For a 25 -year pure endowment of 1 on [math](x)[/math], you are given:

(i) [math]\quad Z[/math] is the present value random variable at issue of the benefit payment

(ii) [math]\operatorname{Var}(Z)=0.10 E[Z][/math]

(iii) [math]{ }_{25} p_{x}=0.57[/math]

Calculate the annual effective interest rate.

• 5.8%
• 6.0%
• 6.2%
• 6.4%
• 6.6%

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

Answer: B

[math]\operatorname{Var}(Z)=0.10 E[Z] \Rightarrow v^{50}{ }_{25} p_{x}\left(1-{ }_{25} p_{x}\right)=0.10 \cdot v^{25}{ }_{25} p_{x}[/math]

[math]\Rightarrow \frac{(1-0.57)}{(1+i)^{50}}=0.10 \times \frac{1}{(1+i)^{25}}[/math]

[math]\Rightarrow(1+i)^{25}=\frac{0.43}{0.10}=4.3 \Rightarrow i=0.06[/math]

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