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]