Optimal purchases of insurance with overlapping risks

The objective of this work is to determine the insurance coverages that an expected utility maximizing consumer will choose under overlapping risks and incomplete insurance markets. Multiple risks can have additive effects, partially overlapping effects, or completely overlapping effects. A house worth {dollar}100,000 that can be destroyed by earthquake or fire suffers only {dollar}100,000 damage when both occur rather than {dollar}200,000 as in additive models.; Markets for insurance against overlapping risks are incomplete if policies are available against all risks, but the policies fail to prorate when more than one risk occurs. This market failure may follow from informational demands of prorating. Alternatively, markets are incomplete if some risks are uninsurable because of moral hazard, adverse selection, or thin markets.; In either case, it is shown that a risk-averse consumer will not choose full insurance coverages that are available at actuarially fair rates. This contradicts the Bernoulli Principle of full insurance at fair prices that follows from risk aversion when the consumer faces just one risk. Other authors have shown such failures from portfolio effects for additive risks. The failures with overlapping risks occur because under incomplete markets, insurance moves some wealth outcomes closer together and others farther apart. When one risk is uninsurable, insurance against another risk lowers the already minimal wealth level when only the uninsurable risk occurs. The consumer loses the property and the insurance premium.; Comparative static results are derived for the case of an uninsurable completely overlapping risk. A more risk averse utility consumer will not generally purchase more insurance. The degrees of risk aversion of the marginal utility function and the utility function determine the sign. Constant absolute and relative risk aversion utility functions lead to less insurance for higher degrees of risk aversion with fair or subsidized prices.; These Bernoulli Principle and comparative statics results raise fundamental issues about the nature of insurance. The results justify studying the basic structure of risks, and they raise questions about previous work that attributes low demand for subsidized insurance to irrationality or misinformation.