You are given:

[math]x[/math]	[math]l_{x}[/math]
60	99,999
61	88,888
62	77,777
63	66,666
64	55,555
65	44,444
66	33,333
67	22,222

[math]a={ }_{3.42 .5} q_{60}[/math] assuming a uniform distribution of deaths over each year of age [math]b={ }_{3.4 \mid 2.5} q_{60}[/math] assuming a constant force of mortality over each year of age Calculate 100,000(a-b).

• -24
• 9
• 42
• 73
• 106

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

Answer: E

Using UDD

[math]l_{63.4}=(0.6) 66,666+(0.4)(55,555)=62,221.6[/math]

[math]l_{65.9}=(0.1)(44,444)+(0.9)(33,333)=34,444.1[/math]

[math]{ }_{3.4 \mid 2.5} q_{60}=\frac{l_{63.4}-l_{65.9}}{l_{60}}=\frac{62,221.6-34,444.1}{99,999}=0.277778[/math]

Using constant force

[[math]] \begin{aligned} l_{63.4} & =l_{63}\left(\frac{l_{64}}{l_{63}}\right)^{0.4}=l_{63}^{0.6} l_{64}^{0.4} \\ & =\left(66,666^{0.6}\right)\left(55,555^{0.4}\right) \\ & =61,977.2 \\ l_{65.9} & =l_{65}^{0.1} l_{66}^{0.9}=\left(44,444^{0.1}\right)\left(33,333^{0.9}\right) \\ & =34,305.9 \\ 3.4 \mid 2.5 & q_{60}=\frac{61,977.2-34,305.9}{99.999} \\ & =0.276716 \end{aligned} [[/math]]

[math]100,000(a-b)=100,000(0.277778-0.276716)=106[/math]

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

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