Free Cash Flow (FCF) formula : EBITDA, EBIT, Net Income and Depreciation Tax Savings…always start with EBITDA


Once in a while, I got a simple question whether in the Free-Cash-Flow (FCF) formula, we need to start with EBITDA or EBIT.

Well, I could say, both will give us the same results. Before we put more details, FCF formula is a concept built under no-debt world, it means there is no debt being used to finance the execution of the entity’s strategy to generate cash flows and profits (returns to shareholders). Under financial terms, it is an unlevered concept.

Additionally, as Free Cash Flow is a cash flow concept, then all components related to non-cash flow, such as depreciation, amortization, provision for bad debt, etc. in the Income Statements, need to be removed out as well. Here we will only consider the depreciation and amortization expenses as illustration.

As many financial analysts love EBITDA, then we could say in most cases, we see EBITDA comes first, and it should. EBIT comes from EBITDA, so it means the door to EBIT is EBITDA. Let me show it for you.

If we start with EBITDA, then the only component is not there is tax that is payable to tax authority.

Then, the formula will be:

Free Cash Flow = EBITDA  – EBIT*Tax(%) – Capex – change in NWC………………….(1)

Bear in mind, that EBIT*Tax(%) is the approximation of the tax expense, as there is a difference between tax and accounting in calculating the depreciation. Companies, in practice, will keep two listing in computing the depreciation expense, one in accordance with the Accounting standards, and the other following Tax rule. This has a consequence, that the tax expense calculation using EBIT*Tax(%) will not be the same with the actual tax expense that ultimately the company has to pay to the tax authority.

Or, if the financial analyst wants to show the size of the depreciation tax shield, that is the tax savings that results from the ability of the Company to deduct depreciation, thus, lowering the Company’s tax expense incurred to the tax authority, which depreciation has positive impact to the FCF, we could show the FCF as follows:

Free Cash Flow = EBITDA (1- Tax(%)) + Depreciation*Tax(%) – Capex – change in NWC….(2)

Note: Depreciation*Tax(%) is again an approximation to the actual tax savings being enjoyed by the Company. If the financial analyst has detailed calculation of tax depreciation, then it might better to use that.

So under Formula (2) we have EBITDA and Depreciation Tax Shield. Knowing the size depreciation tax savings is relevant in certain industry going heavy on depreciable assets.

If you are EBIT lover, then you still start with EBITDA, as shown below.

Free Cash Flow = (EBITDA – Depreciation)(1- Tax(%)) + Depreciation – Capex – change in NWC…(3a) or

Free Cash Flow = EBIT (1-Tax(%)) + Depreciation – Capex – change in NWC ……(3b)

Why we need to add back the depreciation under formula (3) above?

As you could see either (EBITDA- Depreciation) or EBIT, both has included depreciation component which is not a cashflow, this is why we need to put it back by adding depreciation after calculating (EBITDA – Depreciation) (1-Tax(%)) or EBIT (1-Tax(%)).

However, you might a bit confused since there is tax element in the depreciation. To answer that, we could expand formula (3a) to prove that by adding back the depreciation, it will be gone from FCF formula.

Formula (3a)

Free Cash Flow = (EBITDA – Depreciation)(1- Tax(%)) + Depreciation – Capex – change in NWC

We could re-write :

FCF = EBITDA (1-Tax(%)) – Depreciation (1-Tax(%)) + Depreciation – Capex – change in NWC.

FCF = EBITDA (1-Tax(%)) – Depreciation + Depreciation*Tax(%) + Depreciation  – Capex – change in NWC

We could cross out both Depreciation above since it is + and -, to be simpler formula:

FCF = EBITDA (1-Tax(%)) + Depreciation*Tax(%)) – Capex – change in NWC, which is the same with Formula (2) above.

You might ask how about using Net Income, since this is the Net Income that is normally what the financial analysts in the published financial statements by the public listed companies or financial institutions? Net Income could also be used to build the FCF, as all roads lead to Rome. We need to remember that FCF is the unlevered concept, which the financing is 100% by equity, no debt at all. Net Income is a levered concept, so it means we need to bring this levered Net Income to unlevered one. As Net Income is after interest expense, then we need to add this back. However, as interest expense is allowed to be deductible by the tax regulations, to get the assessable income of the company, then we need to consider the after-tax interest expense, instead of only interest expense. As long as EBIT is positive, in general, the presence of the interest expense will give the company the interest tax shield, because it lowers the company’s assessable income for tax purposes.

When we add back Net Income + Interest Expense (1-Tax), then we have:

Net Income + Interest Expense (1-Tax) = EBIT (1-Tax(%))

Once we have EBIT (1-Tax(%)), this will lead us to Formula (3b), which is we just have to add back the Depreciation to get FCF, as shown below.

Free Cash Flow = EBIT (1-Tax(%)) + Depreciation – Capex – change in NWC ……(3b)

So, if we start with Net Income, the whole formula for FCF is as folows:

Free Cash Flow = Net Income + Interest Expense – Interest Expense*Tax(%) + Depreciation – Capex – change in NWC…(3c)

As a closing note, in the valuation using multiple method, EBITDA multiple and FCF multiple might general similar value estimates, when there are no extraordinary capex or investments in net working capital.


FCF = Free Cash Flow (to the Firm), or sometimes it is written as FCFF, to differentiate it from Free Cash Flow to the Debt holders or Equity holder.

EBITDA = Earnings Before Interest, Tax and Depreciation and Amortization, or = Revenue – cash Costs

EBIT = Earnings Before Interest and Tax

Capex = Capital expenditure, the investments in fixed assets or intangible assets

NWC = Net Working Capital

Another thing to do is reconciling FCFF to the Statement of Cash Flows, which just remove out the change in cash balance from FCFF.

The next question, is why we need to remove out the cash balance change from FCFF?

My quick answer typically will be:

Statement of Cash Flows is the Cash Flow Statement which explains away the movement of the cash balance from the beginning of the year to the end of the year, by categorizing the components into 3 : Operations, Investing and Financing. As it is explaining the movement of cash balance, then the cash movement should not be included the component of changes of cash flow.

However, when it comes to FCFF calculation, any addition to the cash balance (higher balance at the end of the year), should be deducted from neutralized, or deducted from the Cash Flow Statement as it is already part of either cash flow from Operations, and/or Investing, and/or Financing.

So the formula to draw from Cash Flow Statement to FCFF will be:

FCFF = Cash Flows from/(used in Operations) +/- Cash Flow from Investing + After-tax interest expense -/+ changes in cash balance (Note : higher balance of cash will be minus, and lower balance at end of year will be plus to this formula).

This “After-tax interest expense” in the above formula needs a footnote as well:

1. This is assumed full-shielded interest expense which EBIT exceeds total interest expense incurred in that period; and

2. this is assumed that interest expense is shown as part of Cash Flows from Operating Activities (Note: IFRS allows the company to put under Operating or Financing Activities). If interest expense is shown as part of Financing Activities, then need to remove the after tax interest expense from the above formula.

Another question which is still related to FCF, EBITDA and valuation is why in the Multiple Valuation, analysts would prefer using EBITDA instead of FCF, though FCF is extensively used in the Discounted Cash Flow valuation.

I could think at least three reasonable arguments for this:

  1. EBITDA is not that volatile as that of FCF. Under the contents of FCF, we could find two “discretionary” expenditures, they are Capital Expenditures and Changes in Net Working Capital. EBITDA could be said is relatively shielded from such discretionary decisions. In early years of many companies, it is normal to see that FCF will be negative, as the Company has to re-invested in capital expenditures and working capital to pursue any positive growth opportunities. I guess, this makes sense as it will take money to make money, right? The speed of investments is higher than the speed that the Company could generate the positive cash flows from operations activities.
  2. If we know that investment in capital expenditures and working capital is so critical then why we don’t just include this, which means FCF will make more sense. Of course, there is a price that we have to pay if we rely on EBITDA multiple. We only can use EBITDA multiple by ignoring capital expenditures and investment in changes in net working capital, if we believe that such investments have no more positive Net Present Value on average and on long term. If we don’t have such belief, then we might come back to FCF to consider, or other Metrics to use.
  3. EBITDA is good since it measures the ‘cash’ earnings being generated by current operations, existing assets-in-place. Here we don’t consider the value of the Company’s new investments. So it is the “as-it-is” valuation.

Feel free to comment.

Startup Random Talk : ROI and IRR, Required Rate of Return

Recently my friend contacted me, asking for a meeting which he will introduce me to his friends who just started a venture, which could be considered as a start-up. He might want to get my comments about that venture.

I met up with them, two young entrepreneurs, and personally I am quite happy that younger generations are now becoming more venturing-spirit and passion to start something from scratch.

They came with a fascinating presentation deck to show me what they are doing. It is after office hour and I was a lot tired to go through that slide deck together with them. So to make my life a bit easier, I just asked them a couple of questions, simple ones, but they are meat.

First, a bit shocking them by politely telling them that I was not too interested in the slides to be presented. To me the real case, is always, about talking with the REAL people, and it means they who are sitting in front of me! They are the ones that are running the business! So it is pretty good to sit with REAL people with the REAL business and with REAL story. I don’t want to skip this opportunity by reading those  slide deck.

I casually started the conversation to ask them, whether they sniff the REAL need being present in the target customers.

This first question is so crucial, since if there is no REAL need that the business might not go anywhere. Of course, you could still catch the fish, but only one (joke!).

Putting the word REAL is to emphasize that the product or service of that venture is what the customers really need, are able to pay and more important, they are willing to pay, to provide some returns to the investment being made.

The product or service might fall into 3 categories:

  • Must have; or
  • Nice to have; or
  • So What?

I am trying to reframe my questions: is the product or service just VITAMINES or is it PAINKILLER ones?

Or is it simply better, faster ones compared to what are already in the market?

They claimed to be the first to market to introduce this product/service. I just put down my pen saying that being first-to-market in this VUCA environment, now seldom matters. The more important is to be first-to-market-fit, in most of cases, will almost be the long-term winner. When Facebook was launched in early 2004, there were already other social network sites, such as Friendster and

We continued our discussion to the investors’ expectation in funding such business.

I stressed out to them that for early stage venture, they might need to rely on 3 Fs’ money : Founders, Families and Friends, or 4 Fs money: Founders, Families, Friends and Fools.

The expectation from the investors might sound shocking to them. For serious early-stage investors, they might target to in between 10x or even 30x returns on their invested capital in risky startups. Which this mean that the investors must be convinced that the venture has the possibility of between 10x to 30x cash-on-cash return (or anticipated  ROI, Return on Investment, here I meant Return as Cash Gross Return – Money In, I am using Gross ROI, to differentiate it from Clean ROI, which “Return” is pure the difference between the money IN and money OUT). Of course, Cash-on-Cash or Gross ROI concept has no time element (it is just a simple ratio between money IN and money OUT. A note: Internal Rate of Returns threw in additional time elements, that is we need to define the years/periods it took to generate that return) and we need to find it out by asking the investors what they are typical investment horizon before they expect their money back to them (either having the venture being acquired by larger strategic company or being exited through IPO to capital market).

Typical early-stage investors might put reasonably that the venture might take 5 to 7 years for a successful venture to get big enough (GROW and SCALE UP) to be seen in the market’s radar for a healthy exit. However, the typical investment fund being raised by Venture Capital (VC, which generally speaking would enter in the later stage of the venture, through their funding series, called Series A, B, B+, C, so on) has a life of 10 years, with money being committed and made during the first 5 years and then the harvest time to be made in the next 5 years.

I am trying to explain this Gross ROI concept using return concept which might be easier to understand and accept.

The anticipated Gross ROI of 10x might scare off the new fledgling business, which might mean, if they got USD 1 million money, then the venture itself is expected to return to the investor’s pocket US$10 million (= cash-on-cash ROI).

Upon knowing that the expected time to exit is 7 years, then we could calculate the anticipated or expected Rate of Return.

So now we have:

  • Gross ROI : 10x
  • Time to exit : 7 years

With these 2 information, we could compute the rate of return on an annual basis:

The easier way to depict this is to get the help from Microsoft Excel, as shown below.

So with Gross ROI of 10x within 7-year exit period, the annual rate of return is approximately 39%.

This 39%-annual-return rate hopefully might not come as a shocking surprise or eye-popping to those young entrepreneurs. This rate is called “investor’s required rate of return“, and it might read as the investor’s target rate of return in the case that the venture will be a big success. So here we don’t have other scenario for non-success story, or in other words, the probability rate for non-success scenario is nil. In other words, it is binary, either 1 (success) or 0 (no success = failure). Of course, logically, the investors won’t even think to invest their money in the first place if it is 0 (no success).

A side note for those math lovers: However, for those that do not like the idea without considering failure scenario, then it is not too difficult to factor that. Using the simple Venture Capital valuation method, let’s do it:

Post-money value = Exit value (expected, of course!) / ( 1 + investor’s required rate of return) ^ (exit period)


V-Post = EV / ( 1 + r ) ^T

Note :

a) probability for survival in each year assuming constant rate = (1- s)

b) d = discount rate without failure risk premium factored in


V-Post = (1 – s )^T * EV / (1 + d)^T

The rest is just secondary-class algebra math, as depicted below.

Here we have :

Investor’s required rate of return equals

discount rate (without failure risk premium)  plus [constant] annual failure risk divided by 1 minus [constant] annual failure risk premium.

If the failure risk is positive, then the investor’s required rate of returns will be higher. For example, discount rate without failure risk premium is 25% and assuming annual failure risk rate is at 10%, then we would get the failure-risk adjusted rate being required to be:

(25% + 10%) / ( 1-10% ) = 38.89%.

Continued with the discussions with those two young entrepreneurs:

The investors here are dealing with private company with its private information. They might SPRAY the money and then PRAY, hoping for the money to be returned (at least for its principal). In the market, all investments that are based on public information, such as investing in public company or public bonds, will have only average rate of return. However, investing in private company with private information should have potentially higher return (and of course the flip side of this high-return investment, there is lurking at every corner of the venture path, the high risk of losing money).

Interrupted with the email exchange I made with another friend:

You might also enjoy these articles on how people misinterpret valuations in this context.

My friend:

I don’t know the fuss about using valuation of startups.

For decades people have discussed and thought about project valuation. What they do in project valuation? They forecast cash flows and discount them to calculate, value, NPV and IRR.

What is the difference for startups? Nothing!


DCF-based method is too early to apply to pre-seed/seed/early stage of such high risky venture. Remember, that probably all those entrepreneurs have is just an idea, a garage to start with and 3 persons (including their 2 dogs). Other than the entrepreneur’s own money (read : maximizing their credit card use), then all he/she has is 3 Fs (Family, Friends and Fools) to rely on.

Market-based discount rate is not there, since this again, a private company. I believe CAPM-based WACC/discount rate could only work for public companies with public information being available to every market participants/players.

So investors that are coming with “sPRAY money and PRAY” have no much choices, other than hinging on their industry knowledge, network reference, and tons of guts, will “punish” such venture with small amount of investment and higher equity participation demand. They are going to apply what we will split into three multiplier discount rate : Dilution, Risk and Return of the Total Exit Value. Exit Value might be guessed wildly using whatever the history in the same industry that they have had, PE x estimated earnings at 5th or 7th year of that venture.

One of the most well-known valuation method for such high-risk venture at its early stage, is Venture Capital Method, introduced by Harvard professor. However, most of angel investors will have their own non-quantitative way to assess the business prospect. Again, here, the investors are betting on the jockey (the founder).

Financial projection which is useful for valuation exercise, becomes an critical instrument for early investors to assess (again, at this seed or pre-seed stage, the emphasis is not on valuation and I had ever read one story from one leading investors that valuation talk will only take 10 minutes max, investor will just say take-or-leave-it, and of course the investors will be quite worried if the entrepreneurs spend more time talking about how the founders could make money out of that venture instead of how to make that venture a successful business), among others:

When the next round financing will come. Usually it could only take 1-2 years after the first round financing. Is there any chance for the venture to attract another investors, and if yes, how much is the ownership retention rate for early investors? If no potential investors are on the horizon, whether the early investors still have “dry powder” to inject for any cash call in order to help the venture to roam thru the valley of deaths until second financiers come.

So my points, the nuances of start-up valuation in many cases are more dynamic. This is why I said that we need to split the discount to whatever the exit value into (1) dilution, (2) risk and (3) return. For example, the dreamed exit value is 1,000. Then the investors will reduce it for (1) dilution to be 500 (assuming he/she will lose half of his/her initial ownership %, and then reduced it for another 200 for risk factors, and ultimately 50 for return expectation. We have end result of post-money valuation which is 250. Dilution, risk and return, the 3 mentioned above, all together lumped into what we call money multiple. (1) x (2) x (3) multiple = money multiple.

Again, this is not perfect, but there is no point to make valuation in this stage for high-risk venture too much. 50% of the ventures will be going belly up, 30% going sideways becoming living deads or zombies, returning if lucky the principal or tiny fraction of that, 19% becoming lifestyle business returning principal plus modest interest and only 1 % to really hit blockbuster exit.

You might notice that the dilution factor is a big reduction to dreamed exit value in my illustration.

From talking with one Finance Professor, I learnt that dilution is like “cost” from the eyes of earlier investors. So though the venture will surely need more investors’ money to coal up the business from pre-seed, seed, growth and scale-up period, the earlier investors will as much as possible try to protect their ownership retention rate to high as possible. There are a lot of ways they put into the agreement to show this intention from fully-rachet anti-dilution clause, conversion with valuation cap, etc. In addition to legal terms, they will make sure that in the ownership cap table, all are reflected on fully-diluted basis to include stock options pool and convertible notes. They don’t want big surprises that could lower significantly their double-digit ownership % to no meaningful %. Again this is back that the exit value is just very rough estimate and 100% no guarantee the venture will eventually reach that successful exit.

Personally I read that here as the gamble is big, betting on the jockey (which could as well lose interest in pursuing his or her dream along the path. — again they are human, and the venture is pretty much without established organization), early investors in many cases do not like to give high post-money valuation. Again, we need to read this cautiously which the investor doesn’t mean they are risk-lover, but here, they have high tolerance for risk taking. Simple math: higher risk, higher return (Modern Portfolio Theory and CAPM). The logic behind this is they are relying on the work of the Law of the Large Numbers in their portfolio.

My friend:

Agree! Of course the idea is not to discuss on valuation with the potential investors. The idea is to help the new entrepreneurs to make a decision when they receive an offer, to give her/him a reference point to accept/reject an offer from the investor.

That simple!


Yes, I agree. Nothing is special for that, just it is more dynamic, since the valuation here is not just about (1) the quality of the founders and the business itself, but it is highly influenced by (2)  the market conditions at the time the financing is raised, and (3) whether there is a heated competition from investors to chase up deals flow, or (4) even whether the investors are a quality one or not. It means in seed fund raising, the entrepreneurs are not necessarily aiming a goal to have the valuation is high as possible, but to ensure that they could reach out the quality investors that could have better networking. As it is not just about growing company, but scaling up the company, then networking really matters. Investors with Region playing field will be much better than investors with only local focus. Quality investors might give lower valuation and open the doors and windows of opportunities to the entrepreneurs from their portfolio, network, etc.

By the way, in my illustration above, I put the return factor as the last, (vs 1. Dilution, 2. Risk), intentionally as you might already guess, that the returns in start-ups could be said entirely driven by the value of the company at the time it exits (either being acquired or through IPO). Again, mapping this into the probability distribution, it will certainly skewed to the right “long right tail“. Possibility is there for the venture to hit extreme return though the chance is also [very] low. Here as depicted below, probability is most high for the investors to hit low to medium returns, either losing some of their investment, or recouping part of the principal. In this “long right tail”, the average returns are higher compared to medium returns.
My friend:
You have a good reason.

The bottom down question is whether it is desirable or not to have an idea of how much value the firm has that others are interested in. I certainly believe yes. Sure that investors will offer what they wish to pay and by sure the valuation will not even be mentioned to them. However, from the point of view of the actual owner there is no doubt that having a reference point as “the value” of the firm before the investors make the offer. If I were one of them, I would like to have an estimation of how much the value of my firm might be.

(to be continued)Categories

Jakarta, 30 May 2021



Let’s say one investor A after long negotiation with you as a founder of a early stage startup, offer you a TERM SHEET, which they will give you US$ 2 million for 10% equity in your startup.

This will mean that the POSTMONEY valuation is:

US$2mio/10% = US$ 20 mio

And the PREMONEY valuation is

US$ 20 mio – US$ 2 mio = US$ 18mio

The next question is how many shares that need to be issued to that investor?

Answering this question, will necessitate the founder to look at his/her Capitalization Table, or called CAP TABLE.

Cap Table will be pretty much a table showing all securities (common share, preferred share for each Series, Warrants, etc.) that you have issued so far, plus the reserve for Employee or Talent Option Pool

As we have known the Premoney valuation, then we could :

(1) Calculate the current share price, and

(2) Then dividing the Investor’s investment against the current share price to come up with the total number of shares being issued to that investor.

Let’s put the above into the Cap Table, as demonstrated below.

So, we could see from the above Cap Table, that founders need to issue 666,667 shares to Series A investors.

Here we noted that the Cap Table should consider the shares that will be issued as well under Employee/Talent Option Pool that will be issued later.

Some notes from the above Cap Table:

First, though it is Preferred Shares that are issued to Series A Investors, and not Common Shares, however in the Cap Table, such Preferred Shares will always be assumed as “as-converted basis”, which means that when determining the right or benefit of preferred stock, it is assumed that the Preferred Share has been converted into some number of common shares.

Second, the Employee or Talent Pool Option that have not been issued yet, will be incorporated into the Cap Table as “fully diluted” basis. Here, the Series A investors, want to know all parties that will have a claim on the startup exit value.

Why in the Cap Table we need to use “fully diluted” assumption and “as-converted basis”, from the perspective of Investor, regardless whether the shares have been issued or not yet, or whether the options has been executed or not?

First, it must be related to risk of the deal.

Rob Johnson, in his paper under the title : Valuing Early-Stage Business: The Venture Capital Method, April 2020) said that such assumptions are necessary in view of the risk of the deal.

He said:

What is important to understand is that the use of such instruments does not actual change the risk of the deal – the capital invested is still at risk; rather these instruments are used to achieve other objectives. The investor will have invested in preferred shares or debt (a) first and foremost to secure simultaneously an equity position in the company (b) while putting most of his/her capital in a senior instrument that achieves the other objective described above (Note: that is to ensure that their investment is in a senior position). For this reason, one must always use the total amount invested – irrespective of what instruments the capital is invested in – to calculate the post-financing and pre-financing valuations.

Second, it must be related to the Exit scenario.

Though risk is definitely one element of this deal, yet, another equally important element to include all instruments (such as non-vested and all non-issued options and shares), I believe it is because all those investors focus on the valuation at a successful exit (otherwise, what is the point to sit there and do such exercise?), and when that times come, logically, all these shares will certainly be issued, vested and valuable. Here Venture Capital method assumes that all equity classes will effectively have the equal claims on the company’s value, although their respective interests might typically have different rights and privileges, which might again translate into differences in exit proceeds per share unit. One thing, which I read, quite common in practice, upon exit (for example, IPO), the terms might require “qualified IPO” meaning that all outstanding preferred stock will be automatically converted to common stock. From the perspective of the buyer at exit, they want to remove all those interests with special privileges and rights.

Third, it must be related to investor’s protection of their interest.

Investor will manage to anticipated dilution in the next rounds of financing. Any unanticipated dilution is shared among all shareholders, whereas anticipated dilution is borne by premoney investors. Thus the insertion of a hypothetical future round is an attempt by current-round investors to hedge future dilution and shift it to the current round’s premoney investors. The same intuition applies to refreshing the incentive stock plan pool in a way that shifts the dilution from shared to premoney investors.

Anticipated (claimed) additional funding necessary to reach a fixed exit value is also that the insertion of an additional future round is actually the insertion of additional costs not currently in the business plan. This indirect insertion is due to using a financing-flows valuation rather than an operating flows valuation like that most common in corporate finance.

Jakarta, 9 January 2021
Rainy day

DERIVATION OF STOCK PRICE WITH NPVGO (Net Present Value of Growth Opportunities)

Dear Readers,

Please find my trying to get the well-known formula of

Stock Price equal to EPS/r + NPVGO

NPVGO, the one I see here is positive NPVGO, meaning that the company will only invest in the projects that will generate return on investment higher that its cost of capital.

If you have any comments or inputs about this formula derivation, feel free to share.

I have not yet finished up with this thinking.

This is only for educational purposes.

It comes up to me that we could solve this issue of its derivation, by having two equations that will lead us to the same results.

(1) Dividend Constant Growth Model =>
Stock Price = Dividends_1/(r – g)
Here we are discounting in growth perpetuity of Dividends (and not Earnings)

(2) Growth Opportunities (GO) Model ==>

Stock Price = Earnings_1/r + NPV of first year investment/(r-g)

Here we are discounting first without growth for Earnings, and then add it the NPV of first year investments discounted at cap rate of (r-g)

Though mathematically I could get that, what is the insight that we could take from the above 2 formula?

GO Model will make more sense since we could link it to the return on investment higher than cost of capital, or the other way around, if the NPV is negative.

Next, will show you how to have both formula mathematically be proven and the example, to see whether they are the same indeed.

So we could see from the above hand-written proof, that both Models, either we are discounting the DIVIDENDS WITH GROWTH, or EARNINGS WITHOUT GROWTH and then add on it the NPV of the GROWTH OPPORTUNITIES (discounted WITH GROWTH), then the results will be the same.

Respondend 1 to Karnen:

The growth opportunities model has the right concepts in mind, but is too vague on the metric to capture the NPV. That will be the (discounted) growth in earnings over the no-growth forecast expressed by Earnings/r. See Chapter 6 : Accrual Accounting and Valuation: Pricing Earnings of the textbook : Financial Statement Analysis and Security Valuation (5th Edition, McGraw-Hill) by Stephen Penman for the full expression.

Respondent 2 to Karnen:

Yes your derivation is correct.
As for insight, it is mainly used to assess the whether the stock price is mainly from current operations or future growth.

Respondent 3 to Karnen:

I think that your problem with this issue stems from the presentation of Gordon formula. The one you are using is ok but not the full one.

If you use the following one: PV = DIV_1/(r-g) = [EPS_1 * (1-b)]/[(r – b*ROE)]

Now, the only step needed is to plug in b =0 for the no growth (and get EPS1/r) and positive b for the growth value.

Karnen to Respondent 3

Noted for that formula, but not the one I am trying to solve.

I have been able to match the Dividends Constant Growth Model against NPV Growth Opportunities Model, under (i) growth situation and(2) return on investment is higher than cost of capital. I attach my (very bad) hand-written doodle to match both models.
The interesting insight during this exercise, I noted that that the cost of capital will be the function of growth rate, as growth is a risky thing, then the higher growth rate, then this will push up the cost of capital as well. The concept that growth is a risky one, came up to me upon reading the books by Prof. Stephen Penman (Financial Statement Analysis and Security Valuation) and Douglas R Emery, John D. Finnerty and John D. Stowe (Corporate Financial Management).

Respondent 4 to Karnen:

It is just a version of decomposing an observable stock price into COV (current operations value) and FGV (future growth value), which assumes COV being measured by capitalising current no growth EPS. Then p – COV gives an estimate of FGV. Typical approach with unrealistic simplifying assumptions regarding COV.



I just finished up reading Appendix 14-A Present Value of Risky Outflows of the textbook Capital Budgeting and Long-Term Financing Decisions by Neil Seitz and Mitch Ellison (Third Edition, Harcourt Brace College Publishers, 1999, page 470 – 473). That Appendix’s example makes a reference to Laurence D. Booth’s article: Correct Procedures for the Evaluation of Risky Cash Outflows, Journal of Financial and Quantitative Analysis 27 (June 1982), page 287-300. Prof. Laurence D. Booth is a name that will be familiar to many finance students (

I am a bit intrigued to read further that article since that Appendix said that the present value of a series of cash outflows is the same as the present value of a series of cash inflows with identical characteristics. This is interesting, since generally speaking, when we do the present value, we just jump to the NET cash flows, but this Appendix shows that underlying this general practice, it is assumed that both cash outflows and cash inflows have the same characteristics, though the book doesn’t really elaborate more about what it means with “having the same characteristics”.

The summary of that Appendix said:

Note that a positive beta on a cash outflow means that the outflow tends to move with revenues, and the variability of the outflow, therefore, decreases risk. A risk-averse decision maker would prefer variable outflows that decrease risk over a known cost with the same expected amount. The higher discount rate applied to risky outflows means a smaller present value of the outflow, which is consistent with the risky outflow being preferred over a riskless outflow.

Likewise, a cash outflow with a negative beta increases risk and, therefore, is less desirable than a riskless outflow. The low discount rate and resulting high net present value reflect the undesirability of the increased risk.

I will be using the example taken from Neil and Mitch’s demonstration of the validity and application of the basic principle of valuation for risky outflows is that the present value of a series of cash outflows is the same as that of the present value of a series of cash inflows with identical characteristics. The intuitive behind this, is that someone’s cash outflow will be another person’s cash inflow. In an equilibrium capital market, the present value of a series of outflows is the present value of those flows to the person receiving them. The above article by Prof. Laurence D. Booth provides a CAPM-based proof and a state-preference-based proof as well as a more general example.

Example 1


  • One-period capital investment (Note: this assumption rings familiar since one of the assumptions underlying the CAPM is that all investors make decisions for a single-period horizon and can revise their portfolios at the end of that horizon, albeit CAPM itself is silent about the length of the holding period. Generally speaking, it is treated as a short period. This assumption might be challenged when we are talking about capital investment, since such investment (i) will yield return over a number of years, and (ii) not always be marketable compared to stock portfolios, and (iii) even it is marketable, will the project owner be able to capture the market value of that projects?)
  • There is only two possible outcomes, State I and State II, each of which with a probability of 50%.
  • The current market value of the market portfolio is $100.
  • The cash flows for the capital investment is identical to the ending value of the market portfolio [Note: in this example, both market portfolio and cash inflows will have the same value of $100 and $130 under State I and State II.]
  • The risk free rate is 10%.

The table below shows the cash flows in each State for the investment and the market portfolio:

With the expected value of market portfolio at the end of the period is $115, then the expected return on the market portfolio is $115/$100 (current market value) – 1 = 15%.

Because the cash INflows from the capital investment are identical or the same to the ending market portfolio value, we could say that the equilibrium present value of the risky INflow stream will be equal to the value of the market portfolio, that is $100, its current market value.

Concerning NET cash flows, since both will have the same value either in State I and State II, that is $20, then it is risk free, and we could discount the NET cash stream using the risk free rate of 10%.

The present value (PV) of a risk-free of NET cash flows = $20/(1+10%) = $18.18.

With PV-out = equilibrium present value of cash OUTflows, then,

PV_out = PV_in (= current market value) – PV_net

PV_out = $100 – $18.18 = $81.82.

Once we have PV_out and the expected value of cash OUTflows, we could get the implied risk-adjusted discount rate (k_out) by discounting it:

PV_out = expected value_out/(1+k_out)

k_out = expected value_out/PV_out – 1 = $95/$81.82 – 1 = 16.11%

As we get 16.11%, we could now confirm that this will be the same risk-adjusted discount rate that would be applied to a series of cash INflows with the same characteristics. We could use CAPM to prove it.

For a series of cash OUTflows, under two possible end-of-period State I and State II, the returns on each State will be as follows:

State I = $80/$81.82 – 1 = -0.0222 (Minus sign), or -2.22%

State II = $110/$81.82 – 1 = +0.3444 (Positive sign), or 34.44%

The market portfolio return for each State will be as follows:

State I = $100/$100 – 1 = 0%

State II = $130/$100 – 1 = 30%

As the beta is the slope coefficient for the relationship between returns for some assets and return for the market portfolio, then the beta for the cash OUTflows is:

beta_out,m = [34.44% – (-2.22%)]/[30% – 0%] = 1.222 (Positive sign)

We now could calculate the required return using this beta_out,m and CAPM standard formulation, as follows:

K_out = Risk free + Beta_out,m (market portfolio return – risk free rate)

K_out = 10% + 1.222 (15% – 10%) = 16.11%

So we could see here that the equilibrium risk-adjusted discount rate is the same, regardless the cash flows is COSTs (for cash OUTflows) or Revenue stream (for cash INflows).

Under Example 1 here, the readers are requested to note that :

A positive beta on a cash OUTflow (in this case, +1.222), means that the OUTflows tend to move IN LINE with

  • the Revenues, in which from State I to State II, revenue is up from $100 to $130, and the cash OUTflows is also moving up from $80 to $110;
  • the variability of the cash OUTflows.

This IN-LINE movement, therefore, DECREASES the risk of the project.

A risk-averse decision maker, when faced with two options: (A) Variable cash OUTflows that decrease risks, or (B) A KNOWN cost with the same expected amount, might prefer Option A.

How do we know that Option A is preferable over Option B?

This is crystal clear if we compare the discount rate of Option A and Option B.

Option A have the discount rate of 16.11%, while Option B have the risk-free discount rate of 10%.

With higher discount rate being applied to the risky cash OUTflows, resulting to a lower PV of the cash OUTflows. The lower the cost (or cash OUTflows), this option will be preferred over a risk-less cash OUTflow.

Example 2


  • We will carry all assumptions from Example 1, but we just change now the direction of the cash OUTflows which will move in the opposite direction from the general market returns, which implicitly this will INCREASE the risk. So when the market returns are up from State 1 to State II, along with the cash INflows going up as well, yet, the cash OUTflows are going down.

Table below displays the cash flows in each State:

Here we need to solve jointly :

  • The present value of the NET cash flows (= PV_net);
  • The beta of the net cash flows (beta_net,m)
  • The required rate of return for the NET cash flows (k_net)

The calculation is as follows:

PV_net = Expected Value of NET cash flows / (1 + k_net) = $20/(1+k_net)

K_net = Risk free rate + beta_net,m (market return – risk free rate) = 5% + beta_net,m (15% – 10%)

Beta_net,m = [($50 – PV_net)/PV_net – (-$10 – PV_net)/PV_net] divided by (30% – 0%) = $200/PV_net

By substituting k_net and beta_net,m above to PV_net, then

PV_net = $20/(1 + (10% + (200/PV_net) x (15% – 10%))

PV_net = $9.09

Once we have the PV_net of $9.09, PV of cash OUTflows will be:

PV_out = PV_in – PV_net = $100 (its current market value) – $9.09 = $90.91.

The implied risk-adjusted discount rate of the cash OUTflows =

PV_out = expected amount / (1 + k_out)

k_out = expected amount/PV_out – 1 = $95/$90.91 – 1 = 4.5%

The beta of this cash OUTflows will be :

State I : ($110/$90.91) – 1 = +20.9988%

State II : ($80/$90.91) – 1 = – 12.0009%

Beta_out,m = (-12.0009% – 20.9988%)/(30% – 0%) = – 1.1 (Minus sign)

The CAPM required rate of return = risk free + beta_out,m (market return – risk free)

K_out = 10% + (-1.1) x (15% – 10%) = 4.5%

Again, we have here the risk-adjusted present value for the risky cash OUTflows that is the same present value for risky cash INflows with identical characteristics.

The readers are requested to note that under Example 2, which we have NEGATIVE Beta for the risky cash OUTflows, this has given rise to increase the risk. When the market is going down, from State II to State I, followed by the decline in the cash INflows, the cash OUTflows shows it has higher costs. This is of course, less desirable compared to a risk-less OUTflows.

When we compare the discount rate of 4.5% vs risk free rate of 10%, then with lower discount rate, this will result in higher net present value (= higher cost), reflecting the un-desirability of the increased risk.

One closing note on Example 1 and Example 2 is that if components of NET cash flows are discounted separately, the sum of the present values (for cash INflows and cash OUTflows) will equal the present value of the NET cash flows.

My understanding, that the discount rate for cash inflows and cash outflows, theoretically, could be different. Only if both shares the same risk factors, then we can use the same discount rate.

Under CAPM world, seems market risk, this single risk, is claimed to impact almost everything of the company’s cash flows, including INflows and OUTflows, and this has implicated that we should use one discount rate for both. When we are talking about cash outflows, this could occur horizontally or vertically. Horizontally, what I meant, is those cash OUTflows during the initial stages of the project, or in many cases, put as Io (Initial Investment) which could take years or months before the project will generate positive cashflows. The risk factors during this initial phases of the project could be said will be quite different from the risk factors impacting the later stages of the project life.

Additionally, cash OUTflows could occur vertically, that is during the period when the project has booked cash INflows, and the project will have costs of manufacturing, production, buying inventory, labor cost, overhead cost, marketing cost, project administration, etc. It is possible that the risk factors around this vertical cash OUTflows will be different from those leading to the cash INflows (which might more market oriented).

This will be logically bringing us to be able:

1. using different discount rate for cash INflows and vertical cash OUTflows

2. using different discount rate for horizontal cash OUTflows (early stage or seed phase of the project)

Respondent 1:

I agree to the general idea that the PV of cash inflow = cash outflow. However, there is an underlying assumption and that is same risk for inflows and outflows.

However, in reality capital outflow will have my risk rating and capital inflow has the payer’s risk rating (I would rather get an inflow from the US government than from a guy at the street). This should be reflected by the risk factors, though.

To make things short: I agree to the statement GIVEN same risk factors, however I do not agree that the risk factors ARE the same.

Karnen to Respondent 1:

In the practical leve, it is not that easy, to say that the risk factors impacting cash inflows and cash outflows are not the same, and this will require different discount rate.

In many cases, the analysts just assume away both cash flows (out and in) are having the same risk factors, and this is why, the NPV formula is applied to the NET cash flows, using one single discount rate.

Respondent 1 to Karnen:

Agree. That is my assumption as well.

Respondent 2:

Well, I would say that there are some generalizations that oversimplify the situation.

They say that an outflow has an equivalent inflow to other person. Yes, in general, but inflows and outflows usually are composed of outflows and inflows of MANY other people. 

For me, a cashflow (in or out) poses a given risk as perceived by me, the investor. This is the risk that is relevant: the ones I perceive. What other parties perceive about the risk of the contrary sign cashflow, is irrelevant to me.

Respondent 3:

A couple of marginal notes.

“a positive beta on a cash outflow means that the outflow tends to move with revenues”. This sounds counterintuitive. Generally, a positive beta means that an asset’s returns (let them be outflows) tend to move with the MARKET. Perhaps, the “beta” should be replaced with “correlation”, shouldn’t it?

“when we do the present value, we just jump to the NET cash flows, but this Appendix shows that underlying this general practice, it is assumed that both cash outflows and cash inflows have the same characteristics” Not sure that the latter statement is correct. We discount an expected net cash flow (a mix of inflows and outflows) at a rate corresponding to its risk, and that does not mean that we assume all components of the net cash flow have the same characteristics. Formally one obtains the same result when splitting the cash flow into parts and discounting them all at, say, the project’s (firm’s) cost of capital, but, in fact, according to corporate finance fundamentals each component should be discounted at it’s specific discount rate, and the weighted average if these rates aggregates into the discount rate  applied to the net cash flow.

Respondent 3:

Yes, that is an interesting issue that you mention and it’s something
I’ve worked out my own explanation for based on microeconomic theory.

Respondent 4:

I totally agree with this.