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 actuar ially 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

Answer: A risk-averse individual generally prefers Policy 1 (covering the complete loss) over Policy 2 (covering only half of any loss incurred). This is because Policy 1 provides a certain outcome of wealth equal to $15,000, whereas Policy 2 involves a gamble with possible wealth outcomes of $15,000 or $17,500. A risk-averse individual values the certainty provided by Policy 1, as it leads to a higher utility according to their concave utility-of-wealth function.

Step-by-step solution

Calculate the cost of actuarially fair insurance

To find the cost of an actuarially fair insurance policy, we will first find the expected loss. The expected loss can be found by multiplying the probability of loss by the amount of loss. In this case, there is a 50% chance of losing $10,000, so the expected loss is: Expected Loss = Probability of Loss × Amount of Loss Expected Loss = 0.5 × \(10,000 = \)5,000 This means the cost of actuarially fair insurance is $5,000.

Interpret the information on a utility-of-wealth graph

A utility-of-wealth graph illustrates the relationship between an individual's wealth and their utility (satisfaction or happiness) from that wealth. For a risk-averse individual, the utility function is concave, meaning they prefer a certain outcome over a gamble with the same expected value. Without insurance, the individual faces a gamble: - With a 50% chance, they keep their $20,000 wealth. - With a 50% chance, they contract the disease and lose \(10,000, leaving them with a wealth of \)10,000. With actuarially fair insurance, the individual pays the premium of \(5,000, guaranteeing that they will not suffer any losses and will have a wealth of \)15,000. A risk-averse individual will prefer the certain outcome of wealth equal to \(15,000 with insurance over the gamble of possibly having wealth of either \)20,000 or $10,000 without insurance. This can be shown on a utility-of-wealth graph where the utility of wealth with insurance is higher than the expected utility of wealth without insurance.

Calculate the cost of the second type of policy

The second type of insurance policy is also actuarially fair, but covers only half of any loss incurred. Since the total expected loss is $5,000, this policy will cover half of that amount: Fair Policy Covering Half of Loss = 0.5 × Expected Loss Fair Policy Covering Half of Loss = 0.5 × \(5,000 = \)2,500 This means the cost of the second type of policy is $2,500.

Compare the two policies and determine which one is preferred

Now we will compare the two policies to determine which one is preferred by the risk-averse individual. With Policy 1 (covering the complete loss), the individual will have a guaranteed wealth of \(15,000 (current wealth of \)20,000 minus insurance premium of $5,000). With Policy 2 (covering half of the loss), in the case of contracting the disease, the individual will have: - A 50% chance of only losing \(5,000 (half of \)10,000), resulting in a wealth of \(15,000 (\)20,000 original wealth - $5,000 loss). - A 50% chance of not losing any wealth and therefore staying at \(20,000 minus the insurance premium of \)2,500, resulting in a wealth of $17,500. Since the risk-averse individual prefers a certain outcome over a gamble with the same expected value, Policy 1 (covering the complete loss) is generally regarded as superior to Policy 2 (covering only half of any loss incurred).