Question

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

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

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) = \(\frac{1}{277}\).

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 = \(20,000 * \frac{1}{277}\) ≈ \(\$72.20\)

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 = \(20,000 * 0.6\%\) = \(20,000 * 0.006\) = \(\$120\)

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.

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.

Key concepts

Probability of Theft

When considering purchasing car theft insurance, understanding the probability of theft is crucial. In Beth's situation, the probability of her car being stolen is calculated as 1 in 277. This can be written as a fraction:

• Probability of theft = \( \frac{1}{277} \)

This probability helps determine the risk involved with owning a particular car without insurance. The probability remains the same regardless of location or car model, as mentioned in the original exercise.
Probability directly impacts how insurance companies calculate premiums. The lower the probability, the lower the associated risk and premium, and vice versa.

Risk-Neutral Behavior

Risk-neutral behavior is an important element in insurance decisions. A risk-neutral person, like Beth in the exercise, judges choices solely on the basis of expected financial outcomes and disregards any potential emotional or psychological comfort from having insurance protection.
Beth will consider purchasing the insurance only if the premium equals the expected loss due to theft; this is because a risk-neutral individual views both scenarios—having the insurance versus not having it—as economically indifferent, provided they cost the same.

• If the expected loss is greater than the premium, a risk-neutral person will buy the insurance as it makes financial sense.
• If the expected loss is less, like in Beth's case where the fair premium is \(\\(72.20\) but the cost is \(\\)120\), they will choose not to purchase it.

Moral Hazard

Moral hazard occurs when having insurance might change an individual's behavior, leading to riskier actions. For Beth, owning insurance could potentially reduce her vigilance about car security. This change in behavior might increase the chance of her car being stolen, as she might park it in less secure areas or be less consistent with locking it.
This altered behavior is a concern for insurance companies, as it can lead to more claims and higher payouts. This potential increase in risky behavior is one reason insurance companies might charge higher premiums than the fair value to mitigate these potential losses.
Moral hazard is a key consideration in the design and pricing of insurance policies to ensure both the insurer and the insured maintain a fair and feasible agreement.

Fair Premium Calculation

The concept of a fair premium is central to the insurance industry. A fair premium is one that exactly covers the expected losses from a certain risk, without any additional charges for overheads, profits, or risk buffer.
In Beth's situation, the fair premium calculation is straightforward. Use the formula:

• Expected Loss = Car's value * Probability of theft = \(20,000 * \frac{1}{277} = \\(72.20\)

The fair premium is \(\\)72.20\), reflecting the average expected loss from theft given the probability. However, real-world premiums are usually higher due to additional company costs and potential moral hazard issues.
Understanding how a fair premium is calculated helps individuals make informed decisions about insurance purchases, particularly in terms of negotiating better rates or understanding the rationale behind pricing.