For a fully discrete whole life insurance of 100 on [math](x)[/math], you are given:

(i) The first year expense is [math]10 \%[/math] of the gross annual premium

(ii) Expenses in subsequent years are [math]5 \%[/math] of the gross annual premium

(iii) The gross premium calculated using the equivalence principle is 2.338

(iv) [math]i=0.04[/math]

(v) [math]\quad \ddot{a}_{x}=16.50[/math]

(vi) [math]{ }^{2} A_{x}=0.17[/math]

Calculate the variance of the loss at issue random variable.

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

Answer: A

The loss at issue is given by:

[math]L_{0}=100 v^{K+1}+0.05 G+0.05 G \ddot{a}_{\overline{K+1}}-G \ddot{a}_{\overline{K+1}}[/math]

[math]=100 v^{K+1}+0.05 G-0.95 G\left(\frac{1-v^{K+1}}{d}\right)[/math]

[math]=\left(100+\frac{0.95 G}{d}\right) v^{K+1}+0.05 G-0.95 \frac{G}{d}[/math]

Thus, the variance is

[[math]] \begin{aligned} \operatorname{Var}\left(L_{0}\right)=[100 & \left.+\frac{0.95(2.338)}{0.04 / 1.04}\right]^{2}\left({ }^{2} A_{x}-\left(A_{x}\right)^{2}\right) \\ & =\left[100+\frac{0.95(2.338)}{0.04 / 1.04}\right]^{2}\left(0.17-\left(1-\frac{0.04}{1.04}(16.50)\right)^{2}\right) \\ & =908.1414 \end{aligned} [[/math]]

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