Consider an entity facing potential liability that has purchased a liability insurance contract providing that it will pay a third party up to an amount

if a judgment is rendered. If the insured could act selfishly, it would prefer an insurance contract that would "diminish": it would specify that in the event a judgment against the insured would, even with the existence of the liability insurance, bankrupt the insured, the insurer should pay nothing. This is so because payments to a third party in such a setting do essentially nothing to help the insured, but drive up the cost of premiums because the insurer must pay. Either as the result of regulation or custom, however, such "diminution" provisions are generally unlawful. This is so because diminution might otherwise prevent victims from being compensated more fully when individuals with liability insurance injure them. This restriction on freedom of contract means, however, that there are some settings in which insureds, even though they might like to transfer risk, find it undesirable to do so.

This Demonstration explores the settings in which "diminution" restricts risk transfer. The model is idealized by considering liability insurance policies that pay the lesser of the amount for which the wrongdoer is liable

and a policy limit

. It further assumes, for purposes of deriving a closed-form expression capable of rapid evaluation, that

is drawn from a mixture distribution

in which a fraction

of the time

is zero (no liability-creating event occurs), and the remaining fraction

of the time

is drawn from a beta distribution with a mean of

and a standard deviation equal to a fraction

of the maximum standard deviation possible for such a beta distribution. The actual amount paid by the wrongdoer, is thus

, but censored to lie between 0 and the amount

of wealth on which a judgment can be executed. The insured pays a premium

multiplied by the mean of

, where

represents a load. The insured seeks to minimize the sum of its premium and its residual risk, where the latter is computed to be equal to the mean of a distribution

, but transformed to subtract the limit of liability

and censored to lie between 0 and

.

differs from

in that, in order to simulate risk aversion, the insured treats the mean loss as being not

but

, where

is a level of risk aversion ranging from 0 to 1.

This Demonstration produces a "region plot" showing for different values of

, the insured's assets available for execution, and for

, the insured's risk aversion, the situations in which full insurance (

) is preferable to no insurance (

). (Some study indicates that partial insurance is never the best decision for the insured.) You choose the remaining parameters, using the controls. When executing this Demonstration on slower computers, you may wish to set the performance goal to "speed"; otherwise "quality" gives somewhat better results.

Here is what you are likely to observe:

• The frequency with which liability-creating events occur turns out to be irrelevant to the decision whether to purchase full insurance.

• Larger mean losses shrink the wealth-risk aversion combinations for which full insurance is desirable.

• More variable losses (a higher standard deviation fraction) shrink the wealth-risk aversion combinations for which full insurance is desirable.

• Higher premium loads shrink the wealth-risk aversion combinations for which full insurance is desirable.

• Holding everything else constant, higher risk aversion decreases (but only up to a point) the minimum level of wealth for which full insurance is desirable.