Project1: Suppose An Individual Has The Following Utility
4430 Project1 Suppose That An Individual Has The Following Utility Fu
Suppose that an individual has the following utility function, U(W) = sqrt(W), where U stands for utility and W for wealth. The individual currently has a net wealth of $400,000. The individual believes there is a 5% chance they will get into a car accident this year. It is expected that a car accident would cost them $100,000 (dropping their overall wealth to $300,000).
a) How much would it cost to purchase an actuarially fair insurance policy to cover all losses from a car accident? (In other words, what are the expected losses)
b) How much would the individual be willing to pay for this policy? (What are the maximum loading fees that an insurance company could charge for this policy)
c) Provide an example utility function for a risk-averse, a risk-neutral, and risk-loving individual.
Paper For Above instruction
The decision-making process under risk often hinges on an individual's attitude toward risk, which can be characterized by their utility function. This paper explores the calculation of insurance costs, willingness to pay, and different risk preferences through utility functions, with a focus on an individual with a specified utility function and probabilistic risk scenario related to car accidents.
Calculating the Actuarially Fair Insurance Premium
The individual's utility function is given by U(W) = √W, and initial wealth W₀ = $400,000. The potential outcomes include either experiencing a car accident costing $100,000, reducing wealth to $300,000 with a 5% probability, or not experiencing an accident, leaving wealth unchanged at $400,000 with a 95% probability. The expected utility without insurance is:
E[U(W)] = 0.95 U(400,000) + 0.05 U(300,000) = 0.95 √400,000 + 0.05 √300,000
Calculating these values:
- √400,000 ≈ 632.46
- √300,000 ≈ 547.72
Thus:
E[U] ≈ 0.95 632.46 + 0.05 547.72 ≈ 601.84 + 27.39 ≈ 629.23
An actuarially fair premium (P) equals the expected loss, which is the probability of the event times the loss:
Expected Loss = 0.05 * $100,000 = $5,000
Therefore, the fair insurance premium per year to cover all expected losses is $5,000.
Willingness to Pay and Maximal Insurance Loading
The maximum amount the individual is willing to pay corresponds to the certainty equivalent (CE)—the amount of wealth that provides the same utility as the expected utility with risk. To find CE, solve for W such that U(W) = E[U]:
U(CE) = 629.23 → √CE = 629.23 → CE ≈ (629.23)^2 ≈ 395,553
The maximum premium (maximum loading fee) is the difference between initial wealth and CE:
Maximum Premium = W₀ - CE = 400,000 - 395,553 = $4,447
This means the individual would be willing to pay up to approximately $4,447 for insurance, which is slightly less than the expected loss due to the risk aversion implied by the utility function.
Examples of Utility Functions for Different Risk Preferences
Risk-averse individual: U(W) = √W – as shown, this utility function reflects diminishing marginal utility of wealth, indicating risk aversion.
Risk-neutral individual: U(W) = W – this utility assumes linearity in wealth, implying the individual weighs expected outcomes equally, regardless of risk.
Risk-loving individual: U(W) = W^2 – representing increasing marginal utility of wealth, indicating preference for risk and potential for higher variance.
Implications and Conclusions
The calculations illustrate that the individual would pay approximately $5,000 for full coverage, equal to expected losses, but their actual willingness to pay is slightly less due to risk aversion. The utility approach demonstrates that risk preferences significantly influence insurance valuation, with risk-averse individuals valuing certainty more highly. Understanding these preferences is vital for insurance companies when designing products and setting premiums, ensuring alignment with consumer risk attitudes.
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