Question

Let X be the damage incurred (in $) in a certain type of accident during a given year. Possible X values are 0, 1,000, 5,000, and 10,000, with probabilities 0.84, 0.09, 0.05, and 0.02, respectively. A particular company offers a $500 deductible policy. If the company wishes its expected profit to be $100, what premium amount should it charge (in dollars)

Answer 1

To achieve an expected profit of $100 on top of expected payouts, given the provided probabilities and damage costs with a $500 deductible, the insurance company should charge a premium of $560.

Step-by-step explanation:

The student is tasked with determining the premium amount that an insurance company should charge to ensure an expected profit of $100, given a $500 deductible policy and certain probabilities for different damage costs. To calculate this, one needs to compute the expected payout by the insurance company and add the expected profit to this amount.

Expected payout is calculated by: Expected Payout = (Damage - Deductible) * Probability, where the Damage values are greater than the Deductible. For this case, the damages considered for the payout become $500, $4,500, and $9,500 after applying the deductible, with their respective probabilities.

The expected payout is thus (500 * 0.09) + (4,500 * 0.05) + (9,500 * 0.02) = 45 + 225 + 190 = $460. To achieve the expected profit of $100, the premium must be: Premium = Expected Payout + Desired Profit, hence the premium should be $460 + $100 = $560.

Answer 2

$560

Step-by-step explanation:

Calculation for what premium amount should it charge

Using this formula to calculate the premium amount

E(Y)=yxP(y)

Let X variable represent the damage that occured because of accident in the year provided

Based on the information given since the amount deductible is $500 while the expected premium charge is $100 then let defined the premium function as,

For X=0

Hence,

Y=X+$100

For X=1,000, 5,000, and 10,000

Y=X-$500+$100

Y=$400

Let the table below represents probability distribution of y

X= 0, 1,000, 5,000, 10,000

Y= 100 600 4,600 9,600

P(y)=0.84, 0.09, 0.05, 0.02,

(1000-400=600)

(5000-400=4,600)

(10,000-400=9,600)

Now let calculate the PREMIUM AMOUNT to be charge Using this formula

E(Y)=yxP(y)

Let plug in the formula

E(Y)=(100 × 0.84)+( 600 × 0.09) + (4,600 × 0.05) +( 9,600 × 0.02)

E(Y)=84+54+230+192

E(Y)=$560

Therefore the premium amount that it should it charge (in dollars) is $560