For a fully continuous 20 -year term insurance policy of 100,000 on (50), you are given:

(i) Gross premiums, calculated using the equivalence principle, are payable at an annual rate of 4500

(ii) Expenses at an annual rate of [math]R[/math] are payable continuously throughout the life of the policy

(iii) [math]\quad \mu_{50+t}=0.04[/math], for [math]t\gt0[/math]

(iv) [math]\delta=0.08[/math]

Calculate [math]R[/math].

• 400
• 500
• 600
• 700
• 800

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

Answer: B

By the equivalence principle,

[[math]] 4500 \bar{a}_{x: \overline{20}}=100,000 \bar{A}_{x: \overline{20}}^{1}+R \bar{a}_{x: \overline{20}} [[/math]]

where

[[math]] \begin{aligned} & \bar{A}_{x: 20 \mid}^{1}=\frac{\mu}{\mu+\delta}\left(1-e^{-20(\mu+\delta)}\right)=\frac{0.04}{0.12}\left(1-e^{-20(0.12)}\right)=0.3031 \\ & \bar{a}_{x: 20 \mid}=\frac{1-e^{-20(\mu+\delta)}}{\mu+\delta}=\frac{1-e^{-20(0.12)}}{0.12}=7.5774 \end{aligned} [[/math]]

Solving for [math]R[/math], we have

[[math]] R=4500-100,000\left(\frac{0.3031}{7.5774}\right)=500 [[/math]]

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