Someone injured in an accident often has two sources of compensation: insurance against at least some of the pecuniary damages suffered in the accident and, at least in some nations, the ability to obtain a "judgment" against the party responsible for the injury ("the third party"). This Demonstration examines the relationship between (1) the indemnity paid by the insurer if the third party is deemed legally responsible for the injury ("the recovery scenario"); (2) the indemnity paid by the insurer if the third party is deemed not legally responsible for the injury ("the no-recovery scenario"); and (3) the "expected utility" and certainty equivalent wealth of the insured. The insured's initial wealth is assumed to be 10 and the pecuniary loss caused by the accident is assumed to be 5.
To operate the Demonstration, you parameterize the utility of wealth functions for the insured if there is no accident and if there is an accident. These parameters may differ because the insured may suffer nonpecuniary losses as a result of the accident and because the insured may ascribe a different marginal utility to wealth following an accident that results in nonpecuniary loss. Specifically, you use sliders to set the coefficients in front of two logarithmic utility functions and, for the accident state, set a value that offsets the accident-state utility function from the no-accident utility function. (Ordinarily, this offset will be nonpositive.)
You choose the probabilities that an accident will occur and the conditional probability of the recovery scenario given that an accident has occurred. From this, the Demonstration computes the (unconditional) probabilities of the recovery scenario and the no-recovery scenario. You choose the amount the insured will be able to recover from the third party if found liable; this amount may well be less than the pecuniary loss of 5 because the third party may not be able fully to pay the judgment. You choose the "load" on the insurance policy: the fraction by which the premium is greater than the expected loss to the insurer simply from paying claims. You choose whether the insured is permitted to "over-insure", that is, recover more from the insurer than the insured's pecuniary losses. Finally, you choose the preferred visualization of the output.
The Demonstration produces a plot showing the utility of the insured in the no-accident and accident states as well as the position of the insured in the no-accident state, the recovery scenario, and the no-recovery scenario after the insured contracts for the levels of indemnity that maximize its expected utility. A dotted yellow horizontal line shows the expected utility and certainty equivalent wealth of the insured given these positions. If you choose "3D plot", the Demonstration also creates a three-dimensional surface showing the expected utility of the insured for varying levels of indemnity in the recovery scenario and the no-recovery scenario. A green dot shows the optimal levels of indemnity. The plot is labeled with a computation of the "optimal subrogation fraction". This fraction is 1 minus the ratio between (a) the net amount the insurer pays the insured in the recovery scenario; and (b) the amount the insurer pays the insured in the no-recovery scenario. The surface is colored according to the subrogation fraction, red representing an area with a high subrogation fraction and blue representing an area with a low subrogation fraction. A tooltip attached to the green optimum point provides additional information. If you choose "statistics", the Demonstration creates a table showing data associated with the optimal insurance contract.


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Both subrogation and reimbursement address situations in which an insured, in addition to a contractual right of indemnity from an insurer, has rights of recovery against some third party; often the party that has injured the insured. With "subrogation", the insurer is permitted to recover from the third party in a judicial proceeding an amount not greater than the amount the insurer previously paid to its insured pursuant to its insurance contract. Any sums thus paid by the third party to the insurer diminish dollar for dollar the amount the third party pays the insured. Often the proceeding in which the insurer asserts a right of subrogation is the same proceeding in which the insured asserts its rights against the third party. With reimbursement, the insurer is permitted to recover from its insured an amount not greater than the amount the insurer previously paid to its insured pursuant to its insurance contract. In general, under a doctrine prevalent in United States law ("the collateral source doctrine"), the existence of any right to subrogation or reimbursement does not affect the amount the third party owes as a result of the third party's wrongdoing.
This Demonstration is based on [1] that explores Sykes's contentions regarding the circumstances under which an insured will want its contract with the insurer to permit "subrogation" or permit reimbursement in the event the insured recovers from the third party. In particular, it examines the circumstances under which the insured would want a "make whole" rule in which the insurer is denied a right of subrogation or reimbursement unless the insured has been made whole for pecuniary and nonpecuniary damages it suffered at the hands of the third party.
Notice that changing the offset in the accident-state utility function appears not to alter the optimal level of coverages in the liable and not-liable scenarios. Marginal utility appears to matter, not absolute utility.
The size and opacity of the points on the top plot reflect the probability that the corresponding state will materialize. The points have tooltips attached that provide additional information.
The green point on the bottom plot has a tooltip attached that shows the value of its , , and coordinates.
A "negative load" on an insurance policy can exist in certain circumstances, among them (a) regulation that sets the price too low relative to expected claims; and (b) risk pooling such that a person with high risk is classified into a pool that has a lower mean level of risk.
Snapshot 1: an increased load results in lower levels of desired indemnity in both the recovery and no-recovery scenarios
Snapshot 2: a scenario in which the third party pays the entire pecuniary loss
Snapshot 3: a scenario in which the third party pays more than the entire pecuniary loss
Snapshot 4: setting the marginal utility of wealth higher in the accident state (though with a negative offset) and prohibiting overinsurance results in the insured wanting little subrogation
Snapshot 5: when the marginal utilities of wealth are equal in the no-accident and accident states and there is no load, the insured wants full insurance in the no-recovery scenario and substantial subrogation in the recovery scenario
Snapshot 6: the output from the Demonstration when visualization is set to "statistics"
[1] A. O. Sykes, "Subrogation and Insolvency," Journal of Legal Studies, 30(2), 2001 pp. 383–399. http://www.jstor.org/stable/724677.


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