In a population initially consisting of [math]75 \%[/math] females and [math]25 \%[/math] males, you are given:

(i) For a female, the force of mortality is constant and equals [math]\mu[/math]

(ii) For a male, the force of mortality is constant and equals [math]1.5 \mu[/math]

(iii) At the end of 20 years, the population is expected to consist of [math]85 \%[/math] females and [math]15 \%[/math] males

Calculate the probability that a female survives one year.

• 0.89
• 0.92
• 0.94
• 0.96
• 0.99

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

Answer: C

The 20-year female survival probability [math]=e^{-20 \mu}[/math]

The 20-year male survival probability [math]=e^{-30 \mu}[/math]

We want 1 -year female survival [math]=e^{-\mu}[/math]

Suppose that there were [math]M[/math] males and [math]3 M[/math] females initially. After 20 years, there are expected to be [math]M e^{-30 \mu}[/math] and [math]3 M e^{-20 \mu}[/math] survivors, respectively. At that time we have:

[[math]]\frac{3 M e^{-20 \mu}}{M e^{-30 \mu}}=\frac{85}{15} \Rightarrow e^{10 \mu}=\frac{85}{45}=\frac{17}{9} \Rightarrow e^{-\mu}=\left(\frac{9}{17}\right)^{1 / 10}=0.938[[/math]]

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

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