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.

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