# Subrogation

Subrogation

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.