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The net present value (NPV) or net present worth (NPW) 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 The time value of money is the widely accepted conjecture that there is greater benefit to receiving a sum of money Money is any item or verifiable record that is generally accepted as payment for goods and services and repayment of debts, ...
. 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.
Time value of money The time value of money is the widely accepted conjecture that there is greater benefit to receiving a sum of money Money is any item or verifiable record that is generally accepted as payment for goods and services and repayment of debts, ...
dictates that time affects the value of cash flows. For example, a lender may offer 99 cents for the promise of receiving $1.00 a month from now, but the promise to receive that same dollar 20 years in the future would be worth much less today to that same person (lender), even if the payback in both cases was equally certain. This decrease in the current value of future cash flows is based on a chosen
rate of return In finance Finance is a term for the management, creation, and study of money In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in t ...
(or discount rate). If for example there exists a
time series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

time series
of identical cash flows, the cash flow in the present is the most valuable, with each future cash flow becoming less valuable than the previous cash flow. A cash flow today is more valuable than an identical cash flow in the futureBerk, DeMarzo, and Stangeland, p. 94. because a present flow can be invested immediately and begin earning returns, while a future flow cannot. NPV is determined by calculating the costs (negative cash flows) and benefits (positive cash flows) for each period of an investment. After the cash flow for each period is calculated, the present value (PV) of each one is achieved by discounting its future value (see
Formula In science Science () is a systematic enterprise that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe."... modern science is a discovery as well a ...

Formula
) at a periodic rate of return (the rate of return dictated by the market). NPV is the sum of all the discounted future cash flows. Because of its simplicity, NPV is a useful tool to determine whether a project or investment will result in a net profit or a loss. A positive NPV results in profit, while a negative NPV results in a loss. The NPV measures the excess or shortfall of cash flows, in present value terms, above the cost of funds. In a theoretical situation of unlimited
capital budgeting Capital most commonly refers to: * Capital letter Letter case (or just case) is the distinction between the letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (or more formally ''minusc ...
, a company should pursue every investment with a positive NPV. However, in practical terms a company's capital constraints limit investments to projects with the highest NPV whose cost cash flows, or initial cash investment, do not exceed the company's capital. NPV is a central tool in
discounted cash flow In finance Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money available ...
(DCF) analysis and is a standard method for using the
time value of money The time value of money is the widely accepted conjecture that there is greater benefit to receiving a sum of money Money is any item or verifiable record that is generally accepted as payment for goods and services and repayment of debts, ...
to appraise long-term projects. It is widely used throughout
economics Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interact ...

economics
,
financial analysis Financial analysis (also referred to as financial statement analysis or accounting analysis or Analysis of finance) refers to an assessment of the viability, stability, and profitability of a business, sub-business or project. It is performed by p ...

financial analysis
, and
financial accounting Financial accounting is the field of accounting Accounting or Accountancy is the measurement, processing, and communication of financial and non financial information about economic entity, economic entities such as businesses and corporat ...
. In the case when all future cash flows are positive, or incoming (such as the
principal Principal may refer to: Title or rank * Principal (academia) The principal is the chief executive and the chief academic officer of a university A university ( la, universitas, 'a whole') is an educational institution, institution of higher ...

principal
and coupon payment of a
bond Bond or bonds may refer to: Common meanings * Bond (finance) In finance Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management ...
) the only outflow of cash is the purchase price, the NPV is simply the PV of future cash flows minus the purchase price (which is its own PV). NPV can be described as the "difference amount" between the sums of discounted cash inflows and cash outflows. It compares the present value of money today to the present value of money in the future, taking inflation and returns into account. The NPV of a sequence of cash flows takes as input the cash flows and a discount rate or discount curve and outputs a present value, which is the current fair price. The converse process in discounted cash flow (DCF) analysis takes a sequence of cash flows and a price as input and as output the discount rate, or
internal rate of return Internal rate of return (IRR) is a method of calculating an investment Investment is the dedication of an asset to attain an increase in value over a period of time. Investment requires a sacrifice of some present asset, such as time, money, ...
(IRR) which would yield the given price as NPV. This rate, called the
yield Yield may refer to: Measures of output/function Computer science * Yield (multithreading) is an action that occurs in a computer program during multithreading * See generator (computer programming) Physics/chemistry * Yield (chemistry), the amou ...
, is widely used in bond trading. Many computer-based
spreadsheet A spreadsheet is a computer application for organization, analysis, and storage of data in table (information), tabular form. Spreadsheets were developed as computerized analogs of paper accounting Worksheet#Accounting, worksheets. The program op ...

spreadsheet
programs have built-in formulae for PV and NPV.


Formula

Each cash inflow/outflow is
discounted Discounting is a financial mechanism in which a debtor A debtor or debitor is a legal entity (legal person) that owes a debt Debt is an obligation that requires one party, the debtor, to pay money or other agreed-upon value to another p ...
back to its present value (PV). Then all are summed such that NPV is the sum of all terms: :\frac where :t is the time of the cash flow :i is the discount rate, i.e. the return that could be earned per unit of time on an investment with similar risk :R_t is the net cash flow i.e. cash inflow – cash outflow, at time ''t''. For educational purposes, R_0 is commonly placed to the left of the sum to emphasize its role as (minus) the investment. The result of this formula is multiplied with the Annual Net cash in-flows and reduced by Initial Cash outlay the present value, but in cases where the cash flows are not equal in amount, the previous formula will be used to determine the present value of each cash flow separately. Any cash flow within 12 months will not be discounted for NPV purpose, nevertheless the usual initial investments during the first year ''R''0 are summed up a negative cash flow. Given the (period, cash flow) pairs (t, R_t) where N is the total number of periods, the net present value \mathrm is given by: :\mathrm(i, N) = \sum_^N \frac For constant cash flow R, the net present value \mathrm is a finite
geometric series In mathematics, a geometric series (mathematics), series is the sum of an infinite number of Summand, terms that have a constant ratio between successive terms. For example, 1/2 + 1/4 + 1/8 + 1/16 + · · ·, the series :\frac \,+\, \frac \,+\, ...
and is given by: : \mathrm(i, N, R) = R \left( \frac \right), \quad i \ne 0 Inclusion of the R_0 term is important in the above formulae. A typical capital project involves a large negative R_0 cashflow (the initial investment) with positive future cashflows (the return on the investment). A key assessment is whether, for a given discount rate, the NPV is positive (profitable) or negative (loss-making). The IRR is the discount rate for which the NPV is exactly 0.


The discount rate

The rate used to discount future cash flows to the present value is a key variable of this process. A firm's
weighted average cost of capital The weighted average cost of capital (WACC) is the rate that a company is expected to pay on average to all its security holders to finance its assets. The WACC is commonly referred to as the firm's cost of capital In economics Economics ...
(after tax) is often used, but many people believe that it is appropriate to use higher discount rates to adjust for risk, opportunity cost, or other factors. A variable discount rate with higher rates applied to cash flows occurring further along the time span might be used to reflect the
yield curve In finance Finance is a term for the management, creation, and study of money In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated ...

yield curve
premium for long-term debt. Another approach to choosing the discount rate factor 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 5% elsewhere, use this discount rate in the NPV calculation to allow a direct comparison to be made between Project A and the alternative. Related to this concept is to use the firm's reinvestment rate. Re-investment rate can be defined as the rate of return for the firm's investments on average. When analyzing projects in a capital constrained environment, it may be appropriate to use the reinvestment rate rather than the firm's weighted average cost of capital as the discount factor. It reflects opportunity cost of investment, rather than the possibly lower cost of capital. An NPV calculated using variable discount rates (if they are known for the duration of the investment) may better reflect the situation than one 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 and the discount rate. For some professional investors, their investment funds are committed to target a specified rate of 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. To some extent, the selection of the discount rate is dependent on the use to which it will be put. If the intent is simply to determine whether a project will add value to the company, using the firm's weighted average cost of capital may be appropriate. If trying to decide between alternative investments in order to maximize the value of the firm, the corporate reinvestment rate would probably be a better choice. Using variable rates over time, or discounting "guaranteed" cash flows differently from "at risk" cash flows, may be a superior methodology but is seldom used in practice. Using the discount rate to adjust for risk is often difficult to do in practice (especially internationally) and is difficult to do well. An alternative to using discount factor to adjust for risk is to explicitly correct the cash flows for the risk elements using
rNPV In finance Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money available ...
or a similar method, then discount at the firm's rate.


Use in decision making

NPV is an indicator of how much value an investment or project adds to the firm. With a particular project, if R_t is a positive value, the project is in the status of positive cash inflow in the time of ''t''. If R_t is a negative value, the project is in the status of discounted cash outflow in the time o ''t''. Appropriately risked projects with a positive NPV could 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 In microeconomic theory Microeconomics (from Greek prefix ''mikro-'' meaning "small" + ''economics'') is a branch of economics Economics () is the social science that studies how people interact with value; in particular, the Produ ...
, 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. A positive net present value indicates that the projected earnings generated by a project or investment (in present dollars) exceeds the anticipated costs (also in present dollars). This concept is the basis for the Net Present Value Rule, which dictates that the only investments that should be made are those with positive NPVs. An investment with a positive NPV is profitable, but one with a negative NPV will not necessarily result in a net loss: it is just that the internal rate of return of the project falls below the required rate of return.


Interpretation as integral transform

The time-discrete formula of the net present value : \mathrm(i,N) = \sum_^ \frac can also be written in a continuous variation : \mathrm(i) = \int_^\infty (1+i)^ \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- respectively
Z-transformed
Z-transformed
cash flow with the
integral operator An integral operator is an operator that involves integration. Special instances are: * The operator of integration itself, denoted by the integral symbol The integral symbol: :∫ (Unicode Unicode is an information technology Technical sta ...
including the complex number ''s'' which resembles to the interest rate ''i'' from the real number space or more precisely ''s'' = ln(1 + ''i''). : F(s) = \left\(s) = \int_0^\infty e^ f(t) \,dt From this follow simplifications known from
cybernetics Cybernetics is a wide-ranging field concerned with regulatory and purposive systems A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influen ...

cybernetics
,
control theory Control theory deals with the control of dynamical system In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contai ...
and
system dynamics System dynamics (SD) is an approach to understanding the nonlinear In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space ...

system dynamics
. Imaginary parts of the
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
''s'' describe the oscillating behaviour (compare with the
pork cycle In economics, the term pork cycle, hog cycle, or cattle cycle describes the phenomenon of cyclical fluctuations of supply (economics), supply and prices in livestock markets. It was first observed in 1925 in pig Market (economics), markets in the ...
,
cobweb theorem
cobweb theorem
, and
phase shift In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...

phase shift
between commodity price and supply offer) whereas real parts are responsible for representing the effect of compound interest (compare with
damping Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples incl ...
).


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: 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: : \mathrm = PV(\text) - PV(\text) In this example: : \mathrm = 68,136.91 - 100,000 : \mathrm = -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). 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). 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: # 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.) # 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. # 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 flow A cash flow is a real or virtual movement of money Money is any item or verifiable record that is generally accepted as payment for goods and services and repayment of debts, such as taxes, in a particular country or socio-economic contex ...
s, and a more realistic
salvage value ''Residual value'' is one of the constituents of a leasing A lease is a contract A contract is a legally binding document between at least two parties that defines and governs the rights and duties of the parties to an agreement. A contract is ...
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 ''R''''t'' 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 Monte Carlo (; ; french: Monte-Carlo , or colloquially ''Monte-Carl'' ; lij, label= Monégasque, Munte Carlu; ) is officially an administrative area of the Principality of Monaco Monaco (; ), officially the Principality of Monaco (french: ...
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 The risk-free rate of return, usually shortened to the risk-free rate, is the rate of return of a hypothetical investment with scheduled payment(s) over a fixed period of time that is assumed to meet all payment obligations. Since the risk-free ...
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 For an individual, a risk premium is the minimum amount of money by which the expected return on a risky asset (such as stock) must exceed the known return on a Risk-free bond, risk-free asset (such as a Treasury bond) in order to induce an individ ...
model can be used to account for the risk premium without compounding its effect on present value. * 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 Internal rate of return (IRR) is a method of calculating an investment Investment is the dedication of an asset to attain an increase in value over a period of time. Investment requires a sacrifice of some present asset, such as time, money, ...
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 Karl Heinrich Marx (; 5 May 1818 – 14 March 1883) was a German philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, M ...

Karl Marx
refers to NPV as
fictitious capital Fictitious capital (German: ''fiktives Kapital'') is a concept used by Karl Marx in his critique of political economy. It is introduced in chapter 25 of the Capital, Volume III, third volume of Capital. Fictitious capital contrasts with what Mar ...
, and the calculation as "capitalising," writing: In
mainstream The mainstream is the prevalent current thought In their most common sense, the terms thought and thinking refer to conscious cognitive processes that can happen independently of sensory stimulation. Their most paradigmatic forms are judging, r ...
neo-classical economics Neoclassical economics is an approach to economics in which the production, consumption and valuation (pricing) of goods and services are driven by the supply and demand In microeconomics Microeconomics is a branch of that studies ...
, NPV was formalized and popularized by
Irving Fisher Irving Fisher (February 27, 1867 – April 29, 1947) was an American economist An economist is a professional and practitioner in the social science Social science is the branch The branches and leaves of a tree. A branch ( or , ...

Irving Fisher
, in his 1907 ''The Rate of Interest'' and became included in textbooks from the 1950s onwards, starting in finance texts.


Alternative capital budgeting methods

*
Adjusted present value Adjusted present value (APV) is a valuation method introduced in 1974 by Stewart Myers. The idea is to value the project as if it were all equity financed ("unleveraged"), and to then add the present value In economics Economics ...
(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 Accounting or Accountancy is the measurement, processing, and communication of financial and non financial information about economic entity, economic entities such as businesses and corporations. Accounting, which has been called the "language ...
(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 Internal rate of return (IRR) is a method of calculating an investment Investment is the dedication of an asset to attain an increase in value over a period of time. Investment requires a sacrifice of some present asset, such as time, money, ...
(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 Payback period in capital budgeting refers to the time required to Recoupment, recoup the funds expended in an investment, or to reach the Break-even (economics), break-even point. Farris, Paul W.; Neil T. Bendle; Phillip E. Pfeifer; David J. Reibst ...
: which measures the time required for the cash inflows to equal the original outlay. It measures risk, not return. *
Real option Real options valuation, also often termed real options analysis,Adam Borison (Stanford University)''Real Options Analysis: Where are the Emperor's Clothes?'' (ROV or ROA) applies option (finance), option Valuation of options, valuation techniques ...
: 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.


See also

*
Profitability index Profitability index (PI), also known as profit investment ratio (PIR) and value investment ratio (VIR), is the ratio of payoff to investment of a proposed project. It is a useful tool for ranking projects because it allows you to quantify the amo ...


References

{{DEFAULTSORT:Net Present Value (NPV) Mathematical finance Investment Engineering economics Management accounting Capital budgeting Valuation (finance)