The total claim amount for a health insurance policy follows a distribution with density function

[[math]] f(x) = \begin{cases} \frac{1}{1000} e^{-x/1000}, \, x \gt 0 \\ 0, \, \textrm{Otherwise.} \end{cases} [[/math]]

The premium for the policy is set at the expected total claim amount plus 100. If 100 policies are sold, calculate the approximate probability that the insurance company will have claims exceeding the premiums collected.

• 0.001
• 0.159
• 0.333
• 0.407
• 0.460

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

Solution: B

A single policy has an exponential distribution with mean and standard deviation 1000. The premium is then 1000 + 100 = 1100. For 100 policies, the total claims have mean 100(1000) = 100,000 and standard deviation 10(1000) = 10,000. Total premiums are 100(1100) = 110,000. The probability of exceeding this number is the probability that a standard normal variable exceeds (110,000 – 100,000)/10,000 = 1. From the tables this probability is 1 – 0.8413 = 0.1587.

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