# Annuity (finance Theory) - Annuity-due

An annuity-due is an annuity whose payments are made at the beginning of each period. Deposits in savings, rent or lease payments, and insurance premiums are examples of annuities due.

 ↓ ↓ ... ↓ payments ——— ——— ——— ——— — 0 1 ... n-1 n periods

Because each annuity payment is allowed to compound for one extra period. Thus, the present and future values of an annuity-due can be calculated through the formula:

$ddot{a}_{overline{n|}i} = (1+i) times a_{overline{n|}i} = frac{1-left(1+iright)^{-n}}{d}$

and

$ddot{s}_{overline{n|}i} = (1+i) times s_{overline{n|}i} = frac{(1+i)^n-1}{d}$

where are the number of terms, is the per term interest rate, and is the effective rate of discount given by .

Future and present values for annuities due are related as:

$ddot{s}_{overline{n}|i} = (1+i)^n times ddot{a}_{overline{n}|i}$

and

$frac{1}{ddot{a}_{overline{n}|i}} - frac{1}{ddot{s}_{overline{n}|i}} = d$

Example: The final value of a 7 year annuity-due with annual interest rate 9% and monthly payments of \$100:

$FV_{due}(0.09/12,7times 12,100) = 100 times ddot{a}_{overline{84}|0.0075} = 11,730.01.$

Note that in Excel, the PV and FV functions take on optional fifth argument which selects from annuity-immediate or annuity-due.

An annuity-due with n payments is the sum of one annuity payment now and an ordinary annuity with one payment less, and also equal, with a time shift, to an ordinary annuity with one payment more, minus the last payment. Thus we have:

(value at the time of the first of n payments of 1)
(value one period after the time of the last of n payments of 1)