For a fully discrete whole life insurance of 100,000 on (45), you are given:

(i) Mortality follows the Standard Ultimate Life Table

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

(iii) Commission expenses are [math]60 \%[/math] of the first year's gross premium and [math]2 \%[/math] of renewal gross premiums

(iv) Administrative expenses are 500 in the first year and 50 in each renewal year

(v) All expenses are payable at the start of the year

(vi) The gross premium, calculated using the equivalence principle, is 977.60

Calculate [math]{ }_{5} V^{e}[/math], the expense policy value at the end of year 5 for this insurance.

• -1070
• -1020
• -970
• -920
• -870

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

Answer: C

Use superscript [math]g[/math] for gross premiums and gross premium policy values.

Use superscript [math]n[/math] (representing "net") for net premiums and net premium policy values.

Use superscript [math]e[/math] for expense premiums and expense policy values.

[[math]] \begin{aligned} P^{g} & =977.60 \text { (given) } \\ P^{e} & =\frac{0.58 P^{g}+450+\left(0.02 P^{g}+50\right) \ddot{a}_{45}}{\ddot{a}_{45}} \\ & =\frac{0.58(977.60)+450+[0.02(977.60)+50] 17.8162}{17.8162}=126.64 \end{aligned} [[/math]]

Alternatively,

[math]P^{n}=\frac{100,000 A_{45}}{\ddot{a}_{45}}=850.97 \quad P^{e}=P^{g}-P^{n}=126.63[/math]

[math]{ }_{5} V^{e}=\left(0.02 P^{g}+50\right) \ddot{a}_{50}-P^{e} \ddot{a}_{50}=[0.02(977.60)+50](17.0245)-126.64(17.0245)=-972[/math]

Alternatively,

[[math]] \begin{aligned} { }_{5} V^{n} & =100,000 A_{50}-P^{n} \ddot{a}_{50} \\ & =100,000(0.18931)-850.97(17.0245)=4443.66 \\ { }_{5} V^{g} & =100,000 A_{50}+\left(50+0.02 P^{g}-P^{g}\right) \ddot{a}_{50} \\ & =100,000(0.18931)+[50+0.02(977.60)-977.60](17.0245)=3471.93 \\ { }_{5} V^{e} & ={ }_{5} V^{g}-{ }_{5} V^{n}=-972 \end{aligned} [[/math]]

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