# Risk Aversion, Load, and Optimal Insurance

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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.

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Contributed by: Seth J. Chandler (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## 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.

## Permanent Citation