An insurance company issues policies covering damage to automobiles. The amount of damage is modeled by a uniform distribution on [0, b]

The policy payout is subject to a deductible of b/10.

A policyholder experiences automobile damage. Calculate the ratio of the standard deviation of the policy payout to the standard deviation of the amount of the damage.

• 0.8100
• 0.9000
• 0.9477
• 0.9487
• 0.9735

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

Solution: E

Without the deductible, the standard deviation is, from the uniform distribution, [math]b/\sqrt{12} = 0.28868b.[/math] Let [math]Y[/math] be the random variable representing the payout with the deductible.

[[math]] \begin{align*} \operatorname{E}(Y) &= \int_{0.1b}^b (y-0.1b) \frac{1}{b} dy = \frac{y^2}{2b} - 0.1y \Big |_{0.1b}^b = 0.5b - 0.1b - 0.005b + 0.01b = 0.405b \\ \operatorname{E}(Y^2) &= \int_{0.1b}^b (y-0.1b)^2 \frac{1}{b} dy = \frac{y^3}{3b} -0.1y^2 + 0.1by \Big |_{0.1b}^b = 0.5b - 0.1b - 0.005b + 0.01b = 0.405b \\ &=\frac{b^3}{3} - 0.1b^2 + 0.01b^2 -0.001b^2/3 + 0.001b^2 - 0.001b^2 = 0.243b^2 \\ \operatorname{Var}(Y) &= 0.243b^2 - (0.405b)^2 = 0.078975b^2 \\ \operatorname{SD}(Y) &= 0.28102b. \end{align*} [[/math]]

The ratio is 0.28102/0.28868 = 0.97347.

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