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}$

 

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