A town has a water treatment plant with capacity to handle $232$ gigalitres per year. The current demand for water per year is $190$ gigalitres. However, this seems to fluctuate over time with an annual volatility of $0\mathrm{.}3.$ Once it reaches capacity the town will need to implement water restrictions. The price for water from this treatment plant is $$255$ for each gigalitre.

A farm next to the water treatment plant has come onto the market and the town is considering buying it to enable future expansion of the treatment plant. The land will cost $$44000$ to purchase and the cost for expanding the treatment plant will be $$354740.$ The farm is unlikely to be available in the future, so the town is considering purchasing it$,$ even though it is not required at present. To determine the feasibility of this line of action, the town council plans to conduct a Real Options analysis. The analysis will be based on a binomial lattice that spans $5$ years and uses $2$ time steps.

The discount rate for building the treatment plant would be $9\%$ and the risk free rate is $2\%.$

All answers should have at least $4$ significant figures.

What is the value of $u?$

What is the value of $d?$

What is the value of $P(u{)}^{\text{:}}$

What is the value of $P(d)?$

The underlying asset is the present value of water treated by the treatment plant. To calculate the present value you may assume that the life of the plant is $40$ years from now and include all water required by the town $($whether above or below the capacity$).$

What is the present value of the fees charged for the water over the future life of the treatment plant based on current demand?

What would be the present value of the fees charged for the water if the treatment plant was at maximum capacity?

Enter the Binomial Lattice for the Underlying Asset values:

$\backslash$table$[[,,],[,,],[,,],[,,],[,,]]$

Enter the binomial Lattice for the Option values:

$\backslash$table$[[,,],[,,],[,,],[,,],[,,]]$

Based on your Real Options analysis, which is the better choice?

Buy the farm

Do not buy the farm