# NPV

Net present value (NPV) measures the net increase in a company’s value resulting from an investment. It equals the difference between the (a) present value of future cash flows of the investment estimated based on an appropriate discount rate and (b) the amount of total initial investment required. Only those investment opportunities that generate a positive NPV are considered and those that maximizes NPV are ultimately selected for investment.

Net present value can be calculated using the Excel NPV funtion or XPNV function or by manually discounting each cash flow to time zero and subtracting the initial investment.

NPV calculation can be expressed algebraically as follows:

$$NPV=\ \frac{{\rm CF}_1}{{(1+r)}^1}+\frac{CF2}{{(1+r)}^2}+\frac{CF3}{{(1+r)}^3}+\ldots+\ \frac{CFn}{\left(1+r\right)^n}-I$$

Where CF stands for net incremental cash flow in a period, r stands for the discount rate and I refers to the initial investment.

Only such cash flows should be considered which are affected by the decision to invest in the project. Costs already incurred, called sunk costs, shouldn't be included.

The net cash flows are the after-tax net operating cash flows of the project which can be worked out as follows:

$$Net\ Cash\ Flows=CI\ -\ CO\ – T$$

CI equals cash inflow, CO stands for cash outflow and T stands for tax amount. Taxes can be worked out by applying the tax rate (t) to the net income which equals cash inflows less operating cash outflows less depreciation expense.

$$Taxes=(CI\ -\ CO\ -\ D)\times t$$

The following formula can be used to directly work out net cash flows:

$$Net\ Cash\ Flows\ =\ (CI\ -\ CO\ -\ D)\times(1\ -\ t)\ +\ D$$

We have first subtracted depreciation to find the net income and then multiplied by (1 – Tax Rate) to get the after-tax income and then added back depreciation to get net cash flows. Depreciation is calculated based on straight-line method by dividing the depreciable amount ($530,000 -$150,000) by the useful life (4).

The initial investment outlay equals total initial investment in new equipment, test runs, etc. minus the after-tax proceeds of any equipment that can be disposed off or used for another project.

If the cash flows are constant each period, which is rare, we can use the present value of annuity formula to work out the present value of future cash flows.

$$NPV\ (Constant\ net\ CFs)\\=Net\ Cash\ Flows\times\frac{1-{(1+r)}^{-n}}{r}-I$$

Where r is the appropriate discount rate, n is the project duration i.e. number of cash flows and I is initial investment.

However, where the cash flows are different, we need to manually discount each cash flow to t=0 and them sum those up to find out present value of all cash flows and then subtract the amount of initial outlay.

Our manual calculation of net present value and Excel NPV assumes that cash flows occur at the period-end. If we want to determine net present value based on the exact date those cash flows occur, we can use Excel XNPV function.

## Example

We will use the same data as in the Example for internal rate of return (IRR). The data is reproduced below:

Your company is interested in a project that will generate cash inflows of $300,000,$350,000, $370,000,$330,000 at the end of each year for 4 years. Cash outflows before taxes amount to 50% of the cash inflows. The project will require purchase of fixed assets of $550,000 which is to be depreciated using straight-line method with a salvage value of$150,000. After the initiation of project, existing equipment with book value of $80,000 can be sold at$100,000. The hurdle rate applicable to the project is 12% and the company’s tax rate is 40%. Working capital of $130,000 is required at time 0 which shall be released at the end of the project. Let’s calculate the net present value. Initial investment equals investment in new equipment plus working capital needed minus after-tax proceeds of any equipment which can be sold or used elsewhere. Initial investment in the example above equals$588,000:

$$Initial\ Investment\\=550,000+130,000\\−(100,000−(100,000−80,000)\times40\%)\\=588,000$$

Net cash flows for Year 1 equals $130,000: $$Net\ Cash\ Flows\ (Year\ 1)\ \\=\ (300,000\ -\ 300,000\times50\%\ -\ 100,000)\times(1\ -\ 40\%)\ \\+\ 100,000\\=130,000$$ Using the same equation, net cash flows for Year 2, Year 3 and Year 4 equal$145,000; $151,000 and$139,000.

Terminal cash flows at the end of project equal the salvage value of equipment plus recoupment of working capital which sums up to \$280,000.

Now, we have a complete stream of project cash flows and initial investment and we can work out NPV using the 12% hurdle rate:

$$NPV\\=\frac{130,000}{{(1+12\%)}^1}+\frac{145,000}{{(1+12\%)}^2}+\frac{151,000}{{(1+12\%)}^3}\\+\frac{139,000+280,000}{\left(1+12\%\right)^4}-588,000\\=17,425$$

## Decision Rule

Since the project’s net present value is higher than zero, it is a good candidate for investment.

Net present value should be used together with other capital budgeting tools such as internal rate of return, payback period and profitability index. Consideration should be given to the capital rationing process which depends on the company's capital budget. Projects should be selected based on their collective net present value given a specific capital budget.

For risk analysis purposes, sensitivity analysis and scenario analysis should be performed.