For fully discrete 30 -payment whole life insurance policies on (x), you are given:

(i) The following expenses payable at the beginning of the year:

	1st Year	Years 2 – 15	Years 16 – 30	Years 31 and after
Per policy	60	30	30	30
Percent of premium	[math]80 \%[/math]	[math]20 \%[/math]	[math]10 \%[/math]	[math]0 \%[/math]

(ii) [math]\quad \ddot{a}_{x}=15.3926[/math]

(iii) [math]\quad \ddot{a}_{x: 15 \mid}=10.1329[/math]

(iv) [math]\quad \ddot{a}_{x: 30 \mid}=14.0145[/math]

(v) Annual gross premiums are calculated using the equivalence principle

(vi) The annual gross premium is expressed as [math]k F+h[/math], where [math]F[/math] is the death benefit and [math]k[/math] and [math]h[/math] are constants for all [math]F[/math]

Calculate [math]h[/math].

• 30.3
• 35.1
• 39.9
• 44.7
• 49.5

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

Answer: D

[[math]] \begin{aligned} & G \ddot{a}_{x: \overline{30}}=\mathrm{APV}[\text { gross premium }]=\mathrm{APV}[\text { Benefits }+ \text { expenses }] \\ & \quad=F A_{x}+\left(30+30 \ddot{a}_{x}\right)+G\left(0.6+0.10 \ddot{a}_{x: 30}+0.10 \ddot{a}_{x: 15}\right) \\ & G=\frac{F A_{x}+30+30 \ddot{a}_{x}}{\ddot{a}_{x: 30 \mid}-0.6-0.1 \ddot{a}_{x: 30}-0.1 \ddot{a}_{x: 15}} \\ & =\frac{F A_{x}+30+30(15.3926)}{14.0145-0.6-0.1(14.0145)-0.1(10.1329)} \\ & =\frac{F A_{x}+491.78}{10.9998} \\ & =\frac{F A_{x}}{10.9998}+\frac{491.78}{10.9998}=\frac{F A_{x}}{10.9998}+44.71 \\ & \Rightarrow h=44.71 \end{aligned} [[/math]]

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