Annuity: A series of equal payments or receipts occurring over a specified number of periods.

Ordinary annuity: A series of equal payments or receipts occurring over a specified number of periods with the payments or receipts occurring at the end of each period.

Annuity due: A series of equal payments or receipts occurring over a specified number of periods with the payments or receipts occurring at the beginning of each period.

WARNING:  While technically correct, the last two definitions shown above can be a bit confusing.  Whether a cash flow appears to occur at the end or the beginning of a period often depends on your perspective. (For example, isn't the end of year 2 also the beginning of year 3?)

Therefore, the real key to distinguishing between an ordinary annuity and an annuity due is the point at which either a future or present value is to be calculated. Remembering the following characteristics should help you to identify the type of annuity that you are dealing with:

1.  For an ordinary annuity, future value is calculated as of the last cash flow, while present value is calculated as of one period before the first cash flow.

2.  For an annuity due, future value is calculated as of one period after the last cash flow, while present value is calculated as of the first cash flow.

The following examples, which make use of a single time line, should help you to identify and value the various annuity patterns:

0        1        2        3        4        5        6  Time line
|--------|--------|--------|--------|--------|--------|
         ¦        R        R        R        R        ¦  Cash flows
         ¦        ¦                          ¦        ¦
        PVA₄     PVAD₄                       FVA₄     FVAD₄

NOTE:
PVA₄ is valued as of one period before the first cash flow, which is at the end of period 1.
PVAD₄ is valued as of the first cash flow, which is at the beginning of period 3.
FVA₄ is valued as of the last cash flow, which is at end of period 5.
FVAD₄ is valued as of one period after the last cash flow, which is also the beginning of period 7.
 

Present Value of an (Ordinary) Annuity (PVA):

        PVA₄ = (R) x (PVIFA_i%,4)

Future Value of an (Ordinary) Annuity (FVA):

        FVA₄ = (R) x (FVIFA_i%,4)

Present Value of an Annuity Due (PVAD):

        PVAD₄ = (R) x (PVIFA_i%,4) x (1 + i%)
or
        PVAD₄ = [(R) x (PVIFA_i%,3)] + R = (R) x (PVIFA_i%,3 + 1)

Future Value of an Annuity Due (FVAD):

        FVAD₄ = (R) x (FVIFA_i%,4) x (1 + i%)
or
        FVAD₄ = (R) x (FVIFA_i%,5) - R = (R) x (FVIFA_i%,5 - 1)
 

TIP:  When valuing annuities due, just remember to due the RIGHT thing.  That is, PVADs and FVADs are determined as of one period to the RIGHT of positions assumed for an ordinary annuity with the same cash flows. So, you can treat them initially as ordinary annuities, find the PVA (or FVA), and simply multiply that answer by (1 + i%), which effectively shifts the earlier answer one period to the right.