The finance industry is a place where uncertainty and risk plays a major role in decisions. Investors

consider risks when making their investment decisions. The investment companies $($like Tasch Portfolio

Management Co$.,$ abbr: TPM$)$ offer financial products considering their risks and potential future profits.

Their main source of profits are the commissions that occurred when the customers purchased their

investment funds$/$plans$.$ Therefore, the products $($investment funds or plans$)$ they offer must be appealing

for the customers.

The financial engineers of the TPM want to introduce new investment plans into the market. There are

three alternatives, product $1,2,3.$ Regarding Plan $1,$ it includes treasury bonds with risk$-$free interest rate

and there is no risk, and the plan is estimated to obtain a net profit of $$1$M $($M denotes millions$).$

Plan $2$ is risky. Returns may be low, medium, or high. In the high case, resulting net profit is estimated

as $ $4$M$.$ In the medium case, the resulting net profit is estimated as $$2$M$,$ or low, in which case the plan

just breaks even.The probabilities for high, medium, and low returns are $0\mathrm{.}1,0\mathrm{.}25$ and $0\mathrm{.}65,$ respectively.

Plan three is extremely risky since it includes cryptocurrencies as investment instruments. There are two

issues related to Plan $3$ which have an impact on return $($due to the behavior of the market$)$ and net profit

$($due to the attitude of the customers$).$ The first one is a problem with the SEC: whether the SEC will see

cryptocurrencies as reliable investments or not. There is a $0\mathrm{.}55$ probability that the SEC will not let

companies invest in them in a short time period $($that is Plan $3$ is not approved by the SEC$).$ In this case,

TPM has to make insurance for the funds invested in Plan $3.$ The addition of insurance in introducing

the plan will lower the returns. That is$,$ the profits will decrease due to low returns and additional

insurance costs.

The second issue is the price, the alternatives are to introduce it either high or low price; the price would

not be set until just before the plan is introduced. Finally, once the plan is introduced, returns can be

either high or low.

Regardless of the price decision of TPM$,$ if SEC approves the Plan $3,$ the probability that the returns are

high and low are $0\mathrm{.}6$ and $0\mathrm{.}4,$ respectively.

If SEC does not approve the Plan $3$ and insurance is made, investors may not find the plan appealing.

Also, the market learning SEC$$ s decision will not favor the cryptocurrencies and the returns are high

and low with probabilities $0\mathrm{.}2$ and $0\mathrm{.}8,$ respectively.

If the SEC approves Plan $3$ and if the TPM decides to set a low price and the return is high, net profit is

$$2$M$.$ If the return is low the net profit is $$1$M$.$ If the TPM sets a high price, net profit is $$4$M if the return

is high and $$0$M if the return is low.

If the SEC does not approve Plan $3$ and the prices are set low, the profits are estimated as $$1$M and

$$0\mathrm{.}5$M when the return is high and low, respectively. If the prices are set high, the profits are $$2\mathrm{.}5$M and

$-$$$0\mathrm{.}5$M for the high and low return scenarios, respectively.

a$.(20$ points$)$ Draw a complete decision tree for this problem.

b$.(20$ points$)$ Solve the decision tree constructed in part $($a$)$ to obtain the optimal decision strategy.