For a special fully discrete whole life insurance of 1,000 on (45), you are given:

(i) The net premiums for year [math]k[/math] are:

[[math]] \left\{\begin{array}{cc} P, & k=1,2, \ldots, 20 \\ P+W, & k=21,22, \ldots \end{array}\right. [[/math]]

(ii) Mortality follows the Standard Ultimate Life Table

(iii) [math]\quad i=0.05[/math]

(iv) [math]{ }_{20} V[/math], the net premium policy value at the end of the [math]20^{\text {th }}[/math] year, is 0

Calculate [math]W[/math].

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

Answer: D

[math]{ }_{20} V=0==\gt1000 A_{65}=(P+W) \times \ddot{a}_{65}[/math]

At issue, present value of benefits must equal present value of premium, so:

[math]1000 A_{45}=P \ddot{a}_{45}+W_{20} E_{45} \times \ddot{a}_{65}[/math]

[math]354.77=(P+W)(13.5498) \Rightarrow P+W=26.182674 \Rightarrow P=26.182674-W[/math]

[math]151.61=17.8162 P+W(0.35994)(13.5498)[/math]

[math]151.61=17.8162(26.182674-W)+W(0.35994)(13.5498)[/math]

[math]\Rightarrow W=24.33447[/math]

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