An annuity is simply a series of regular cash flows either received or paid periodically. An annuity due is one in which the cash flows occur at the beginning of each period.

Life insurance, pension payments, motor vehicle leases, property rents, are all forms of annuities.

## Cash Flow Diagram

The diagram below represents an annuity due of 4,000 a period for 3 periods. Notice how each receipt is the same and occurs at the **beginning** of each period. This is in contrast to an ordinary annuity where the cash flows occur at the end of a period.

Period | 0 | 1 | 2 | 3 |
---|---|---|---|---|

Cash flow | ↑ 4,000 | ↑ 4,000 | ↑ 4,000 |

The period for an annuity can be of any length, week, month year etc. providing all periods are of the same length.

## Future Value of an Annuity Due

Each of the cash flows can be regarded as a single lump sum cash flow and the future value of an annuity due can be calculated using the future value of a lump sum formula, by compounding each cash flow forward to the end of year 3.

In the above example, assuming a periodic discount rate of 6%, at the end of year 3 the future value of an annuity due would be shown as follows:

FV = FV cash flow 1 + FV cash flow 2 + FV cash flow 3 FV = 4,000 x (1 + 6%)^{3}+ 4,000 x (1 + 6%)^{2}+ 4,000 x (1 + 6%)^{1}FV = 13,498.46

The cash flow received at the beginning of year 1 is compounded forward 3 years to the end of year 3, likewise the cash flow received at the beginning of year 2 is compounded forward 2 years to the end of year 3, and finally the cash flow received at the beginning of year 3 is compounded forward 1 year to the end of year 3.

Clearly this calculation can be performed for any number of periods and cash flows, however to avoid repetitive time, consuming calculations, it can be shown mathematically that the future value of an annuity due is given by the formula shown below.

It should be noted that, rearranged, this formula is simply the formula for the future value of an ordinary annuity (highlighted in red), compounded forward one more period by the term (1 + i).

`FV = (1 + i) x Pmt x ( (1 + i)`^{n} - 1 ) / i

This results from the fact that for an annuity due the cash flows occur at the beginning of the period, one period earlier than those of an ordinary annuity, and so one additional period of compounding takes place.

Using the above annuity due as an example, the future value of the annuity due at the end of year 3 can be calculated using the formula as follows:

Pmt = 4,000 i = 6% n = 3 FV = Pmt x (1 + i) x ( (1 + i)^{n}- 1 ) / i FV = 4,000 x (1 + 6%) x ( (1 + 6%)^{3}- 1 ) / 6% FV = 13,498.46

The future value of annuity due forms the basis of many time value of money calculations. The formula allows any series of regular cash flows (Pmt) to be compounded forward a number of periods (n) at a discount rate of i per period.

## About the Author

Chartered accountant Michael Brown is the founder and CEO of Plan Projections. He has worked as an accountant and consultant for more than 25 years and has built financial models for all types of industries. He has been the CFO or controller of both small and medium sized companies and has run small businesses of his own. He has been a manager and an auditor with Deloitte, a big 4 accountancy firm, and holds a BSc from Loughborough University.