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

Answer:

35.35%

Step-by-step explanation:

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

$E(X) = \frac{10,000 +1,000}{2} \\E(X) =\ 5,500$

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:

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

The percentage reduction on the claim payment is:

$P=(1-\frac{E_2(X)}{E(X)})*100 \\P=(1-\frac{3,555.56}{5,500})*100\\P=35.35\%$

There was a 35.35% reduction.