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A risk adverse agent

In state of nature 1 the individual has income w, whereas in state of nature 2 the individual's income is y < w. The probabilities that these states will occur are (1 - p) and p, respectively. The individual can purchase insurance before the state of nature is known; an increase in income of s in state 2 can be purchased by a reduction in income of &#8719;s in state 1. Prove that a risk-averse von Neumann-Morgenstern individual will over-insure, fully-insure, or under-insure according as the insurance is available at a price &#8719; lower than, equal to, or higher than the actuarially fair price.

&#8719; - symbol pi

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In state of nature 1 the individual has income w, whereas in state of nature 2 the individual's income is y < w. The probabilities that these states will occur are (1 - p) and p, respectively. The individual can purchase insurance before the state of nature is known; an increase in income of s in state 2 can be purchased by a reduction in income of &#8719;s in state 1. Prove that a risk-averse von Neumann-Morgenstern individual will over-insure, fully-insure, or under-insure according as the insurance is available at a price &#8719; lower than, equal to, or higher than the actuarially fair price.

&#8719; - symbol pi
Part 1: A risk adverse agent offered actuarially fair insurance choose to insure fully

Suppose the total loss to the property in the event of the loss is T and the value of the property insured is q (where q<=T). The probability of loss is p.
T = w-y (change in worth if the loss event takes place)
Let the premium rate is &#8719;
Thus, premium = &#8719;q
The ...

Solution Summary

A risk adverse agent is assessed.

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