An auto insurance policy has a deductible of 1 and a maximum claim payment of 5. Auto loss amounts follow an exponential distribution with mean 2.

Calculate the expected claim payment made for an auto loss.

• [math]0.5e^{−2} − 0.5e^{−12}[/math]
• [math]2e^{-1/2}-7e^{-3}[/math]
• [math]2e^{-1/2}-2e^{-3}[/math]
• [math]2e^{-1/2}[/math]
• [math]3e^{-1/2}-2e^{-3}[/math]

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

Solution: C

[[math]] \begin{align*} \int_1^6(x-1)0.5e^{-0.5x} dx + \int_{6}^{\infty} 5(0.5)e^{-0.5x} dx &= -(x-1)e^{-0.5x} \Big |_1^6 + \int_1^6 e^{-0.5x} dx - 5e^{-0.5x} \Big |_0^{\infty} \\ &= -5e^{-3} + 0 -2e^{-0.5x} \Big |_1^6 + 5e^{-3}\\ &= -2e^{-3} + 2e^{-1/2}. \end{align*} [[/math]]

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