For a special fully discrete whole life insurance of 100,000 on (40), you are given:

(i) The annual net premium is [math]P[/math] for years 1 through [math]10,0.5 P[/math] for years 11 through 20, and 0 thereafter

(ii) Mortality follows the Standard Ultimate Life Table

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

Calculate [math]P[/math].

• 850
• 950
• 1050
• 1150
• 1250

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

Answer: D

[math]100,000 A_{40}=P\left[\ddot{a}_{40: \overline{10}}+0.5_{10} \ddot{a}_{40: \overline{10}}\right][/math]

[math]P=\frac{100,000 A_{40}}{\ddot{a}_{40: \overline{10}}+0.5_{10 \mid} \ddot{a}_{40: 10}}=\frac{100,000(0.12106)}{8.0863+0.5(4.9071)}=\frac{12,106}{10.53985}=1148.59[/math]

where

[math]{ }_{10} \ddot{a}_{40: 10}={ }_{10} E_{40}\left[\ddot{a}_{50: 10}\right]=0.60920[8.0550]=4.9071[/math]

There are several other ways to write the right-hand side of the first equation.

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