Problem $2$: Project Alpa has two phases. You may invest in the first, in both, or in neither. The first phase requires an investment of $$100$ today. One year later, Alpha will deliver either $$120$ or $$80,$ with equal probability. At that time, $($after phase $1$ payout has been received$)$ you can invest an additional $$100$ for phase $2.$ One year later phase $2$ pays out either $20\%$ more cash than phase $1$ actually delivered, or $($equally likely$)20\%$ less. For investments in this business, your company normally applies a $10\%$ hurdle rate.

a$.$ How much would Project Alpha be worth if it offered only phase $1$ cash flows, without the phase $2$ opportunity.

b$.$ How much would the phase $2$ opportunity be worth if you had to choose today once and for all, whether or not to invest in it$?$

c$.$ Assuming you can wait to decide about phase $2,$ what is the total value of Project Alpha? Should you invest the first $$100?$

d$.$ Project Omega has exactly the same structure as Project Alpha, and the same systematic risk, but somewhat different cash flows. For $$100$ investment today, Omega delivers in phase $1$ either $$140$ or $$60,$ with equal probability. Phase $2$ requires an additional $$100$ investment and delivers either $40\%$ more or $40\%$ less than phase $1$ did. What is the total value of Project Omega? Should you invest the first $$100?$

e$.$ Compare these two projects. Which is riskier? Which is more valuable? which has a higher fraction of its value accounted for by "growth options," i$.$e$.$ the phase $2$ opportunity? Assuming both were undertaken, would you finance Alpha and Omega differently? How and why?