Rachel Duncan is the CEO of Dyad Pharmaceuticals, and she faces a decision about her company$$s newest and most promising drug candidate: NZT$-48.$ Clinical trials have been completed, and the team awaits an approval decision from the U$.$S$.$ Food and Drug Administration $($FDA$)$ before it can be marketed. In the meantime, Ms$.$ Duncan receives an offer from a global biotech company, International Genetics Incorporated $($InGen$)$ to license NZT$-48$ from Dyad. $($Licensing here means that InGen provides capital to help the development and launch process of the new drug, but will share the profits with Dyad once the drug enters the market.$)$

With InGen$$s offer, Ms$.$ Duncan has a decision to make right now, before the FDA$$s decision. She can accept InGen$$s offer. If FDA approves the new drug, then Ms$.$ Duncan estimates that her company$$s valuation will reach $$4$ billion; if FDA does not approve, with InGen$$s injected capital, Ms$.$ Duncan estimates that her company would still value at $$2$ billion. If Ms$.$ Duncan rejects InGen$$s offer and brings the new drug to market by her own company, should FDA approves the drug, then Dyad may reach the valuation of $$7$ billion $($since it would not need to share the profits with InGen$)$; however, if FDA rejects the drug, Dyad$$s valuation may tumble to $$1$ billion.

a$)$ Let p denote the probability for FDA to prove the new drug. Draw a decision tree to represent the situation that Ms$.$ Duncan faces, and identify all the strategies.

b$)$ Ms$.$ Duncan and her team estimate that p $=0\mathrm{.}4.$ Suppose that Ms$.$ Duncan is risk averse, with a risk tolerance of $$0\mathrm{.}7$ billion.

$($b$\mathrm{.}1)$ Using the exponential utility function, calculate the risk premiums of all the strategies identified in Part $($a$).$

$($b$\mathrm{.}2)$What decisions should Ms$.$ Duncan make? $($Please explain your answer.$)$

c$)$ Now a third choice is presented to Ms$.$ Duncan. Umbrella Corp., an international conglomerate, just proposed to purchase the right of the new drug from Dyad $($that is$,$ Dyad can transfer everything related to the new drug to Umbrella Corp., and pocket the profits from the sale$).$ In this case, FDA$$s final approval is irrelevant to Dyad $($since Umbrella Corp wants to purchase the right before FDA$$s approval to keep the purchase price relatively low$).$ The final purchase price, of course, will be subject to negotiation, and before the final price is known, Ms$.$ Duncan estimates that her company$$s valuation would be uniformly distributed between $$2\mathrm{.}5$ and $$3\mathrm{.}5$ billions, should the sale go through.

$($c$\mathrm{.}1)$ Assume that Ms$.$ Duncan$$s utility function remains the same as in Part $($b$),$ calculate the certainty equivalent $($an approximation is fine$)$ of this third choice.

$($c$\mathrm{.}2)$ Out of all the choices that Ms$.$ Duncan has, which one should she take $($assume p $=0\mathrm{.}4)?($Please explain your answer. Be careful: while the potential valuation is uniformly distributed, it does not mean that the corresponding utility is uniformly distributed. There are different ways to answer this question. We do not care which way you choose, so long as it is based on sound theories with correct mathematical calculations.$)$