as well:
FV of an annuity chart
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:
