An insurance company sells an auto insurance policy that covers losses incurred by a policyholder, subject to a deductible of 100. Losses incurred follow an exponential distribution with mean 300.

Calculate the 95th percentile of losses that exceed the deductible.

• 600
• 700
• 800
• 900
• 1000

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

Solution: E

This is a conditional probability. The solution is

[[math]] \begin{align*} 0.95 = \operatorname{P}[ X \leq p | X \gt 100 ] &= \frac{P[100 \leq X \leq p ]}{\operatorname{P}[X \gt 100]} = \frac{F(p) - F(100)}{1-F(100)} \\ &= \frac{1 − e^{-p /300} − 1 + e^{−100/300}}{1-1 + e^{-100/300}} \\ &= {e ^{-100/300} - e^{-p/300}}{e^{-100/300}} \\ &= 1 - e^{-(p-100)/300} \\ 0.05 &= e^{-(p-100)/300 } \\ -2.9957 &= -(p-100)/300 \\ p &= 999 \end{align*} [[/math]]

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