An annuity due and an ordinary annuity are two types of annuities that differ primarily by the timings of the payments.

Both are widely used in the financial markets but the use of ordinary annuity mechanisms is more common.

Let’s discuss what ordinary annuities are, annuity due, how these types of annuities work, and their key differences with some examples.

## What is an Ordinary Annuity?

An ordinary annuity is a series of payments made after a payment interval for a specified period. Annuities can be for a fixed period (finite) or can be made for unending periods (infinite).

Annuity payment intervals can vary by length from daily, weekly, monthly, quarterly, and semi-annually to annually. Annuity payments can also be made at the end of the annuity period.

Common examples of ordinary annuities include payments on bonds, saving certificates, preference shares, etc. These payments are made after a certain period and at the end of the payment interval.

## What are the Characteristics of an Ordinary Annuity?

An ordinary annuity must have the following characteristics:

- The payment interval should be the same throughout the length of the annuity period. It should have equal and consistent intervals.
- The payment amount should be fixed for each payment interval.
- Each payment should be made at the end of the payment interval.

The cash flows arising from an ordinary annuity must be equal and fixed. That is one reason regular interest payments made by banks on saving accounts are not considered annuities.

The present value (PV) of annuities can change depending on the length of the annuity and the interest rate charged—generally, the lower the interest rate the higher the PV of an annuity and vice versa.

It also means that the PV of an ordinary annuity can change with a change in the interest rate during the period.

Therefore, both the receiver and the payer would like to use a fixed interest rate and know the total PV of all future annuity payments.

## How to Calculate the Present Value (PV) of Ordinary Annuity?

The present value of an ordinary annuity can be calculated with the help of the formula given below:

**PV of Ordinary Annuity = C1 x ((1 – (1 + i) ^ -n) / i)**

C1 = Cash Flow or payment received at the end of the first interval.

And i = Interest rate, n = number of intervals

We can use the number of periods as they are or convert them into annual payments. However, the interval should be consistent with the interest rate.

So, if we use semi-annual payment intervals, we must also adjust the annual interest rate to semi-annual terms.

The value of an ordinary annuity increases as the interest rates reduce as investors wish to invest in prevailing annuities offering higher returns than the lower market interest rates.

## Example

Suppose a corporate bond pays $ 30,000 annually for the next 10 years at an interest rate of 6% to investors. We can use the formula above to calculate the present value of this ordinary annuity.

**PV of Ordinary Annuity = C1 x ((1 – (1 + i) ^ -n) / i)**

PV of Ordinary Annuity = 30,000 x ((1- (1+6%) ^-10/6%))

PV of Ordinary Annuity = 30,000 x 7.360087 = $ 220,802

As discussed above, if the interest rate increases, the PV of the annuity will decrease. Suppose the interest rate on the bond changes to 10%.

Then,

PV of Ordinary Annuity = 30,000 x ((1- (1+10%) ^-10/10%))

PV of Ordinary Annuity = $ 184,338

## What is Annuity Due?

An annuity due refers to a series of equal and consistent payments received at the beginning of the payment interval for a specified period.

So, an annuity due is primarily different from an ordinary annuity by the payment timings. As the name suggests, the payment is due immediately or in advance rather than at the end of the payment interval.

Payment instruments making advance payments to investors are examples of annuity-due instruments.

For example, payments received at the beginning of a month for a house rent are an example of an annuity due.

## What are the Characteristics of Annuity Due?

The characteristics of an annuity due are also similar to an ordinary annuity:

- The payment interval should be the same throughout the annuity period. It should have equal and consistent intervals.
- The payment amount should be fixed for each payment interval.
- Each payment should be made at the beginning of the payment interval.

Lease, rental, and interest payments paid in advance are examples of annuity-due instruments. However, in the financial markets, ordinary annuity instruments are more common.

Receivers would generally prefer payments through annuity-due arrangements as the payment received is early.

Likewise, payers would like to use the ordinary annuity terms for a delay in the payment mechanism.

## How to Calculate the Present Value (PV) of Annuity Due?

All other factors being equal, the present value of an annuity due will be higher than an ordinary annuity as the payments are received earlier, particularly, the first payment is received immediately.

We can calculate the PV of an annuity due with the help of the following formula:

**PV of Ordinary Annuity = C1 x [(1 – (1 + i) ^ -n) / i] × (1+i)**

C1 is the cash flow at the beginning of the payment interval for the “n” number of intervals, and “i” is the interest rate.

The formula can also be rearranged as:

**PV of Ordinary Annuity = C1 + C1 x [(1 – (1 + i) ^ -n-1) / i]**

As we do not need to compound the value of the first cash flow since it is received on day one.

## Example

Let us revise our example above for the same values and calculate the present value of the future cash flows if the bond works as an annuity due this time.

PV of Ordinary Annuity = C1 + C1 x [(1 – (1 + i) ^ -n-1) / i]

PV of Ordinary Annuity = 30,000 + 30,000 x [(1 – (1 + 6%) ^ -9) /6%]

PV of Ordinary Annuity = 30,000 + 30,000 x 6.8016

PV of Ordinary Annuity = 30,000 + 204,050

PV of Ordinary Annuity = $ 234,050.

Like an ordinary annuity, the PV of an annuity will also decrease if the interest rate increases. Suppose the interest rate for the annuity above rises to 10%. Then,

PV of Ordinary Annuity = 30,000 + 30,000 x [(1 – (1 + 10%) ^ -9) /10%]

PV of Ordinary Annuity = 30,000 + 30,000 x 5.7590

PV of Ordinary Annuity = $ 202,770.

## Ordinary Annuity Vs. Annuity Due – Key Differences

Let us now briefly summarize the key differences between an ordinary annuity and an annuity due.

**Payment Timing**

The prime difference between both annuity types is the payment timing for each payment interval.

An ordinary annuity payment is at the end of a payment interval while for an annuity due it is in the beginning.

**Present Value**

If all factors are equal for both types of annuities, the present value of an annuity due will be higher than an ordinary annuity.

**Preference**

Receivers of annuities generally prefer a due annuity mechanism as they receive interval payments quickly.

Also, the PV of an annuity due is generally higher than an ordinary annuity. Likewise, payers of annuities would prefer an ordinary annuity mechanism.