Future Value of Ordinary Annuity

by Obaidullah Jan, ACA, CFA

An ordinary annuity is a finite stream of equal equidistant cash flows that occur in arrears.

It’s 1st January 2018 and you have decided to save $1,000 each month for next three months to save enough money to start your MBA program. But you can make your first deposit only on 31st January 2018, the date you receive you next salary and your last deposit will be on 31 December 2020. This series of deposits constitute an (ordinary) annuity because it is a finite i.e. there are 36 deposits, the cash flows are equal i.e. you are saving $1,000 each month and there is equal time between cash flows i.e. one month.

You must be wondering how much you will have saved by 31st December 2018 if your bank pays 5% on deposits. There are several ways you can do such computation. The most basic but time-consuming is to manually grow each cash flow to 31st December 2020 using the following equation:

$$ FV=PMT\times\left(1+\frac{r}{m}\right)^{n\times m} $$

Where PMT is the monthly deposit i.e. $1,000, r is the annual percentage interest rate, n is the total number of years for which the payment will grow, it is 3 years in case of first deposit and 0 in case of the last deposit, and m is the total number of deposits per year.

Formula

You can calculate the future value of annuity using the following direct formula:

$$ FV=PMT\times\frac{\left(1+\frac{r}{m}\right)^{n\times m}-1}{\frac{r}{m}} $$

Alternatively, you can use Excel FV function. FV function syntax is: FV(rate, nper, pmt, [pv], [type]). Where rate is the periodic interest rate (i.e. r/m), nper is the total number of cash flows, pmt is the amount of cash flows, [pv] is an optional argument. [type] is also an optional argument that has value of 0 or 1, 0 being the default value specifying an ordinary annuity and 1 specifying an annuit due.

Example

Using the formula above we can calculate your accumulated balance as at 31st December 2020 as follows:

$$ FV=$1,000\times\frac{\left(1+\frac{5\%}{12}\right)^{3\times12}-1}{\frac{5\%}{12}}=$38,753 $$