Dax Inc. has decided to go public and hired an underwriter to help it with the process. After talking with some regular clients who are experts in dog sled freight, the underwriter decides that the true value of the firm's equity will either be $80 million with probability 0.6 or $50 million with probability 0.4. The underwriter has decided to sell 8 million shares, but also needs to compute the appropriate offer price. There is a group of uninformed investors willing to submit bids for 10 million shares, as long as their expected profit is not negative. These uninformed investors know the probability distribution of firm values stated above, but do not know the true value of Dax Inc. as informed investors do. These informed investors are willing to order 5 million shares of the IPO if the offer price is lower than the true value.

(a) Calculate the price per share in the "good" state of the world, the price per share in the "bad" state of the world, and, hence, the expected price per share.

(b) Uninformed investors will only participate if their expected profit is not negative. That is, at the extreme their expected gain in the "good" state of the world is equal to their expected loss in the "bad" state. Set up an equation to compute the equilibrium offer price that the underwriter should set so that uninformed investors are willing to submit bids for the IPO, and solve it.
[NOTE: Expected gain/loss in each state of the world
= (Proportion of shares allocated to the uninformed investors in that state)
× Probability of that state occurring
× Dollar gain/loss

(c) Calculate, in percentage terms, how much the underpricing is relative to the expected price per share.

(d) Repeat the analysis of parts (a)-(c), but this time assume that the true value of the firm's equity will either be $100 million with probability 0.6 or $20 million with probability 0.4. What effect does increased uncertainty in possible outcomes have on underpricing?

(e) Repeat the analysis of parts (a)-(c) with the original expected valuations, but this time assume that the group of uninformed investors is willing to submit bids for 25 million shares (the informed investors are still only willing to order 5 million shares of the IPO if the offer price is lower than the true value). What effect does an increased proportion of uninformed investors have on underpricing?

(f) Based on parts (d) and (e), what are the real-world implications of this model?