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# Risk Aversion, Load, and Optimal Insurance

The "certainty equivalent wealth" of a person facing a loss with probability is the inverse utility of the probability-weighted average of their positions in the loss and no-loss states. Using an insurance market and contract law, a person (now acting as an "insured") is often able to alter their wealths in the loss and no-loss states. They do this by entering into an insurance contract with an insurer who will pay the insured a fraction ("the indemnity fraction") of in the loss state in exchange for a premium in both states. The net payments from the insurer to the insured are thus {, } in the loss and no-loss states, respectively.
This Demonstration explores the relationship between the certainty equivalent wealth of the insured and this indemnity fraction , on the assumption that the premium charged by the insurer is equal to the expected payments to the insured multiplied by a loading factor, denoted as . You set the size of the loss, the probability of a loss, the load factor, and the degree of risk aversion of the insured. The Demonstration then draws a graph showing the relationship between the indemnity fraction and certainty equivalent wealth. It also draws a line showing the optimal indemnity fraction bounded between 0 and 2. (Under some circumstances the mathematically optimal indemnity fraction might be outside these bounds, but the insurable interest doctrine and market practicalities would make it difficult for the insured actually to purchase such a contract.)

### DETAILS

Snapshot 1: For small losses and a significant load, even an insured with a high degree of risk aversion would prefer a contract that obligates the insurer to pay only a small fraction of any loss suffered by the insured. The value of is thus well below 1.
Snapshot 2: Low risk aversion coupled with even a moderate load also substantially reduces the value of to well below 1.
Snapshot 3: Negative load produces a contract calling for overinsurance , although the extent of overinsurance is not terribly high when the insured's risk aversion is also high.
The utility functions in this Demonstration satisfy constant relative risk aversion.
A somewhat more difficult optimization problem exists if insurance contracts are constrained to lie only in a less regular region of the {, } domain. The problem studied here assumes only that . In some circumstances, however, the insured might have to select contracts that satisfy but also satisfy or an even more complex constraint. In some of these circumstances, may be zero, that is, no insurance transaction will be consummated.

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