An insurance company writes a policy to the effect that an...

An insurance company writes a policy to the effect that an amount of money $A$ must be paid if some event $E$ occurs within a year. If the company estimates that $E$ will occur within a year with probability $p,$ what should it charge the customer in order that its expected profit will be 10 percent of $A ?$

Key Concepts

Risk Assessment

Risk assessment involves evaluating the likelihood and potential impact of uncertain events. In the context of insurance, it includes estimating the probability of a claim event occurring and analyzing the potential financial loss associated with that event.

Profit Margin

Profit margin in insurance represents the percentage over the expected payout that an insurer includes in the premium to achieve a desired level of profit. This margin ensures that the insurer not only covers the expected claims but also earns an additional profit on the policies issued.

Expected Value

Expected value is a fundamental concept in probability that represents the weighted average of all possible outcomes of a random event. It is used in insurance to compute the average expected cost of claims by multiplying the probability of an event with its corresponding payout.

Insurance Premium

The insurance premium is the amount charged to a customer for coverage under an insurance policy. It is determined by considering the expected cost of potential claims, based on the probability of those claims, and adding an additional amount to account for profit and administrative expenses.

Transcript

00:01 In this exercise, we are given a probability distribution for the value of the damage incurred in car accidents in a given year.

00:09 There is an insurance company that offers a $500 deductible, and if the insurance company wishes to have a profit of $100, we are asked to find what premium the insurance company should charge.

00:25 So what should the insurance company charge its customers if it wishes to have an expected profit of $100? so we have the distribution for the dollar amounts for the damage for car accidents, but if the insurance company has a $500 deductible, that means that they are paying out $500 less for any positive claims, so any claims greater than zero.

00:53 So we could define a second random variable.

00:59 We can call the payout, and it's equal to the damage amount.

01:16 Minus the deductible, and that's for damage greater than zero.

01:38 Otherwise it's zero.

01:56 So we can show the probability distribution for this variable.

02:08 We have zero.

02:15 Now we'll have 500.

02:17 So i'm just taking $1 ,000 and subtracting 500, so we get 500.

02:24 Now take $5 ,000, subtract 500.

02:27 That's a payout of 4 ,500.

02:30 And 10 ,000 minus 500.

02:44 So this is the probability distribution for the amount that the insurance company will have to pay its customer.