The amount of a claim that a car insurance company pays out follows an exponential distribution. By imposing a deductible of d, the insurance company reduces the expected claim payment by 10%.

Calculate the percentage reduction on the variance of the claim payment.

• 1%
• 5%
• 10%
• 20%
• 25%

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

Solution: A

Let [math]X[/math] denote the amount of a claim before application of the deductible. Let [math]Y[/math] denote the amount of a claim payment after application of the deductible. Let [math]λ[/math] be the mean of [math]X[/math], which because [math]X[/math] is exponential, implies that [math]λ^2 [/math] is the variance of [math]X[/math] and [math]\operatorname{E} ( X^2 ) = 2λ^2. [/math]

By the memoryless property of the exponential distribution, the conditional distribution of the portion of a claim above the deductible given that the claim exceeds the deductible is an exponential distribution with mean [math]λ[/math] . Given that [math]\operatorname{E} (Y ) = 0.9λ [/math] , this implies that the probability of a claim exceeding the deductible is 0.9 and thus [math]\operatorname{E}(Y^2) - 0.9(2\lambda^2) = 1.8\lambda^2. [/math] Then

[[math]]\operatorname{Var}(Y) = 1.8\lambda^2 -(0.9 \lambda)^2 = 0.99 \lambda ^2. [[/math]]

The percentage reduction is 1%.

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