# von Neumann-Morgenstern utility function - risk averse agent

1- An agent, with wealth 50, faces a probability 0.2 of a loss 35. The agent is offered insurance at a premium rate of 0.25. The agent has the von Neumann-Morgenstern utility function, u=lnx, where x is wealth. How much insurance should the agent buy?

2 - Show that a risk averse agent offered terms worse than actuarially fair will not choose to insure fully?

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#### Solution Preview

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Question (1)

An agent, with wealth 50, faces a probability 0.2 of a loss 35. The agent is offered insurance at a premium rate of 0.25. The agent has the von Neumann-Morgenstern utility function, u=lnx, where x is wealth. How much insurance should the agent buy?

Suppose the risk cover is L. Then the agent will choose the risk cover such that his expected utility is maximized.

The wealth because of the insurance contract is as follows

In the event of no actual loss

y1= 50-0.25*L ---------Equation 1

In the event of actual loss

y2= 50-35+L-0.25*L = 15+0.75*L ----------Equation 2

Since the probability of loss is 0.2, the expected utility from the two states will be ...

#### Solution Summary

Shows mathematically and in simple English how a risk averse decision maker will take a decision to insure.