Annuities |

Ordinary?
Due? What do I do? |

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**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

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`

**NOTE:**

PVA_{4} is valued as of one period before the
first cash flow, which is at the **end of period 1.**

PVAD_{4 }is valued as of the first cash flow,
which is at the **beginning of period 3.**

FVA_{4} is valued as of the last cash flow, which
is at **end of period 5.**

FVAD_{4} 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):**

PVA_{4} = (R) x (PVIFA_{i%,4})

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

** **FVA_{4} =
(R) x (FVIFA_{i%,4})

**Present Value of an Annuity Due (PVAD):**

** **PVAD_{4} =
(R) x (PVIFA_{i%,4}) x (1 + i%)
**or**
** **PVAD_{4}
= [(R) x (PVIFA_{i%,3})] + R = (R) x (PVIFA_{i%,3} + 1)

**Future Value of an Annuity Due (FVAD):**

** **FVAD_{4} =
(R) x (FVIFA_{i%,4}) x (1 + i%)
**or**
** **FVAD_{4}
= (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.

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