exercise:0131ae68ce

Jan 20'24

For a fully discrete whole life insurance of [math]B[/math] on [math](x)[/math], you are given:

(i) Expenses, incurred at the beginning of each year, equal 30 in the first year and 5 in subsequent years

(ii) The net premium policy value at the end of year 10 is 2290

(iii) Gross premiums are calculated using the equivalence principle

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

(v) [math]\quad \ddot{a}_{x}=14.8[/math]

(vi) [math]\quad \ddot{a}_{x+10}=11.4[/math]

Calculate [math]{ }_{10} V^{g}[/math], the gross premium policy value at the end of year 10.

Jan 20'24

Answer: E

[math]V_{10}=2,290=B\left(1-\frac{\ddot{a}_{x+10}}{\ddot{a}_{x}}\right)=B\left(1-\frac{11.4}{14.8}\right) \Rightarrow B=9,968.24[/math]

[math]G \ddot{a}_{x}=25+5 \ddot{a}_{x}+B \times A_{x}[/math]

[math]A_{x}=1-d \ddot{a}_{x}=1-\left(\frac{0.04}{1.04} \times 14.8\right)=0.430769231[/math]

[math]G \times 14.8=25+5 \times 14.8+9,968.24 \times 0.430769231[/math]

[math]\Rightarrow G=296.82[/math]

[math]{ }_{10} V^{g}=9,968.24 A_{x+10}+5 \ddot{a}_{x+10}-296.82 \ddot{a}_{x+10}[/math]

[math]A_{x+10}=1-d \ddot{a}_{x+10}=1-\left(\frac{0.04}{1.04} \times 11.4\right)=0.561538462[/math]

[math]{ }_{10} V^{g}=9,968.24 \times 0.561538462+5 \times 11.4-296.82 \times 11.4[/math]

[math]\Rightarrow{ }_{10} V^{g}=2,270.80[/math]

Alternatively, the expense net premium is based on the extra expenses in year 1, so [math]P^{e}=(30-5) / 14.8=1.68919[/math]

[math]{ }_{10} V^{e}=0-1.68919(11.4)=-19.26[/math]

[math]{ }_{10} V^{g}={ }_{10} V^{n}+{ }_{10} V^{e}=2290-19.26=2270.74[/math]