The initial outlay represents all cash flows occurring at time 0, including the direct purchase price of the asset and any costs necessary to get it ready for use.

For the swizzle-stick machine, the purchase price of $587,000 and shipping and installation costs of$23,000 sum to $610,000, which forms the depreciable basis of the asset.

An additional $25,000 increase in net working capital is also required at time 0; this is a financing requirement that is not depreciated and has no immediate tax implications.

Since there is no old asset being replaced, there are no after-tax proceeds from an old asset sale to offset the initial costs, leading to a total initial outlay of $635,000 ($587,000 + $23,000 +$25,000).

Differential cash flows are the annual incremental cash flows generated by the project, calculated by creating a statement of cash flow for each year.

In this example, revenues ($175,000), operating costs ($25,000), and depreciation expense are constant each year, simplifying the calculation to a single annual differential cash flow.

Annual depreciation is calculated using simplified straight-line depreciation: the depreciable basis of $610,000 divided by the five-year life, resulting in an annual depreciation expense of$122,000.

To find incremental earnings before taxes (EBT), subtract incremental costs ($25,000) and depreciation ($122,000) from incremental revenue ($175,000), yielding$28,000.

Taxes are then calculated on EBT at the 40% marginal rate ($28,000$\times$0.4 =$11,200), resulting in incremental earnings after taxes (EAT) or net income of $16,800 ($28,000 - $11,200).

Finally, since depreciation is a non-cash expense, it must be added back to EAT to determine the actual differential cash flow: $16,800 (EAT) +$122,000 (depreciation) = $138,800 per year for years 1-5.

Terminal cash flows encompass all cash flows occurring in the project's last year, including the final year's differential cash flow and cash flows related to asset disposal and working capital recapture.

The machine will have a book value of $0 at the end of year 5, as the entire$610,000 depreciable basis will have been depreciated ($122,000/year$\times$ 5 years).

The machine is expected to be sold for a salvage value of $75,000, which, since the book value is$0, represents a taxable capital gain of $75,000.

Taxes on this capital gain are $75,000$\times$0.4 (40$30,000, resulting in an after-tax cash inflow from the sale of $45,000 ($75,000 - $30,000).

The initial $25,000 investment in net working capital is recaptured at the end of the project, providing an additional cash inflow.

The total terminal cash flow is the sum of the after-tax salvage value ($45,000) and the recapture of NWC ($25,000), equaling $70,000.

The total cash flow at time 5 is the sum of the final year's differential cash flow ($138,800) and the terminal cash flow ($70,000), resulting in $208,800.

Assuming the project's risk is similar to the firm's overall risk, the project's required rate of return (discount rate) is set equal to the firm's cost of capital, which is 15% in this case.

The project's cash flows are summarized as: an initial outlay of $635,000 at time 0, differential cash flows of$138,800 for years 1 through 4, and a combined cash flow of $208,800 at time 5 (which includes the year 5 differential cash flow and the terminal cash flows).

These cash flows are then used to calculate the Net Present Value (NPV) or other capital budgeting metrics to determine whether the project should be accepted or rejected.

The timeline of cash flows is: Time 0: -$635,000; Time 1:$138,800; Time 2: $138,800; Time 3:$138,800; Time 4: $138,800; Time 5:$208,800.

To calculate NPV, we input the project's cash flows and the discount rate into a financial calculator or spreadsheet. The initial outlay is entered as a negative value at time 0, representing an outflow.

Subsequent annual cash flows are entered for each period. For example, if a project has annual cash flows of $138,800 for years 1 through 4, these are entered sequentially.

The total terminal cash flow, which occurs at the very end of the project's life (e.g., year 5), is also entered. This flow often combines the final operating cash flow with any salvage value or recovery of net working capital.

Once all cash flows and the discount rate (e.g., 15%) are entered, the financial calculator computes the NPV. In our example, with an initial outlay of –$635,000, annual cash flows of$138,800 for years 1-4, and a terminal cash flow of $208,800 in year 5, at a 15$134,918.50.

Mathematically, the NPV is calculated as the sum of the present values of all cash flows, both inflows and outflows, over the project's life, as shown by the formula: $NPV = \sum_{t=0}^{n} \frac{CF_t}{(1+r)^t}$, where $CF_t$ is the cash flow at time $t$, $r$ is the discount rate, and $n$ is the project's life.

A negative Net Present Value, such as –$134,918.50 in our example, indicates that the project is expected to destroy firm value. This means that the present value of the project's expected cash inflows is less than the present value of its expected cash outflows.

Conversely, a positive NPV would suggest that the project is expected to increase firm value, as its inflows outweigh its outflows in present value terms.

Based solely on the NPV criterion, a project with a negative NPV should be rejected because it does not meet the firm's minimum required rate of return and would diminish shareholder wealth.

However, it is crucial to understand that while NPV provides a strong quantitative basis, it is not the only factor in a capital budgeting decision. There are numerous qualitative and strategic issues that can influence the final accept/reject decision, even if the numerical analysis suggests rejection.

Despite these broader considerations, if we rely strictly on the numerical NPV analysis, a project yielding a negative NPV, like the one in our example, would lead to a recommendation for rejection.