Billingham Packaging is considering expanding its production capacity by purchasing a new machine, the $\backslash ($ X C$-750\backslash ).$ The cost of the $\backslash ($ X C$-750\backslash )$ is $\backslash (\backslash$$ $2\mathrm{.}76\backslash )$ million. Unfortunately, installing this machine will take several months and will partially disrupt production. The firm has just completed a $\backslash (\backslash$$ $45,000\backslash )$ feasibility study to analyze the decision to buy the XC $-750,$ resulting in the following estimates:

$-$ Marketing: Once the XC$-750$ is operational next year, the extra capacity is expected to generate $\backslash (\backslash$$ $10\mathrm{.}05\backslash )$ million per year in additional sales, which will continue for the $10-$year life of the machine.

$-$ Operations: The disruption caused by the installation will decrease sales by $\backslash (\backslash$$ $5\mathrm{.}06\backslash )$ million this year. As with Billingham's existing products, the cost of goods for the products produced by the XC$-750$ is expected to be $\backslash (69\backslash \%\backslash )$ of their sale price. The increased production will also require increased inventory on hand of $\backslash (\backslash$$ $1\mathrm{.}13\backslash )$ million during the life of the project, including year $0.$

$-$ Human Resources: The expansion will require additional sales and administrative personnel at a cost of $\backslash (\backslash$$ $2\mathrm{.}03\backslash )$ million per year.

$-$ Accounting: The XC$-750$ will be depreciated via the straight$-$line method over the $10-$year life of the machine. The firm expects receivables from the new sales to be $\backslash (15\backslash \%\backslash )$ of revenues and payables to be $\backslash (11\backslash \%\backslash )$ of the cost of goods sold. Billingham's marginal corporate tax rate is $\backslash (30\backslash \%\backslash ).$

a$.$ Determine the incremental earnings from the purchase of the XC$-750.$

b$.$ Determine the free cash flow from the purchase of the XC$-750.$

c$.$ If the appropriate cost of capital for the expansion is $\backslash (9\mathrm{.}9\backslash \%\backslash ),$ compute the NPV of the purchase.

d$.$ While the expected new sales will be $\backslash (\backslash$$ $10\mathrm{.}05\backslash )$ million per year from the expansion, estimates range from $\backslash (\backslash$$ $8\mathrm{.}15\backslash )$ million to $\backslash (\backslash$$ $11\mathrm{.}95\backslash )$ million. What is the NPV in the worst case? In the best case?

e$.$ What is the break$-$even level of new sales from the expansion? What is the breakeven level for the cost of goods sold?

f$.$ Billingham could instead purchase the XC$-900,$ which offers even greater capacity. The cost of the XC$-900$ is $\backslash (\backslash$$ $3\mathrm{.}94\backslash )$ million. The extra capacity would not be useful in the first two years of operation, but would allow for additional sales in years $3$ through $10.$ What level of additional sales $($above the $\backslash (\backslash$$ $10\mathrm{.}05\backslash )$ million expected for the XC $-750)$ per year in those years would justify purchasing the larger machine?