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)

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