For fully discrete whole life insurance policies of 10,000 issued on 600 lives with independent future lifetimes, each age 62 , you are given:

(i) Mortality follows the Standard Ultimate Life Table

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

(iii) Expenses of [math]5 \%[/math] of the first year gross premium are incurred at issue

(iv) Expenses of 5 per policy are incurred at the beginning of each policy year

(v) The gross premium is [math]103 \%[/math] of the net premium.

(vi) [math]{ }_{0} L[/math] is the aggregate present value of future loss at issue random variable

Calculate [math]\operatorname{Pr}\left({ }_{0} L\lt40,000\right)[/math], using the normal approximation.

• 0.75
• 0.79
• 0.83
• 0.87
• 0.91

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

Answer: B

Net Premium [math]=10,000 A_{62} / \ddot{a}_{62}=10,000(0.31495) / 14.3861=218.93[/math]

[math]G=218.93(1.03)=225.50[/math]

Let [math]{ }_{0} L^{*}[/math] be the present value of future loss at issue for one policy.

[[math]] \begin{aligned} { }_{0} L^{*} & =10,000 v^{K+1}-(G-5) \ddot{a}_{\overline{K+1}}+0.05 G \\ & =10,000 v^{K+1}-(225.50-5) \frac{1-v^{K+1}}{d}+0.05(225.50) \\ & =(10,000+4630.50) v^{K+1}-4630.50+11.28 \\ & =14,630.50 v^{K+1}-4619.22 \end{aligned} [[/math]]

[math]E\left({ }_{0} L^{*}\right)=14,630.50 A_{62}-4619.22=14,630.50(0.31495)-4619.22=-11.34[/math]

[math]\operatorname{Var}\left({ }_{0} L^{*}\right)=(14,630.50)^{2}\left({ }^{2} A_{62}-A_{62}^{2}\right)=(14,630.50)^{2}\left(0.12506-0.31495^{2}\right)=5,536,763[/math]

Let [math]{ }_{0} L[/math] be the aggregate loss for 600 such policies.

[[math]] E\left({ }_{0} L\right)=600 E\left({ }_{0} L^{*}\right)=600(-11.34)=-6804 [[/math]]

[math]\operatorname{Var}\left({ }_{0} L\right)=600 \operatorname{Var}\left({ }_{0} L^{*}\right)=600(5,536,763)=3,322,057,800[/math]

[math]\operatorname{StdDev}\left({ }_{0} L\right)=3,322,057,800^{0.5}=57,637[/math]

[math]\operatorname{Pr}\left({ }_{0} L\lt40,000\right)=\Phi\left(\frac{40,000+6804}{57,637}\right)=\Phi(0.81)=0.7910[/math]

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