View source for Discounted cash flow

{{Short description|Method of valuing a project, company, or asset}}
{{multiple issues |
{{tone|date=September 2015}}
{{More citations needed|date=January 2010}} 
}}
{{Use dmy dates|date=August 2021}}The '''discounted cash flow''' ('''DCF''') analysis, in [[financial analysis]], is a method used [[Valuation (finance)|to value]] a [[security (finance)|security]], project, company, or [[financial asset|asset]], that incorporates the [[time value of money]]. 
Discounted [[cash flow]] analysis is widely used in investment finance, [[real estate developer|real estate development]], [[corporate financial]] management, and [[patent valuation]]. Used in industry as early as the 1700s or 1800s, it was widely discussed in financial economics in the 1960s, and U.S. courts began employing the concept in the 1980s and 1990s. <!-- Related is the [[carbon discounted cash flow]], which integrates [[climate change|climate-related]] considerations. -->

==Application==
{| class="wikitable floatright" | width="250"
|- style="text-align:center;"
| Main Elements
|-
|
On a very high level, the main elements in valuing a corporate by Discounted Cash Flow are as follows; see [[Valuation using discounted cash flows]], and graphics below, for detail:
* '''Free Cash Flow Projections:''' Projections of the amount of Cash produced by a company's business operations after paying for operating expenses and capital expenditures.<ref name=":0">{{Cite web|url=http://www.streetofwalls.com/finance-training-courses/investment-banking-technical-training/discounted-cash-flow-analysis/|title=Discounted Cash Flow Analysis {{!}} Street of Walls|website=streetofwalls.com|access-date=2019-10-07}}</ref>
* '''Discount Rate:''' The cost of capital (Debt and Equity) for the business. This rate, which acts like an interest rate on future Cash inflows, is used to convert them into current dollar equivalents.
* '''Terminal Value:''' The value of a business at the end of the projection period (typical for a DCF analysis is either a 5-year projection period or, occasionally, a 10-year projection period).<ref name=":0" />
|}
[[File:DCFMDPD.gif|thumb|[[Flowchart]] for a typical DCF valuation, with each step detailed in the text (click on image to see at full size)]]
[[File:DCFM Calculator.JPG|thumb|Here, a [[spreadsheet]] valuation, uses [[Free cash flow]]s to estimate stock's [[Fair Value]] and measure the [[sensitivity analysis|sensitivity]] of [[Weighted average cost of capital|WACC]] and [[Dividend discount model|Perpetual growth]] ]]

In '''discount cash flow analysis''', all future cash flows are estimated and [[Discounting|discounted]] by using [[cost of capital]] to give their [[present value]]s (PVs). The sum of all future cash flows, both incoming and outgoing, is the [[net present value]] (NPV), which is taken as the value of the cash flows in question;<ref>{{Cite web |url = http://www.wallstreetoasis.com/finance-dictionary/what-is-a-discounted-cash-flow-DCF |title = Wall Street Oasis (DCF)|access-date = 5 February 2015|website = Wall Street Oasis}}</ref>
see aside.

For further context see {{slink|Valuation (finance)|Valuation overview}}; 
and for the mechanics see [[valuation using discounted cash flows]], which includes modifications typical for [[startup]]s, [[private equity]] and [[venture capital]], [[corporate finance]] "projects", and [[mergers and acquisitions]].

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 (finance)|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 [[Ancient Egyptian mathematics|Egyptian]] and [[Babylonian mathematics]] 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.<ref>Susie Brackenborough, et al., The Emergence of Discounted Cash Flow Analysis in the Tyneside Coal Industry c.1700-1820. The British Accounting Review 33(2):137-155
DOI:10.1006/bare.2001.0158</ref>

Discounted cash flow valuation is differentiated from the accounting [[book value]], which is based on the amount paid for the asset.<ref>{{Cite book|title = Otto Eduard Neugebauer, The Exact Sciences in Antiquity (Copenhagen :Ejnar Mukaguard, 1951) p. 33|last = [[Otto Eduard Neugebauer]], The Exact Sciences in Antiquity (Copenhagen :Ejnar Mukaguard, 1951) p. 33|publisher = Dover Publications|year = 1969|isbn = 978-0-486-22332-2|location = US|pages = 33}}</ref> Following the [[Wall Street Crash of 1929|stock market crash of 1929]], discounted cash flow analysis gained popularity as a valuation method for [[capital stock|stock]]s. [[Irving Fisher]] in his 1930 book ''The Theory of Interest'' and [[John Burr Williams]]'s 1938 text ''[[The Theory of Investment Value]]'' first formally expressed the DCF method in modern economic terms.<ref>Fisher, Irving. "The theory of interest." ''New York'' 43 (1930).</ref>

==Mathematics==
===Discounted cash flows===
The discounted cash flow formula is derived from the [[Time_value_of_money#Present_value_of_a_future_sum|present value formula for calculating the time value of money]]

:<math>DCF = \frac{CF_1}{(1+r)^1} + \frac{CF_2}{(1+r)^2} + \dotsb +
\frac{CF_n}{(1+r)^n}</math>

and [[Compound interest|compounding]] returns:

:<math>FV = DCF \cdot (1+r)^n</math>.
Thus the discounted present value (for one cash flow in one future period) is expressed as:

:<math>DPV =  \frac{FV}{(1+r)^n}</math>

where
* ''DPV'' is the discounted present value of the future cash flow (''FV''), or ''FV'' adjusted for the delay in receipt;
* ''FV'' is the [[Real versus nominal value (economics)|nominal value]] of a cash flow amount in a future period (see [[Mid-year adjustment]]);
* ''r'' is the [[interest rate]] or discount rate, which reflects the cost of tying up [[Capital (economics)|capital]] and may also allow for the risk that the payment may not be received in full;<ref>{{cite web |url=http://data.gov.uk/sib_knowledge_box/discount-rates-and-net-present-value |title=Discount rates and net present value |publisher=Centre for Social Impact Bonds |access-date=28 February 2014 |archive-url=https://web.archive.org/web/20140304094708/http://data.gov.uk/sib_knowledge_box/discount-rates-and-net-present-value |archive-date=4 March 2014 |url-status=dead }}</ref>
* ''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:

:<math>DPV = \sum_{t=0}^{N} \frac{FV_t}{(1+r)^{t}}</math>

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]] figure. If the amount to be paid at time&nbsp;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]].

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:

:<math>DPV= \int_0^T  FV(t) \, e^{-\lambda t} dt = \int_0^T \frac{FV(t)}{(1 + r)^t} \, dt\,,</math>

where <math>FV(t)</math> is now the ''rate'' of cash flow, and <math>\lambda = \ln(1+r)</math>.

==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]] 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]] ([[Risk-free interest rate|risk-free rate]]) – according to the theory of [[time preference]], investors would rather have cash immediately than having to wait and must therefore be compensated by paying for the delay.
# [[Risk premium]] – 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 [[economic model|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 "[[market portfolio|overall market's]]"; see {{slink|Capital asset pricing model|Asset-specific required return}} and {{slink|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]], 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".

The terminology "[[expected return]]", although formally the [[expected value|mathematical expected value]], is often used interchangeably with the above, where "expected" means "required" or "demanded" by investors.

The method may also be modified by industry, for example various formulae have been proposed when choosing a discount rate [[healthcare economics|in a healthcare setting]];<ref>{{Cite journal|last1=Lim|first1=Andy|last2=Lim|first2=Alvin|date=2019|title=Choosing the discount rate in an economic analysis|journal=Emergency Medicine Australasia|language=en|volume=31|issue=5|pages=898–899|doi=10.1111/1742-6723.13357|pmid=31342660|s2cid=198495952|issn=1742-6723}}</ref>
similarly in a [[Valuation_(finance)#Valuation_of_mining_projects|mining setting]], where risk-characteristics can differ (dramatically) by [[mineral rights|property]]. <ref>[[Queen's University at Kingston|Queen's University]] minewiki (N.D.). [https://minewiki.engineering.queensu.ca/mediawiki/index.php/Discount_rate "Discount rate"]</ref>

==Methods of appraisal of a company or project==
For these [[Valuation (finance)|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]] of the company. However the assumptions used in the appraisal (especially the equity discount rate and the [[cash flow forecast|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]] 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 {{slink|Outline of finance|Financial modeling}}.

===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]] 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]] approach (WACC)
** Derive a weighted cost of the capital obtained from the various sources and use that discount rate to discount the unlevered free 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){{Clarify|date=February 2009}}
** 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.<ref>{{cite book
  | last = Pratt | first = Shannon |author2=Robert F. Reilly|author3=Robert P. Schweihs  | title = Valuing a Business | publisher = McGraw Hill | series = McGraw-Hill Professional  | year = 2000  | url = https://books.google.com/books?id=WO6wd8O8dsUC&q=shannon+pratt  | isbn = 0-07-135615-0 }}
</ref>

==Shortcomings==
{{Further|Dividend discount model#Problems with the constant-growth form of the model}}
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.<ref>{{Cite work|last1=Chan|first1=Louis K.C.|last2=Karceski|first2=Jason|last3=Lakonishok|first3=Josef|date=May 2001|title=The Level and Persistence of Growth Rates|location=Cambridge, MA|doi=10.3386/w8282|doi-access=free}}</ref> (See [[Stock valuation#Growth rate]] and [[Sustainable growth rate#From a financial perspective]].) <br/>In other terms, using DCF models is problematic due to the [[problem of induction]], 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.<ref>{{Cite web|url=https://www.sec.gov/fast-answers/answersmperfhtm.html|title=SEC.gov {{!}} Mutual Funds, Past Performance|website=sec.gov|access-date=2019-05-08}}</ref><br/>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 [[Real estate appraisal#The income approach|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.<ref>{{Cite book|last1=Reilly|first1=Robert F.|last2=Schweihs|first2=Robert P.|date=2016-10-28|title=Guide to Intangible Asset Valuation|doi=10.1002/9781119448402|isbn=9781119448402|s2cid=168737069 }}</ref>
#* Early-stage Technology Companies: [[Startup company#Valuations|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.<ref>{{Citation|chapter=Measuring and Managing Value in High-Tech Start-ups|date=2015-09-12|pages=285–311|publisher=John Wiley & Sons, Inc.|isbn=9781119200154|doi=10.1002/9781119200154.ch18|title=Valuation for M&A}}</ref>
# '''Discount rate estimation''': Traditionally, DCF models assume that the [[capital asset pricing model]] 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.<ref>{{Cite journal|last1=Fama|first1=Eugene F.|last2=French|first2=Kenneth R.|date=2003|title=The Capital Asset Pricing Model: Theory and Evidence|journal=SSRN Working Paper Series|doi=10.2139/ssrn.440920|s2cid=12059689 |issn=1556-5068}}</ref> various other models are proposed (see [[asset pricing]]), 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 value (finance)|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.<ref>{{Cite book|title=Integrated management : how sustainability creates value for any business|last=Sroufe|first=Robert|isbn=978-1787145627|oclc=1059620526|date = 5 October 2018}}</ref> This criticism, true for all valuation techniques, is addressed through an approach called "IntFV" discussed below.
<!-- == Example ==
TOO MUCH
To show how discounted cash flow analysis is performed, consider the following example.

: John Doe buys a house for $100,000. Three years later, he expects to be able to sell this house for $150,000.

Simple subtraction suggests that the value of his profit on such a transaction would be $150,000 − $100,000 = $50,000, or 50%. If that $50,000 is [[amortization (business)|amortized]] over the three years, his implied annual return (known as the [[internal rate of return]]) would be about 14.5%. Looking at those figures, he might be justified in thinking that the purchase looked like a good idea. 1.145<sup>3</sup> x $100,000 = $150,000, approximately.

However, since three years have passed between the purchase and the sale, any cash flow from the sale must be discounted accordingly. At the time John Doe buys the house, the three-year [[United States Treasury security#Treasury note|US Treasury Note]] rate is 5% per annum. Treasury notes are generally considered to be inherently less risky than real estate, since the value of the note is guaranteed by the US government and there is a [[liquidity|liquid]] market for the purchase and sale of T-notes. If he had not put his money into buying the house, he could have invested it in the relatively safe T-Notes instead. This 5% per annum can, therefore, be regarded as the [[risk-free interest rate]] for the relevant period (three years).

Using the DPV formula above (FV=$150,000, i=0.05, n=3), that means that the value of $150,000 received in three years actually has a [[present value]] of $129,576 (rounded off). In other words, we would need to invest $129,576 in a T-bond now to get $150,000 in three years almost risk-free. This is a quantitative way of showing that money in the future is not as valuable as money in the present ($150,000 in three years is not worth the same as $150,000 now; it is worth $129,576 now).

Subtracting the purchase price of the house ($100,000) from the [[present value]] results in the [[net present value]] of the whole transaction, which would be $29,576 or a little more than 29% of the purchase price. Another way of looking at the deal as the excess return achieved (over the risk-free rate) is (114.5 - 105)/(100 + 5) or approximately 9.0% (still very respectable).

But what about risk? We assume that the $150,000 is John's best estimate of the sale price that he will be able to achieve in three years time (after deducting all expenses). There is a lot of uncertainty about house prices, and the outcome may end up higher or lower than this estimate. (The house John is buying is in a "good neighborhood", but market values have been rising quite a lot lately and the real estate market analysts in the media are talking about a slow-down and higher interest rates. There is a probability that John might not be able to get the full $150,000 he is expecting in three years due to a slowing of price appreciation, or that loss of liquidity in the real estate market might make it very hard for him to sell at all.

Under normal circumstances, people entering into such transactions are [[risk-averse]], that is to say that they are prepared to accept a lower expected return for the sake of avoiding risk. See [[Capital asset pricing model]] for a further discussion of this. For the sake of the example (and this is a gross simplification), let us assume that he values this particular risk at 5% per annum (we could perform a more precise probabilistic analysis of the risk, but that is beyond the scope of this article). Therefore, allowing for this risk, his expected return is now 9.0% per annum (the arithmetic is the same as above). And the excess return over the risk-free rate is now (109 - 105)/(100 + 5) which comes to approximately 3.8% per annum.

That return rate may seem low, but it is still positive after all of our discounting, suggesting that the investment decision is probably a good one: it produces enough profit to compensate for tying up capital and incurring risk with a little extra left over. When investors and managers perform DCF analysis, the important thing is that the net present value of the decision after discounting all future cash flows at least be positive (more than zero). If it is negative, that means that the investment decision would actually ''lose'' money even if it appears to generate a nominal profit. For instance, if the expected sale price of John Doe's house in the example above was not $150,000 in three years, but ''$130,000'' in three years or $150,000 in ''five'' years, then on the above assumptions buying the house would actually cause John to ''lose'' money in present-value terms (about $3,000 in the first case, and about $8,000 in the second). Similarly, if the house was located in an undesirable neighborhood and the [[Federal Reserve Bank]] was about to raise interest rates by five percentage points, then the risk factor would be a lot higher than 5%: it might not be possible for him to predict a profit in discounted terms even if he thinks he could sell the house for ''$200,000'' in three years.

In this example, only one future cash flow was considered.  For a decision which generates multiple cash flows in multiple time periods, all the cash flows must be discounted and then summed into a single [[net present value]]. See [[#Methods of appraisal of a company or project]] for cases where multiple periods are considered. -->

== 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|Integrated Management]] approach to reporting, that expands DCF or Net Present Value to Integrated Future Value (IntFV).<ref>{{Cite book|title=One Report: Integrated Reporting for a Sustainable Strategy|url=https://archive.org/details/onereportintegra00eccl_0|url-access=registration|last1=Eccles|first1=Robert|last2=Krzus|first2=Michael|publisher=Wiley|year=2010|isbn=9780470587515 }}</ref>

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]] (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.
<ref>{{Cite journal|last=Sroufe|first=Robert|date=July 2017|title=Integration and Organizational Change Towards Sustainability|url=https://www.researchgate.net/publication/318126290|journal=Journal of Cleaner Production|volume=162|pages=315–329|via=Research Gate|doi=10.1016/j.jclepro.2017.05.180}}</ref>

==See also==
{{div col|colwidth=22em}}
*[[Adjusted present value]]
*[[Capital asset pricing model]]
*[[Capital budgeting]]
*[[Cost of capital]]
*[[Debt ratio]]
*[[Economic value added]]
*[[Enterprise value]]
*[[Financial report|Financial reporting]]
*[[Flows to equity]]
*[[Forecast period (finance)]]
*[[Free cash flow]]
*[[Internal rate of return]]
*[[Market value added]]
*[[Net present value]]
*[[Owner earnings]]
*[[Patent valuation]]
*[[PVGO|Present value of growth opportunities]]
*[[Residual income valuation]]
*[[Terminal value (finance)]]
*[[Time value of money]]
*[[Valuation using discounted cash flows]]
*[[Weighted average cost of capital]]
{{div col end}}

==References==
{{Reflist}}

==Further reading==
*{{cite book | author=International Federation of Accountants |url=https://www.ifac.org/system/files/publications/files/Project-Appraisal-Using-DCF.pdf |archive-url=https://web.archive.org/web/20190414093226/https://www.ifac.org/system/files/publications/files/Project-Appraisal-Using-DCF.pdf |archive-date=2019-04-14 |url-status=live |title=Project Appraisal Using Discounted Cash Flow | year=2008| author-link=International Federation of Accountants }}
*{{cite book | last=Copeland | first=Thomas E. |author2=Tim Koller|author3=Jack Murrin | title=Valuation: Measuring and Managing the Value of Companies | publisher=[[John Wiley & Sons]] | location=New York | year=2000 | isbn=0-471-36190-9| title-link=Valuation: Measuring and Managing the Value of Companies }}
*{{cite book | author=Damodaran, Aswath | title=Investment Valuation: Tools and Techniques for Determining the Value of Any Asset | publisher=[[John Wiley & Sons]] | location=New York | year=1996 | isbn=0-471-13393-0| author-link=Damodaran, Aswath | title-link=Investment Valuation: Tools and Techniques for Determining the Value of Any Asset }}
*{{cite book | author=Rosenbaum, Joshua |author2=Joshua Pearl | title=Investment Banking: Valuation, Leveraged Buyouts, and Mergers & Acquisitions | publisher=[[John Wiley & Sons]] | location=Hoboken, NJ | year=2009 | isbn=978-0-470-44220-3}}
*{{cite book | author=James R. Hitchnera | title=Financial Valuation: Applications and Models | publisher=[[Wiley (publisher)|Wiley Finance]] | year=2006 | isbn=0-471-76117-6}}
*{{cite book | author=Chander Sawhney|url=http://corporatevaluations.in/static-1047-22-oth%20-Articles%20and%20Research%20Hub |title=Discounted Cash Flow – The Prominent Income Approach to Valuation| publisher=corporatevaluations.in  | year=2012 }}
<!-- strange redirect
*[http://www.iacam.org/ International Association of CPAs, Attorneys, and Management (IACAM)] (Free DCF Valuation E-Book Guidebook) -->

==External links==
*[https://wealthyeducation.com/how-to-calculate-intrinsic-value/ Calculating Intrinsic Value Using the DCF Model], wealthyeducation.com
*[https://wealthyeducation.com/how-to-calculate-terminal-value/ Calculating Terminal Value Using the DCF Model], wealthyeducation.com
*[http://ocw.mit.edu/courses/nuclear-engineering/22-812j-managing-nuclear-technology-spring-2004/lecture-notes/lec03slides.pdf Continuous compounding/cash flows], ocw.mit.edu
<!-- *[http://www.wacc.biz Monography about DCF (including some lectures on DCF)]. extremely technical-->
*[https://web.archive.org/web/20080110115513/http://www.thestreet.com/university/personalfinance/10385275.html Getting Started With Discounted Cash Flows]. ''[[TheStreet.com|The Street]]''.

{{Corporate finance and investment banking}}
{{Authority control}}

[[Category:Cash flow]]
[[Category:Engineering economics]]
[[Category:Corporate finance]]
[[Category:Valuation (finance)]]