In$-$depth, step$-$by$-$step, solve the following question for 'Capital Budgeting: Discounted Cash Flow Analysis':

$1.$ A company buys a machine for $$500,000$ and depreciates it on a straight$-$line basis over a five$-$year period for tax purposes. The investment would result in cash cost savings of $$200,000$ per year, before taxes, for five years. At the end of five years, it was estimated that the machine would be sold for $$75,000.$ The gain on the sale of the machine would be taxed at a $40\%$ rate. Is the investment in the machine attractive in economic terms, given all of the cash flows? Please assume that the cash flows occur at the end of each year, that the tax rate is $40\%,$ and that the appropriate discount rate is $8\%.$ What is the net present value? the internal rate of return? the payback period?

$2.$ Schmidt A$.$G$.$ is considering the replacement of three hand$-$loaded block milling machines with an automatic milling machine. The three hand$-$loaded machines are only three years old and were purchased at a total cost of DM $300,000.$ The useful life of the machines at the time of their purchase was estimated to be fifteen years. The salvage value at the end of the fifteen years was estimated to be zero. Schmidt A$.$G$.$ can continue to use the three hand loaded machines for their remaining twelve years. The machines would continue to be depreciated at a rate of DM $20,000$ per year $($the original DM $300,000$ divided by the total useful life of fifteen years$).$ The depreciation expense would reduce taxable income and, therefore, tax payments. Schmidt A$.$G$.$ is taxed at a $40\%$ rate. Alternatively, Schmidt A$.$G$.$ can replace the three hand$-$loaded machines with an automatic milling machine. The new machine would have the same capacity as the combined capacity of the three hand$-$loaded machines, would have a twelve year useful life, would be depreciated for tax purposes at a rate of DM $40,000$ per year for twelve years, and would have zero salvage value. Cost of the automatic milling machine is DM $480,000.$ The automatic machine would result in pre$-$tax labor savings, including benefits, of DM $135,000$ per year. Other out$-$of$-$pocket cash savings were estimated at DM $25,000$ per year, before taxes. Based on the charge made for each square meter of floor space, the machining department would save DM $3,000$ in the annual charge for space. No alternative use of the space was anticipated. If Schmidt acquires the automatic milling machine, it will sell the three hand$-$loaded machines immediately for a total price of DM $100,000.$ The loss of DM $140,000($the book value of DM $240,000$ at the end of the third year minus the sale price of DM $100,000)$ resulting from the sale will be a tax$-$deductible expense. No inflation is anticipated. Schmidt A$.$G$.$ uses a discount rate of $7\%$ to evaluate cost reduction projects. Is the investment in the automatic milling machine economically attractive? $($i$)$ What are the actual, after$-$tax cash flows for each of the two alternatives? $($ii$)$ What is the net present value of the actual cash flows for each of the two alternatives?