Question

In Problem \(8.5,\) Ms. Fogg was quite willing to buy insurance against a 25 percent chance of losing \(\$ 1,000\) of her cash on her around-the-world trip. Suppose that people who buy such insurance tend to become more careless with their cash and that their probability of losing \(\$ 1,000\) rises to 30 percent. What is the actuarially fair insurance premium in this situation? Will Ms. Fogg buy insurance now? (Note: This problem and Problem 9.3 illustrate moral hazard.)

Short answer

Answer: The actuarially fair insurance premium for a 30% probability of losing $1,000 is $300. Ms. Fogg's decision to purchase insurance depends on her risk tolerance, as her expected loss with or without insurance is the same (\$300). If she is risk-averse, she will likely buy insurance; if she is risk-neutral or risk-seeking, she may choose not to buy insurance.

Step-by-step solution

Calculate the actuarially fair insurance premium

To determine the actuarially fair insurance premium, we can use the formula: Actuarially fair insurance premium = Probability of loss × Amount of loss In our case, the probability of loss has increased to 30%, and the amount of loss remains $1,000. Actuarially fair insurance premium = 0.3 × \(1,000 = \)300

Evaluate Ms. Fogg's decision in purchasing insurance

To determine if Ms. Fogg would still buy insurance, we need to compare her expected loss without insurance and her expected loss with insurance. 1. Without insurance: Expected loss = Probability of loss × Amount of loss Expected loss = 0.3 × \(1,000 = \)300 2. With insurance: Expected loss = Insurance premium + (Probability of loss × Excess amount) Since the insurance premium is actuarially fair, and the excess amount is zero, the expected loss with insurance is equal to the insurance premium. Expected loss with insurance = $300 The expected loss with or without insurance remains the same (\(300). The decision to purchase insurance depends on Ms. Fogg's risk tolerance. If she is risk-averse and prefers paying the premium to avoid the potential of a higher loss, she will buy insurance. If she is risk-neutral or risk-seeking and can bear the risk of potentially losing all \)1,000, she may choose not to buy insurance.

Key concepts

Actuarially Fair Insurance Premium

An actuarially fair insurance premium is a concept from economics and insurance that helps determine the true cost of an insurance policy. It's calculated by multiplying the probability of an event happening by the amount of loss that event would cause. This ensures the insurance company collects enough premiums to cover potential payouts.

In Ms. Fogg's case, with a 30% probability of losing $1,000, the actuarially fair insurance premium is 0.3 multiplied by $1,000, which equals $300. This calculation shows the premium amount that would fairly cover the average expected loss, without any added cost for administrative expenses or the insurer’s profit. It's a useful benchmark for understanding whether the insurance premium being charged is reasonable or inflated beyond the risk being insured.

This situation illustrates a critical insurance principle: the premium must reflect the underlying risk. When the probability of loss changes, as it did for Ms. Fogg (from 25% to 30%), the actuarially fair premium adjusts accordingly to reflect the increased risk.

Risk Aversion

Risk aversion describes a person's tendency to prefer certainty over uncertainty, even if potentially rewarding. Individuals who are risk-averse would rather avoid uncertainty by making choices that offer predictable outcomes, rather than gambling on possibilities with high variance in potential results.

In deciding whether to purchase an insurance policy, a risk-averse individual like Ms. Fogg might opt to pay the actuarially fair premium. This eliminates the chance of a huge financial loss in case the unfortunate event occurs. The idea is to trade the anxiety of potential larger loss for the certainty of a smaller known cost, which is the premium.

Risk aversion is fundamental in understanding why people buy insurance: they prefer the certainty of paying a premium over the uncertain prospect of incurring a substantial loss. For Ms. Fogg, buying insurance becomes a matter of balancing her risk aversion against the cost of the premium and the potential for loss. If her preference is for financial security, she might willingly accept the actuarially fair premium as a rational choice.

Probability of Loss

The probability of loss is a measure of the likelihood that a certain adverse event, such as losing cash during a trip, will occur. This concept is crucial for determining insurance premiums and assessing risk.

In Ms. Fogg's scenario, the probability of loss initially was 25%, but because of a phenomenon called moral hazard, it increased to 30% when she considered buying insurance. Moral hazard occurs when having insurance leads someone to take more risks, because they feel financially protected.

Understanding the probability of loss helps in the realistic assessment of risk and the formulation of insurance offerings. When there's a rise in the probability, as with Ms. Fogg, it affects the actuarially fair premium, increasing the cost of coverage to match the higher risk. For prospective buyers, knowing this probability enables smarter decisions by weighing the risk against the cost of insurance, ensuring it aligns with their financial strategy and risk tolerance.