I see in many valuation books, there is no mention at all about the Non-Controlling Interest/NCI (or formerly known as Minority Interest) as part of the Equity component in the WACC. Though it might be obvious for some analysts, it could raise a bit confusion upon reading the balance sheet, and they noted that NCI be shown as part of Equity (Note : Formerly, shown separately from the Equity. In general, the minority interest is displayed above Equity section in the past). The NCI shares are also floated and listed in the stock exchange, so by taking Outstanding Shares x Market Price/share, it is pretty much the total of that calculation has included NCI value.

However, I noted that IFRS (International Financial Reporting Standard) 3 on Business Combination is a bit different in valuing  NCI [for the purposes of determining the goodwill], in which the valuation of the NCI could be made by either: (i) Fair Value Method or (ii) Proportionate Share of the acquiree/target’s [fair value] identifiable net assets. Assuming (ii) is followed, will that mean, it is possible to have the market value of the Equity in the WACC = Controlling Shares x Market Price/Share + Proportionate Share of the Acquiree/Target’s [Fair Value] identifiable net assets (for NCI)?




When you build the discount rate of WACC. The debt you are going to use is Debt or Debt minus Cash (=Net Debt)?

The latter is becoming more popular knowing the fact that many companies now (started this trend is the companies in technology sector, such as Apple, Microsoft, etc.) that maintain large cash balances in excess of their cash operating needs. Much of the reason is for acquisition purposes as part of inorganic growth strategy, which require quick decisions and cash is the currency that is a lot easier to exchange in the negotiation table.

However, I am not really in big favour going for using Net Debt, knowing that:


  • In reality, that excess cash is not used for debt repayment and the debt covenant doesn’t require to have early repayment/retirement.
  • The market risk and  yield for cash is different with that of debt.
  • Investors/analysts might be more concerned about the risk underlying the company’s operating business value and not the enterprise value (Debt + Equity – Cash, if we define so).
  • Cash balance is quite fluctuating (and unpredictable) from one period to another period, depending upon the realization and implementation of the execution in the acquisitions.

So I prefer to use “Debt” only as the component in the WACC, and address the risk of Cash separately. Meaning once we have estimated the company’s enterprise value (note: we need to use noncash working capital in coming up with the unlevered free cash flow, or ideally, if possible, to include only operating cash into the working capital), then we could deduct Excess Cash from the Enterprise Value.

Comments from Edward Bodmer (Finance Energy Institute)

I totally agree with you with respect to credit analysis — I don’t think you can call cash as negative debt for things like Debt/EBITDA.

But for equity analysis, consider the following:

One company like apple has billions of cash

Another company with the same operating risks has no cash.

When you measure beta or even volatility of stock prices, the company with no cash should have a higher beta and you can even say that the beta of the equity that you see is the weighted average of the beta on cash (zero) and the underlying asset beta.

If you are using free cash flow without interest income, then you should use a WACC that does not have the downward effect of the cash.

I understand if you disagree with this, but I go even further and say that to get the WACC and beta on free cash flows you should adjust for everything. What I meant is that if you have an associated company that is not in EBITDA (and therefor not in FCF) then you should find the beta of that company (the associated investment) and adjust for the beta like with the cash balance.  You could also try the same with subordinated debt etc.

Comments from Ignacio Velez-Pareja:

Agree with you!

Moreover, I should tell you that all the trash behind net debt is what Aswath Damodaran calls “potential dividends” [Karnen’s Note: the readers who are interested in learning more about Potential Dividends vs Actual Dividends in Valuation of the Firm, could google the papers related to this topic under https://papers.ssrn.com). What does it mean? Well, distributing cash and quasi cash items, but keeping it in the Balance Sheet! I think we have debated this issue before.

Listen, my Golden Rule is that you model what you think it is going to happen in the future. This is, if you repay debt in advance, you reflect that in your Cash Budget and in your cash flows. That simple.

Hence, my answer to your question is use the debt that is in the Balance Sheet. Forget of net debt. If you don’t, then reflect that in the Balance Sheet and the Cash Flows.

When I have lots of invested cash I will have interest income. Remember that equity has a residual income. When I see the Cash Balance, there will be some income at module 5 where you find superavits or zero.When you use the indirect method to arrive to the Cash Flows (CFs), you should start from EBIT+OI (other income that includes interest received). The generation of CF is mainly the operating items (EBIT). I don’t see the need to weigh beta with zero beta. Do you weigh beta with beta debt? Of course not!

I agree with you, except that equity value is PV(CFE at Ke) or PV(CCF at Ku)-Debt. The cash on hand is part of the equity value. I don’t follow you when you define EV subtracting cash. EV is D+E, no more, no less.

What do you think?




I have made 5 videos showing the consistent formulas that we need to use for Tax Shield discount rate, Ke (Cost of Levered Equity), which will give us the same computed value result.

In those videos, I use the assumption of TS discount rate being discounted at Ku (Cost of Unlevered Equity).

In a nutshell, we need to be consistent in which formulas to use..otherwise, the resulted value will not be the same…trust me!

Kd that should be used in the WACC for CCF Kd WITHOUT (1-Tax). Once we apply this, it is so easy to see that the WACC is now Ku….. This is why Ignacio Velez-Pareja (the co-author of the Principles of Cash Flows Valuation book) kept saying that the simplest thing to do the valuation if we don’t have much time, we just apply Ku as the discount rate to CCF.

No circularity….and we have more time to focus on building the better forecast for FCF and TS.

Now I could see WACC applied to FCF is not the best option…easily leading to incorrect TS which might not always be there for the company at tax loss situation, and the constant leverage assumed.

Notes from Ignacio Velez-Pareja: (Note: I put in italics)

For clearness, when you wish to use the name WACC. it is better to say WACC for the FCF or for the CCF. For Ku as discount rate for TS, in the first case, it is Kd(1-T)D%+KeP% or BETTER, Ku-TS_t/V-t-1. Please throw out the first formula for WACC for the FCF to the trash. In the second case, CCF, WACC is Ku = KdD%+KeE%. Throw out the last formula  to the trash. Just use Ku and that’s all.

We have 5 methods: 3 of them have circularity and 2 don’t. No circularity: APV and PV of CCF at Ku. Circularity: PV of CFE at Ke, “general ” WACC and textbook WACC. 

You are 100% right when you say that using Ku and CCF gives you time to devote to make a better forecast and models. 

Please notice that Ku = KdD%+KeE% is true ONLY when Ku is the discount rate for TS. Look at the other tabs/sheets for other discount rates (Kd, Ke or any number) and see the formulas you have worked on in the case of Ku as discount rate of TS and the others. Try calculating Ku = KdD%+KeE% for each case and you will notice that the ONLY case when it is identical to Ku is when Ku is the discount rate for the TS.

In the literature, it seems to me that people mix cases (discount rate for TS) even for perpetuities. For perpetuities, the case of Ke is as follows:

  1. For Ku Ke = Ku + (Ku-Kd)D/E for finite cash flows AND perpetuities.
  2. For Kd, in general, and for finite CFs: Ke= Ku + (Ku-Kd)[Dt-1/Et-1 – VTSt-1/ Et-1] .  For perpetuities. remember that VTS = KdDT/Kd=DT, hence, Ke= Ku + (Ku-Kd)[Dt-1/Et-1 – (KdDT/Kd)t-1/ Et-1] = Ku + (Ku-Kd)[Dt-1/Et-1 – (DT)t-1/ Et-1]. And then you have the popular Ke formulation for perpetuity (that many wrongly use for finite CFs): Ke= Ku + (Ku-Kd)[Dt-1/Et-1 – (KdDT/Kd)t-1/ Et-1] = Ku + (Ku-Kd)(1-T)Dt-1/Et-1
  3. Those many that use (or used) Ke=Ku + (Ku-Kd)(1-T)Dt-1/Et-1 were authors such as Brealey and Myers, just to mention one pair of Holy Cows in finance books. 


Back to my videos, in the next videos, I will use the TS discount rate at Kd (Cost of Debt).




Clifford S. Ang, CFA dari Compass Lexecon pada tanggal 8 Februari 2017 memposting satu artikel berjudul: Estimating Debt Betas and Beta Unlevering Formulas (Note : terlampir dalam bentuk pdf.)

http://quickreadbuzz.com/2017/02/08/estimating-debt-betas-beta-unlevering-formulas/ (diakses pada tanggal 19 Februari 2017)


Cukup menarik untuk dibaca, menyinggung Hamada, Benninga-Sarig dan Pablo Fernandez.

Sebagai catatan kaki:

The relationship between leverage and equity betas was developed by R. Hamada in “The Effect of the Firm’s Capital Structure on the Systematic Risk of Common Stocks,” Journal of Finance 27(2) (1972): 435–452, and by M. Rubinstein in “A Mean-Variance Synthesis of Corporate Financial Theory,” Journal of Finance 28(1) (1973): 167–181.

Hamada formula assume Kd (cost of debt) as the discount rate for Tax Shield. The formula shown in the Clifford’s Article by having (1-T) in the below equation, is in the perpetuity context (not to be applied to the finite cash flow)

Beta_Levered = Beta_Unlevered + (Beta_Unlevered – Beta_Debt)*(1-Tax Rate)*Debt/Equity

Then I am going to show here how to derive this Hamada’s “beta-levering-and-unlevering formula” and the assumption Hamada used for Tax Shield (TS) discount rate in coming up with that very well known formula (herinafter referred to as Hamada formula).

It’s again critical to mention that Hamada formula is built under Perpetuity context and it is not appropriate to be applied to Finite streams of cash flows context.

Let’s start the roller coster trip:

  • FCF= Free Cash Flow
  • TS = Tax Shield [cash flow]
  • CFD = Cash Flows to the Debtholders
  • CFE = Cash Flows to the Equityholders
  • Kl = [Required/Expected] Return to Levered Equity
  • Ku = [Required/Expected] Return to Unlevered Equity
  • Psi = Discount rate for Tax Shield
  • Kd = Return to the Debtholders
  • Vl = [Market] Value of the company under Levered Equity
  • Vu = [Market] Value of the company under Unlevered Equity
  • El = [Market] Value of the company’s Levered Equity
  • D = [Market] Value of the Debt
  • Vts = [Market] Value of the Tax Shield
  • T = Tax Rate
  • Rf = Risk Free Rate
  • E(rm) = Expected Market Portfolio Return

Under M&M World with Tax:

FCF  + TS  = CFD  + CFE


Under Perpetuity, the above equation be translated into:

(Ku * Vu)  + (Psi * Vts) = (Kd * D) + (Ke * El)

(Ke * El) = (Ku * Vu) + (Psi * Vts) – (Kd * D)

Since Vu = El + D – Vts, then we could replace Vu above with :

(Ke * El) = (Ku * (El + D – Vts) + (Psi * Vts) – (Kd * D)

Rearranging the above equation to become:

(Ke * El) = (Ku * El) + [(Ku – Kd) * D] – [(Ku – Psi) * Vts]


Ke  = Ku + [(Ku – Kd) * D/El) – [(Ku – Psi) * Vts/El)

Since Vts = (T * Kd * D)/Psi, then we could substitute Vts above to :

Ke = Ku + [(Ku – Kd) * D/El] – [(Ku – Psi) * (T*Kd*D)/Psi/El)

If we ASSUME that Psi (Discount Rate for Tax Shield) = Kd (Cost of Debt), then we could simplify the above formula to:

Ke = Ku + [(Ku – Kd) * D/El] – [(Ku – Kd) * (T*D)/El],


Ke = Ku + (Ku – Kd) * (1-T) * D/El…………………..(this is I named Formula A), that is the relationship between Return to Levered Equity with Unlevered Equity, with a positive relationship with the Debt to Equity Ratio

Please put a note, that in Formula A above, we could see (1-T) present, a case we find under Perpetuity context.

Now, we move to the CAPM world, with its very very famous formulas

Ke = Rf + Beta_equity * [E(rm) – Rf]

Assuming similar CAPM relationship could be applied to Kd and Ku as well, then we could put the relationship as follows:

Kd = Rf + Beta_debt * [E(rm) – Rf]

Ku = Rf + Beta_Unlevered_equity * [E(rm) – Rf]  ……(let’s call this Formula B)

Let’s we subtract Kd from Ku, then,

Ku – Kd = (Rf + Beta_Unlevered_equity * [E(rm) – Rf]) – (Rf + Beta_debt * [E(rm) – Rf]), then

Ku – Kd = Beta_Unlevered_equity * [E(rm) – Rf]) – Beta_debt * [E(rm) – Rf]), or

Ku – Kd = (Beta_unlevered_equity – Beta_debt) * [E(rm) – Rf]) …..(let’s call this Formula C)

We have now three Formulas:

Ke = Ku + (Ku – Kd) * (1-T) * D/El (Formula A)

Ku = Rf + Beta_Unlevered_equity * [E(rm) – Rf]  (Formula B)

Ku – Kd = (Beta_unlevered_equity – Beta_debt) * [E(rm) – Rf]) (Formula C)

If we substitute Formula B and Formula C into Formula A, then we are going to see this relationship:

Ke = (Rf + Beta_Unlevered_equity * [E(rm) – Rf] ) – (Beta_unlevered_equity – Beta_debt) * [E(rm) – Rf]) * (1-T) * D/El

If we clean up the above formula, then

Ke = Rf + Beta_unlevered_equity + (Beta_unlevered_equity – Beta_debt)*(1-T) * [E(rm) – Rf]

Or we could put

Ke = Rf + [Bu + (Bu-Bd)*(1-T)* D/El] * [E(rm) – Rf]

Please compare the above equation with the CAPM well known formula that:

Ke = Rf + Beta_equity * [E(rm) – Rf],

Then we could see

Beta_levered_ equity = Bu + (Bu – Bd) * (1-T) * D/El

Assuming Beta_debt = 0, then, the above equation will be seen :

Beta_levered_equity = Bu + Bu * (1-T) * D/El, or

Beta_levered_equity = Bu (1+(1-T)D/El)

This is exactly what Hamada’s formula  gives us the insight into this relationship between Beta_levered_equity and Beta_unlevered_equity, which is very useful in doing the unlevering and levering the beta for many valuation analysts, since Beta_unlevered_equity is UNOBSERVABLE IN THE MARKET.

However, watch out, there are strong assumptions behind this formula to use:

  1. That formula is constructed assuming FCF is perpetual and time constant
  2. The discount rate for Tax Shield = Cost of Debt (which not all finance scholars have such agreement)
  3. Beta for debt = zero


Jakarta, February 2017

quickreadbuzz.com-Estimating Debt Betas and Beta Unlevering Formulas


It might be of interest to you in understanding the crucial role of TS (Tax Shield) discount rate in deriving a correct cost of capital formula….


Edward Bodmer

Hello Karnen:


I have become obsessed with you questions and ideas.  I think that the WACC issues can be derived by simple models and clear thinking concepts rather than dense formulas.  I am making simple models to demonstrate various ideas and also videos.  I will be very interested in your thoughts on this.



I have attached a couple of files and articles on tax shields and cost of capital etc.


The first file demonstrates that when the target capital structure does not equal the current capital structure, discounting free cash flows at the WACC (which does not change in theory when the capital structure changes) gives the same answer as computing a new cost of equity with changing capital structure.


The second file is the beginning of work on the tax effects of interest.


The other attachments are articles on WACC and taxes etc that I am reviewing for our analysis.





The (required) return to levered equity will depend as well on the assumption we put on the discount rate for Tax Shield. Some use cost of debt, or return to unlevered equity or between. You could google a whole bunch of finance papers discussing this issue.


WACC is a general formula, the most important what we are going to put into the return to levered equity (Ke). Knowing that TS discount rate assumption is also part of Ke then different assumption for TS discount rate will impact the Ke.


Edward Bodmer


I have been working very hard on your issue and I am attaching the set of videos I have made on this.


Interest Shield Exercise https://www.youtube.com/watch?v=G6BBvAFHJGo
Interest Shield Exercise https://www.youtube.com/watch?v=9LKppeYudVU
Un-lever and Re-lever with Tax Shield https://www.youtube.com/watch?v=jaLOx3RJFkc
Theory of Debt Beta and WACC https://www.youtube.com/watch?v=n_csEtMHveA
Mechanics of Debt Beta and WACC https://www.youtube.com/watch?v=Bvp0ruVYBSw
Unlever and Re-lever with Varying WACC https://www.youtube.com/watch?v=7ny67y-EpR8
Growth Rate, Tax Shields and WACC https://www.youtube.com/watch?v=oKNmQ9fimZc
Circularity, WACC and Ku https://www.youtube.com/watch?v=2EPd8_yi_0M
Growth and WACC Bias
Target Capital Structure without Taxes
Target Capital Structure with Tax Shield


January 2017

Consistent valuation of project Nyborg Discount Rates and Tax Nyborg Nyborg The value of tax shields IS equal to the present value of tax shields Valuing the Debt Tax Shield Cooper Nyborg Tax-Adjusted Discount Rates with Investor Taxes and Risky Debt Consistent methods of valuing companies by DCF