A risk has a loss amount that has a Poisson distribution with mean 3. An insurance policy covers the risk with an ordinary deductible of 2. An alternative insurance policy replaces the deductible with coinsurance [math]\alpha[/math], which is the proportion of the loss paid by the policy, so that the expected cost remains the same.

Calculate [math]\alpha [/math].

• 0.22
• 0.27
• 0.32
• 0.37
• 0.42

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

Key: E

[[math]] \begin{aligned} \operatorname{E}[ X \wedge 2) &= 1 f (1) + 2[1 − F (1)] = 1 f (1) + 2[1 − f (0) − f (1)] \\ &= 1(3e^{−3} ) + 2(1 − e^{−3} − 3e^{−3} ) = 2 − 5e^{−3} = 1.75 \end{aligned} [[/math]]

Cost per loss with deductible is

[[math]] \operatorname{E}( X ) − \operatorname{E}( X \wedge 2) = 3 −1.75 = 1.25 [[/math]]

Cost per loss with coinsurance is [math] \alpha \operatorname{E}( X ) = 3\alpha [/math]

Equating cost: [math]3\alpha = 1.25 \Rightarrow \alpha = 0.42 [/math]

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