For a special fully discrete whole life insurance, you are given:

(i) The death benefit is [math]1000(1.03)^{k}[/math] for death in policy year [math]k[/math], for [math]k=1,2,3 \ldots[/math]

(ii) [math]\quad q_{x}=0.05[/math]

(iii) [math]\quad i=0.06[/math]

(iv) [math]\quad \ddot{a}_{x+1}=7.00[/math]

(v) The annual net premium for this insurance at issue age [math]x[/math] is 110

Calculate the annual net premium for this insurance at issue age [math]x+1[/math].

• 110
• 112
• 116
• 120
• 122

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

Answer: C

Let [math]P[/math] be the annual net premium at [math]x+1[/math]. Also, let [math]A_{y}^{*}[/math] be the expected present value for the special insurance described in the problem issued to [math](y)[/math].

[math]P \ddot{a}_{x+1}=1000 \sum_{k=0}^{\infty}(1.03)^{k+1} v_{k \mid}^{k+1} q_{x+1}=1000 A_{x+1}^{*}[/math]

We are given

[math]110 \ddot{a}_{x}=1000 \sum_{k=0}^{\infty}(1.03)^{k+1} v^{k+1}{ }_{k} q_{x}=1000 A_{x}^{*}[/math]

Which implies that

[math]110\left(1+v p_{x} \ddot{a}_{x+1}\right)=1000\left(1.03 v q_{x}+1.03 v p_{x} A_{x+1}^{*}\right)[/math]

Solving for [math]A_{x+1}^{*}[/math], we get

[[math]] A_{x+1}^{*}=\frac{\frac{110}{1000}[1+v(0.95)(7)]-1.03 v(0.05)}{1.03 v(0.95)}=0.8141032 [[/math]]

Thus, we have

[math]P=\frac{1000(0.8141032)}{7}=116.3005[/math]

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