For whole life annuities-due of 15 per month on each of 200 lives age 62 with independent future lifetimes, you are given:

(i) [math]\quad i=0.06[/math]

(ii) [math]\quad A_{62}^{(12)}=0.4075[/math] and [math]{ }^{2} A_{62}^{(12)}=0.2105[/math]

(iii) [math]\quad \pi[/math] is the single premium to be paid by each of the 200 lives

(iv) [math]S[/math] is the present value random variable at time 0 of total payments made to the 200 lives

Using the normal approximation, calculate [math]\pi[/math] such that [math]\operatorname{Pr}(200 \pi \gt S)=0.90[/math].

• 1850
• 1860
• 1870
• 1880
• 1890

Answer: E

Let [math]X_{i}[/math] be the present value of a life annuity of [math]1 / 12[/math] per month on life [math]i[/math] for [math]i=1,2, \ldots, 200[/math].

Let [math]S=\sum_{i=1}^{200} X_{i}[/math] be the present value of all the annuity payments.

[math]E\left[X_{i}\right]=\ddot{a}_{62}^{(12)}=\frac{1-A_{62}^{(12)}}{d^{(12)}}=\frac{1-0.4075}{0.05813}=10.19267[/math]

[math]\operatorname{Var}\left(X_{i}\right)=\frac{{ }^{2} A_{62}^{(12)}-\left(A_{62}^{(12)}\right)^{2}}{\left(d^{(12)}\right)^{2}}=\frac{0.2105-(0.4075)^{2}}{(0.05813)^{2}}=13.15255[/math]

[math]E[S]=(200)(180)(10.19267)=366,936.12[/math]

[math]\operatorname{Var}(S)=(200)(180)^{2}(13.15255)=85,228,524[/math]

With the normal approximation, for [math]\operatorname{Pr}(S \leq M)=0.90[/math]

[math]M=E[S]+1.282 \sqrt{\operatorname{Var}(S)}=366,936.12+1.282 \sqrt{85,228,524}=378,771.45[/math]

So [math]\pi=\frac{378,771.45}{200}=1893.86[/math]

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