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

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. Will you have enough money to retire in 20 years?

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).

Future Value of an Annuity Due (FVad)

The Future Value of an Annuity Due is identical to an ordinary annuity except that each payment occurs at the beginning of a period rather than at the end. Since each payment occurs one period earlier, we can calculate the present value of an ordinary annuity and then multiply the result by (1 + i).

FVad = FVoa (1+i)

Where:

FVad = Future Value of an Annuity Due

FVoa = Future Value of an Ordinary Annuity

i = Interest Rate Per Period

Example: What amount will accumulate if we deposit $5,000 at the beginning of each year for the next 5 years? Assume an interest of 6% compounded annually.

PV = 5,000
i = .06
n = 5

FVoa = 28,185.46 (1.06) = 29,876.59

Year	1	2	3	4	5
Begin	0	5,300.00	10,918.00	16,873.08	23,185.46
Deposit	5,000.00	5,000.00	5,000.00	5,000.00	5,000.00
Interest	300.00	618.00	955.08	1,312.38	1,691.13
End	5,300.00	10,918.00	16,873.08	23,185.46	29,876.59

Combined Formula

You can also combine these formulas and the future value of a single amount formula into one.

Concepts