You are given the following extract of ultimate mortality rates from a two-year select and ultimate mortality table:

[math]x[/math]	[math]q_{x}[/math]
50	0.045
51	0.050
52	0.055
53	0.060

The select mortality rates satisfy the following: - [math]q_{[x]}=0.7 q_{x}[/math] - [math]q_{[x]+1}=0.8 q_{x+1}[/math]

You are also given that [math]i=0.04[/math].

Calculate [math]A_{[50]: 3]}^{1}[/math].

• 0.08
• 0.09
• 0.10
• 0.11
• 0.12

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

Answer: D

[math]A_{[50]: 3]}^{1}=v q_{[50]}+v^{2} p_{[50]} q_{[50]+1}+v^{3} p_{[50]} p_{[50]+1} q_{52}[/math]

where: [math]v=\frac{1}{1.04}[/math]

[math]q_{[50]}=0.7(0.045)=0.0315[/math]

[math]p_{[50]}=1-q_{[50]}=0.9685[/math]

[math]q_{[50]+1}=0.8(0.050)=0.040[/math]

[math]p_{[50]+1}=1-q_{[50]+1}=0.960[/math]

[math]q_{52}=0.055[/math]

So: [math]A_{[50]: 3]}^{1}=0.1116[/math]

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