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.
