For a fully discrete whole life insurance of 10,000 on (45), you are given:

(i) Commissions are [math]80 \%[/math] of the first year premium and [math]10 \%[/math] of subsequent premiums. There are no other expenses

(ii) Mortality follows the Standard Ultimate Life Table

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

(iv) [math]{ }_{0} L[/math] denotes the loss at issue random variable

(v) If [math]T_{45}=10.5[/math], then [math]{ }_{0} L=4953[/math]

Calculate [math]\mathrm{E}\left[{ }_{0} L\right][/math].

• -580
• -520
• -460
• -400
• -340

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

Answer: D

If [math]T_{45}=10.5[/math], then [math]K_{45}=10[/math] and [math]K_{45}+1=11[/math].

[math]{ }_{0} L=10,000 v^{K_{45}+1}-G(1-0.10) \ddot{a}_{K_{45}+1}+G(0.80-0.10)=10,000 v^{11}-0.9 G \ddot{a}_{\overline{11}}+0.7 G[/math]

[math]4953=10,000(0.58468)-0.9 G(8.72173)+0.7 G[/math]

[math]G=(5846.8-4953) /(7.14956)=125.01[/math]

[math]E\left({ }_{0} L\right)=10,000 A_{45}-(1-0.1) G \ddot{a}_{45}+(0.8-0.1) G[/math] [math]=(10,000)(0.15161)-(0.9)(125.01)(17.8162)+(0.7)(125.01)[/math]

[math]E\left({ }_{0} L\right)=-400.87[/math]

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