You are given:

i)[math] q_{80}=0.04[/math]

ii)[math] q_{81}=0.06[/math]

iii)[math]q_{82}=0.08[/math]

iv) Deaths between ages 80 and 81 are uniformly distributed

v) Deaths between ages 81 and 82 are subject to a constant force of mortality

Calculate the probability that a person aged 80.6 will die between ages 81.1 and 81.6.

• 0.0294
• 0.0296
• 0.0298
• 0.0300
• 0.0302

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

Answer: C

Let [math]l_{80}=1000 \Rightarrow l_{81}=960 \Rightarrow l_{82}=902.4[/math]

[[math]]\textrm{Answer} \, =\frac{l_{81.1}-l_{81.6}}{l_{80.6}}=\frac{(960)^{0.9}(902.4)^{0.1}-(960)^{0.4}(902.4)^{0.6}}{(0.4)(1000)+(0.6)(960)}=0.02978[[/math]]

The value for [math]l_{80}[/math] is arbitrary. Any other starting value gives the same result.

Another form for the survivors between 81 and 82 would be

[math]\mu=-\ln (902 / 960)=0.0095833[/math];

[math]l_{81.1}=960 \times e^{-0.1 \times 0.0095833}=959.08 ;[/math]

[math]l_{81.6}=960 \times e^{-0.6 \times 0.0095833}=954.496[/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.