Net present value

Net present value (NPV) is a standard method for the financial appraisal of long-term projects. Used for capital budgeting, and widely throughout economics, it measures the excess or shortfall of cash flows, in present value (PV) terms, once financing charges are met. By definition,

NPV = Present value of net cash flows. For its expression, see the formula section below.

Formula

Each cash inflow/outflow is discounted back to its PV. Then they are summed. Therefore

${\mbox{NPV}}=\sum _{t=1}^{n}{\frac {C_{t}}{(1+r)^{t}}}-{C_{0}}$

Where

t - the time of the cash flow

n - the total time of the project

r - the discount rate

$C_{t}$ - the net cash flow (the amount of cash) at time t.

$C_{0}$ - the capital outlay at the beginning of the investment time ( $t$ = 0 )

For more information on how to calculate the PV of a dollar or of a stream of payments, see time value of money.

The discount rate

Choosing an appropriate discount rate is crucial to the NPV calculation. A good practice of choosing the discount rate is to decide the rate which the capital needed for the project could return if invested in an alternative venture. If, for example, the capital required for Project A can earn five percent elsewhere, use this discount rate in the NPV calculation to allow a direct comparison to be made between Project A and the alternative. Obviously, NPV value obtained using variable discount rates with the years of the investment duration is more reflecting to the real situation than that calculated from a constant discount rate for the entire investment duration. Refer to the tutorial article written by Samuel Baker for more detailed relationship between the NPV value and the discount rate.

For some professional investors, their investment funds are committed to target a specified rate o f return. In such cases, that rate of return should be selected as the discount rate for the NPV calculation. In this way, a direct comparison can be made between the profitability of the project and the desired rate of return.

The rate used to discount future cash flows to their present values is a key input of this process. Most firms have a well defined policy regarding their capital structure. So the weighted average cost of capital (after tax) is appropriate for use with all projects. Alternately, higher discount rates can be used for more risky projects. Another method is to apply higher discount rates to cash flows occurring further along the time span, to reflect the yield curve premium for long-term debt.

Reinvestment rate

There are assumptions made about what rate of return is realized on cash that is freed-up before the end of the project. In the NPV model it is assumed to be reinvested at the discount rate used. This is appropriate in the absence of capital rationing. In the IRR model, no assumption is made about the reinvestment rate of free cash, which tends to exaggerate the calculated values. Some people believe that if the firm's reinvestment rate is higher than the Weighted Average Cost of Capital, it becomes, in effect, an opportunity cost and should be used as the discount rate.

What NPV tells

With a particular project, if $C_{t}$ is a positive value, the project is in the status of cash inflow in the time of t. If $C_{t}$ is a negative value, the project is in the status of cash outflow in the time of t. Appropriately risked projects with a positive NPV should be accepted. This does not necessarily mean that they should be undertaken since NPV at the cost of capital may not account for opportunity cost, i.e. comparison with other available investments. In financial theory, if there is a choice between two mutually exclusive alternatives, the one yielding the higher NPV should be selected. The following sums up the NPV's various situations.

If...	It means...	Then...
NPV > 0	the investment would add value to the firm	the project should be accepted
NPV < 0	the investment would subtract value from the firm	the project should be rejected
NPV = 0	the investment would neither gain nor lose value for the firm	the project could be accepted because shareholders obtain required rate of return. This project adds no monetary value. Decision should be based on other criteria, e.g. strategic positioning or other factors not explicitly included in the calculation.

Example

X corporation must decide whether to introduce a new product line. The new product will have startup costs, operational costs, and incoming cash flows over six years. This project will have an immediate (t=0) cash outflow of $100,000 (which might include machinery, and employee training costs). Other cash outflows for years 1-6 are expected to be $5,000 per year. Cash inflows are expected to be $30,000 per year for years 1-6. All cash flows are after-tax, and there are no cash flows expected after year 6. The required rate of return is 10%. The present value (PV) can be calculated for each year:

T=0 -$100,000 / 1.10^0 = -$100,000 PV.

T=1 ($30,000 - $5,000)/ 1.10^1 = $22,727 PV.

T=2 ($30,000 - $5,000)/ 1.10^2 = $20,661 PV.

T=3 ($30,000 - $5,000)/ 1.10^3 = $18,783 PV.

T=4 ($30,000 - $5,000)/ 1.10^4 = $17,075 PV.

T=5 ($30,000 - $5,000)/ 1.10^5 = $15,523 PV.

T=6 ($30,000 - $5,000)/ 1.10^6 = $14,112 PV.

The sum of all these present values is the net present value, which equals $8,881. Since the NPV is greater than zero, the corporation should invest in the project.

The same example in an Excel formulae:

• NPV(rate,net_inflow)+initial_investment
• PV(rate,year_number,yearly_net_inflow)

File:PV example.jpg

File:DiscreteCF.jpg

More realistic problems would need to consider other factors, generally including the calculation of taxes, uneven cash flows, and salvage values as well as the availability of alternate investment oportunities.

Common Pitfalls

If some (or all) of the $C_{t}$ have a negative value, then paradoxical results are possible. For example, if the $C_{t}$ are generally negative late in the project (eg, an industrial or mining project might have clean-up and restoration costs), then an increase in the discount rate can make the project appear more favourable. Some people see this as a problem with NPV. A way to avoid this problem is to include explicit provision for financing any losses after the initial investment, ie, explicitly calculate the cost of financing such losses.

Another common pitfall is to adjust for risk by adding a premium to the discount rate. Whilst a bank might charge a higher rate of interest for a risky project, that does not mean that this is a valid approach to adjusting a net present value for risk, although it can be a reasonable approximation in some specific cases. One reason such an approach may not work well can be seen from the foregoing: if some risk is incurred resulting in some losses, then a discount rate in the NPV will reduce the impact of such losses below their true financial cost. A rigorous approach to risk requires identifying and valuing risks explicitly, e.g. by actuarial or Monte Carlo techniques, and explicitly calculating the cost of financing any losses incurred.

Yet another issue can result from the compounding of the risk premium. R is a composite of the risk free rate and the risk premium. As a result, future cash flows are discounted by both the risk free rate as well as the risk premium and this effect is compounded by each subsequent cash flow. This compounding results in a much lower NPV than might be otherwise calculated. The certainty equivalent model can be used to account for the risk premium without compounding its effect on present value.

Influence of currency system

Currency systems that include demurrage alter the effective cost of capital and lead to an increased NPV emphasis on long term returns. While such currency systems are atypical in the modern world they were prevalent in earlier eras when commodities formed the basis of private currencies.

Alternative capital budgeting methods

• payback period: which measures the time required for the cash inflows to equal the original outlay. It measures risk, not return.
• cost-benefit analysis : which includes issues other than cash, such as time savings.
• real option method : which attempts to value managerial flexibility that is assumed away in NPV.
• internal rate of return (IRR): which calculates the rate of return of a project without making assumptions about the reinvestment of the cash flows (hence internal)
• modified internal rate of return - similar to Internal Rate of Return, but it makes explicit assumptions about the reinvestment of the cash flows. Sometimes called Growth Rate of Return

Applications of NPV

• NPV Methodology, Examples, Limitations.
• Using NPV to calculate share prices.
• Calculating Net Present Value.

See also

• Rate of return on investment
• Capital budgeting
• Cost of capital
• Discounted cash flow
• Internal rate of return
• Real versus nominal value

External links

• Online NPV calculator