For a select and ultimate mortality model with a one-year select period, you are given:

(i) [math]\quad p_{[x]}=(1+k) p_{x}[/math], for some constant [math]k[/math]

(ii) [math]\quad \ddot{a}_{x: n}=21.854[/math]

(iii) [math]\quad \ddot{a}_{[x]: n]}=22.167[/math]

Calculate [math]k[/math].

• 0.005
• 0.010
• 0.015
• 0.020
• 0.025

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

Answer: C

[math]\ddot{a}_{[x]: n]}=1+v p_{[x]} \ddot{a}_{x+1: n-1]}=1+(1+k)\left(v p_{x} \ddot{a}_{x+1: n-1}\right)=1+(1+k)\left(\ddot{a}_{x: n]}-1\right)[/math]

Therefore, we have

[math]k=\frac{\ddot{a}_{[x]: n]}-1}{\ddot{a}_{x: n]}-1}-1=\frac{21.167}{20.854}-1=0.015[/math]

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