Suppose that an auto insurance company has collected the following data on drive

Question

Suppose that an auto insurance company has collected the following data on drivers in the 20-25 age group. For simplicity, we assume that when a car accident does happen, the amount the insurance company has to pay out in claims is one of $1000, $2000, $3000, $5000, or $10000. (This is a big simplification, and actual insurance companies do a more intensive version of this computation, but ultimately this is the set of computation they are doing). Describe an ideal data set of values (these are the costs) for the experiment of selecting a driver at random, and nothing the cost to the insurance company to insure a driver in this age group (in the mathematical sense of "expected")? Compute this in two different ways. In the long run, what will happen if the company charges every driver in this age group $1, 800 per year? What must the company charge each driver to make this a fair game, i.e, in order to except to break even in the long run? What should the company charge such drivers in order to have an expected profit of $300 for the year, per insured driver?

Explanation / Answer

7a. An ideal data set representing the costs to the insurance company of the driver’s accidents should be based on the expected damage claim to be made in case of an actual accident. The insurance company needs to analyze the individual probabilities of different categories of accidents and the likely damage claims, which will act as an appropriate data set for further analysis.

7b. The expected cost to the insurance company to insure a driver in this age group can be calculated as:
     Expected Cost = 0.35*0 + 0.21*1000 + 0.16*2000 + 0.15*3000 + 0.11*5000 + 0.02*10000
                               = 0 + 210 + 320 + 450 + 550 + 200
                               = $1,730
The expected cost can also be calculated in terms of relative frequency. Suppose we assume that the total number of drivers insured in a year = 100. Therefore, the number of accidents in the different categories will be 35, 21, 16, 15, 11 and 2 respectively. Therefore, the expected cost will be calculated as:
     Expected cost = (35*0 + 21*1000 + 16*2000 + 15*3000 + 11*5000 + 2*10000) / 100
                               = $1730

7c. In the long run, if the company charges each driver $1800 per year, it will make an expected profit of ($1800 - $1730) = $70 per driver per year.

7d. In order to make this a fair game, i.e. to break even in the long run, the expected cost should equal the amount charged from each driver. Therefore, the company must charge each driver $1730.

7e. To have an expected profit of $300 per year per driver, the company should charge each driver an amount of ($1730 + $300) = $2030.

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