DEFINITION

An annuity refers to a series of payments made at regular intervals of time (monthly, quarterly, annually, etc.)
An annuity consisting of equal amounts of payments is called a level annuity, whereas one with varying amounts in each instalment, is called a variable (increasing or decreasing) annuity.

An annuity can also be classified on the basis of timing of each of its payments. An annuity consisting of payments made at the beginning of each time period is called an annuity due. Whereas one with payments made the end of each time interval is called an annuity immediate.

Also, an annuity with a fixed term, which is known in advance, is called annuity certain. An annuity consisting of payments made forever is called a perpetuity.

**FORMULAS AND DERIVATION**

Annuity immediate:-

Present value:

\[a_{\overline{n|}}=\ \frac{1-\ v^n}{i}\]

where, \(~\mathrm{v=\ }\left(1+i\right)^{-1}\)

Future value:

\[s_{\overline{n|}}=\ \frac{{\left(1+i\right)}^n-1}{i}\]
\[\ \ \ \ \ =a_{\overline{n|}}\ \times \ {\left(1+i\right)}^n\]

Annuity due:-

Present value:

\[{\ddot{a}}_{\overline{n|}}=\ \frac{1-v^n}{d}\]

Future value:

\[{\ddot{s}}_{\overline{n|}}=\ \frac{{\left(1+i\right)}^n-1}{d}\]
\[\ \ \ \ \ \ =\ {\ddot{a}}_{\overline{n|}}\times \ {\left(1+i\right)}^n\]

where, \(d=iv\)

Continuously payable annuity:-

Present value:

\[{\overline{a}}_{\overline{n|}}=\ \frac{1-v^n}{\delta }\]

Future value:

\[{\overline{s}}_{\overline{n|}}=\ \frac{{\left(1+i\right)}^n-\ 1}{\delta }\]
\[\ \ \ \ \ ={\overline{a}}_{\overline{n|}}\ \times \ {\left(1+i\right)}^n\]

where, \(\delta =\ {\mathrm{ln} \left(1+i\right)\ }\)

Annuities payable p times a year:-

Annuity immediate:-

Present value:

\[a^{\left(p\right)}_{\overline{n|}}=\ \frac{1-v^n}{i^{\left(p\right)}}\]

Future value:

\[s^{\left(p\right)}_{\overline{n|}}=\ \frac{{\left(1+i\right)}^n-1}{i^{\left(p\right)}}\]

where, \(i^{\left(p\right)}=p\left[{\left(1+i\right)}^{{1}/{p}}-1\right]\)

Annuity due:-

Present value:

\[{\ddot{a}}^{\left(p\right)}_{\overline{n|}}=\ \frac{1-v^n}{d^{\left(p\right)}}\ \]

Future value:

\[{\ddot{s}}^{\left(p\right)}_{\overline{n|}}=\ \frac{{\left(1+i\right)}^n-1}{d^{\left(p\right)}}\]

where, \(d^{\left(p\right)}=p\left[1-{\left(1-d\right)}^{{1}/{p}}\right]\)

Present value of perpetuities:-

Annuity immediate:-

\[a_{\overline{\infty |}}=\ \frac{1}{i}\]

Annuity due:-

\[{\ddot{a}}_{\overline{\infty |}}=\ \frac{1}{d}\]

Continuously payable perpetuity:-

\[{\overline{a}}_{\overline{\infty |}}=\ \frac{1}{\delta }\]