For a fully discrete whole life insurance of 10,000 on [math](x)[/math], you are given:

(i) Deaths are uniformly distributed over each year of age

(ii) The net premium is 647.46

(iii) The net premium policy value at the end of year 4 is 1405.08

(iv) [math]q_{x+4}=0.04561[/math]

(v) [math]\quad i=0.03[/math]

Calculate the net premium policy value at the end of 4.5 years.

• 1570
• 1680
• 1750
• 1830
• 1900

Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Answer: E

[[math]] \begin{aligned} { }_{4.5} V & =v_{0.5}^{0.5} p_{x+4.5} V+v_{0.5}^{0.5} q_{x+4.5} b, \text { where } b=10,000 \text { is the death benefit during year } 5 \\ { }_{0.5} q_{x+4.5} & =\frac{{ }_{0.5} q_{x+4}}{1-{ }_{0.5} q_{x+4}}=\frac{0.5(0.04561)}{1-0.5(0.04561)}=0.02334 \\ { }_{0.5} p_{x+4.5} & =0.97666 \\ { }_{5} V & =\frac{\left({ }_{4} V+P\right)(1.03)-q_{x+4} b}{p_{x+4}} \\ { }_{5} V & =\frac{(1,405.08+647.46)(1.03)-0.04561(10,000)}{0.95439}=1,737.25 \\ { }_{4.5} V & =(1.03)^{-0.5}(0.97666)(1,737.25)+(1.03)^{-0.5}(0.02334)(10,000) \\ & =1,671.81+229.98=1,902 \end{aligned} [[/math]]

[math]{ }_{4.5} V[/math] can also be calculated recursively:

[[math]] { }_{0.5} q_{x+4}=0.5(0.04561)=0.02281 [[/math]]

[math]{ }_{4.5} V=\frac{(1,405.08+647.16)(1.03)^{0.5}-0.02281(10,000) /(1.03)^{0.5}}{1-0.02281}=1,902[/math]

The interest adjustment on the death benefit term is needed because the death benefit will not be paid for another one-half year.

Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.