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Chapter 20: Problem 6

From 1990 to 2013,1 in approximately every 277 cars produced in the United States was stolen. Beth owns a car worth \(\$ 20,000\) and is considering purchasing an insurance policy to protect herself from car theft. For the following questions, assume that the chance of car theft is the same in all regions and across all car models. a. What should the premium for a fair insurance policy have been in 2013 for a policy that replaces Beth's car if it is stolen? b. Suppose an insurance company charges \(0.6 \%\) of the car's value for a policy that pays for replacing a stolen car. How much will the policy cost Beth? c. Will Beth purchase the insurance in part b if she is risk-neutral? d. Discuss a possible moral hazard problem facing Beth's insurance company if she purchases the insurance.

Short Answer

Expert verified
Answer: The fair premium for the insurance policy is approximately $72.20. Based on her risk-neutral behavior, Beth will not purchase the insurance since the insurance company charges higher than the fair premium at $120.

Step by step solution

01

Calculate the probability of car theft

The probability of car theft is 1 in approximately every 277 cars. So we can represent this probability as: P(car theft) = \(\frac{1}{277}\).
02

a. Step 2: Calculate the premium for a fair insurance policy

For a fair insurance policy, the premium charged must be equal to the expected loss due to car theft. The car's worth is \(\$20,000\), and the probability of theft is \(\frac{1}{277}\). So the expected loss is: Expected Loss = Car's worth * Probability of car theft Expected Loss = \(20,000 * \frac{1}{277}\) Now, compute the premium for the fair insurance policy: Premium = Expected Loss = \(20,000 * \frac{1}{277}\) ≈ \(\$72.20\)
03

b. Step 3: Calculate the policy cost charged by the insurance company

The insurance company charges \(0.6 \%\) of the car's value for the policy. To find the cost of the policy, multiply the car's worth by the percentage: Policy Cost = Car's worth * 0.6% Policy Cost = \(20,000 * 0.6\%\) = \(20,000 * 0.006\) = \(\$120\)
04

c. Step 4: Determine if Beth will purchase the insurance

Beth is risk-neutral, meaning she will only purchase the insurance if the premium equals the expected loss. We calculated the fair premium as \(\$72.20\), while the insurance company charges \(\$120\). Since the actual cost is higher than the fair premium, Beth will not purchase the insurance.
05

d. Step 5: Discuss the possible moral hazard problem

A moral hazard problem arises when the presence of insurance changes the behavior of the insured, potentially leading to riskier actions. In this case, if Beth purchases the insurance, she might be less vigilant about securing her car or parking it in safe areas. This behavior could increase the probability of her car being stolen, which in turn could lead to a higher payout from the insurance company, causing losses for the insurance company. This moral hazard problem may be a contributing factor to why the insurance company is charging a higher premium than the calculated fair premium.

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Most popular questions from this chapter

You have \(\$ 1,000\) that you can invest. If you buy Ford stock, you face the following returns and probabilities from holding the stock for one year: with a probability of 0.2 you will get \(\$ 1,500\); with a probability of 0.4 you will get \(\$ 1,100\); and with a probability of 0.4 you will get \(\$ 900 .\) If you put the money into the bank, in one year's time you will get \(\$ 1,100\) for certain. a. What is the expected value of your earnings from investing in Ford stock? b. Suppose you are risk-averse. Can we say for sure whether you will invest in Ford stock or put your money into the bank?

You own a company that produces chairs, and you are thinking about hiring one more employee. Each chair produced gives you revenue of \(\$ 10\). There are two potential employees, Fred Ast and Sylvia Low. Fred is a fast worker who produces ten chairs per day, creating revenue for you of \(\$ 100\). Fred knows that he is fast and so will work for you only if you pay him more than \(\$ 80\) per day. Sylvia is a slow worker who produces only five chairs per day, creating revenue for you of \(\$ 50 .\) Sylvia knows that she is slow and so will work for you if you pay her more than \$ 40 per day. Although Sylvia knows she is slow and Fred knows he is fast, you do not know who is fast and who is slow. So this is a situation of adverse selection. a. Since you do not know which type of worker you will get, you think about what the expected value of your revenue will be if you hire one of the two. What is that expected value? b. Suppose you offered to pay a daily wage equal to the expected revenue you calculated in part a. Whom would you be able to hire: Fred, or Sylvia, or both, or neither? c. If you know whether a worker is fast or slow, which one would you prefer to hire and why? Can you devise a compensation scheme to guarantee that you employ only the type of worker you prefer?

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