You are given that mortality follows Makeham's Law with the following parameters:

[[math]] \begin{array}{ll} \text { i) } & A=0.004 \\ \text { ii) } & B=0.00003 \\ \text { iii) } & c=1.1 \end{array} [[/math]]

Let [math]L_{15}[/math] be the random variable representing the number of lives alive at the end of 15 years if there are 10,000 lives age 50 at time 0 .

Calculate [math]\operatorname{Var}\left[L_{15}\right][/math].

• 1,317
• 1,328
• 1,339
• 1,350
• 1,361

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

Answer: E

The distribution is binomial with 10,000 trials.

[[math]] \begin{aligned} & \operatorname{Var}\left[L_{15}\right]=n p q=10,000\left({ }_{15} p_{50}\right)\left({ }_{15} q_{50}\right) \\ & { }_{15} p_{50}=e^{\left[-A(15)-\frac{B}{\ln C} c^{50}\left(c^{15}-1\right)\right]}=0.837445 \\ & { }_{15} q_{50}=1-{ }_{15} p_{50}=0.162555 \end{aligned} [[/math]]

[math]\operatorname{Var}\left[L_{15}\right]=10,000(0.837445)(0.162555)=1361.3[/math]

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

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

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