Question

richard is purchasing car insurance. he has two options: a high deductible policy with a lower yearly premium or a low deductible policy with a higher yearly premium. the high deductible policy has a yearly premium of $1,397.00 and a deductible of $2,697.00 that must be paid in the case of an auto accident. the low deductible policy has a yearly premium of $2,794.00 and a deductible of $1,294.00 that must be paid in the case of an auto accident. both policies pay, on average, $10,571.00 in the case of an auto accident.
suppose the probability that richard will be involved in an auto accident in the given year is 0.07. given this probability, which policy should richard choose in order to be the most cost efficient?
a. the high deductible policy is the most cost efficient because the expected value of the high deductible policy is less than the expected value of the low deductible policy, therefore, richard should choose the high deductible policy.
b. the high deductible policy is the most cost efficient because the expected value of the high deductible policy is greater than the expected value of the low deductible policy, therefore, richard should choose the high deductible policy.
c. the low deductible policy is the most cost efficient because the expected value of the low deductible policy is greater than the expected value of the high deductible policy, therefore, richard should choose the low deductible policy.
d. the low deductible policy is the most cost efficient because the expected value of the low deductible policy is less than the expected value of the high deductible policy, therefore, richard should choose the low deductible policy.

Explanation:

Step1: Calculate expected value for high - deductible policy

The formula for expected value \(E(X)\) is \(E(X)=p\times(\text{cost if accident})+(1 - p)\times(\text{cost if no accident})\).
For the high - deductible policy:
The cost if accident is \(1397+2897=4294\) (premium + deductible).
The cost if no accident is \(1397\).
\(p = 0.07\), \(1-p=0.93\)
\(E_{high}=0.07\times4294 + 0.93\times1397\)
\(E_{high}=0.07\times4294+0.93\times1397=300.58+1299.21 = 1599.79\)

Step2: Calculate expected value for low - deductible policy

For the low - deductible policy:
The cost if accident is \(2794 + 1294=4088\) (premium+deductible).
The cost if no accident is \(2794\).
\(E_{low}=0.07\times4088+0.93\times2794\)
\(E_{low}=0.07\times4088 + 0.93\times2794=286.16+2608.42=2894.58\)

Answer:

A. The high deductible policy is the most cost efficient because the expected value of the high deductible policy is less than the expected value of the low deductible policy, therefore, Richard should choose the high deductible policy.