# INTRODUCTION TO ANNUITY — MEANING, TYPES AND FORMULAS

An annuity is a series of equal payments with a fixed frequency.

It is a sequence of equal payments made at an equal period.

It is an investment in which payments are made regularly over a long period.

for example, if your rent a house and promises to make a series of payments over an agreed period, you have created an annuity.

The time between these regular payments is called the payment period and the time from the beginning of the first payments to the last payments is called the term of the annuity

## Types Of Annuity

There are three types of annuity namely: Annuity due, ordinary annuity and perpetuities.

1. Annuity due: This is an annuity in which payments are made at the beginning of the period.

It is a series of regular payments made at the beginning of each period.

For example, if the payment period is weekly, payment will be made at the beginning of the week.

2. Ordinary annuity: This is an annuity where payments are made at the end of the period.

It is a series of regular payments made at the end of each period.

For example, if the payment period is monthly, payments are made at the 28-31 of each month.

Most mortgage payments are ordinary annuities.

3. Perpetuities: This is an annuity that has an indefinite term of the annuity.

In other words, perpetuity is an annuity where regular payments begin on a fixed date and continue indefinitely.

Needless to say, Perpetuity is also known as a perpetual annuity

Perpetuities are very rare, and we will not be covering them in this post.

Annuity due and ordinary annuity are common types of annuity, hence, it makes sense to look at the differences between this two, which are:

1. Annuity due is a regular series of payments made at the beginning of the period whereas an ordinary annuity is a regular series of payments made at the end of the period.

2. The cash flows of the annuity due occur one period earlier than the cash flows of a ordinary annuity.

3. Annuity due usually have a higher present value than an ordinary annuity.

Having looked at the types of annuity, we move on to the present and future value of the annuity at a due and ordinary annuity.

## Present Value Of An Annuity

This is the amount of money that a stream of equal future payments is worth now.

As there are two main of annuity, the present value of each type of annuity is calculated differently.

### Present Value Of An Ordinary Annuity

This is the amount of money that a stream of equal payment made at the end of the period is worth now if it is to be paid in a single lump sum.

It is calculated as follows:

$$PV=C\left[\frac{1-\left(1+\frac{i}{m}\right)^{-n m}}{\frac{i}{m}}\right]$$

Where C is the regular cash flow

i is the interest rate expressed as a percentage

m is the number of compounding interest in one year

n is the number of years

### Present Value Of The Annuity Due

This is the amount of money that a stream of equal payment made at the beginning of the period is worth now if it is to be paid in a single lump sum.

It is calculated as follow:

$$PV=C\left[\frac{1-\left(1+\frac{i}{m}\right)^{-nm}}{\frac{i}{m}}\right]\left(1+\frac{i}{m}\right)$$

## Future Value Of An Annuity

This is the value of a stream of equal payment at a certain date in the future.

Like present value, the future value of the annuity is calculated differently

### Future value of an ordinary annuity

This is the future value of a stream of end-of-period equal payments at a certain date.

It is calculated as follow:

$$FV=C\left[\frac{\left(1+\frac{i}{m}\right)^{n m}-1}{\frac{i}{m}}\right]$$

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### Future Value Of An Annuity Due

This is the future value of a stream of beginning-of-period equal payments at a certain date.

It is calculated as follow:

$$FV=C\left[\frac{\left(1+\frac{i}{m}\right)^{n m}-1}{\frac{i}{m}}\right]\left(1+\frac{i}{m}\right)$$

That will be all for now. In the next post, We will be solving some examples on the future value of an annuity due.