For a semi-continuous 20 -year endowment insurance of 100,000 on (45), you are given:

(i) Net premiums of 253 are payable monthly

(ii) Mortality follows the Standard Ultimate Life Table

(iii) Deaths are uniformly distributed over each year of age

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

Calculate [math]{ }_{10} V[/math], the net premium policy value at the end of year 10 for this insurance.

• 38,100
• 38,300
• 38,500
• 38,700
• 38,900

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

Answer: A

Let [math]P=0.00253[/math] be the monthly net premium per 1 of insurance.

[[math]] \begin{aligned} { }_{10} V & =100,000\left[\frac{i}{\delta} A_{55: 10 \mid}^{1}+A_{55: 10 \mid}-12 P \ddot{a}_{55: 10 \mid}^{(12)}\right] \\ & =100,000[1.02480(0.02471)+0.59342-(12)(0.00253)(7.8311)] \\ & \approx 38,100 \end{aligned} [[/math]]

Where

[[math]] \begin{aligned} A_{55: \overline{10}}^{1} & =A_{55: \overline{10}}-{ }_{10} E_{55}=0.61813-0.59342=0.02471 \\ A_{55: 10 \mid} & ={ }_{10} E_{55}=0.59342 \\ \ddot{a}_{55: \overline{10}} & =8.0192 \\ \ddot{a}_{55: 10 \mid}^{(12)} & =\alpha(12) \ddot{a}_{55: \overline{10}}-\beta(12)\left[1-{ }_{10} E_{55}\right] \\ & =1.00020(8.0192)-0.46651(1-0.59342)=7.8311 \end{aligned} [[/math]]

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