Chapter 3 Varying Annuity

Increasing annuity immediate

\[(Ia)_{\overline{n}\mid}=\frac{\ddot{a}_{\overline{n}\mid}-nv^n}{i}\] \[(Is)_{\overline{n}\mid}=(1+i)^n(Ia)_{\overline{n}\mid}\]

Increasing annuity due

\[(I\ddot{a})_{\overline{n}\mid}=(1+i)(Ia)_{\overline{n}\mid}\]

\[(I\ddot{s})_{\overline{n}\mid}=(1+i)(Is)_{\overline{n}\mid}\]

Decreasing annuity immediate

\[(Da)_{\overline{n}\mid}=\frac{n-a_{\overline{n}\mid}}{i}\]

Compound increasing annuity immediate

\[(Ca)_{\overline{n}\mid i}=\frac{(C\ddot{a})_{\overline{n}\mid i}}{1+i}=\frac{\ddot{a}_{\overline{n}\mid j}}{1+i}\neq{a}_{\overline{n}\mid j},\] where \(j=(i-r)/(1+r).\)

Compound increasing annuity due

\[(C\ddot{a})_{\overline{n}\mid i}=\ddot{a}_{\overline{n}\mid j},\] where \(j=(i-r)/(1+r).\)

\(m\)-thly payable increasing annuity immediate

\[(Ia)^{(m)}_{\overline{n}\mid}=\frac{i}{i^{(m)}}(Ia)_{\overline{n}\mid}\]

\(m\)-thly payable increasing annuity due

\[(I\ddot{a})^{(m)}_{\overline{n}\mid}=\frac{d}{d^{(m)}}(I\ddot{a})_{\overline{n}\mid}\]

\(m\)-thly payable varying annuity

The above equations are also applied to decreasing annuity and compound increasing annuity. In general we have

\[V^{(m)}_{\overline{n}\mid}=\frac{i}{i^{(m)}}V_{\overline{n}\mid}\] \[\ddot{V}^{(m)}_{\overline{n}\mid}=\frac{d}{d^{(m)}}\ddot{V}_{\overline{n}\mid}\]

Continuous payable varying annuity

\[\bar{V}_{\overline{n}\mid}=\frac{i}{\delta}V_{\overline{n}\mid}=\frac{d}{\delta}\ddot{V}_{\overline{n}\mid}\]

Increasing perpetuity

\[(Ia)_{\overline{\infty}\mid}=\frac{1}{di}\]

Continuous payable increasing perpetuity

\[(I\bar{a})_{\overline{\infty}\mid}=\frac{1}{d\delta}\]

Present value of a general varying annuity with payment rate of \(\rho(t)\)

\[\int_0^{\infty}\rho(t)\exp\left(-\int_0^t\delta(s)ds\right)dt\]

Cumulative value of a general varying annuity with payment rate of \(\rho(t)\)

\[\int_0^{T}\rho(t)\exp\left(\int_t^T\delta(s)ds\right)dt\]

Continuously increasing annuity with payment rate of \(t\)

\[(\bar{I}\bar{a})_{\overline{n}\mid}=\int_0^nte^{-\delta t}dt=\frac{\bar{a}_{\overline{n}\mid}-nv^n}{\delta}\]

Continuously increasing perpetuity

\[(\bar{I}\bar{a})_{\overline{\infty}\mid}=\frac{1}{\delta^2}\]

Continuously decreasing annuity with payment rate of \(n-t\)

\[(\bar{D}\bar{a})_{\overline{n}\mid}=\int_0^n (n-t) e^{-\delta t}dt=\frac{n-\bar{a}_{\overline{n}\mid}}{\delta}\]

Key relations