annuity-due
Some payments are made up front rather than at the end. Rent is the classic example: you pay for the coming month before you live in it, on the first day, not the last. When equal payments arrive at the beginning of each period, the arrangement is an annuity-due. It is the same stream as an annuity-immediate, just shifted one period earlier — and that small shift, because money earns interest, makes it worth slightly more.
The present value of an annuity-due paying 1 at the start of each of n periods, written a-double-dot-angle-n, equals (1 minus v to the power n) divided by d, where d is the rate of discount. Because each payment is one period earlier, the annuity-due relates to the annuity-immediate by a single accumulation factor: a-double-dot-angle-n equals (1 + i) times a-angle-n, and likewise the accumulated value s-double-dot-angle-n equals (1 + i) times s-angle-n. Insurance premiums and pension annuities in payment are usually modelled as annuities-due because benefits and premiums tend to be paid at the start of each year.
The distinction matters enormously in life-contingent work, where the standard life annuity is an annuity-due (the retiree is paid at the start of each year they are alive). Getting the timing wrong by one period systematically misprices the contract. A reliable check: an annuity-due is always worth a factor of (1 + i) more than the otherwise identical annuity-immediate, never less, because every dollar arrives one period sooner.
You pay 1,200 in rent at the start of each year for 3 years. At 5 percent, the present value is 1,200 times a-double-dot-angle-3, which equals 1,200 times (1 + v + v squared), about 3,428 — more than the same payments made at year-end.
Payments fall at the start of each period; the first one is at time zero.
An annuity-due of n payments and an annuity-immediate of n payments cover different time windows: the due version starts at time 0 and ends at time n-1, the immediate at time 1 and ends at time n.