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

(i) Premiums and benefits:

	First 20 years	After 20 years
Premium Rate	[math]3 P[/math]	[math]P[/math]
Benefit	[math]1,000,000[/math]	500,000

(ii) [math]\quad \mu_{x+t}=0.03, t \geq 0[/math]

(iii) [math]\delta=0.06[/math]

Calculate [math]P[/math] using the equivalence principle.

• 10,130
• 10,190
• 10,250
• 10,310
• 10,370

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

Answer: D

[math]\operatorname{EPV}([/math] Premiums [math])=\operatorname{EPV}([/math] Benefits [math])[/math]

[math]\mathrm{EPV}([/math] Premiums [math])=3 P \bar{a}_{x}-2 P_{20} E_{x} \bar{a}_{x+20}[/math]

[[math]] \begin{aligned} & =3 P(1 / \mu+\delta)-2 P\left(e^{-20(\mu+\delta)}\right)(1 / \mu+\delta) \\ & =3 P(1 / 0.09)-2 P e^{-1.8}-1 / 0.09 \\ & =29.66 P \end{aligned} [[/math]]

[math]\operatorname{EPV}([/math] Benefits [math])=1,000,000 \bar{A}_{x}-500,000{ }_{20} E_{x} \bar{A}_{x+20}[/math]

[[math]] \begin{aligned} & =1,000,000(\mu / \mu+\delta)-500,000 e^{-20(\mu+\delta)} \mu / \mu+\delta \\ & =1,000,000(0.03 / 0.09)-500,000 e^{-1.8} 0.03 / 0.09 \\ & =305,783.5 \end{aligned} [[/math]]

[math]29.66 P=305,783.5[/math]

[[math]] P=\frac{305,783.5}{29.66} [[/math]]

[[math]] P=10,309.62 [[/math]]

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