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# Property Coinsurance

Property insurance frequently contains a "coinsurance clause" stating that, if the insured values its property at some fraction of its actual value , and is less than a coinsurance percentage (often 80%), the insurer may reduce its payment to of the loss. By way of example, if a house were insured for \$140,000, which was a fraction 0.7 of its actual value of \$200,000, and the coinsurance percentage were 0.8, the insurer would have to pay only \$70,000 following an insured \$80,000 loss to the insured. This provision is intended to deter deliberate underinsurance of property, which can work to the insurer's disadvantage when premiums are a linear function of the value the insured places on the property.
This Demonstration permits exploration of work done by Professor Itzhak Venezia involving whether a knowledgeable insured would prefer such a coinsurance provision to an alternative payment structure. More specifically, it compares an insurance policy with valuation fraction α and coinsurance percentage to an alternative insurance policy that had no coinsurance requirement but whose valuation percentage β was lower than α. In the "basic controls" bank, the user sets , , and with sliders. In the "advanced controls" bank, the user determines whether statistics on the distribution of wealth will be computed. Doing so will significantly reduce the responsiveness of the other controls. The user selects alternative parameters to the transformed beta distribution of wealth. The user selects the initial wealth of the insured, the actual value of the nsured property and whether any multiplicative load is placed on the premium for purposes of computing the statistics of the wealth distribution.
Mathematica outputs graphs showing insurance payments and wealth as a function of loss and a graph showing the cumulative distribution of wealth under both insurance policies. It further output a table showing the value of α that equilibrates premiums under the two insurance policies and, if "advanced computations" is checked, statistics on the distribution of wealth that results when this equilibrating value of α is used. The statistics confirm that for the same money, the insured has lower risk when using a policy with a coinsurance requirement.

### DETAILS

The policies will have the same premium if the areas under the curves in the bottom-most wealth distribution graphic are equal to each other. Assuming that the insured is risk averse and the premiums for the policies are the same, the insured will prefer the less "disperse" wealth distribution curve, which ends up always being the curve generated by an insurance policy with a coinsurance requirement. Professor Venezia's original article setting forth this model may be found in The Journal of Risk and Insurance, 55(2) 1988 pp. 307-314. The graphics go black if the constraint is violated.
The top panel contains an inset identifying the fraction of property value insured when coinsurance is present that will cost the insured the same premium as the policy with a lower fraction of property value insured but without coinsurance. The policy with coinsurance will yield a risk averse insured greater utility than the equally priced policy without coinsurance.
Snapshot 1: a different loss distribution function and a low coinsurance requirement
Snapshot 2: the same loss distribution function as in Snapshot 1 but with a high coinsurance requirement and low α
Snapshot 3: a loss distribution function with a high first parameter and low second parameter

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