For a 40 -year endowment insurance of 10,000 issued to (25), you are given:

(i) [math]\quad i=0.04[/math]

(ii) [math]\quad p_{25}=0.995[/math]

(iii) [math]\quad \ddot{a}_{25: \overline{20}}=11.087[/math]

(iv) [math]\quad \ddot{a}_{25: 40}=16.645[/math]

(v) The annual level net premium is 216

(vi) A modified net premium reserving method is used for this policy, where the valuation premiums are: - A first year premium equal to the first year net cost of insurance, - Level premiums of [math]\beta[/math] for years 2 through 20, and - Level premiums of 216 thereafter.

Calculate [math]\beta[/math].

• 140
• 170
• 200
• 230
• 260

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

Answer: D

We have Present Value of Modified Premiums [math]=[/math] Present Value of level net premiums

[math]v q_{x}+\beta\left(\ddot{a}_{25: \overline{20}}-1\right)+P \cdot{ }_{20} E_{25} \cdot \ddot{a}_{45: \overline{20}}=P \ddot{a}_{25: 40}[/math]

[math]\Rightarrow \beta=\frac{P\left(\ddot{a}_{25: \overline{40}}\right)-P \cdot{ }_{20} E_{25} \cdot \ddot{a}_{45: \overline{20}}-v q_{x}}{\ddot{a}_{25: 20 \mid}-1}=\frac{P \ddot{a}_{25: \overline{20}}-v q_{x}}{\ddot{a}_{25: 20 \mid}-1}[/math]

We are given that [math]P=0.0216[/math]

[math]\Rightarrow \beta=\frac{0.0216(11.087)-(1.04)^{-1}(0.005)}{11.087-1}=0.023265[/math]

For insurance of [math]10,000, \beta=233[/math].

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