The graph of the density function for losses is:

Calculate the loss elimination ratio for an ordinary deductible of 20.

• 0.20
• 0.24
• 0.28
• 0.32
• 0.36

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

Key: E

[[math]] f(x) = \begin{cases} =0.01, \quad 0 \leq x \leq 80 \\ =0.01-0.00025(x-80)=0.03-0.00025 x, \quad 80 \lt x \leq 120 \\ \end{cases} [[/math]]

[[math]]\begin{aligned} \operatorname{E}(x) &=\int_{0}^{80} 0.01 x d x+\int_{80}^{120}\left(0.03 x-0.00025 x^{2}\right) d x \\ & =\left.\frac{0.01 x^{2}}{2}\right|_{0} ^{80}+\left.\frac{0.03 x^{2}}{2}\right|_{80} ^{120}-\left.\frac{0.00025 x^{3}}{3}\right|_{80} ^{120} \\ & =32+120-101.33=50.66667 \\ \end{aligned} [[/math]]

[[math]] \begin{aligned} \operatorname{E}(X-20)_{+} &=\operatorname{E}(X)-\int_{0}^{20} x f(x) d x-20\left[1-\int_{0}^{20} f(x) d x\right] \\ &=50.6667-\left.\frac{0.01 x^{2}}{2}\right|_{0} ^{20}-20\left(1-\left.0.01 x\right|_{0} ^{20}\right) \\ & =50.6667-2-20(0.8) \\ &=32.6667 \end{aligned}[[/math]]

Loss Elimination Ratio

[[math]]=1-\frac{32.6667}{50.6667}=0.3553[[/math]]

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