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Moral hazard is the proclivity of insureds to decrease their level of care once they have insurance. Unchecked moral hazard can hinder insurance markets and, on occasion, cause loss to third parties affected by and uncompensated for losses. Two traditional methods for control of moral hazard are incomplete insurance and conditioning indemnity on observations about the level of care taken by the insured. This Demonstration simulates the situation of an insured who has an initial wealth of 20 but faces the possibility of a loss of 15. The idea is to assume various levels of σ (the accuracy with which the insurer can measure the care taken by the insured) and try to find the levels of indemnity, premium and care condition that result in the highest level of insured wealth that doesn't result in the insurer's profit becoming negative. (Insurers avoid contracts that will result in their losing money.) See what level of care proves optimal and the associated level of insured wealth for each "optimal" insurance contract you create. How does the "optimal contract" vary with the accuracy with which the insurer can determine the level of care taken by the insured?
Contributed by: Seth J. Chandler (March 2011)
Open content licensed under CC BY-NC-SA
Snapshot 1: violation of the overinsurance requirement
Snapshot 2: violation of the non-negative profit requirement
Snapshot 3: high level of error in the measurement of care
The probability of a loss is , where is the level of care. The probability that the insured will be indemnified for loss is , where is the standard of care required by the insurance contract and is the standard deviation of error in the insurer's measurement of care. The insured is assumed to have a square root utility function that thus exhibits risk aversion and decreasing marginal utility of wealth. The idea is to select three features of the insurance contract: (1) the indemnity amount; (2) the premium; and (3) the care condition that will maximize the certainty equivalent wealth of the insured according to the formula given below. The graphic shows the way in which the level of care taken by the insured affects certainty equivalent wealth. The graphic also includes a dashboard displaying the state of various parameters and derivative variables. Violation of various usual requirements, such as the non-negativity of the insurer's profit and the existence of overinsurance, are signaled by various color changes in the graphic. The idea is to set a standard error of care level and then find the contract that maximizes the certainty equivalent wealth of the insured. Working in this multidimensional space proves trickier than many would imagine.
Wolfram Demonstrations Project
Published: March 7 2011