An automobile insurance company issues a one-year policy with a deductible of 500. The probability is 0.8 that the insured automobile has no accident and 0.0 that the automobile has more than one accident. If there is an accident, the loss before application of the deductible is exponentially distributed with mean 3000.

Calculate the 95th percentile of the insurance company payout on this policy.

• 3466
• 3659
• 4159
• 8487
• 8987

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

Solution: B

The 95^th percentile is in the range when an accident occurs. It is the 75^th percentile of the payout, given that an accident occurs, because (0.95 − 0.80)/(1 − 0.80) = 0.75. Letting x be the 75^th percentile of the given exponential distribution, [math]F (x) =1-e^{-x/3000} = 0.75 [/math], so x = 4159. Subtracting the deductible of 500 gives 3659 as the (unconditional) 95 percentile of the insurance company payout.

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