变额年金


一般形式的连续变额现金流现值与终值

设在时刻t 时的付款率\(\rho_{t}\),利息力为\(\delta_{t}\)
现值：

\[\int_{a}^{b} \exp \left(-\int_{0}^{t} \delta_{s} d s\right) \rho_{t} d t \]

终值：

\[\int_{a}^{b} \exp \left(\int_{t}^{T} \delta_{s} d s\right) \rho_{t} d t \]

5-10：\(\int_{5}^{10}\left(1.2 t^{2}+2 t\right) \exp \left[-\int_{5}^{t}\left(0.0006 s^{2}+0.001 s\right) \mathrm{d} s\right] \mathrm{d} t=382.88\)
0-5：\(382.88 \exp \left[-\int_{0}^{5}(0.004 t+0.01) \mathrm{d} t\right]=346.44\)