For a fully continuous whole life insurance of 1 on [math](x)[/math], you are given:

(i) [math]\quad L[/math] is the present value of the loss at issue random variable if the premium rate is determined by the equivalence principle

(ii) [math]\quad L^{*}[/math] is the present value of the loss at issue random variable if the premium rate is 0.06

(iii) [math]\delta=0.07[/math]

(iv) [math]\quad \bar{A}_{x}=0.30[/math]

(v) [math]\quad \operatorname{Var}(L)=0.18[/math]

Calculate [math]\operatorname{Var}\left(L^{*}\right)[/math].

• 0.18
• 0.21
• 0.24
• 0.27
• 0.30

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

Answer: E

In general, the loss at issue random variable can be expressed as:

[[math]] L=\bar{Z}_{x}-P \times \bar{Y}_{x}=\bar{Z}_{x}-P \times\left(\frac{1-\bar{Z}_{x}}{\delta}\right)=\bar{Z}_{x} \times\left(1+\frac{P}{\delta}\right)-\frac{P}{\delta} [[/math]]

Using actuarial equivalence to determine the premium rate:

[[math]] P=\frac{\bar{A}_{x}}{\bar{a}_{x}}=\frac{0.3}{(1-0.3) / 0.07}=0.03 [[/math]]

[math]\operatorname{Var}(L)=\left(1+\frac{P}{\delta}\right)^{2} \times \operatorname{Var}\left(\bar{Z}_{x}\right)=\left(1+\frac{0.03}{0.07}\right)^{2} \times \operatorname{Var}\left(\bar{Z}_{x}\right)=0.18[/math]

[math]\operatorname{Var}\left(\bar{Z}_{x}\right)=\frac{0.18}{\left(1+\frac{0.03}{0.07}\right)^{2}}=0.088[/math]

[math]\operatorname{Var}\left(L^{*}\right)=\left(1+\frac{P^{*}}{\delta}\right)^{2} \times \operatorname{Var}\left(\bar{Z}_{x}\right)=\left(1+\frac{0.06}{0.07}\right)^{2}(0.088)=0.304[/math]

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