A company provides a death benefit of 50,000 for each of its 1000 employees. There is a 1.4% chance that any one employee will die next year, independent of all other employees. The company establishes a fund such that the probability is at least 0.99 that the fund will cover next year’s death benefits.

Calculate, using the Central Limit Theorem, the smallest amount of money, rounded to the nearest 50 thousand, that the company must put into the fund.

• 750,000
• 850,000
• 1,050,000
• 1,150,000
• 1,400,000

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: D

Let X denote the number of deaths next year, and S denote life insurance payments next year. Then [math]S = 50000X [/math], where [math]X \sim \textrm{Bin}(1000,0.014) [/math]. Therefore,

[[math]] \operatorname{E}(S) = E(50, 000 X ) = 50,000(1000)(0.014) = 700,000 [[/math]]

[[math]] \operatorname{Var}(D) = \operatorname{Var}(50,000 X ) = 50,000^2 (1000)(0.014)(0.986) = 34,510, 000, 000 [[/math]]

[[math]] \operatorname{StdDev}(S) = 185,769. [[/math]]

The 99th percentile is

700,000+185,769(2.326)= 1,132,099, 

which rounds to 1,150,000.

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.