For a special fully discrete 2 -year endowment insurance on [math](x)[/math], you are given:

(i) The death benefit for year [math]k[/math] is [math]25,000 k[/math] plus the net premium policy value at the end of year [math]k[/math], for [math]k=1,2[/math]. For year 2 , this net premium policy value is the net premium policy value just before the maturity benefit is paid

(ii) The maturity benefit is 50,000

(iii) [math]\quad p_{x}=p_{x+1}=0.85[/math]

(iv) [math]\quad i=0.05[/math]

(v) [math]\quad P[/math] is the level annual net premium

Calculate [math]P[/math].

• 27,650
• 27,960
• 28,200
• 28,540
• 28,730

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

Answer: D

[math]{ }_{1} V=\left({ }_{0} V+P\right)(1+i)-\left(25,000+{ }_{1} V-{ }_{1} V\right) q_{x}=P(1+i)-(25,000) q_{x}[/math]

[math]{ }_{2} V=\left({ }_{1} V+P\right)(1+i)-\left(50,000+{ }_{2} V-{ }_{2} V\right) q_{x+1}=50,000[/math]

[math]\left(\left(P(1+i)-25,000 q_{x}\right)+P\right)(1+i)-50,000 q_{x+1}=50,000[/math]

[math]((P(1.05)-25,000(0.15))+P)(1.05)-50,000(0.15)=50,000[/math]

Solving for [math]P[/math], we get

[math]P=\frac{61,437.50}{2.1525}=28,542.39[/math]

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