Talk:Discounted cash flow

Would it be possible to add an example to this page? After reading it's still kind of unclear. Iflipti 04:29, 22 May 2005 (UTC)

• Response from User:Ertyqway, 21 April 2006: Done.

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I would like to see a correlation of discounted cash flow to the fair value of a share of stock. I have often heard of discounted cash flow being used as a way to value a stock however I am having trouble equating this to stock price.--69.249.154.41 01:44, 30 October 2005 (UTC)

Resonse to the above comment: Equity analysts develop their own proprietary models of what they expect the dividends and future stock price to be. Then those are discounted to the present using the DCF model. When one compares their model to the market price of a share, this is a source of buy or sell recommendation because they may not be the same. Of course, the model is only as good as the forecasted dividends and predicted dtock price.

Discount cash flow versus nominal cash flow

Which cash flow is best to use for evaluating real estate investment discount or nominal cash flow. Why?

Response: Expected future cash flows that do to the owner in nominal (not real) terms. In real estate, this is normally called operating income. These are the cash receipts to the owner from which he or she derrives value.

• Response from User:Ertyqway, 21 April 2006: Future cash flows for real estate transaction should only be considered in discounted, not nominal, terms. In fact, a real estate pro forma is not considered complete without a full DCF analysis. A complete DCF for a real estate transaction must include all expected cash flows, positive and negative. This not only includes net operating income, but also tax shelter value, built-up equity, and capital appreciation. For a real estate development project, further cash flows must be included in the analysis: construction costs, marketing costs, proceeds from sale, commissions, permit fees, and a whole host of other cash flows. When I do DCF for my own projects, the analysis matrix is typically many pages long, broken out quarterly or monthly depending on the detail required.

Article substantially revised on 21 APR 2006

I have substantially re-written the article to include the entire DCF formula and a very simple example showing how DCF analysis is generally done and what it means. Also, the original article limited the calculation of the discount factor to opportunity cost only, while not really discussing where opportunity cost numbers come from or what they really mean. A discussion of risk factors (commonly used in real estate DCF) has been included. The article has also been linked to the "real estate" category, where it is often referenced.

For further discussion: there appear to be several finance articles that all cover similar ground. Among these are discounted cash flow, future value, present value, and net present value. These subjects are all intimately related to one another (especially DCF and NPV), and should perhaps be collected together somehow.

Ertyqway

Thanks for the awesome article, this so saved me mega time in my capital costs class for engineering.

Ditto. Article is bangin'. Financial markets exam in two weeks. Keep up the good work

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I would disagree with merging DCF with NPV (although I note that the above suggestion does not specifically demand this, I just want to be clear). DCF is a much broader topic than NPV, and covers the basic concept of time value of money, which is relevant to a range of applications. One of these is NPV.

NPV is a specific application of DCF analysis that is used to evaluate projects based on a pre-determined 'hurdle rate'. IRR and even Stern Stewart's Economic Value Added methodologies also implicitly use the DCF concept, while being distinct from NPV.

I would suggest that instead of merging DCF and NPV, perhaps merging present value and future value into NPV and linking to NPV and IRR in the DCF article would make sense. I believe this will make clear that these articles (and the ideas and methods they relate to) are conceptually and practically distinct, but intimately related. —Preceding unsigned comment added by 139.149.31.232 (talk) 20:55, 15 September 2007 (UTC)

Large re-write March 2007 - opinions?

I have moved comments from user to bottom of this page, as traditional.--Gregalton 13:54, 1 April 2007 (UTC)

I have several problems with that article. It starts with an introduction that describes what is going on. Then we have a section Mathematics that contains some formulaes without giving the reader a clue qhat is going on (what has this to do with cash flows, values or cost of capital?). The exmaple (in my opinion) complicates the stoty even more - how do we proceed if we have different cash flows? Is that approach always allowed? Does it lead to the correct value? Then one citation and that's it. Since I am interested in that subject I take the freedom to erase the formulas. Examples are fine if you know what is going on, otherwise they can confuse people: This example is particular flawed since it calculated the net present value of an uncertain income stream with riskless interest rates. Where are all those numbers coming from? Can they be choosen arbitrarily? Or is there a rule that forces us to take particular interest rates? And I can add a lot more questions... User:al64 29 March 2007 (UTC)

I disagree with most of this evaluation but would like opinions of others. I think the content (not from me) was previously better and more complete, and examples useful. I previously reverted as I think this gets order wrong, definition before history, as well as blanking most of page. I think additional content addressing the question posed by al64 above would be useful, but do not see how the modified text addresses any of these questions. And in general, would suggest that we should be more cautious about deleting content, as opposed to adding detail, references and refining existing. (Ok, there are plenty of exceptions). Comments from other users above have been positive, which leads me to believe this is not an article where wholesale change is needed - although a lot of work is needed.--Gregalton 13:54, 1 April 2007 (UTC)

OK, let me put it this way: Let us talk about what should be added to the text. We had an example, but the problem was that this example got us to believe we understood something although it raised in fact more questions than it gave answers (where were those numbers coming from, when can we in fact calculate like the authors suggested, is such a calculation always successfull...). What should be added to that text or the topic? For example, a long text about CAPM is in my opinion misleading, since CAPM is an equilibrium model about one period (although there are some extensions to multiperiod) and valuation is in fact a multiperiod problem etc. Some sentences about cost of capital should be moved to the topic cost of capital and the same applies to cost of equity and cost of debt. For me, Discounted Cash Flow is just like a headline, that now is to be distribbuted into several sections. Al64 2 April 2007 (UTC)

I must have missed something - I don't see any mention of CAPM in this article. What does that have to do with it? Please, feel free to edit and add, and correct, but I'm going to go back to the previous version. Despite its weaknesses, it was more complete.

And I'm not sure I understand what it is you're suggesting in concrete terms. Each of your points can simply be addressed directly by editing into the existing text.--Gregalton 15:11, 2 April 2007 (UTC)

OK, I am commenting to the current version:

• It starts with time value of money. The first line has an interest rate i that is somehow changed to a rate d. Why? What is the relation between those two?

• The next line gives us a simple NPV formula. This formula makes only sense for certain cash flows, not for uncertain ones (than you need expectations). I do not see this assumption anywhere in that section.

• It is worthless having a formula for certain cash flows and no formula for uncertain cash flows. What changes with uncertainty?

• The last sentence says that this can be used as net present value. That is wrong: Net present value is the difference between market price minus present value (DPV).

• The internal rate of return is then mentioned. As anyone who hast studied business or finance should know, this is a very bad advice if cash flows change sign.

• The example of buing a house the comparison is not correct (complicating the example): you should compare renting and buying, hence you have to add fictious rent payments for the house. I suggest that a much easier example would be appropriate.

• Also the didactics is bad: The example starts with the wrong calculation. As people from teaching will tell you the better way to present a story is to start with the simplest example possible (one period I would suggest),present a correct calculation, then say something about why the wrong way to calculate is indeed wrong and then continue to the more complicated cases.

• Then a risk factor of 5% is added. Where is that risk factor comming from? Is there a theory for that or is it just taken from the air? If it is beyond the scope, why is no reference given?

• The main idea underlying the net present value is arbitrage. So why not saying it? The word arbitrage does not even appear.

• In the main text some links appear: Adjusted present value, Total cash flow, Flow to equity. Why are they mentioned again on the bottom? Some links are not mentioned there - why?

Al64 4 April 2007 (UTC)

So what about our discussion? Those who prefered the older version should say something - or am I right? Then why not going back to my version?

Al64 14 April 2007 (UTC)

Your points are rather detailed. Why not edit, bit by bit, rather than massive wholesale rewrite? That is the usual approach (unless objectively false). I agree with some of your points. Keep in mind that there is usually not "one mind" behind an article like this, and hence some things are just a result of the wikipedia process. I agree (for example) that the housing section is not the best illustration - any generic asset would be more simple and not raise the questions you have raised. As another point, you mention starting with a simple example - great, why not put one in? As a general suggestion, start by improving sections, then adding sections and references in particular, and remove only when clearly necessary. And before removing, preferably suggest removing a particular section on the talk page first. Just my opinion. And welcome.--Gregalton 18:35, 14 April 2007 (UTC)

Al64, I understand some of your concerns, but as the person who wrote the vast majority of the original text you are critiquing, I have to disagree with many of your points. The formulae and example given are intended to illustrate the basic concept of discounted cash flow analysis, not deliver an extended doctoral dissertation on analysis methods. Every analyst has a different way of applying the logic of DCF, and it would be impractical to cover them all here. To address your points:

• Fundamentally, the logic of DCF is derived from the time value of money, exactly as the article discusses. Thus, the formulae are essential to illustrating the logic of the discount method in general, and not just for certain cash flows. The point of discounting is that future cash flows do not have the same value as present cash flows. This is due to opportunity cost over time and uncertainty. The formulae should remain, though clarification is welcome (so long as it really clarifies and does not obfuscate).

• In the article as presently written, uncertain cash flows are dealt with via the addition of a "risk factor" to the discount rate. I described it that way because my background is in real estate finance and that's how the subject is usually treated in that milieu. Treating uncertainty in DCF analysis is far more of an art than a science, and as I mention in the article, the calculation of risk factors (the specific analysis of uncertainty even if probalistic) is far beyond the scope of a general encyclopedia article. It might be a good subject for it's own entry on Wikipedia, but it doesn't belong here. If you'd like to propose further elaboration on what the risk factor is supposed to do (i.e. deal with uncertainty) you are welcome to do so as long as this stays an encyclopedia article.

• I tried to create a simple example that most laypeople could understand without difficulty given a basic level of numeracy. Does it leave out a lot of detail? Sure it does. That's not the point. A good example should treat a subject most people are at least cursorily familiar with in a simple way to illustrate the point being made. Also, the calculation issue has gone back and forth here. People keep changing the numbers around so I can't really vouch for their accuracy anymore. The example could certainly be improved, but it should not be made more complicated, nor should it be made so generic that it's difficult to figure out how to apply.

• The main idea of DCF/NPV is NOT arbitrage, as you say; it is the analysis of projected, unrealized, and uncertain cash flows in present-value terms in order to evaluate courses of action and the value of decisions in present time.

I hope that helps you understand why the article is written the way it's currently written. I am opposed to the wholesale rewrite you propose, but I do think refinement and clarification would help the article a lot.

Ertyqway 8:36AM PDT April 18, 2007

Ertyqway, as far as I can see it there are two counterarguments:

• Discounting is about time value of money, not the idea of arbitrage. OK, I might have put it wrong: If there is any arbitrage, the idea of NPV becomes useless. Image, you can undertake an arbitrage strategy. Then you are simply able to print money. It does not make sense then to speak of a value of a project, since if you do not have enough money you just print again some more. So arbitrage is necessary for DCF.

• Including risk in DCF is an art. I disagree. You are a practicioneer, I am a scientist and I have worked on DCF (for one english book: www.wacc.de [1]). So how do we proceed now: Is wikipedia devoted to scientists (including risk is science) or devoted to practice (including risk is art)? For those of you who think this does not matter look at chapter 3.2 in the above mentioned book where my coauthor and I talk about including income tax into valuation, something German CPA's are supposed to do (almost) by law! The difference can boil down to values as different as X or 5X.

You do not say anything about the minor details (presentation and so forth). I still believe that the article needs an entire overwork. Al64 25 April 2007 (UTC)

Al64, are you using the term "arbitrage" in its common form of applying to the process of taking advantage of market inefficiencies and price asymmetries to make a profit on a parallel trade? If so, I'm not really sure what that has to do with DCF. The primary purpose of DCF is the analysis of future or uncertain cash flows, not present-time trading strategies in price-inefficient markets. If it's some other, more idiosyncratic meaning, perhaps there's a simpler way to describe the point you're trying to make.

Also, the "practice/science" dichotomy you're trying to set up here is false. If it works in practice, it must to some degree be true. If it's true, it must to some degree work in practice. Risk must be included in any complete analysis (as must be taxes, which are not uncertain to any significant degree within reasonable time frames). There is no question about that. However, the methods by which the risk is accounted in the analysis, particularly where the uncertainty surrounding it is high, relies very much on judgment calls by the analyst and not some a priori method dictated ex cathedra as "science." That's the part that makes it an art, and that's also the part that you can't easily put in a Wikipedia article or a book. For instance, we can use Bayesian probability distributions to handle risk assessment in many cases, but it's not always (or even often) appropriate. Market events tend to follow power laws rather than normal distributions and Black Swans are not unheard of. How do you choose which method to use? When do you chuck the normal distribution and account for the six sigma event? If the treatment of risk is really as straightforward ("scientific") as you indicate, why does everybody do it differently, often with much success? Ultimately, you've got to fall back on judgment, and judgment includes its own set of risks.

Maybe you have answers to all these questions in your book. I don't know (though please forgive me for doubting it very much). If so, I encourage you to try and distill them into layperson's language and include them here.

User:Ertyqway:Ertyqway May 1, 2007

What shall I say? Goto my website www.wacc.info [2] to see my answers to your questions. What you ask me to do is exactly what I wanted with the re-write of this article. (Concering arbitrage: DCF is about tax advantages from debt and these can be prized using arbitrage arguments. That is why I stressed this connection.)

Al64 May 1st, 2007 (UTC)

Question regarding DCF

For a while now, I've had a question regarding DCF that I've never seen addressed anywhere. I'm wondering if any mathematicians or financial wizards here are smart enough to figure it out. If so, it might be a caveat that is worth adding to the article.

Imagine there are two companies, ABC Corp and XYZ Corp. They both trade at the same price per share, earn $1.00 per share (assume FCF and earnings are equal), grow earnings at 7.5% per year, and have no debt. However, there is a difference between the two companies. ABC Corp has a 15% ROE and pays out a 50¢ dividend, while XYZ Corp has a 7.5% ROE and pays out no dividend.

Clearly, ABC Corp is the better investment because the only way XYZ Corp can match its growth is by not paying a dividend. So, the ABC Corp investor gets both 7.5% growth and a dividend, while the XYZ Corp investor only gets 7.5% growth. DDM would value ABC Corp as being more valuable that XYZ Corp. Using traditional DCF, however, the two companies would be valued the same.

(If you think that the two companies actually are the same value, then imagine that XYZ Corp also starts to pay out a 50¢ dividend. Its growth rate would drop to 3.25%. Would you still consider them equal?)

Does anyone know how a DCF calculation can properly address the difference in value between ABC Corp and XYZ Corp? The answer should be based on accurate mathematics, rather than answers that "feel right". --JHP 05:37, 15 May 2007 (UTC)

First, very good question. I had to mull over this a while. And there may be more than one answer - one non-facetious answer is that it is not clear to me that the discount rate should be the same in both cases.

It seems to me that the better answer to the problem you posed is a specification error in the question, in the sense that you have specified the ROE to be exogenous and the earnings to be fixed and equal to FCF. As formulated, earnings/FCF (and growth) depends entirely on the amount (and growth) of equity.

If free cash flow is specified correctly, it should be the amount of free cash flow after investment required to meet the growth rate (there is endogeneity). Otherwise, for ABC it would make sense (if the investment is less than the cost of risk/capital) to invest all returns back - the 15% ROE is guaranteed (because it is specified this way). If this were the case, the earnings would grow faster for ABC than for XYZ and so would the DCF valuation.

In this sense, the dividend is a component of free cash flow - it represents "excess" cash flow, surplus to the equity needed to drive the 7.5% growth, whereas more of the earnings of XYZ are required for reinvestment.

The accounting reconciliation of the earnings and cash flow are more difficult, and probably inherently over-simplified - it's a hypothetical model after all.

The deeper answer (based on the same reasoning) is that there is a limit to the amount of reinvestment that can generate the 7.5% growth and associated return on equity. For example, because of any number of reasons, any investment above this level would generate diminishing returns (less growth per dollar reinvested in equity). So, ABC can reinvest anything above the dividend level but no more without reducing ROE.

Most importantly (the analysis can be varied) for these two companies earnings and free cash flow cannot be the same as specified. Or alternatively, the ROE cannot be specified as exogenous. Or to get to the requirements of the specification: ROE, FCF and earnings cannot all be specified externally as fixed.

I hope this makes sense, and I'm sure there are better answers, but it seems to me this is the crux of the answer: in your theoretical case, the earnings and free cash flows of these two companies cannot be the same.--Gregalton 20:02, 16 May 2007 (UTC)

I should have pointed out that I am looking at it from the perspective of an investor who has no influence on management. I view the cost of capital as the investor's cost of capital. Pretending that index funds don't exist, we could view the cost of capital as being that of the return on a money market account. I guess I'm seeing things a bit more from a John Burr Williams perspective, rather than a modern MBA's perspective (although JBW was concerned with dividends, not FCF). Unfortunately, my understanding of accounting is limited to "Finance and Accounting for Non-Financial Managers". I see FCF as simply operating earnings minus capital expenditures. I am viewing the growth rate as ROE x (1 - dividend-payout ratio). Does that change any of your assumptions? --JHP 13:19, 19 May 2007 (UTC)

So, my take on it with three answers of increasing levels of complexity. All boil down to the fact that DCF requires some probability/likelihood that cash will/can be paid to investors. Part of the problem is that the question as specified is artificial - companies don't usually have unlimited capacity to reinvest at a given return, there are diminishing returns or other constraints. (Or put more simply, if the firm's cost of capital is that of a money market account and it can generate 15% returns, it should invest infinitely, which is clearly not possible)

1. Discount rate should be lower for the dividend-paying company. Any valuation based on cash flows/certainty of cash flows will assign less risk to the investment that generates cash flows to the investor sooner rather than later, particularly if there is no ability to influence whether/when such funds will be received (as in your case).
2. Define and analyze the free cash flow more carefully. As currently defined, all of the growth is financed by reinvesting earnings, and in this sense is not free cash flow. Carried to an extreme, this gives an absurd result (there are no free cash flows), so there has to be some ability to specify the problem in a way that makes sense. One way to look at it is to require (for the purpose of analysis) that both companies pay equal dividends, and see where that leads. Clearly the one company can only generate growth at the current level by never paying a dividend.
3. The more complicated but complete answer: Specify some terms by which both companies will pay out cash. For a simple example, say XYZ must pay all of its earnings out starting at some date in the future, and pay out all earnings every year from then on. Use 7.5% as the discount rate for convenience, which gives the same valuation as a perpetuity of $1 starting today. Apply the same approach to ABC, and you get two distinct cash flow streams: $0.50 from now through year X, and the same stream as XYZ above. In other words, ABC's valuation will always be higher by the net present value of the stream of dividends to the date when full payout begins (the value of the full payout starting in the future should be the same).

So using the last, I can't give you a definitive answer of what the difference in value is, as assumptions have to be made. But you can easily demonstrate (mathematically) that ABC's value is always higher than XYZ's (except for the case where all earnings are paid out from day 1, i.e. no dividends ever and no ability for ABC to exploit its higher ROE capability). Any investment where the value of ABC is always equal to or greater than XYZ is clearly preferable.

I've referred to the math in shorthand form, so hope this is clear. Thanks for posing an interesting question.--Gregalton 04:29, 23 May 2007 (UTC)

Together with a colleague I have looked at a question that might be similar to the one above - although it carries to story probably too far. Look at a company where its managers always find enough projects with positive NPV, that lives infinitely and that follows Jensen's advice and will therefore not distribute a dividend. Since dividends are zero, the company must be worthless - which sounds absurd. We believe that this is a shortcoming of the DCF method that (up to now) only says something about cost of capital, cash flows and values and cannot say anything about the relation between investment projects and values or cost of capital. These relations must be described with something from beyond Discounted Cash Flow.

Al64 May 28th, 2007 (UTC)

I partly agree with your point, but partly not. Clearly a company that will never pay out cash and cannot be made to pay out cash in any form under any circumstances is worthless. Under other circumstances - where the possibility of cash being paid out is greater than zero - DCF will give a result, and a positive one, but perhaps of questionable accuracy (if the likelihood of cash actually being distributed cannot be estimated).

In the cases mentioned above, where presumably the securities are publicly traded and the possibility of a buy-out offer is real (there is no controlling shareholder, for example, and management is unable to block such a process), DCF is commonly used and may be at least partly realistic and useful. Any divergence of the value from DCF could be realised by a significantly deep-pocketed investor (rival, private equity, whatever). Dividends or growth can be normalized (and hence FCF put on comparable terms). Although different assumptions may be made and result in different valuations, if the securities are traded the price (should) converge to a price (or price range) that reflects the aggregation of those valuation estimates (cash flow and discount rate). Major divergences can be exploited by outright purchase (transaction, oppportunity and other factors may require the divergence to reach a certain magnitude before this happens, though).

Part of my point above is that the process still requires estimations/assumptions - defining in advance what the earnings are does not tell us precisely what the free cash flows are or when they occur. Even the way the question is structured does not tell us exactly what the shares should cost (or that they should have the same price), because the shares that have the long-term potential to pay more (to a purchaser willing to make the investment/effort to control the company and cash flows) should cost more, unless those payouts really are fixed irrevocably. The intuition that the one with higher ROE should cost more is unambiguously true (except as mentioned), because they can support higher payments to shareholders (ceteris parabus) than the other company for a given level of growth.

So while DCF has weaknesses, in many cases it's still useful. I don't think it means that it 'can't say anything about the relation between investment projects and values', just that a) some problems cannot be solved using only DCF; and b) valuations may differ depending on estimates (guesses/judgment calls) by investors using the same inputs. Informed use of DCF also involves sensitivity analysis and other valuation methods to check the results.--Gregalton 15:14, 28 May 2007 (UTC)

Different Solution: This had me going for a minute or two... however the answer is mathematically simple and is all about definition of free cashflow (FCF). The FCF that you should be using is the cashflow after maintenance or replacement capital expenditure - i.e. the capital expenditure that is required to keep the business going. Also you should deduct (but not in perpetuity) any growth capital expenditure.

Based on the way you have phrased the question, it appears to imply two options:

Option 1

• Debt / Cash remains at zero (i.e. any earnings not paid out is re-invested)
• The ROE's remain constant (i.e. each marginal dollar earns the same as the previous dollar)
• The re-investment of capital is required to keep the businesses operating
• The earnings grows at 7.5% - which is an outcome of the above. i.e. ABC has equity in year 1 of $6.67 which earns a 15% return to give $1.00. ABC then grows that equity by reinvesting half each year, and so grows its equity by 7.5% each year. XYZ is the same, but it needs to invest all of its profits to grow its equity by 7.5%.

So, in this case one company needs to invest $1 per year and the other $0.50, and so the FCF should be $0.50 for ABC and $0.00 for XYZ. This would mean that XYZ is worthless unless you make the assumption that at some stage this company must accept a lower growth rate and return capital.

Option 2

• Debt / Cash remains at zero (i.e. any earnings not paid out is re-invested)
• The ROE's remain constant (i.e. each marginal dollar earns the same as the previous dollar)
• The earnings are equal to the FCF - which implies that excluding re-investment that earnings would not grow at all.
• The earnings grows at 7.5% because there is growth capital expenditure invested.

So, in this case you should do a DCF of how many years that the re-investment occurs for, and then keep earnings constant beyond that period. THEN, you need to deduct the NPV of all of the capital that has been invested, for ABC this will be lower than XYZ and so ABC will have a higher valuation. Damien74 23:45, 14 November 2007 (UTC)

Guys, this is so simple. This is a classic trick question. Accountants and engineers look at statements and not the real world and so they seem to make this mistake a lot. You are looking at the ROE as a measure of performance on the market value of the equity. In order for ROE to have any such relevance, you need to used a reconstructed balance sheet adjusted to market values. If there is no debt, then the equity on the un-reconstructed balance sheet will equal the book value of the assets. If the two firms have identical incomes and identical income growth, then they are performing identically from a valuation standpoint. If FCF = income, then there is no depreciation expense. Your calculation that XYZ paying dividends would change its return on equity is wrong. It earns income. No debt, so its all income to equity. Divide income by equity, get ROE of 7.5 %. It earns income, then pays half of the income as dividends, still have the same income, they will have less current assets, less retained earnings--will eventually increase ROE, not cut it by half, except for the fact that actual market value of assets go down ex-dividend, there should be no change in valuation. Since the two companies have the same share price with no debt, then they have the same WAAC and thus the market value of the assets must be the same. The only real difference between the two companies is that ABC has a lower book value to its assets than XYZ, which has nothing to do with valuation. In the future, as retained earnings continue to accumulate, the ROE of XYZ will continue to deteriorate, but its value will continue to climb as its assets grow.KTrimble (talk) 09:35, 6 April 2008 (UTC)

Mathematics

The claim "(1+i)^(-t) can of course also be expressed as exp(-it)" is only true in the limit of $t\rightarrow \infty$, with $i\rightarrow 0$ such that $i\times t$ remains constant, no? In any other case it's an approximation.