Suppose that you purchase a new car for $23,000 and wish to insure it for full replacement value if it is stolen and 75% replacement value if damaged beyond repair due to natural c... Suppose that you purchase a new car for $23,000 and wish to insure it for full replacement value if it is stolen and 75% replacement value if damaged beyond repair due to natural causes. If there is a 4% chance that the car will be stolen and a 1% chance of the car being damaged beyond repair due to natural causes within the next year, what should the yearly premium be for this insurance policy with the company's expected profit on this policy being $1200?
Understand the Problem
The question is asking to calculate the yearly premium for a car insurance policy. The car costs $23,000. The insurance covers full replacement if stolen (4% chance) and 75% replacement if damaged beyond repair due to natural causes (1% chance). The insurance company expects a $1200 profit. We need to calculate the premium that covers these expected payouts and the profit.
Answer
The yearly premium should be $2292.50.
Answer for screen readers
The yearly premium should be $2292.50.
Steps to Solve
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Calculate the expected payout for theft The car costs $23,000 and there's a 4% chance it will be stolen. The expected payout is the cost of the car multiplied by the probability of theft. $$ \text{Expected payout for theft} = 0.04 \times $23,000 = $920 $$
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Calculate the expected payout for damage due to natural causes The car costs $23,000 and the insurance covers 75% of the replacement value if damaged. There's a 1% chance of this happening. The expected payout is 75% of the car's cost multiplied by the probability of damage. $$ \text{Expected payout for damage} = 0.01 \times 0.75 \times $23,000 = $172.50 $$
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Calculate the total expected payout The total expected payout is the sum of the expected payout for theft and the expected payout for damage. $$ \text{Total expected payout} = $920 + $172.50 = $1092.50 $$
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Calculate the premium The premium should cover the total expected payout plus the insurance company's desired profit. $$ \text{Premium} = \text{Total expected payout} + \text{Profit} = $1092.50 + $1200 = $2292.50 $$
The yearly premium should be $2292.50.
More Information
The premium calculation ensures the insurance company covers its expected losses (payouts) and achieves its desired profit margin.
Tips
A common mistake is to forget to multiply the damage payout by 75% since the insurance only covers that percentage, and also to forget to add the profit. Another mistake could be calculating percentage incorrectly.
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