A pharmaceutical firm has just patented a new painkiller, which is superior to existing products. Today is $31$ December $2050$ and the new drug, named Expill, will be able to capture $20\%$ of the painkiller market until the patent expires at the end of $2060.$ After $2060,$ generic drugs with the same substance will be allowed for sale so that Expill will no longer make profit for the company. The following information is also given:

The size of the painkiller market in terms of annual sales will be $$50$ billion in $2051.$

According to industry experts, the market is expected to grow at $5\%$ per year from $2052$ onwards that is$,$ annual sales in $2052,2053,$ and so on are expected to be $$50(1\mathrm{.}05)$ billion, $$50(1\mathrm{.}05{)}^2$ billion, and so on$.$

However, the breakup of European Union $($EU$)$ is currently being discussed and the breakup, if happens, will affect economy. If EU is dissolved, the size of the painkiller market will remain as $$50$ billion $($i$.$e$.,$ no growth$).$ The EU member countries will make the decision by the end of $2051,$ one year from now.

Manufacturing Expill in full scale requires a one$-$time investment cost $$7$ billion.

Annual cash flows from Expill are expected to be $10\%$ of its sales.

Assume that all cash flows are uncorrelated with the market portfolio, and take place at the end of each year. The risk$-$free rate equals $10\%.$

Required:

I. There is a traded security whose payoff depends solely on the EU breakup decision in December $2051.$ Its payoff will be $$100$ if the decision is to break up EU and zero otherwise. The price of this security is $$36\mathrm{.}364$ today $(31$ December $2050).$ Calculate the probability that justifies the current price of the security in a risk$-$neutral world.

II$.$ Calculate the expected NPV for two scenarios: $(1)$ the firm starts manufacturing Expill now, i$.$e$.,$ from the $1$st of January $2051$; and $(2)$ the firm waits for one year to start from the $1$st of January $2052.$

III. The two NPVs calculated in Part $($ii$)$ should be different. Identify the sources of difference in the NPVs and discuss them.