Priya has $$10,000$ that she wishes to invest for one year. She has narrowed her choices down to one of the following three actions:

a$1$: Buy bonds of Rockstar Mining Ltd$.$ This pays $18\%$ interest, unless Rockstar goes bankrupt, in which case Priya will lose $25\%$ of her principal and receive no interest.

a$2$: Buy GICs, paying $3\mathrm{.}6\%$ interest.

a$3$: Buy shares of Wright Corp. The payoff from this investment depends on the state of the economy. If the economy is $$Good$,$ the investment will increase by $30\%.$ However, if the economy is $$Bad$,$ the investment could decrease by $40\%.$

Priya assesses her prior probability of Rockstar Mining Ltd$.$ going bankrupt as $0\mathrm{.}25.$ She also assesses her prior probability that the economy will be good as $0\mathrm{.}40.$ Priya$$s utility for money is given by the square root of the amount of her gross payoff. If she buys the GICs her gross payoff will be $$10,360.$

Required:

a$)$ Based on her prior probabilities, which action should Priya take? Show your calculations, taken to $2$ decimal places.

b$)$ Before making a final decision, Priya decides she needs more information. She obtains Rockstar Mining$$s current financial statements and examines its debt$-$to equity ratio. This ratio can be either $$HI$$ or $$LO$.$ Upon calculating the ratio, Priya observes that it is HI$.$ Priya also finds out that the probability that a firm with a $$HI$$ debt$-$to$-$equity ratio will go bankrupt in a year is $0\mathrm{.}75.$ Non$-$bankrupt firms are equally likely to have HI or LO debt$-$to$-$equity ratios.

Priya also examines the financial statements of companies belonging to the same industry as Wright Corp and finds that most companies reported healthy profits. Upon further research, Priya finds that if the state of the economy was $$Good$,$ the probability that such companies would report healthy profits is $0\mathrm{.}8.$ On the other hand, if the economy were $$Bad$,$ the probability that companies would report similarly healthy profits was only $0\mathrm{.}15.$

Calculate Priya$$s posterior probabilities regarding her investment decision.

c$)$ Which action should Priya take now? Show your calculations, taken to two decimal places.

d$)$ Ignore your answer in part $[$c$]$ above. Assume that Priya decides to invest her money in the GICs, in spite of the possibility of earning higher returns from the other two investments. Explain her decision using prospect theory $.$

e$)$ If Priya was risk neutral, and her utility is equal to her net expected payoff $[$i$.$e$.$ weighted average of expected returns$],$ which investment option will she prefer? Show your calculations $.$