Question

Suppose there is a \(50-50\) chance that a risk-averse individual with a current wealth of \(\$ 20,000\) will contract a debilitating disease and suffer a loss of \(\$ 10,000\) a. Calculate the cost of actuarially fair insurance in this situation and use a utility-of-wealth graph (such as shown in Figure 7.1 ) to show that the individual will prefer fair insurance against this loss to accepting the gamble uninsured. b. Suppose two types of insurance policies were available: (1) a fair policy covering the complete loss; and (2) a fair policy covering only half of any loss incurred. Calculate the cost of the second type of policy and show that the individual will generally regard it as inferior to the first.

Short answer

Explain your answer based on the utility theory of risk-averse individuals. Answer: A risk-averse individual would generally regard an insurance policy covering the complete loss as superior. This is because, based on the utility theory of risk-averse individuals, they prefer to minimize their exposure to the risk of loss. In this scenario, the insurance policy covering the complete loss provides better protection and ensures higher utility of wealth for the individual compared to the policy covering only half of the loss.

Step-by-step solution

Calculate the expected loss

First, we need to find the expected loss from the debilitating disease. Given that there is a 50-50 chance of the individual contracting the disease and hence, suffering a loss of $10,000. The expected loss can be computed as follows: Expected Loss = Probability of contracting disease * Loss = 0.5 * $10,000 = $5,000

Calculate the cost of actuarially fair insurance

Actuarially fair insurance means that the insurance premium is equal to the expected loss. In this case, the cost of actuarially fair insurance can be calculated as: Cost of fair insurance = Expected Loss = $5,000

Illustrate the utility-of-wealth graph

Since the individual is risk-averse, their utility function is concave. We can illustrate this situation with the following graph: - On the x-axis, plot wealth ranging from \(10,000 to \)20,000 - On the y-axis, plot the utility of wealth - Draw a concave utility function curve - Mark the initial wealth of the individual ($20,000) on the x-axis, and the corresponding utility on the curve - Also mark the utility points if the individual contracts the disease with and without insurance - The graph should show a higher utility of wealth for the individual when they have insurance compared to when they are uninsured Based on the utility graph, the risk-averse individual would prefer to purchase the fair insurance against the loss rather than accepting the gamble uninsured, as it provides them a higher utility of wealth. #Part B#

Calculate the cost of fair insurance covering the complete loss

Since the individual already has a fair insurance policy covering the complete loss for a premium of $5,000 (from Part A), we can use this value to compare it with the second type of insurance policy. Cost of fair insurance covering complete loss = $5,000

Calculate the cost of fair insurance covering only half of the loss

Now, let's calculate the premium of the fair insurance policy covering only half of any loss incurred. To do this, we need to find the expected loss covered by the insurance policy: Expected Loss covered by the policy = 0.5 * $5,000 = $2,500 Since this is an actuarially fair insurance policy, the premium is equal to the expected loss covered by the policy: Cost of fair insurance covering half of the loss = Expected Loss covered by the policy = $2,500

Compare the two insurance policies

Now, we can compare the two insurance policies: 1. Fair insurance policy covering the complete loss: Cost = $5,000 2. Fair insurance policy covering half of the loss: Cost = $2,500 Since the individual is risk-averse, they would prefer to minimize their exposure to the risk of loss. In this case, the first insurance policy (covering the complete loss) provides better protection for the individual compared to the second policy (covering only half of the loss). Therefore, the risk-averse individual would generally regard the first policy as superior to the second one.

Key concepts

Risk Aversion

A risk-averse individual prefers certainty over uncertainty when it comes to their wealth. This means they would rather have a guaranteed outcome instead of taking a gamble, even if the expected value of the gamble is the same. This type of person dislikes risk and prefers to minimize it whenever possible.
For example, in the case of our individual with $20,000 wealth and the potential for a debilitating disease that could cost them $10,000, a risk-averse nature would lead them to seek insurance. This insurance would shield them from the financial blow of the disease, effectively turning an uncertain situation into a more predictable one.
In the insurance market, risk-averse individuals drive demand for policies that provide security, even if it means paying a premium. They value the peace of mind that comes with being protected against potential losses.

Utility of Wealth

Utility of wealth is a concept used to describe the satisfaction or wellbeing an individual derives from their wealth. Risk-averse individuals usually have a concave utility of wealth function, meaning they experience diminishing returns in utility as their wealth increases.
The idea can be visualized with a graph where utility is plotted against wealth. A concave curve in this graph indicates that as wealth increases, each additional dollar provides less utility than the previous one. This explains why a risk-averse individual values an insurance policy: it helps to maintain their utility by preventing significant drops due to potential losses.
In our scenario, the person with $20,000 wealth and facing a $10,000 potential loss would have a utility that drops significantly if they incur the loss without insurance. With insurance, they maintain a more stable utility as they avoid such a drastic wealth reduction.

Actuarially Fair Insurance

Actuarially fair insurance is a policy where the premium paid is equal to the expected loss. This ensures that the insurer isn't making a profit on the policy, and theoretically, neither is the insured losing out. For a policy to be actuarially fair, it matches the statistical risk that the individual faces.
For instance, if there's a 50% chance of a $10,000 loss for our individual, then the expected loss is $5,000 (0.5 probability * $10,000 loss). Thus, actuarially fair insurance would cost exactly $5,000 to cover the entire expected risk.
This type of insurance is particularly attractive to risk-averse individuals because it offers them a break-even point where they neither gain nor lose financially relative to their risk. It's a key concept in understanding how insurance can be priced to reflect the actual risk someone faces.

Expected Loss

Expected loss is a fundamental concept in insurance economics, representing the average potential loss an individual might face. It is calculated as the product of the probability of an event occurring and the monetary loss if that event occurs.
In the case of our example, the expected loss was determined by multiplying the 50% chance of a $10,000 loss, resulting in an expected loss of $5,000. This figure is crucial as it forms the basis for determining the price of fair insurance.
Understanding expected loss helps both insurers and insureds better grasp the risks involved and thus appropriately plan either insurance coverage or pricing strategies. It's a measure of the financial risk someone avoids by purchasing insurance.

Insurance Policies

Insurance policies are agreements that provide financial protection against losses. They are essential tools for risk management, especially for risk-averse individuals looking to safeguard their wealth against unforeseen events.
There are different types of insurance policies, each with varying degrees of coverage. In our scenario, two types were considered: one covering the complete loss and another covering only half. The full coverage policy required a $5,000 premium, matching the expected loss. In contrast, the half-coverage policy had a lower premium of $2,500, since it covered only half of the potential loss.
Risk-averse individuals typically prefer policies that offer more comprehensive protection, even at a higher cost. This preference is based on their aversion to risk, as fuller coverage means less financial uncertainty and more stability in their wealth.