Insurance and Precautions
A prospective insured facing an insurable risk generally chooses how much insurance to purchase against the risk and the level of precautions to take to avoid the risk materializing. Increasing the precautions reduces the probability of a potentially costly accident but costs money. Increasing the insurance reduces the loss to the insured from an accident but increases premiums. The insurer attempts to control the insured's choices due to the twin threats of moral hazard—reduced precautions by the insured after purchasing large amounts of insurance—and adverse selection—increased purchase of insurance by the insured who suspects, usually correctly, that the risk is beyond that reflected in the premium. The insurer generally addresses moral hazard by assessing, after an accident, whether the insured has been "grossly negligent" or failed to meet some other contractually specified standard. If so, the insurer refuses to indemnify the insured. The insurer generally addresses adverse selection (and also the potential for heightened moral hazard) through its underwriting processes by refusing to issue policies where the amount of insurance requested is excessive. Neither of these conditions can generally be enforced with perfect accuracy. Increasing the accuracy of determining whether these conditions have been satisfied also costs money.[more]
This Demonstration models the demand for insurance and the optimal level of costly precautions when the insurer is permitted to condition payment—but in a way that costs money—on the insured taking a required level of precautions and the insured not overinsuring. You select the two major characteristics of the insured: the "baseline accident probability" for the probability that the insured will suffer an accident if the insured fails to take any precautions, and the insured’s level of "risk aversion". You then specify the two conditions the insurer imposes: the maximum amount of insurance the insured can purchase and the level of precautions the insured is required to take. You also specify the prices to purchase a unit of insurance and to increase the level of precautions by one unit. Advanced controls let you determine (1) the accuracy with which compliance with the conditions is assessed and (2) the costs (denoted as and ) faced by the insurer in achieving this accuracy.
The Demonstration responds with a plot showing certainty equivalent wealth as a function of the level of insurance purchased and the level of precautions taken. You can choose to color the surface according to its height or according to the probability that the insurer will pay the insured in the event of an accident. A red dot shows the optimal combination of precaution and insurance along with the associated certainty equivalent wealth. By default, the Demonstration also presents a table showing various statistics that result when insurance and precautions are optimized. By unchecking the "compute details" checkbox, you can obtain speedier redraws of the surface, although doing so results in omission of the statistics table.
Want a challenge? Try to set the controls so that the premium the insured pays is close to the premium needed for the insurer to break even. Try doing it when the cost of determining compliance with the conditions is high.[less]
The variables used in the formulas employed by this Demonstration are defined as follows:
is the baseline probability of an accident, that is, the probability of an accident if the insured takes no precautions against it.
is a measure of the risk aversion of the insured, where higher levels of correspond to higher levels of risk aversion; the insured is assumed to have a constant absolute level of risk aversion (CARA).
is the maximum amount of insurance that may lawfully be purchased.
is the accuracy with which it is determined whether the amount of insurance purchased exceeded the maximum.
is the minimum amount of precautions that the insured must take before the insured has a right to payment in the event of an accident.
is the accuracy with which it is determined whether the amount of precautions taken was under the minimum.
is the price per unit of insurance.
is the price per unit of precaution.
is the amount of insurance purchased.
is the level of precautions taken.
The model is scaled so that an accident causes a loss of 1.
The utility of the insured is . The function of , , has the property that equals 1 at 1, that its derivative at 1 is 1, and that, for the ratio of its first derivative to its second derivative (absolute risk aversion) is equal to .
The expected utility of the insured is computed based on the mathematical expectation of a transformed distribution: , where , , and are random variables. The random variable is distributed according to a Bernoulli distribution with a parameter usually equal to the cumulative distribution function of a normal distribution with mean and standard deviation evaluated at , the level of precautions taken. When the standard deviation of this normal distribution goes to the limit of zero, the Bernoulli parameter is 1 if the level of precautions equals or exceeds the "standard" of and is 0 otherwise. The random variable is subject to a somewhat similar distribution. It is distributed according to a Bernoulli distribution with a parameter usually equal to the cumulative distribution function of a normal distribution with mean and standard deviation , evaluated at , the level of insurance purchased. When the standard deviation of this normal distribution goes to the limit of zero, the Bernoulli parameter is 0 if the level of precautions equals or exceeds the "standard" of and is 0 otherwise. The random variable (used to prevent negative insurance) is distributed according to a Bernoulli distribution with parameter 1 if , the level of insurance purchased, is greater than 1 and a parameter of zero otherwise.
The certainty equivalent wealth of the insured is the inverse of its expected utility , where is expected utility. The certainty equivalent wealth is the amount of money that, if held with absolute certainty, would yield the insured the same expected utility as the "lottery" it faces as a result of the dependence of its utility on random variables. Use of limits is necessary to address cases where risk aversion is zero or one.
Many of the slider controls have nonlinear responses in order to gain the greatest "resolution" within the zones of interest while providing for the controls to take on relatively extreme values. This interface methodology can be used in conjunction with slider zoom to provide even greater resolution.
Tooltips on the controls and table elements of the output provide additional information.
The insurer may also prevent excessive insurance purchases by declining to pay or limiting the amount it will pay if it discovers that the amount of insurance the insured has purchased is excessive. Coordination provisions in an insurance contract, such as those requiring pro rata payment in the event of multiple insurance against the same event, provide one vehicle whereby the insurer attempts to prevent excessive purchase. See J. W. Stempel, Stempel on Insurance Contracts, 3rd. ed., 2007. A more complex approach to the situation modeled in this Demonstration would permit the insurer to use both conventional underwriting and "post-claims" underwriting to determine whether the insured had purchased excessive insurance. Such post-claims underwriting might encompass assessments after an accident whether the insured, given other insurance the insured might have in place, had purchased insurance that resulted in being better off after an accident than if no accident occurred.
Snapshot 1: the surface is colored according to indemnity probability rather than certainty equivalent wealth
Snapshot 2: a high price of precautions relative to precaution conditions results in purchase of no insurance
Snapshot 3: a reduction in the precaution condition permits purchase of insurance even when precautions are costly
Snapshot 4: use of advanced controls shows an effect of high costs in accurately determining whether the precaution condition has been satisfied
Snapshot 5: an insured with a high baseline risk relative to the premium purchases a large amount of insurance and takes minimal precaution where the precaution condition is low
Snapshot 6: an insured with a high baseline risk relative to the premium purchases no insurance and takes high precautions where the precaution condition is high
Snapshot 7: an insured with a low baseline risk relative to the premium charged purchases no insurance but takes high levels of precaution (well beyond what the insurer would have required)