Wilbur is an airline pilot who currently has income of ...

Wilbur is an airline pilot who currently has income of $\$ 60,000 .$ If he gets sick and loses his flight medical certificate, he loses his job and has only $\$ 10,000$ income. His probability of staying healthy is $0.6,$ and his probability of getting sick is 0.4 . Wilbur's utility function is given in the accompanying table. a. What is the expected value of Wilbur's income? b. What is Wilbur's expected utility? Wilbur thinks about buying "loss-of-license" insurance that will compensate him if he loses his flight medical certificate. c. One insurance company offers Wilbur full compensation for his income loss (that is, the insurance company pays Wilbur $\$ 50,000$ if he loses his flight medical certificate), and it charges a premium of $\$ 40,000 .$ That is, regardless of whether he loses his flight medical certificate, Wilbur's income after insurance will be $\$ 20,000$. What is Wilbur's utility? Will he buy the insurance? d. What is the highest premium Wilbur would just be willing to pay for full insurance (insurance that completely compensates him for the income loss)?

Wilbur is an airline pilot who currently has income of $\$ 60,000 .$ If he gets sick and loses his flight medical certificate, he loses his job and has only $\$ 10,000$ income. His probability of staying healthy is $0.6,$ and his probability of getting sick is 0.4 . Wilbur's utility function is given in the accompanying table.
a. What is the expected value of Wilbur's income?
b. What is Wilbur's expected utility? Wilbur thinks about buying "loss-of-license" insurance that will compensate him if he loses his flight medical certificate.
c. One insurance company offers Wilbur full compensation for his income loss (that is, the insurance company pays Wilbur $\$ 50,000$ if he loses his flight medical certificate), and it charges a premium of $\$ 40,000 .$ That is, regardless of whether he loses his flight medical certificate, Wilbur's income after insurance will be $\$ 20,000$. What is Wilbur's utility? Will he buy the insurance?
d. What is the highest premium Wilbur would just be willing to pay for full insurance (insurance that completely compensates him for the income loss)?

Key Concepts

Risk Aversion

Risk aversion is a behavioral trait where an individual prefers a certain outcome over a risky one with the same expected value. Risk-averse individuals are willing to pay a premium or forgo some potential gain in order to avoid uncertainty and secure a more predictable outcome.

Utility Function

A utility function maps levels of wealth or income into levels of satisfaction or happiness. It reflects the decision-maker's individual risk preferences—whether they are risk-averse, risk-neutral, or risk-seeking—and is a key component in assessing choices under uncertainty.

Insurance Premium Determination

Determining an insurance premium involves calculating the maximum amount an individual is willing to pay to eliminate or reduce risk. This calculation often relies on the individual’s expected utility and certainty equivalent, allowing them to assess how much they value the trade-off between a secure income and a risky one.

Expected Value

The expected value is the weighted average of all possible outcomes, where each outcome is multiplied by its probability. In decision-making under uncertainty, it provides a measure of the central tendency or the 'average' outcome that one might expect if the situation were repeated many times.

Expected Utility

Expected utility extends the concept of an expected value by applying a utility function to monetary outcomes, reflecting an individual's preferences and attitudes towards risk. It represents the average utility across different states of the world, weighted by their corresponding probabilities, and is used to make decisions in uncertain environments.

Transcript

0:00 Ms.

00:01 Fogg is planning around the world trip on which she plans to spend $10 ,000.

00:09 Now, the utility for the trip is a function of how much she actually spends on it, and the utility is given of uy is equal to lny.

00:22 So, first question, if there is a 25 % probability that ms.

00:34 Fogg would lose, she'll lose $1 ,000 of a cash on the trip what is the trip expected utility so there's a 25 % chance of losing money so therefore the expected utility would be eu of the trip is equal to 0 .25 multiplied by you so first of all if she if she loses 1 ,000 she will have 10 ,000 minus 1000 right so that will be 9 ,000 so it's a review of 9 ,000 plus the remaining 75 % multiplied by u of 10 ,000 because it's 25 % chance that should lose it and 75 % chance that should not so this would be equals to 0 .25 multiplied by then plus 0 .75 multiplied by 10 ,000 so then this is equals to 9 .184 so this is the expected utility of the trip the next question is suppose that she can buy insurance against losing $1 ,000 so by purchasing travelers check but then actually actually real premium of $250 sure that i expected this is a higher if she purchases this insurance than if she faces the chances of losing the 1 ,000 without insurance so you want to show that expected utility will be higher which are purchasing the insurance right so the insurance premium means 250 therefore our wealth after buying insurance to be where our wealth after insurance will be equals to 9 ,750 because you're going to take it away from the 10 ,000 right? so this is how much you're going to have left.

03:27 So then the expected utility of insurance because to the utility of 975 is equal to 9750.

03:42 Then this will give you 9 .185...

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