# Annuities Basic Formulae

### Simple Annuities

$a_{\overline{n}\lvert } = \frac{1-v^n}{i}$   $a_{\overline{\infty}\lvert } = \frac{1}{i}$

$\ddot{a}_{\overline{n}\lvert}= \frac{1-v^n}{d}$     $\ddot{a}_{\overline{\infty}\lvert } = \frac{1}{d}$

$\overline{a}_{\overline{n}\lvert}= \frac{1-v^n}{\delta}$     $\overline{a}_{\overline{\infty}\lvert } = \frac{1}{\delta}$

The future value counterparts can be derived by multiplying the PV with $(1+i)^n$

### Increasing and Decreasing Annuities

$PV_{o}= (P+\frac{Q}{i}) a_{\overline{n}\lvert } + (-\frac{nQ}{i})v^n$

$(Ia)_{\overline{n}\lvert } = \frac{\ddot{a}_{\overline{n}\lvert}-nv^n}{i}$     $(Ia)_{\overline{\infty}\lvert } =\frac{1+i}{i^2}$

$(I\ddot{a})_{\overline{n}\lvert } = \frac{\ddot{a}_{\overline{n}\lvert}-nv^n}{d}$      $(I\ddot{a})_{\overline{\infty}\lvert } =\frac{1}{d^2}$

$(I\overline{a})_{\overline{n}\lvert } = \frac{\ddot{a}_{\overline{n}\lvert}-nv^n}{\delta}$     $(I\overline{a})_{\overline{\infty}\lvert } =\frac{1}{\delta d}$

$(\overline{I}\overline{a})_{\overline{n}\lvert } = \frac{\overline{a}_{\overline{n}\lvert}-nv^n}{\delta}$     $(\overline{I}\overline{a})_{\overline{\infty}\lvert } =\frac{1}{\delta^2}$