# A risk averse agent offered actuarially fair insurance

Important Note:

Please try to use mathematical notation as much as you can to demonstrate your answer, but don't forget to carefully define each step you make.

Question

(a) Define "risk averse".

(b) Why does a risk averse agent offered actuarially fair insurance choose to insure fully?

(c) What does the agent choose if the terms are worse than actuarially fair?

(d) Show there are either one or two kinds of equilibrium in competitive insurance markets, depending on how equilibrium is defined.

(e) Explain, in the context of competitive insurance markets, what is meant by: "Separating equilibrium" and "Pooling equilibrium". What assumptions are required for each of these types of equilibrium, in turn, to exist?

#### Solution Preview

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Question

(a) Define "risk averse".

A risk averse is an attitude of the decision taker where he values a risky alternative at less than its expected value and seeks premium for taking risk. For example, a decision maker having a utility curve of w^0.5 is a risk averse person where w is the level of wealth.

Suppose, the person is given two choices

1. To take an amount of $100 for sure

2. To play a lottery and Take an amount of $0 with 50% probability and $200 with 50% probability

The expected value of the payoff in both cases is same however, for the risk averse person following the utility curve of w^0.5, the utility of first option is 100^0.5=10.00

Where as for the second option is =0.5*(0)^0.5+0.5*(200)^0.5 = 7.07

To be neutral between the two options, the risk averse person would expect to receive an amount of $400 if he wins the lottery with 50% probability.

Risk aversion corresponds to a strictly concave utility function.

(b) Why does 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 x.

Let the premium rate is p

Thus premium = pq

The wealth as a consequence of the insurance contract is

In the event of no actual loss

y1= y-pq ---------Equation 1

where y is the initial wealth

In the event of actual loss

y2= y-T-pq+q = y-T+(1-p)q ----------Equation 2

Increase in the amount q will reduce the y1 and increase the y2.

Since the probability of loss is x, the expected income to from the two states will be

(1-x)*(y-pq)+x*( y-T+(1-p)q)

=y-x*T+q((x-p) ----------Equation 3

An ...

#### Solution Summary

Explains the concepts of risk, uncertainty and asymmetry of information with mathematical equations as well as in simple English.