The most widely used utility function takes form of:

=() (), 2

where is called risk aversion coefficient. It reflects the risk aversion level of any given investors. One method to estimate investors' risk aversion coefficient is through experiment. Carol has taken one such experiment in a behavior science research center. Carol documented her experience stepby step as follows.

Step 1. She was asked to write down how much her net worth is. Net worth equals the sum of all the savings and cash in hands that are immediately disposable. She only has $20,000 in her savingaccount that time, so she wrote down $20,000.

Step 2. After she submitted her net worth value, she was presented with a fair coin toss bet, betting with all her net worth. The rule is, if the head shows, she wins $200. If the tail shows, she loses $200. Then, she was asked whether she would participate in this bet. Carol thought for a while,and decided to answer no.

Step 3. Next, she was asked how much the researcher has to pay her in order to induce her to take the bet. She did some quick calculations and told the organizer that she has to be paid at least $20.

The researcher told her that she has successfully completed the experiment and her risk aversion coefficient has been estimated. Carol was quite puzzled by how the researcher did the estimation. After taking FINA 2320 lectures, you can solve Carol's puzzle by answering the following questions.

A) (10') The coin toss bet, betting with all her net worth can be viewed as an investment with all her savings. Please calculate the expected return and the return variance of this bet or investment in Step 2.

B) (10') If in an alternative world, Carol accepts the bet in Step 2. What range does her risk aversion fall in? Is she a mean-variance investor?

Hint: when she rejects the bet, the utility value derived from this process is zero.

C) (20') In Step 3, Carol decided that she has to be paid at least $20 for her to take the bet. It means that if she were paid even infinitesimal less than $20, she will reject the bet, but on the other side, if she were paid infinitesimal more than $20, she will happily accept the bet. Please help Carol to calculate her risk aversion coefficient .