Let's say you are the CEO of a Twitter company that works in the film business. You're thinking of starting a new project that involves writing a script from scratch or buying one. A completed script might run into the millions of dollars and result in a real production. Still, a vast percentage of scripts are dropped. Indeed, Twitter is aware that just a select number scripts will show promise ahead of time, but they are unsure of which ones. To find a few good scripts, they therefore cast a broad net and commissioned numerous of them. Furthermore, Twitter needs to be brutal when it comes to bad scripts because the costs involved are so small compared to the enormous losses that result from making a bad movie.

A few fortunate scripts will then proceed to production, where tens of millions of dollars may be spent. However, the studios are equally brutal in this situation, based on the film's box office performance. $$5$ million is required for the beginning development of a script and must be paid at date zero. There are three conceivable consequences in the event that this investment is made. Based on previous script developments that were comparable to this one, the probability of each of the scenarios occurring has been assessed. More specifically, there is a $10\%$ chance of finding a medium script, a $5\%$ chance of developing a good script, and an $85\%$ chance that the produced script is too bad to proceed. If the script development process is successful $($that is$,$ if you have created a good or medium$-$length script$),$ you may choose to invest at the outset of the $$50$ million film's production. Following production, you ought to use the major media to promote the film.

These $$70$ million worth of advertising expenses must be paid for by date three. The box office earnings are still unknown, though. If the script was successful, you should expect high revenue at the beginning on date $4,$ which will be equal to an annual FCF of $$63$ million for perpetuity $($based on an estimated $75\%$ probability$).$ Alternatively, you should expect low box office receipts $($with a $25\%$ probability$)$ that will only amount to an annual FCF of $$10\mathrm{.}5$ million for perpetuity beginning on date $4.$ Or your script was a medium one and then you will get high box office revenues starting at date $4,$ that consists in a yearly FCF of $$13\mathrm{.}5$ million for perpetuity $($with an estimated probability of $20\%)$ or you will get low box office revenues $($with a $80\%$ probability$)$ that only consist in a yearly FCF of $$10\mathrm{.}5$ million for perpetuity starting at date $4.$ You have estimated that the cost of capital for this project is $15\%$ and that it will remain constant over all the period. As a consequence, you have the following five potential outcomes if you invest in the project: $$ Either the developed script is too bad to be continued, $$ Or you have developed a good script that leads to high box office revenues, $$ Or you have developed a good script that leads to low box office revenues, $$ Or you have developed a medium script that leads to high box office revenues, $$ Or you have developed a medium script that leads to low box office revenues. You now have to answer to the following questions.

Questions :

$**1.$ Compute the expected NPV of the project without real options. You may consider all the scenarios; compute the Present Value of the Cash$-$Flows for each scenario and finally, the expected NPV$.$

$**2-$ Compute the expected NPV of the project including real options. The Real option embedded in this project is the possibility to abandon the project and not to invest the $$50$ million at date $1($the production phase$)$ if the expected NPV of a medium script is negative $($including the probabilities of the scenarios $4$ and $5$: i$.$e$.,$ that the medium script leads to high or low box office revenues$).$ Notice however that in order to determine whether the script is of good or medium quality, you have to invest $$5$ million in the initial development phase at date $0.$

$**3-$ Compute the value of the real options for this project.

$****$This is related to question number $4******$ ::::

Assume that, due to external shocks, there are changes in:

$$ The probability to develop a good scenario that will change from $5\%$ to $11\%,$

$$ The probability to develop a medium scenario that will change from $10\%$ to $4\%$

$$ The probability to get high box office revenues if you have developed a good scenario that will change from $75\%$ to $35\%$

$$ The probability to get low box office revenues if you have developed a good scenario that will change from $25\%$ to $65\%$

$$ The probability to get high box office revenues if you have developed a medium scenario that will change from $20\%$ to $10\%$ and

$$ The probability to get low box office revenues if you have developed a medium scenario that will change from $80\%$ to $90\%.$

$4-$ Compute the expected NPV of the project without real options AND the expected NPV of the project including real options and compute the value of the real options for this project $($with exter