Question

a) Rachel has a bicycle valued at £100. If Rachel locks the bicycle up everywhere she goes, it
still has a probability of being stolen of 20%. What does it mean to say that Rachel is risk
natural? If Rachel is risk neutral, what is the expected value to her of owning the bicycle?

b) If Rachel buys a bike insurance policy for £25 that will reimburse her for the value of the
bike if stolen, she eliminates the risk of owning a bike. Then the expected value of the insured
bike is 0.8*(£100-£25) + 0.2*(£100-£25) = £75. Under what circumstances would Rachel be
prepared to buy the insurance policy?

c) Calculate the expected profit of offering insurance on the bicycle, given the values stated
above.

d) Now assume that Rachel is part of a ‘low risk group’ of customers, each of whose probability
of having their bike stolen is 0.2. Assume that there is also a high risk group whose probability
of having their bike stolen is 0.4. Assume that while the low risk group are prepared to buy an
insurance contract for £25 or less, the high risk group are just prepared to buy the contract for
£45 or less. What happens if the firm tries to sell insurance at a price of £35? Who buys the
contract and what is the profit on the contract? What problem does this outcome illustrate?

Answer 1

A. A person is risk neutral when the decision they are taking depends solely on the expected return of the investment, and has no role of risk in it. Its a purely rational decision.

So, if Rachel is risk neutral, her expected value of owning a bike would be

=Probability of not losing the bike*value of not losing the bike+probability of losing the bike*value of losing the bike

=.8*100+.2*0

=80

B. It is clea that the value derived from insurance, 75, is lower than the value derived from now having the insurance.

So, Rachel will only buy the insurance when she is a risk-averse person. A risk aversion values the fact that there is no risk. So, even though the purely economical value of having insurance is $5 less than of not having it, a risk averse person might still want it if they value removal of risk as higher than $5.

C. The insurance costs 25. That is the revenue. The expected loss is

.8*0+.2*100=20.

So,

Profit of insurance=25-20=5.

D. It the cost of insurance is 35, it is clear that low risk group would not buy it. On the other hand, the high risk group would buy the insurance at 35 as it is lower than the value they think they get from insurance, which is 45.

The revenue from selling this insurance would be 35. The cost would be

.4*100+.6*0

=40.

Profit=35-40=-5.

This illustrates the problem of adverse selection, where the the buyer of insurance knows whether they are high risk or low risk, but the insurer doesnt.