POSCO Steel is considering a major investment in a new plant with a capacity of $1$ million tons. It is

anticipated that the plant will cost $ $1$ billion to build and generate expected after$-$tax cashflows of $ $100$

million a year for $20$ years. There is substantial uncertainty about these expected cash flows $$ the standard

deviation in annual cash flows is $40\%-$ and the cost of capital is $10\%.$ As CEO of the company, you are

concerned about the risk and you are considering an alternative plan. You will invest in a smaller plant with a

capacity of $500,000$ tons, which will cost $ $500$ million to build and generate expected cashflows of $ $50$

million a year for $20$ years. However, you will have the ability to double the capacity and the expected

annual cashflows for the remaining project life anytime at the end of year $5$ by spending an extra $ $500$

million. You can assume that the cost of capital remains $10\%$ and that the treasury bond rate is $3\%.$ Should

you invest in this alternative plan? Support your answer with numerical analysis.