Future Value of Annuity Due

Annuity due is an annuity in which the cash flows occur at the start of each period. Due to the advance nature of cash flows, each cash flow is subject to the compounding effect for one additional period when compared to an otherwise similar (ordinary) annuity. The future value of an annuity due is higher than the future value of an (ordinary) annuity by the factor of one plus the periodic interest rate.

Let’s say you want to invest $1,000 each month for 5 years to accumulate enough money for an MBA program. There are sixty total payments in your annuity. If you deposit the $1,000 dollars right on the day you decide to invest, the first deposit will growth for full 60 months. Alternatively, if you deposit the amount at the end of the month, the first deposit will growth for only 59 months. This one period difference persists for all cash flows.

Formula

It follows from the difference in an (ordinary) annuity and an annuity due that we can get the future value of an annuity due by growing the present value of an annuity with the same terms (periodic payment, periodic interest rate and total number of payments) over one more period. This can be expressed as follows:

$$ FV\ of\ Annuity\ Due\ = FVA\ \times (1+i) $$

Where FVA stands for future value of an (ordinary) annuity and i stands for periodic interest rate (i.e. annual percentage rate divided by total number of compounding periods per year).

Substituting FVA with the formula for FV for an (ordinary) annuity, we get:

$$ FV\ of\ Annuity\ Due\ =PMT\times\ \frac{{(1+i)}^n-1}{i}\times\ (1+\ i) $$

Where PMT is the periodic cash flow in the annuity due, i is the periodic interest rate and n is the total number of payments.

If you don’t know the formula, you can work out the future value by individually growing each payment in the annuity due using the following formula and then summing all the component present values up:

$$ FV=PV\times{(1+i)}^n $$

Where n is the relevant number of periods for which each cash flow must grow, starting from 60 in the above example and down to 1 for the last cash flow.

You can also use Excel FV function to find future value of an annuity due. FV function syntax is FV(rate, nper, pmt, [pv], [type]). You need to specify 1 in the [type] argument to get Excel to treat the series as an annuity due instead of an (ordinary) annuity.

Example

If you invest $1,000 at the end of each month for 5 years at 10%, your money will be worth

$$ FV\ (assuming\ deposit\ in\ arrears)\\=$1,000\times\frac{{(1+\frac{10\%}{12})}^{60}-1}{\frac{10\%}{12}}\\=$77,437 $$

If you invest the amount at the start of each month, each cash flow will grow for one more period, the compounding effect of which can be captured by multiplying the above value with 1 plus periodic interest rate.

$$ FV\ (assuming\ deposit\ in\ advance)\\=$1,000\times\frac{{(1+\frac{10\%}{12})}^{60}-1}{\frac{10\%}{12}}\times\ (1+\frac{10\%}{12})\\=$78,082 $$

You can calculate the above value in Excel by entering the following function: FV(10%/12,60,-1000,0,1).

Written by Obaidullah Jan, ACA, CFA and last modified on