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The discounted cash flow (DCF) analysis is a method in
finance Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline ...
of valuing a
security Security is protection from, or resilience against, potential harm (or other unwanted coercive change) caused by others, by restraining the freedom of others to act. Beneficiaries (technically referents) of security may be of persons and socia ...
, project, company, or
asset In financial accounting, an asset is any resource owned or controlled by a business or an economic entity. It is anything (tangible or intangible) that can be used to produce positive economic value. Assets represent value of ownership that c ...
using the concepts of 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 now rather than an identical sum later. It may be seen as an implication of the later-developed concept of time preference. The ...
. Discounted
cash flow A cash flow is a real or virtual movement of money: *a cash flow in its narrow sense is a payment (in a currency), especially from one central bank account to another; the term 'cash flow' is mostly used to describe payments that are expected ...
analysis is widely used in investment finance,
real estate development Real estate development, or property development, is a business process, encompassing activities that range from the renovation and re- lease of existing buildings to the purchase of raw land and the sale of developed land or parcels to oth ...
, corporate financial management and patent valuation. It was used in industry as early as the 1700s or 1800s, widely discussed in financial economics in the 1960s, and became widely used in U.S. courts in the 1980s and 1990s.

# Application

To apply the method, all future cash flows are estimated and discounted by using
cost of capital In economics and accounting, the cost of capital is the cost of a company's funds (both debt and equity), or from an investor's point of view is "the required rate of return on a portfolio company's existing securities". It is used to evaluate ne ...
to give their
present value In economics and finance, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has inte ...
s (PVs). The sum of all future cash flows, both incoming and outgoing, is the
net present value 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 ...
(NPV), which is taken as the value of the cash flows in question; see aside. For further context see valuation overview; and for the mechanics see valuation using discounted cash flows, which includes modifications typical for
startup A startup or start-up is a company or project undertaken by an entrepreneur to seek, develop, and validate a scalable business model. While entrepreneurship refers to all new businesses, including self-employment and businesses that never inten ...
s,
private equity In the field of finance, the term private equity (PE) refers to investment funds, usually limited partnerships (LP), which buy and restructure financially weak companies that produce goods and provide services. A private-equity fund is both a t ...
and
venture capital Venture capital (often abbreviated as VC) is a form of private equity financing that is provided by venture capital firms or funds to startups, early-stage, and emerging companies that have been deemed to have high growth potential or which ...
,
corporate finance Corporate finance is the area of finance that deals with the sources of funding, the capital structure of corporations, the actions that managers take to increase the value of the firm to the shareholders, and the tools and analysis used to ...
"projects", and
mergers and acquisitions Mergers and acquisitions (M&A) are business transactions in which the ownership of companies, other business organizations, or their operating units are transferred to or consolidated with another company or business organization. As an aspec ...
. Using DCF analysis to compute the NPV takes as input cash flows and a discount rate and gives as output a present value. The opposite process takes cash flows and a price (present value) as inputs, and provides as output the discount rate; this is used in bond markets to obtain the yield.

# History

Discounted cash flow calculations have been used in some form since money was first lent at interest in ancient times. Studies of ancient Egyptian and
Babylonian mathematics Babylonian mathematics (also known as ''Assyro-Babylonian mathematics'') are the mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the centuries following the fall of Babylon in 539 BC. Babyl ...
suggest that they used techniques similar to discounting future cash flows. Modern discounted cash flow analysis has been used since at least the early 1700s in the UK coal industry. Discounted cash flow valuation is differentiated from the accounting
book value In accounting, book value is the value of an asset according to its balance sheet account balance. For assets, the value is based on the original cost of the asset less any depreciation, amortization or impairment costs made against the asset. T ...
, which is based on the amount paid for the asset. Following the
stock market crash of 1929 The Wall Street Crash of 1929, also known as the Great Crash, was a major American stock market crash that occurred in the autumn of 1929. It started in September and ended late in October, when share prices on the New York Stock Exchange coll ...
, discounted cash flow analysis gained popularity as a valuation method for
stock In finance, stock (also capital stock) consists of all the shares by which ownership of a corporation or company is divided.Longman Business English Dictionary: "stock - ''especially AmE'' one of the shares into which ownership of a compan ...
s.
Irving Fisher Irving Fisher (February 27, 1867 – April 29, 1947) was an American economist, statistician, inventor, eugenicist and progressive social campaigner. He was one of the earliest American neoclassical economists, though his later work on debt d ...
in his 1930 book ''The Theory of Interest'' and
John Burr Williams John Burr Williams (November 27, 1900 – September 15, 1989) was an American economist, recognized as an important figure in the field of fundamental analysis, and for his analysis of stock prices as reflecting their " intrinsic value". He is ...
's 1938 text '' The Theory of Investment Value'' first formally expressed the DCF method in modern economic terms.

# Mathematics

## Discounted cash flows

The discounted cash flow formula is derived from the present value formula for calculating the time value of money :$DCF = \frac + \frac + \dotsb + \frac$ and
compounding In the field of pharmacy, compounding (performed in compounding pharmacies) is preparation of a custom formulation of a medication to fit a unique need of a patient that cannot be met with commercially available products. This may be done for me ...
returns: :$FV = DCF \cdot \left(1+r\right)^n$. Thus the discounted present value (for one cash flow in one future period) is expressed as: :$DPV = \frac$ where * ''DPV'' is the discounted present value of the future cash flow (''FV''), or ''FV'' adjusted for the delay in receipt; * ''FV'' is the
nominal value In economics, nominal value is measured in terms of money, whereas real value is measured against goods or services. A real value is one which has been adjusted for inflation, enabling comparison of quantities as if the prices of goods had not ...
of a cash flow amount in a future period (see Mid-year adjustment); * ''r'' is the
interest rate An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...
or discount rate, which reflects the cost of tying up
capital Capital may refer to: Common uses * Capital city, a municipality of primary status ** List of national capital cities * Capital letter, an upper-case letter Economics and social sciences * Capital (economics), the durable produced goods used f ...
and may also allow for the risk that the payment may not be received in full; * ''n'' is the time in years before the future cash flow occurs. Where multiple cash flows in multiple time periods are discounted, it is necessary to sum them as follows: :$DPV = \sum_^ \frac$ for each future cash flow (''FV'') at any time period (''t'') in years from the present time, summed over all time periods. The sum can then be used as a
net present value 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 ...
figure. If the amount to be paid at time 0 (now) for all the future cash flows is known, then that amount can be substituted for ''DPV'' and the equation can be solved for ''r'', that is the
internal rate of return Internal rate of return (IRR) is a method of calculating an investment’s rate of return. The term ''internal'' refers to the fact that the calculation excludes external factors, such as the risk-free rate, inflation, the cost of capital, or fi ...
. All the above assumes that the interest rate remains constant throughout the whole period. If the cash flow stream is assumed to continue indefinitely, the finite forecast is usually combined with the assumption of constant cash flow growth beyond the discrete projection period. The total value of such cash flow stream is the sum of the finite discounted cash flow forecast and the Terminal value (finance).

## Continuous cash flows

For continuous cash flows, the summation in the above formula is replaced by an integration: :$DPV= \int_0^T FV\left(t\right) \, e^ dt = \int_0^T \frac \, dt\,,$ where $FV\left(t\right)$ is now the ''rate'' of cash flow, and $\lambda = \ln\left(1+r\right)$.

# Discount rate

The act of discounting future cash flows asks "how much money would have to be invested currently, at a given rate of return, to yield the forecast cash flow, at its future date?" In other words, discounting returns the
present value In economics and finance, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has inte ...
of future cash flows, where the rate used is the cost of capital that ''appropriately'' reflects the risk, and timing, of the cash flows. This "required return" thus incorporates: #
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 now rather than an identical sum later. It may be seen as an implication of the later-developed concept of time preference. The ...
( risk-free rate) – according to the theory of
time preference In economics, time preference (or time discounting, delay discounting, temporal discounting, long-term orientation) is the current relative valuation placed on receiving a good or some cash at an earlier date compared with receiving it at a later ...
, investors would rather have cash immediately than having to wait and must therefore be compensated by paying for the delay. #
Risk premium A risk premium is a measure of excess return that is required by an individual to compensate being subjected to an increased level of risk. It is used widely in finance and economics, the general definition being the expected risky return less ...
– reflects the extra return investors demand because they want to be compensated for the risk that the cash flow might not materialize after all. For the latter, various models have been developed, where the premium is (typically) calculated as a function of the asset's performance with reference to some macroeconomic variable - for example, the CAPM compares the asset's historical returns to the " overall market's"; see Capital asset pricing model#Asset-specific required return and Asset pricing#General Equilibrium Asset Pricing. An alternate, although less common approach, is to apply a "fundamental valuation" method, such as the " T-model", which instead relies on accounting information. (Other methods of discounting, such as
hyperbolic discounting In economics, hyperbolic discounting is a time-''inconsistent'' model of delay discounting. It is one of the cornerstones of behavioral economics and its brain-basis is actively being studied by neuroeconomics researchers. According to the disco ...
, are studied in academia and said to reflect intuitive decision-making, but are not generally used in industry. In this context the above is referred to as "exponential discounting".) Note that the terminology " expected return", although formally the mathematical expected value, is often used interchangeably with the above, where "expected" means "required" or "demanded" in the corresponding sense. The method may also be modified by industry, for example different formulae have been proposed when choosing a discount rate in a healthcare setting.

# Methods of appraisal of a company or project

For these valuation purposes, a number of different DCF methods are distinguished today, some of which are outlined below. The details are likely to vary depending on the
capital structure In corporate finance, capital structure refers to the mix of various forms of external funds, known as capital, used to finance a business. It consists of shareholders' equity, debt (borrowed funds), and preferred stock, and is detailed in the ...
of the company. However the assumptions used in the appraisal (especially the equity discount rate and the projection of the cash flows to be achieved) are likely to be at least as important as the precise model used. Both the income stream selected and the associated
cost of capital In economics and accounting, the cost of capital is the cost of a company's funds (both debt and equity), or from an investor's point of view is "the required rate of return on a portfolio company's existing securities". It is used to evaluate ne ...
model determine the valuation result obtained with each method. (This is one reason these valuation methods are formally referred to as the Discounted Future Economic Income methods.) The below is offered as a high-level treatment; for the components / steps of business modeling here, see .

## Equity-approach

* Flows to equity approach (FTE) ** Discount the cash flows available to the holders of equity capital, after allowing for cost of servicing debt capital ** Advantages: Makes explicit allowance for the cost of debt capital ** Disadvantages: Requires judgement on choice of discount rate

## Entity-approach

*
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 of the tax shield of debt – and other ...
approach (APV) ** Discount the cash flows before allowing for the debt capital (but allowing for the tax relief obtained on the debt capital) ** Advantages: Simpler to apply if a specific project is being valued which does not have earmarked debt capital finance ** Disadvantages: Requires judgement on choice of discount rate; no explicit allowance for cost of debt capital, which may be much higher than a risk-free rate *
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. Importantly, it is dictated by ...
approach (WACC) ** Derive a weighted cost of the capital obtained from the various sources and use that discount rate to discount the cash flows from the project ** Advantages: Overcomes the requirement for debt capital finance to be earmarked to particular projects ** Disadvantages: Care must be exercised in the selection of the appropriate income stream. The net cash flow to total invested capital is the generally accepted choice. * Total cash flow approach (TCF) ** This distinction illustrates that the Discounted Cash Flow method can be used to determine the value of various business ownership interests. These can include equity or debt holders. ** Alternatively, the method can be used to value the company based on the value of total invested capital. In each case, the differences lie in the choice of the income stream and discount rate. For example, the net cash flow to total invested capital and WACC are appropriate when valuing a company based on the market value of all invested capital.

# Shortcomings

The following difficulties are identified with the application of DCF in valuation: # Forecast reliability: Traditional DCF models assume we can accurately forecast revenue and earnings 3–5 years into the future. But studies have shown that growth is neither predictable nor persistent. (See Stock valuation#Growth rate and Sustainable growth rate#From a financial perspective.)
In other terms, using DCF models is problematic due to the
problem of induction First formulated by David Hume, the problem of induction questions our reasons for believing that the future will resemble the past, or more broadly it questions predictions about unobserved things based on previous observations. This infere ...
, i.e. presupposing that a sequence of events in the future will occur as it always has in the past. Colloquially, in the world of finance, the problem of induction is often simplified with the common phrase: past returns are not indicative of future results. In fact, the SEC demands that all mutual funds use this sentence to warn their investors.
This observation has led some to conclude that DCF models should only be used to value companies with steady cash flows. For example, DCF models are widely used to value mature companies in stable industry sectors, such as utilities. For industries that are especially unpredictable and thus harder to forecast, DCF models can prove especially challenging. Industry Examples: #* Real Estate: Investors use DCF models to value commercial real estate development projects. This practice has two main shortcomings. First, the discount rate assumption relies on the market for competing investments at the time of the analysis, which may not persist into the future. Second, assumptions about ten-year income increases are usually based on historic increases in the market rent. Yet the cyclical nature of most real estate markets is not factored in. Most real estate loans are made during boom real estate markets and these markets usually last fewer than ten years. In this case, due to the problem of induction, using a DCF model to value commercial real estate during any but the early years of a boom market can lead to overvaluation. #* Early-stage Technology Companies: In valuing startups, the DCF method can be applied a number of times, with differing assumptions, to assess a range of possible future outcomes—such as the best, worst and mostly likely case scenarios. Even so, the lack of historical company data and uncertainty about factors that can affect the company's development make DCF models especially difficult for valuing startups. There is a lack of credibility regarding future cash flows, future cost of capital, and the company's growth rate. By forecasting limited data into an unpredictable future, the problem of induction is especially pronounced. # Discount rate estimation: Traditionally, DCF models assume that the
capital asset pricing model In finance, the capital asset pricing model (CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a well-diversified portfolio. The model takes into acc ...
can be used to assess the riskiness of an investment and set an appropriate discount rate. Some economists, however, suggest that the capital asset pricing model has been empirically invalidated. various other models are proposed (see
asset pricing In financial economics, asset pricing refers to a formal treatment and development of two main pricing principles, outlined below, together with the resultant models. There have been many models developed for different situations, but correspo ...
), although all are subject to some theoretical or empirical criticism. # Input-output problem: DCF is merely a mechanical valuation tool, which makes it subject to the principle " garbage in, garbage out." Small changes in inputs can result in large changes in the value of a company. This is especially the case with terminal values, which make up a large proportion of the Discounted Cash Flow's final value. # Missing variables: Traditional DCF calculations only consider the financial costs and benefits of a decision. They do not include the environmental, social and governance performance of an organization. This criticism, true for all valuation techniques, is addressed through an approach called "IntFV" discussed below.

# Integrated future value

To address the lack of integration of the short and long term importance, value and risks associated with natural and social capital into the traditional DCF calculation, companies are valuing their environmental, social and governance (ESG) performance through an Integrated Management approach to reporting, that expands DCF or Net Present Value to Integrated Future Value (IntFV). This allows companies to value their investments not just for their financial return but also the long term environmental and social return of their investments. By highlighting environmental, social and governance performance in reporting, decision makers have the opportunity to identify new areas for value creation that are not revealed through traditional financial reporting. As an example, the social cost of carbon is one value that can be incorporated into Integrated Future Value calculations to encompass the damage to society from greenhouse gas emissions that result from an investment. This is an integrated approach to reporting that supports Integrated Bottom Line (IBL) decision making, which takes
triple bottom line The triple bottom line (or otherwise noted as TBL or 3BL) is an accounting framework with three parts: social, environmental (or ecological) and economic. Some organizations have adopted the TBL framework to evaluate their performance in a broader ...
(TBL) a step further and combines financial, environmental and social performance reporting into one balance sheet. This approach provides decision makers with the insight to identify opportunities for value creation that promote growth and change within an organization.

*
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 of the tax shield of debt – and other ...
*
Capital asset pricing model In finance, the capital asset pricing model (CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a well-diversified portfolio. The model takes into acc ...
*
Capital budgeting Capital budgeting in corporate finance is the planning process used to determine whether an organization's long term capital investments such as new machinery, replacement of machinery, new plants, new products, and research development projects ...
*
Cost of capital In economics and accounting, the cost of capital is the cost of a company's funds (both debt and equity), or from an investor's point of view is "the required rate of return on a portfolio company's existing securities". It is used to evaluate ne ...
* Debt ratio * Economic value added *
Enterprise value Enterprise value (EV), total enterprise value (TEV), or firm value (FV) is an economic measure reflecting the market value of a business (i.e. as distinct from market price). It is a sum of claims by all claimants: creditors (secured and unsecured) ...
*
Financial modeling Financial modeling is the task of building an abstract representation (a model) of a real world financial situation. This is a mathematical model designed to represent (a simplified version of) the performance of a financial asset or portfolio o ...
* Flows to equity * Forecast period (finance) *
Free cash flow In corporate finance, free cash flow (FCF) or free cash flow to firm (FCFF) is the amount by which a business's operating cash flow exceeds its working capital needs and expenditures on fixed assets (known as capital expenditures). It is that porti ...
*
Internal rate of return Internal rate of return (IRR) is a method of calculating an investment’s rate of return. The term ''internal'' refers to the fact that the calculation excludes external factors, such as the risk-free rate, inflation, the cost of capital, or fi ...
* Market value added *
Net present value 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 ...
* Patent valuation * Present value of growth opportunities * Residual income valuation * Terminal value (finance) *
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 now rather than an identical sum later. It may be seen as an implication of the later-developed concept of time preference. The ...
* Valuation using discounted cash flows *
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. Importantly, it is dictated by ...

* * * * * *