Question

Suppose the claim size of an auto collision insurance, X, is uniformly distributed on the interval $1,000 to $10,000. What is the percent reduction in the expected claim payment by the insurer if a policyholder has a $2,000 deductible

Answer 1

35.35%

Explanation:

If there were no deductibles, the expected claim payment would be:

`E(X) = (10,000 +1,000)/(2) \\E(X) =\$5,500`

If the collision insurance claim is under $2,000, then the insurer would not pay anything, but if X > $2,000, then the insurer would pay X - $2,000. The new expected value is:

`E_2(X)=(2,000-1,000)/(10,000-1,000) *0+(10,000-2,000)/(10,000-1,000)*((2,000-2,000)+(10,000-2,000))/(2) \\E_2(X)=(8)/(9)*(0+8,000)/(2)\\ E_2(X)=\$3,555.56`

The percentage reduction on the claim payment is:

`P=(1-(E_2(X))/(E(X)))*100 \\P=(1-(3,555.56)/(5,500))*100\\P=35.35\%`

There was a 35.35% reduction.