For a fully discrete whole life insurance of 1 on (50), you are given:

(i) Expenses of 0.20 at the start of the first year and 0.01 at the start of each renewal year are incurred

(ii) Mortality follows the Standard Ultimate Life Table

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

(iv) Gross premiums are determined using the equivalence principle.

Calculate the variance of [math]L_{0}[/math], the gross loss-at-issue random variable.

• 0.023
• 0.028
• 0.033
• 0.0038
• 0.043

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

Answer: C

Let [math]\pi[/math] be the annual premium, so that [math]\pi \ddot{a}_{50}=A_{50}+0.01 \ddot{a}_{50}+0.19[/math]

[math]\Rightarrow \pi=\frac{A_{50}+0.19}{\ddot{a}_{50}}+0.01=\frac{0.18931+0.19}{17.0245}+0.01=0.03228[/math]

Loss at issue: [math]L_{0}=v^{k+1}-(\pi-0.01) \ddot{a}_{\overline{k+1}}\left(1-v^{k+1}\right) / d+0.19[/math]

[[math]] \begin{aligned} \Rightarrow \operatorname{Var}\left[L_{0}\right] & =\left(1+\frac{(\pi-0.01)}{d}\right)^{2}\left({ }^{2} A_{50}-A_{50}^{2}\right) \\ & =(2.15467)\left(0.05108-0.18931^{2}\right) \\ & =(2.15467)(0.015242) \\ & =0.033 \end{aligned} [[/math]]

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