Show and explain why risk averse individuals will generally refuse actuarially fair bets and purchase insurance that allows them to avoid participating in fair bets.

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Question 3:

(i).        Show and explain why risk averse individuals will generally refuse actuarially fair bets and purchase insurance that allows them to avoid participating in fair bets.

(ii).        Discuss some of the problems that have beset the von Neumann-Morgenstern model of choice under uncertainty.

(i)

There are three possible attitudes to risk: risk averse, risk neutral and risk loving.  Each of these attitudes to the outcome of an uncertain activity are different.  If  an individual is risk averse then he will only accept a gamble if the expected monetary value (EMV) of said gamble is greater than the certain monetary equivalent (CME).  In other words if this individual was given the choice of a certain amount of money guaranteed and a risky game that would yield the same expected return, this individual would choose the guaranteed sum of money (or CME), since he prefers to avoid risk.  It follows that a risk neutral individual is indifferent between such a gamble and a guaranteed amount of money, if CME equals EMV, and a risk lover requires CME to be greater than EMV not to accept the gamble.      

If indifference curves are drawn for these three attitudes to risk – as below – the degrees of risk averse or loving can be shown by their distance from the straight (risk neutral) line.  

                Indifference Curves for Money and Probability of Risk

As far as a risk averse individual’s preferences concerning a fair bet are concerned it has already been stated that he will not accept, since he wishes to avoid ‘unnecessary’ gambles.  This can be proven mathematically as well as by ad hoc methods by means of an example.  

If an individual is offered a fair gambles, with a 50-50 chance of winning or losing, but with gamble A offering the chance to win (or lose) £x.  It can be seen that for gamble A, the EMV is £0, since:

EMV         =          (p)x        +        (1 – p)y         

p        =        probability winning the gamble

1 – p        =        probability not winning the gamble

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x        =        payoff from winning the gamble

y        =        cost of losing the gamble

Where the payoff equals the cost, x = y, and the EMV is equal to zero.

If the utility from the individuals’ present wealth is U(W*), then using basic mathematics and geometric representation it can be shown that the utility for a risk averse individual who accepts a gamble (with EMV equal to zero) can be represented as below:

UA(W*)        =        (p)U(W* + x)                 +        (1 - p)U(W* - x)

=        (1/2)U(W* + x)         +        (1/2)U(W* - x)

where UA(W*) is the individual’s current wealth plus the outcome of the ...

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