For a fully discrete whole life insurance of 1000 on [math](x)[/math], you are given:

(i) The following expenses are incurred at the beginning of each year:

	Year 1	Years 2+
Percent of premium	[math]75 \%[/math]	[math]10 \%[/math]
Maintenance expenses	10	2

(ii) An additional expense of 20 is paid when the death benefit is paid

(iii) The gross premium is determined using the equivalence principle

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

(v) [math]\quad \ddot{a}_{x}=12.0[/math]

(vi) [math]{ }^{2} A_{x}=0.14[/math]

Calculate the variance of the loss at issue random variable.

• 14,600
• 33,100
• 51,700
• 70,300
• 88,900

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

Answer: E

1,020 in the solution is the 1,000 death benefit plus the 20 death benefit claim expense.

[[math]] A_{x}=1-d \ddot{a}_{x}=1-d(12.0)=0.320755 [[/math]]

[math]G \ddot{a}_{x}=1,020 A_{x}+0.65 G+0.10 G \ddot{a}_{x}+8+2 \ddot{a}_{x}[/math]

[math]G=\frac{1,020 A_{x}+8+2 \ddot{a}_{x}}{\ddot{a}_{x}-0.65-0.10 \ddot{a}_{x}}=\frac{1,020(0.320755)+8+2(12.0)}{12.0-0.65-0.10(12.0)}=35.38622[/math]

Let [math]Z=v^{K_{x}+1}[/math] denote the present value random variable for a whole life insurance of 1 on [math](x)[/math].

Let [math]Y=\ddot{a}_{\overline{K_{x}+1}}[/math] denote the present value random variable for a life annuity-due of 1 on [math](x)[/math].

[[math]] \begin{aligned} L & =1,020 Z+0.65 G+0.10 G Y+8+2 Y-G Y \\ & =1,020 Z+(2-0.9 G) Y+0.65 G+8 \\ & =1,020 v^{K_{x}+1}+(2-0.9 G) \frac{1-v^{K_{x}+1}}{d}+0.65 G+8 \\ & =\left(1,020+\frac{0.9 G-2}{d}\right) v^{K_{x}+1}+\frac{2-0.9 G}{d}+0.65 G+8 \end{aligned} [[/math]]

[math]\operatorname{Var}(L)=\left[{ }^{2} A_{x}-\left(A_{x}\right)^{2}\right]\left(1,020+\frac{0.9 G-2}{d}\right)^{2}[/math]

[[math]] \begin{aligned} & =\left(0.14-0.320755^{2}\right)\left(1,020+\frac{0.9(35.38622)-2}{d}\right)^{2} \\ & =0.037116(2,394,161) \\ & =88,861 \end{aligned} [[/math]]

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