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.

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

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