# Net present value

In finance, the net present value (NPV) or net present worth (NPW)[1] applies to a series of cash flows occurring at different times. The present value of a cash flow depends on the interval of time between now and the cash flow. It also depends on the discount rate. NPV accounts for the time value of money. It provides a method for evaluating and comparing capital projects or financial products with cash flows spread over time, as in loans, investments, payouts from insurance contracts plus many other applications.

## Interpretation as integral transform

The time-discrete formula of the net present value

${\displaystyle \mathrm {NPV} (i,N)=\sum _{t=0}^{N}{\frac {R_{t}}{(1+i)^{t}}}}$

can also be written in a continuous variation

${\displaystyle \mathrm {NPV} (i)=\int _{t=0}^{\infty }(1+i)^{-t}\cdot r(t)\,dt}$

where

r(t) is the rate of flowing cash given in money per time, and r(t) = 0 when the investment is over.

Net present value can be regarded as Laplace-[6][7] respectively Z-transformed cash flow with the integral operator including the complex number s which resembles to the interest rate i from the real number space or more precisely s = ln(1 + i).

${\displaystyle F(s)=\left\{{\mathcal {L}}f\right\}(s)=\int _{0}^{\infty }e^{-st}f(t)\,dt}$

From this follow simplifications known from cybernetics, control theory and system dynamics. Imaginary parts of the complex number s describe the oscillating behaviour (compare with the pork cycle, cobweb theorem, and phase shift between commodity price and supply offer) whereas real parts are responsible for representing the effect of compound interest (compare with damping).

## Example

A corporation must decide whether to introduce a new product line. The company will have immediate costs of 100,000 at t = 0. Recall, a cost is a negative for outgoing cash flow, thus this cash flow is represented as −100,000. The company assumes the product will provide equal benefits of 10,000 for each of 12 years beginning at t = 1. For simplicity, assume the company will have no outgoing cash flows after the initial 100,000 cost. This also makes the simplifying assumption that the net cash received or paid is lumped into a single transaction occurring on the last day of each year. At the end of the 12 years the product no longer provides any cash flow and is discontinued without any additional costs. Assume that the effective annual discount rate is 10%.

The present value (value at t = 0) can be calculated for each year:

Year Cash flow Present value
T = 0 ${\displaystyle {\frac {-100,000}{(1+0.10)^{0}}}}$ −100,000
T = 1 ${\displaystyle {\frac {10,000}{(1+0.10)^{1}}}}$ 9,090.91
T = 2 ${\displaystyle {\frac {10,000}{(1+0.10)^{2}}}}$ 8,264.46
T = 3 ${\displaystyle {\frac {10,000}{(1+0.10)^{3}}}}$ 7,513.15
T = 4 ${\displaystyle {\frac {10,000}{(1+0.10)^{4}}}}$ 6,830.13
T = 5 ${\displaystyle {\frac {10,000}{(1+0.10)^{5}}}}$ 6,209.21
T = 6 ${\displaystyle {\frac {10,000}{(1+0.10)^{6}}}}$ 5,644.74
T = 7 ${\displaystyle {\frac {10,000}{(1+0.10)^{7}}}}$ 5,131.58
T = 8 ${\displaystyle {\frac {10,000}{(1+0.10)^{8}}}}$ 4,665.07
T = 9 ${\displaystyle {\frac {10,000}{(1+0.10)^{9}}}}$ 4,240.98
T = 10 ${\displaystyle {\frac {10,000}{(1+0.10)^{10}}}}$ 3,855.43
T = 11 ${\displaystyle {\frac {10,000}{(1+0.10)^{11}}}}$ 3,504.94
T = 12 ${\displaystyle {\frac {10,000}{(1+0.10)^{12}}}}$ 3,186.31

The total present value of the incoming cash flows is 68,136.91. The total present value of the outgoing cash flows is simply the 100,000 at time t = 0. Thus:

${\displaystyle \mathrm {NPV} =PV({\text{benefits}})-PV({\text{costs}})}$

In this example:

${\displaystyle \mathrm {NPV} =68,136.91-100,000}$
${\displaystyle \mathrm {NPV} =-31,863.09}$

Observe that as t increases the present value of each cash flow at t decreases. For example, the final incoming cash flow has a future value of 10,000 at t = 12 but has a present value (at t = 0) of 3,186.31. The opposite of discounting is compounding. Taking the example in reverse, it is the equivalent of investing 3,186.31 at t = 0 (the present value) at an interest rate of 10% compounded for 12 years, which results in a cash flow of 10,000 at t = 12 (the future value).

A simplified cash flow model shows the expected cash flow. Each value is calculated with the discount rate. The simplified approach is used in project management simulation SimulTrain.

The importance of NPV becomes clear in this instance. Although the incoming cash flows (10,000 × 12 = 120,000) appear to exceed the outgoing cash flow (100,000), the future cash flows are not adjusted using the discount rate. Thus, the project appears misleadingly profitable. When the cash flows are discounted however, it indicates the project would result in a net loss of 31,863.09. Thus, the NPV calculation indicates that this project should be disregarded because investing in this project is the equivalent of a loss of 31,863.09 at t = 0. The concept of time value of money indicates that cash flows in different periods of time cannot be accurately compared unless they have been adjusted to reflect their value at the same period of time (in this instance, t = 0).[2] It is the present value of each future cash flow that must be determined in order to provide any meaningful comparison between cash flows at different periods of time. There are a few inherent assumptions in this type of analysis:

1. The investment horizon of all possible investment projects considered are equally acceptable to the investor (e.g. a 3-year project is not necessarily preferable vs. a 20-year project.)
2. The 10% discount rate is the appropriate (and stable) rate to discount the expected cash flows from each project being considered. Each project is assumed equally speculative.
3. The shareholders cannot get above a 10% return on their money if they were to directly assume an equivalent level of risk. (If the investor could do better elsewhere, no projects should be undertaken by the firm, and the excess capital should be turned over to the shareholder through dividends and stock repurchases.)

More realistic problems would also need to consider other factors, generally including: smaller time buckets, the calculation of taxes (including the cash flow timing), inflation, currency exchange fluctuations, hedged or unhedged commodity costs, risks of technical obsolescence, potential future competitive factors, uneven or unpredictable cash flows, and a more realistic salvage value assumption, as well as many others.

A more simple example of the net present value of incoming cash flow over a set period of time, would be winning a Powerball lottery of $500 million. If one does not select the "CASH" option they will be paid$25,000,000 per year for 20 years, a total of $500,000,000, however, if one does select the "CASH" option, they will receive a one-time lump sum payment of approximately$285 million, the NPV of \$500,000,000 paid over time. See "other factors" above that could affect the payment amount. Both scenarios are before taxes.

## Common pitfalls

• If, for example, the Rt are generally negative late in the project (e.g., an industrial or mining project might have clean-up and restoration costs), then at that stage the company owes money, so a high discount rate is not cautious but too optimistic. 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, that is, 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 following: if some risk is incurred resulting in some losses, then a discount rate in the NPV will reduce the effect 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.[citation needed]
• Another issue with relying on NPV is that it does not provide an overall picture of the gain or loss of executing a certain project. To see a percentage gain relative to the investments for the project, usually, Internal rate of return or other efficiency measures are used as a complement to NPV.
• Non-specialist users frequently make the error of computing NPV based on cash flows after interest. This is wrong because it double counts the time value of money. Free cash flow should be used as the basis for NPV computations.

## History

Net present value as a valuation methodology dates at least to the 19th century. Karl Marx refers to NPV as fictitious capital, and the calculation as "capitalising," writing:[8]

The forming of a fictitious capital is called capitalising. Every periodically repeated income is capitalised by calculating it on the average rate of interest, as an income which would be realised by a capital at this rate of interest.

In mainstream neo-classical economics, NPV was formalized and popularized by Irving Fisher, in his 1907 The Rate of Interest and became included in textbooks from the 1950s onwards, starting in finance texts.[9][10]

## Alternative capital budgeting methods

• Adjusted present value (APV): adjusted present value, is the net present value of a project if financed solely by ownership equity plus the present value of all the benefits of financing.
• Accounting rate of return (ARR): a ratio similar to IRR and MIRR
• Cost-benefit analysis: which includes issues other than cash, such as time savings.
• Internal rate of return (IRR): which calculates the rate of return of a project while disregarding the absolute amount of money to be gained.
• Modified internal rate of return (MIRR): similar to IRR, but it makes explicit assumptions about the reinvestment of the cash flows. Sometimes it is called Growth Rate of Return.
• Payback period: which measures the time required for the cash inflows to equal the original outlay. It measures risk, not return.
• Real option: which attempts to value managerial flexibility that is assumed away in NPV.
• Equivalent annual cost (EAC): a capital budgeting technique that is useful in comparing two or more projects with different lifespans.