Question: Is the Net Present Value (NPV) of Project will be equal to the NPV to Equityholders?

Prof. Peter DeMarzo (Stanford Graduate School of Business)


Hi Karnen,

We demonstrate this in Chapter 18 of the text (Corporate Finance), where we show that discounting FCF using WACC is equivalent to discounting Free Cash Flow to Equity using equity cost of capital.


That said, I find that in practice when people use the equity method the usually make mistakes in computing the FCF to Equity (because they don’t correctly account for the debt dynamics), so in that sense FCF/WACC or APV approach is generally much more reliable!

Ignacio Velez-Pareja:


Dear Karnen,

I understand that you are referring to NPVequity = NPVfirm. It is YES, you can see these papers:


Market Value Calculation and the Solution of Circularity Between Value and the Weighted Average Cost of Capital WACC (A Note on the Weighted Average Cost of Capital WACC)

(This has the original paper plus the published paper.)

Ignacio Velez-Pareja   Joseph Tham

Revista de Administração Mackenzie (RAM), Vol. 10, No. 6, pp 101-131 November-December 2009


Company Valuation in an Emerging Economy – Caldonia: A Case Study
Ignacio Velez-Pareja  Joseph Tham 

The Valuation Journal, Vol. 5, No. 2, pp. 4-45, 2010


Constructing Consistent Financial Planning Models for Valuation
Ignacio Velez-Pareja

IIMS Journal of Management of Science, Vol. 1, January-June 2010 (Inaugural Issue) pp 1-26


For proper WACC calculation Return to Basics: Cost of Capital Depends on Free Cash Flow
Ignacio Velez-Pareja

The IUP Journal of Applied Finance, Vol. 16, No. 1, pp. 27-39, January 2010


Applicability of the Classic WACC Concept in Practice
M. A. Mian Ignacio Velez-Pareja

Latin American Business Review, Vol. 8, No. 2, pp.19-40, 2007


Ignacio Velez-Pareja (in commenting the reply from Prof. Peter DeMarzo):

I understand their problem. Apparently, they use variable WACC and hence, variable Ke, BUT the problem is with the CFE calculations. Hard to believe.

Well, I read the part of Chapter 18 (of Corporate Finance textbook by Jonathan Berk and Peter DeMarzo).

They show the simple case (that when modeling and reality is not that simple) where they assume constant leverage.

They say that NOT assuming constant is a problem to find the amount of debt. It depends on how they make the financial forecasting. Using my methods (that we have discussed at lenght), you forecast financial needs! Trying to act as a blind user of the FCF departing from an Income Statement the analyst have NO CLUE of what is going on related to the financing of the firm. How come they could say that (either explicitly or implicitly) in a Corporate Finance book? Isn’t that relevant for managing the firm? That makes no sense.

What do you think?

I forgot to tell you that they cheated because they use the value from the value of FCF method to calculate debt. That is not serious. You have to define debt for each method independently. They also define interest from results obtained for the FCF. And they do this for APV and for PV(ECF).

Sorry for not telling you before… I forgot. That is a real failure and lack of academic rigor.

Haven’t you seen the step by step example where I define value for each method independently?

In short, to have in a forecast a constant leverage at market prices is the most difficult task. They show that as if it were very easy and it is not. You should read a paper I have about that:


I am not really against using constant leverage. Some companies have not always ups and downs of its market value of equity. So in certain periods it might work well. I don’t think valuation is about being accurate. Saying that we could give numbers that are reliable, will be questioned easily. This happened all the time with reports produced by big boys in finance. It might read as fraudulent.

However, being transparent and consistent might be good to use as you suggested in your papers.

Ignacio Velez-Pareja:

I am not against using constant leverage neither. The issue is to do it correctly. The issue is if you wish to use the tool just to do a blind and quick valuation without really knowing how things will be happening, if you wish to have a real managerial tool for tracking something relevant as leverage and if you wish to track down what is going on in the firm to achieve the goal of some given leverage.

With a tool like the one proposed by most textbooks like Corporate Finance (Jonathan Berk & Peter DeMarzo), you will never achieve that. Unless you just need a quick and dirty way to arrive to a magical number… like a NPV today to say yes or not to accept or reject the project.

Tell me which is the managerial tool for doing that tracking proposed by corporate finance textbook writers?

Going back to the use of FCF and WACC, they are just saying look, the only reliable method is it. The others (APV, CFE) are like self fulfilling prophecies! They “show” that the methods have identical results, because they depart from the first value (from FCF) to define the variables of the second (APV and CFE) and, Bingo! they yield the same result! That is cheating, my dear friend.

Going back to the beginning of our exchange of ideas, remember that you started it with the question of telling them about NPV equity identical to NPV firm. Yes, it is true but doing what they do is not the proper way to proof it. See, please,

V = D + E (mkt values)
NPV = V – Bvalue Assets
NPV = PV(CFD) + PV(CFE) – D – Ebv
NPV = PV(CFE) -Ebv = PV(FCF)- Book Value Assets only if D=PV(CFD).


Go to pages 629-640 (of Corporate Finance textbook by Jonathan Berk and Peter DeMarzo).

Ask them what should we do if we decide to use ONLY the APV or the CFE method. Assume that we say ok, we in our firm value ONLY with CFE because them and Velez-Pareja and Tham have shown that all methods yield the same results. Shall we, as suggested in those pages, do it with FCF first to estimate debt?

I insist, doing that is not that simple.

Please download this (bilingual) example:

I use it to teach my students how to calculate Values and NPV’s with different methods and I show they are all independent from each other (this means that discount rates, say, Ku, Ke, WACC and values are estimated independently for each method). And they match. For constant leverage, it is a little more difficult and it is explained in the paper I sent these days.

If you construct a set of financial statements it is almost impossible to get constant leverage in terms of market value.

Book value constant leverage may be possible. Even this may not be easy.

In fact as an exercise it would be good to try and construct a set of consistent financial statements with constant leverage, based on market value.

If you can do this, then you have truly understood valuation!!


“Market value” for many companies is an elusive concept, easier said than having the same measuring line falling into an agreement among many people.

Constant leverage  I believe no such thing in this real world, except in Excel and math world. As long as it doesn’t oscillate too high or too down, we need to bite it off as relatively constant. Of course, relatively constant could be read as constant.

Joe Tham (co-author of Principles of Cash Flow Valuation with Ignacio Velez-Pareja)

We try to measure market value. Sure it is elusive. Based on the assumptions we try to estimate market value.

In our models we DO NOT have constant leverage.

In our models, we use variable leverage.

You should make your argument against authors who ASSUME constant leverage.

Please read our approach carefully and check that you understand our principles.

However, in our models we can construct ANY kind of leverage, constant or variable.

Please check if other authors can do VARIABLE leverage.

We need no approximations in our models.

We use the exact leverage that you specify.

Why bite off as approximate when it is NOT necessary?

Please ask your other friends.

Ignacio Velez-Pareja:

Dear Karnen,

Well, almost all of our papers on valuation relate to variable leverage. However, I have to say that it is not previously defined (a predefined leverage is possible and that would be a generalization of the constant leverage case as in the paper I sent you these days). It is the result of the interaction of cash flows, values, debt levels, Ku and Kd when you use the popular Ke formulation (Ke_t = Ku_t + (Ku_t – Kd_t)D_t-1/E_t-1), for instance. Notice that Ke at t depends on value at t-1 and it is there where circularity appears.

If you go to my page you can search there for papers like WACC depends on FCF, or A Note on the Weighted Average Cost of Capital. In addition, ALL that is in the CFV book that you have read so carefully, comes from those initial papers. The same regarding the step by step example I sent you few days ago (


Another finance professors that I sent him up your papers, seems to me does’t really catch up your ideas and talking something that might be preached many times in the current textbook.

I believe and suggest you need to publish your corporate finance textbook in English. Nowadays it’s becoming rarer for people to read papers in the midst of so many papers published and it becomes harder for us to separate gold from dull.

Will try to look back at your excel file.

Ignacio Velez-Pareja:

I see what you mean. However, all the ideas in CFV book has been taken from papers we developed prior to that book. The Spanish book is based on the ideas published in the CFV book; at least what we have related to the financial model and the cost of capital. This Spanish book is an improvement to the CFV book in the sense that we have made an effort to explain step by step, for instance, the development of the financial model. Yes, that might be interesting to translate it into English.

Yes, most papers deal with finite cash flows that is what happens in reality. Do you work with perpetuities in your consulting activity?

Tell me which papers you or your colleagues don’t understand. I will try to explain them.

What I can say is that the underlying ideas in our papers/books are the M&M ideas. For instance, we use an equilibrium equatios for cashflows and values:


VUn + VTS = D + E


  • FCF = Free Cash Flows
  • TS = Tax Shield
  • CFD = Cash Flow to Debtholders
  • CFE  = Cash Flow to Equityholders
  • VUn = (Market) Value of Unlevered Firm
  • VTS = Value of Tax Shield
  • D = (Market) value of Debt
  • E = (Market) value of Equity

These two previous equations are from M&M papers.

The same happens with the fomulas for WACC and Ke. They have been developed in CFV book and in a previous paper.

Part of the problem is that we think that many people doesn’t have clear ideas on what the tax shields are. If that is not clear, tehy will not understand the derivations of general expressions for different cost of capital. These formulas are derived based upon the two equation above and a basic tenet of finance: PV_t = (PV_t+1 + CF_t+1)/(1+ i_t+1)

This formula is in some of the recent messages I sent you.

I wish to understand why finite is harder to understand than perpetuities. To me, perpetuities are an oversimplification and at the same time, a Pandora Box, because you might find some surprises…

Best regards



Dear Prof. Peter DeMarzo,

I read Chapter 17 : Does Debt Policy Matter? Brealey Myers Allen, where it is said,

Capital structure can be irrelevant even when debt is risky.

Do you agree with that above statement?

One of strong assumptions of the MM propositions is Debt is default risk free, so cost of debt will be only related to time premium (and no default risk premium).

Prof. Peter M. DeMarzo:

Yes of course, risky debt does not change MM, as we explain in Chapter 14 under Figure 14.1.  See the figure in that chapter, which includes risky debt. (


I noted that it is Stiglitz (1969) and Rubinstein (1973) that have shown that the conclusions concerning the total value of company do not change as compared to the findings derived by Modigliani and Miller under assumptions about free of risk debt (Modigliani and Miller 1958, 1963, 1966. Note: MM 1958 assumed away distress by allowing the firm to issue risk-free debt). However, the debt cost will be changed.


Stiglitz J (1969) A re-examination of the Modigliani–Miller theorem. America Economic Review 59 (5):784–793  and its Comments on Stiglitz’s Reexamination of the Modigliani Miller Theorem by David T. Whitford (1980)

Rubinstein M (1973) A mean–variance synthesis of corporate financial theory. Journal of  Finance 28:167–181


Peter M. DeMarzo:

Yes, those are useful references on the topic.  I will suggest adding them to further readings.


Reading both of those papers, Stiglitz and Rubinstein took different route in incorporating risky debt into the cost of capital. Stiglitz used a state preference framework and Rubinstein applied a mean-variance approach. However, both authors gave us the same results that risky debt has no impact on value, the same conclusion gave by MM Proposition. Those 2 papers are not really easy to follow, yet, the simple idea in the assumptions of MM Theorem that there are no costs to bankruptcy i is much simpler. Meaning without the bankruptcy cost, then it doesn’t make much different whether the firm could issue debt at risk-free rate or at riskier one.


Dear Prof. Ivo Welch (*),

Hi, I am Karnen from Indonesia. I am currently reading your Corporate Finance textbook (third edition). Overall, I like the book. (

However, on Chapter 17 : Taxes and Capital Structure, page 550, seems to me you are indicating that all three valuation methods (APV, WACC and FTE) will not give us the same result.

You put there : Properly applied, all three methods should provide similar – though not necessarily the exact same – answers.

This statement of course is not right. All three valuation methods should give us the exact same answer (up to 0.00). This is very clear and proven already in many papers, which I enclose herewith for  your reading.

1. Taggart (1991): Consistent Valuation and Cost of Capital Expressions with Corporate and Personal Taxes.

2. Papers written by Joseph Tham and Ignacio Velez Pareja (you could easily download their papers via Both authors also wrote the book : Principles of Cash Flow Valuation: An Integrated Market-Based Approach ( in details how all those three valuation methods should result in the same value.

First, of course, we need to determine what the discount rate that we are going to use to discount the Tax Shield (could be Cost of Unlevered equity, Cost of Debt, etc.) then using the correct formula for Cost of Levered Equity, then all these three valuation methods will give us the exact same answer. No question about that.. Those authors have proven it.

I do hope there will be a revision to your Corporate Finance textbook in fifth edition.

Kind regards

Jakarta, 16 November 2018

Prof. Ivo Welch:

hi karnen—thanks for your note.  how many of your MBA students will understand the nuances of the discount rate on the tax shield?  and if there are any other imperfections in the market, such as investor market segmentation, how perfect will it remain?






Dear Prof. Ivo,

Still I see, it is very important to state something correctly though students might not understand it when we said it. Some students will develop their curiosity and some don’t. However, for those students with growing curiosity, at least we have given them correct understanding early on. You could put that in the Companion to the book or Appendix.

I have gone through a handful of Corporate Finance textbooks (Stephen Ross & Jeffrey, Brealey & Myers & Allen, Brigham, Titman, Jonathan Berk & Peter DeMarzo, Ivo, etc.) and noted that it is Jonathan Berk & Peter DeMarzo explains away the concept of taxes and capital structure in a better way. You might need to look into their book, I guess, the best in the market. The concept of discount rate for Tax Shield is put there for students with higher curiosity.

I hope you don’t mind, if, I come back to you with more observation as I am reading more of your chapters.


Dear Prof. Ivo,

I sent herewith the Excel files, proofing all valuation methods (even 6 here) all leads to the same value (up to 0.00000000), so precise.

Again, the formulas were initially proposed by Taggart (1989), which paper I have ever sent it to you, and then elaborated more (using baby steps) by Ignacio Velez-Pareja and Joseph Tham.

Kind regards


Dear Prof. Ivo Welch,

I send you herewith the full-blown statements forecast including the Valuation using WACC. The assumption for the Tax Shield discount is Ku (Cost of Unlevered Equity). We could use simpler formula Capital Capital Flows (CCF) as suggested by RS Ruback (2000) in which both the Unlevered Cash Flows and Tax Shield are discounted using Ku. If used consistently, any methods will give us the exact answer, as long as we define the Tax Shield Discount and use correctly the formula for the Cost of Levered Equity.

In many corporate finance textbooks, many authors just jump to Income Statement and then build Free Cash Flows. Personally I found this a bit confusing to audience since it does not always tell us what happens to Balance Sheet and Cash Flow Statement.

The technique for balancing lies in the Cash Flow Waterfall.



Prof. Ivo Welch:

hi S—I will take a look at your materials next quarter when I will be teaching the course again.



Ignacio Velez-Pareja:

Dear Karnen

Well, I think what we do in the model I sent days ago, is the same approach of the Cash Flow WaterFall. Could you give a look to that file?

From the Cash Flow Statement or Cash Budget Statement, that has 5 modules, I get directly the Cashflows as follows:

If you see the Module 3, it is what the firm pays to debt owners. Hence, the CFD (Cash Flow to Debtholders) is just the Net Cash Balance (NCB) of Module 3 multiplied by -1. The same with the NCB of Module 4: this NCB multiplied by -1 you get the CFE (Cash Flow to Equityholders). Hence the sum of CFE and CFD you GET the CCF (Capital Cash Flows) that is CFD+CFE. To get the FCF (Free Cash Flows), simply you subtract the TS (Tax Shield).

In the model, you have to define the TS as

In Excel

TS =Max(T*Min(EBIT+OI,FE),0) where OI is other income and FE is financial expense. With this formula you get three cases:

1) EBIT+OI (Other Income)>FE (Financial Expenses) ==> TS=T*FE

2) 0<EBIT+OI<FE ==> TS= T*(EBIT+OI)

3) EBIT+OI <0 ==> TS=0

Best regards

Ignacio Velez-Pareja:

Listen, my dear Karnen: I forgot to mention that they might be thinking on perpetuities and not in finite cashflows when they think of constant D or D%. Once they are freed from the perpetuities idea, they must realize the problem of constant debt/D%.

Again, if they are confronted with how do they forecast financial statements and how do they keep constant D/D% they will understand the issue.

Remember that value and cost of capital depend on cashflows. When I say that it means that say, Ke depends on the value of E and E depends on the CFE.

(*)  Prof. Ivo Welch is J. Fred Weston Distinguished Professor of Finance and Economics
at the Anderson Graduate School of Management at UCLA.


Potential dividends versus actual cash flows in firm valuation



I read the joint paper of Ignacio Velez-Pareja (IVP) with CA Magni, Potential Dividends vs Actual Cash Flows in Firm Valuation.

My comments:

There are no examples shown by authors how to get Actual Cash Flows…as this is about Forecast then there is such thing called Actual.

Probably what I capture from this paper. The authors want that the investment policy and payout policy for the excess cash is explicitly spelled out. Instead of assuming all Excess Cash be distributed as dividends.

The investment policy for excess cash will have the same impact as all distributed as dividends if the excess cash be put into zero NPV activities. For this statement. I am not really clear since in many valuation exercises during forecast mostly return is higher that WACC (abnormal then slowing down to normal or competitive margin).

Zero NPV activities also raise a it is hard to imagine the company want to invest in such thing.

Excess cash in many cash be retained for “temporary” reason, for example, in Apple, to anticipate legal claims, to execute Acquisition, etc. Ultimately, it will go down to shareholders..this could be seen in Microsoft and Apple cases.

Excess cash should not be a norm in many companies. It could be high in certain period if there is a potential liquidity crises in financial market as management decides to keep the excess cash for a time being until the threat is lifted out.

As forecast is for long term view. Will this potential cash flow vs actual one will be a big difference to valuation? Or the error of not doing will have big impact? To answer this. Again I don’t see any example in numbers be shown by the authors.




The idea of that paper on potential dividends is simply that when calculating CFs you should take into account working capital including cash in hand and excess cash as ST investments if any.



I guess excess cash is not a spontaneous one, like Trade receivables or any operating cash.  This is a discretionary element. If it is considered too big, management could distribute it by increasing the payout ratio.




Yes, it can. But I wonder if you as analyst should decide that. It is a decision of the board of directors. 

If you are an analyst and forecast the CFs probably you don’t acummulate cash permanently. Hence, in general, cash and ST investments shoud be included in the working capital and forget the idea promoted by Damodaran of Potential Dividends.

When forecasting you can (as I do in my model) define a cash hold policy, a distribution policy (payment of dividends or Net Income), and a mechanism (see module 5 in our model) of investing temporary any excess cash that could be found. The policy IS NOT or SHOULD NOT to accumulate cash, in general. In any case, if the firm decides to hold cash it is a decision that destroys value and should be reflected in the CFs and hence in Value. 

In the model we have 5 modules: 

Module 1. Operating income and expenses

Module 2. Investment module

Module 3 Financing module: defines amounts of debt to be taken and payments of principal and interest

Module 4 Equity module:  amount od equity to be invested by owners and dividends payments

Module 5 Excess cash definition.

In Mod 3 and 4 we define the size of the deficit and we increase it by the amount of cash in hand, otherwise, your minimum end cash would be zero. Mod 5 defines if there is superavit that is decreased by the minimum cash desired. Otherwise the minimun cash in hand would be zero.

Minimum cash is defined by policy. The greater the cash in hand, the lower firm value. Defining the level of cash should be an issue to be formally studied. You can explore the firm history to see what final cash is compared, say with revenues or with expenses and based on that you define the policy as a % of one of them



Including excess cash into working capital movement comes with a cost. The amount of working capital then doesn’t make sense anymore…too big?

About the BOD (Board of Directors)’s decision, the shareholders could push the BOD to distribute the excess cash in dividends or share buy-backs. Remember FCF theory behind Acquisitions and Buy-outs in mid 1980s. The company holds too much cash.

Anyway, the issue is how to determine the elements of unlevered cash flows generated by the project, assuming that the project is financed entirely by equity.

Is excess cash of part of that?

Seems so..if this is generated by Operating Assets entirely.



Yes, few months ago I sent the complete model.

Except ST debt, you can model the project with 100% equity and this yields zero LT debt in Module 3. If you wish not to include ST, I will have to double check the model. I have to try it.

The behavior of the model is that when there is debt, then no excess cash to invest. It would make no sense to invest excess cash while paying interest on debt.



In reality, there is a cash called compensating balance in which company keeps paying interest on debt while keeping deposit in the same bank. Or, a company has a borrowing facility with a bank which loan facility requires the borrower to have their cash management be handled by the same bank lender, which means the borrower has to keep their current account and any excess money with the same bank.

So this is not uncommon thing.




Yes, that could be a condition from the bank. However, if the bank requires the same amount as collateral it makes no sense to borrow money from it.




November 2018





Why it is so important to have an assumption that debt is risk free and the interest rate on debt is the risk free rate?

I keep noting that many corporate valuation textbooks and papers have such explicit assumption.

This assumption is also part of MM Theory.

If debt is risk free that Kd = Rf, which in many cases, analysts will use government bonds as a reference. Why is not using corporate bonds since we are valuing firm (public or private)?

Will this assumption be used because we want to use Book Value of Debt instead of Market Value of Debt? If the Debt is not risk free, what is the impact to WACC or firm valuation?

Kd is only assumed at risk free is a lot confusing.

Kd should be Risk Free + Business Risk Premium

Ke = Risk Free + Business Risk Premium + Financial Risk Premium

Then how come we assume away Kd = Risk Free only?

Risk free debt is very common assumption though I don’t know why is so important to stress this.

See Hamada (1972) paper. He has this assumption as well, risk free here means default free..if this is what it means, then government bond or AAA corporate bonds will be the representative of Default free debt.

Hamada model assumes that tax shield is riskless ( is default risk free or riskless the same?) and thus each period’s tax deduction arising from interest payment shud be discounted back to date 0 at the risk free rate. This implies beta of debt tax shield in the Hamada model is zero.

I resolved this risk free rate assumption for debt.

Risk free here means default risk free, but still includes business risk premium.

Why is default risk free assumption crucial?

It is because we are going to use EXPECTEd rate of return to discount EXPECTED cash flows.

To be EXPECTED rate of return, then it should be default risk free, otherwise it becomes PROMISED rate of return.

For example, a bank wants to lend  $100 with expected rate of return of 10%, then the actual interest rate the bank will charge the borrower will depend its default probability. The higher the prob % then the higher the promised interest rate it will be charging.


Ignacio Velez-Pareja (Columbia)’s comments

In fact, Kd = Rf + Debt Risk Premium. You can see that in the DB from World Bank. It was a surprise for me because in our model (the xls model I sent you) I said that Kd was estimated as in CAPM: Rf + DRP (debt risk premium). I know that we have been using the wording Kd as risk free, but that is not exact. ANY debt has a risk, but it is lower that the implied risk in Ke