4. Sangeeta owns an equity stake in a biotech firm that is waiting for FDA approval for a new cancer treatment. Specifically, she owns 10,000 shares in the company. If the treatment is approved, then her 10,000 shares are worth a total of $160,000. If the treatment is not approved, then the total value of her shares is only $10,000. The probability that the FDA will approve the treatment is 0.2. Her only income is the value of her stake in this company. Sangeeta's utility of income, x, is represented by the function u(x)=(x)0.5 where x is the total value of her shares in the company.

a) What is Sangeeta's expected income? What is her expected utility? Illustrate Sangeeta's expected utility in a diagram.

b) What is the lowest price that Sangeeta would be willing to sell all of her equity (ie all of her shares)? Illustrate your answer in your diagram above.

c) What is the relationship between the certainty equivalent and your answer to part (b)?

Suppose that Sangeeta can sell individual shares in the company. If the FDA approves the treatment then each share will be worth $16, and if it does not then the share will only be worth $1. The probability of approval remains at 0.2. Let P be the price that she can sell a share in the company.

d) If the purchase of a share is a fair bet, then what must be the price P?

e) If the purchase of a share is a fair bet, then how many shares will she want to sell?

f) If Sangeeta can sell as many shares as she would wish, then what will be her income in each of the two states? What would be her (approximate) expected utility? Graph these values in your diagram for part (a).