## Steven Clark's "Future Value of an Ordinary Annuity"

## TIME VALUE OF MONEY: ORDINARY ANNUITIES

So far we’ve looked at the Time Value of Money: Present & Future of fixed amounts and expanded that understanding into Time Value of Money: Financial Tables. However, it’s time to move past fixed amounts and discuss more complex cash flow streams.

### The Two Types of Annuity

The definition of an annuity is a series of periodic cash flows (inflows or outflows) of equal amounts over a specified period of time. It is common for these cash flows to be annual or monthly.

There are two types of annuity that you need to understand. An Ordinary Annuity is where those cash flows appear at the end of each period. An Annuity Due is where those cash flows appear at the beginning of the period.

For example, if you were to pay $2,000 per year at the end of every year spanning four years, subject to five per cent interest, you would be paying an Ordinary Annuity. This is an Ordinary Annuity because the same cash flows occur at the end of each equal period.

If those cash flows occurred at the beginning of each period – $2,000 per year on 1 January – they would be regarded as an Annuity Due.

It is important to note that all things being equal an Annuity Due will hold greater value than an Ordinary Annuity because the payments accrue an extra period of interest (due to immediate investment rather than deferred investment).

### Future Value of an Ordinary Annuity

If we run with the example and plug the figures into our equation for calculating the Future Value of an Ordinary Annuity – where PMT is the size of the regular payment and FVIFA_{i,n} is the Future Value Interest Factor co-ordinate on the FVIFA Time Value of Money Finanical Table:

Future Value of an Ordinary Annuity = PMT * (FVIFA_{i,n})

This would be calculated using the FVIFA Time Value of Money Financial Tables:

- Future Value of an Ordinary Annuity = $2,000 * (FVIFA
_{0.05,4}) - Future Value of an Ordinary Annuity = $2,000 * 4.310
- Future Value of an Ordinary Annuity = $8,620

### Present Value of an Ordinary Annuity

In the same manner it is easy to calculate the Present Value of an Ordinary Annuity using the formula:

Present Value of an Ordinary Annuity = PMT * (PVIFA_{i,n})

The calculation would follow through as:

- Present Value of an Ordinary Annuity = $2,000 * (PVIFA
_{0.05,4}) - Present Value of an Ordinary Annuity = $2,000 * 3.546
- Present Value of an Ordinary Annuity = $7,092

To rationalise that Present Value of the Ordinary Annuity you can look at the Future Value of the Ordinary Annuity and re-calculate its Present Value as a single amount. The results are strikingly similar.

- Present Value of a Fixed Amount = Future Value * (PVIF
_{i,n}) - Present Value of a Fixed Amount = $8,620 * (PVIF
_{0.5,4}) - Present Value of a Fixed Amount = $8,620 * 0.823
- Present Value of a Fixed Amount = $7,094.26

Again, note that Ordinary Annuities have cash flows that appear *at the end of each time period*.

### The Next Step: Annuities Due

In the fourth installment in this Time Value of Money series I will outline the calculations for Annuities Due. These calculations may appear mundane and beneath your concern but if you’re a manager or business owner you should take the time to understand them. They offer a toolkit for more effective decision making.

I would also advise you to pick up any decent copy of a managerial finance textbook to expand on this knowledge.