# Time Value of Money: Annuity

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Types of Annuity

## Defining Annuity

Annuity = series of cash flows of the same value occurring at equal intervals

## Types of Annuity

- Ordinary annuity = cash flows occur at the end of each period
- Annuity due = cash flows occur at the beginning of each period
- Perpetuity = never-ending sequence of cash flows = cash flows occur at the end of each period indefinitely

Formula for the future value of an ordinary annuity:

\(FV_N=A\times \Bigl[\frac{(1+r)^N-1}{r}\Bigr]\)

Formula for the future value of an annuity due:

\(FV_N=A\times \Bigl[\frac{(1+r)^N-1}{r}\Bigr]\times(1+r)\)

Formula for the present value of an ordinary annuity:

\(PV=A\times \Bigl[\frac{1-\frac{1}{(1+r)^N}}{r}\Bigr]\)

Formula for the present value of an annuity due:

\(PV=A\times \Bigl[\frac{1-\frac{1}{(1+r)^N}}{r}\Bigr]\times(1+r)\)

Formula for the present value of a perpetuity:

\(PV=\frac{A}{r}\)

Where:

- \(FV_N\) – future value
- PV – present value
- r – periodic interest rate
- N – number of periods
- A – annuity amount

### Algorithm for Solving TVM Problems

When you are asked to do a time value of money question, it's best if you follow a defined sequence of steps. After reading a TVM question:

- determine what you need to calculate,
- establish whether you are dealing with a single payment or an annuity,
- check if it is an ordinary annuity or annuity due,
- put the cash flows on a time line,
- apply relevant formulas and – using your calculator – solve the problem,
- interpret the results when necessary.

If you are using only the TVM worksheet for time value of money problems, you should read our next post in which we show how you can use the CF and NPV worksheets to do time value of money problems.