Future Value of Annuity Due
An 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 us 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.
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 ordinary 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 = FV of Ordinary Annuity × (1 + i)
Where i stands for periodic interest rate, i.e. the 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 ×||(1 + i)n - 1||× (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 for future value of a single sum and then summing all the component present values up:
FV = PV × (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.
If you invest $1,000 at the end of each month for 5 years at 10%, your money will be worth
|FV (Ordinary Annuity) = $1,000 ×||(1 + 10%/12)5×12 - 1||= $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 (Annuity Due) = $77,437 × (1 + 10%/12) = $78,082
You can calculate the above value in Excel by entering the following function: FV(10%/12,60,-1000,0,1).