Prof. Damodaran in his second edition of Investment Valuation, page 194, Chapter 8 : Estimating Risk Parameters and Costs of Financing, shows that:


if all the firm’s risk is borne by the stockholders (i.e., the beta of debt is zero), and debt has a tax benefit to the firm, then,


B_Levered = B_Unlevered [ 1 + (1-t) (D/E)]


B_levered = Levered beta for equity in the firm

B_unlevered = Unlevered beta of the firm (i.e., the beta of the firm without any debt)

t = corporate tax rate

D/E = Debt-to-equity ratio (market value)

Additionally, the author gave a footnote to the above formula:

This formula was originally developed by Hamada in 1972. There are two common modifications. One is to ignore the tax effects and compute the levered beta as:

B_levered = Beta_unlevered * [1 + D/E]

If debt has market risk (i.e., its beta is greater than zero), the original formula can be modified to take this into account. If the beta of debt is Beta_debt, the beta of equity can be written as :

B_levered = Beta_unlevered * [ 1 + (1-t) (D/E)] – Beta_debt (1-t) (D/E)

Comments from IVP upon discussing this formula shown in many of Prof. Damodaran’s valuation books:

Ignacio Velez-Pareja (IVP):

The formula for unlevering beta is for perpetuity AND when the discount rate of TS is Kd.

In his tables from his (Damodaran) site, the unlevered beta is calculated with only one term that contains the (1-T) factor.

What I do is to estimate somehow, present Ku and deflate it. Then I assume a constant ku (real).

As we usually “forecast” inflation rate assuming that economic planning activities (our Central Bank, for example) try to reach a target inflation rate we increase/decrease the actual inflation rate.

With the “forecasted” inflation we inflate ku (real) and obtain our “forecasted” Ku (nominal).

Regarding the use of unlevered beta we normally assume the same unlevered beta for perpetuities.

For me, the MOST IMPORTANT task is to estimate cashflows. We do the best to estimate rates but we consider more relevant to estimate cashflows.

In any case, remember that we present our results with MCS and simple sensitivity analysis and show third parties not a single value, but a Range or a Distribution of Values.


Then how to calculate Ku (Cost of Unlevered Equity)?


Well, the standard and direct way is to estimate Bu from some source, say Damodaran. From there you can also estimate the ERP and look around for your real interest rate (defated governments bonds) and estimate future inflation rate. With this and using CAPM you can get Ku = Rf + Bu x ERP.

There is a collection of formulas for WACC_FCF, WACC_CCF and Ke and they depend on the assumption of the discount rate for TS, the tax savings or tax shields. Attached you will find a table where there is a summary for the 3 discount rates for 3 assumptions for discounting TS: Kd, Ku and Ke.

Most of them have circularity: the rate today depends on the value of yesterday. At this time you know how to handle that, right?

I use Ku as discount rate for TS and the different cost of capital are
WACC_FCF_t = Ku_t – TS_t/V_t-1
WACC_CCF_t = Ku_t
Ke_t = Ku_t + (Ku_t-Kd_t)D_t-1/E_t-1


The second has no circularity as you can notice.As I said yesterday to forget US Market rates and use subjectivity for estimating rates/betas, in some cases, let me tell you what I think we can do for estimating Ku/Ke for non-traded firms.

Explanation of my approach to estimate Ku or Ke asking the investor:

1.    CAPM is a tool designed to estimate what an anonymous investor expects to earn as a minimum, when you are dealing with traded firms (usually you don’t have access to the equity investors). That is the reason to make regtressions using public information of prices and indexes.

2.    Most of our real cases are for non-traded firms and for many of them you have ACCESS to the investor.

3.    Prepare yourself and the investor to think on an unlevered project/firm, because what you are looking for is Ku or Bu.

4.    Given 1) and 2), find out if the investor is or is not completely diversified. In many cases she is not. Hence, you might consider to think she is assuming total risk and should estimate a kind of total beta in case you use CAPM.

5.    Make the investor conscious on how diversified she is and what it means in terms of risk.

6.    When asking the investor how much she is willing to earn as a minimum, the most probable answer is a VERY high rate. Then you have to start trying to lower it to her minimum. A possible approach to do that is to show her some local market returns in different but public issues/investments 

7.    When trying to find her minimum make her aware that the higher his rate, the lower the value of her equity.

8.    Make the investor aware how high are the market rates with and without risk just to make her to choose something that makes sense.

9.    After some trials you might reach to a subjective minimum. This subjective estimate will be composed of your country risk free rate. Rf and a risk premium that implies your country ERP. 

10. Given 9), calculate the implicit beta in the estimation. Rf and country ERP could be found either in specific country information (Central Bank or similar sources for Rf) and Damodaran for ERP (beware of not double counting Country Risk). Country Risk is needed if you start from US Bonds (Remember that EMBI has embedded local Rf) not if you start from your local Rf.   

11. Compare that implicit beta with total beta and levered/unlevered beta from, say, Damodaran.

12. Trust on 9), but if you need to “negotiate” Ku, or what is the same, Bu, trust 10) and/or 11). Eventually you might have to discuss with the counterpart (in case you are raising funds for the project, for instance) in terms of beta taken by him from Damodaran or another similar source.

13. Why Ku or Bu? Because you might use Ke_t= Kut + (Kut – Kdt)Dt-1/Et-1 or without some sub indexes, Ke = Ku + (Ku-Kd)Dt-1/Et-1. Ask Roberto Decourt about how easy is to do that. However, if you as lazy as I am, use Ku and Capital Cash Flow, CCF, to estimate firm/project value and that’s it. This is identical to PV(FCF at WACC) and to (PV(CFE at Ke) + D).

14. Why do I suspect from using betas from Damodaran for emerging markets? Remember that in those emerging markets are included giants such as China and India. And within a local concern, in Latin America, you have Brazil added to Colombia, Peru, Ecuador, etc. I know that Damodaran has apart giants such as China and India and you might wish to do some, what I call, “data massage” to exclude them based upon averages, but I think is to add salt and lemon to the wound as we say in Spanish in the sense that there are enough simplifications when using those betas. Let me tell you that for instance, in our sotckmarket we have very few industries (no more that 5-10), while we know that in reality (counting the non-traded firms) we have about 100+. Damodaran has about the same number of industries.   

After reading or listening Pablo’s attack to CAPM wouldn’t you accept a subjective approach to defining Ku/Ke/beta? In any case, remember that Pablo makes a survey mainly to teachers, that might pick out the beta from the thin air to illustrate their examples in their lectures…

You get first Ku and from there you get Ke. Ku is for me as the origin of everything. It is a shame that Damodaran has not implemented a method to define Ku independently for Beta for the Ke.

What is better, to use the levered/unlevered beta from Damodaran for emerging markets that includes China, Brazil, Colombia, Peru, India, and so on or to ask the investor?

Remember, the great improvement of CAPM is to try to guess the beta of an unknown anonymous investor. Do you trust on that? I think it is not better that trying to get a good estimate of the investor when you can sit with whom, you could look to her eyes and try to find the minimum beta (discount rate) he/she is interested in asssuming. 



Other author, for example, Prof. Peter DeMarzo, in his Corporate Finance textbook, on Chapter 18 : Capital Budgeting and Valuation with Leverage, Section 18.3 The Adjusted Present Value Method, gave a formula

Ku (cost of unlevered equity) = (E/(E+D) * Ke (=cost of levered equity) + (D/(D+E) * Kd (cost of debt)) = Pretax WACC ……..(Formula 18.6)


The above formula is only correct under the assumption that  Ku is the discount rate for Tax Shield. the above 18.6 formula doesn’t work if the discount rate for Tax Shield, let’s say Cost of Debt, or any discount rate between Cost of Unlevered Equity and Cost of Debt.

By definition, WACC_Before _Tax should be Ku! And this happens only when you assume Ku as the discount rate of TS.



Will Ku mean more stabile compared to WACC?


Ku is reflective of the project risk, and with using only 1 Ku, will that mean we assume away the project risk is constant both in the finite forecast period and terminal period. Something sounds not making sense?



Ku is the unlevered cost of equity. This means that its risk is not affected by leverage. What I do is to recognize only one cause of variation of Ku: inflation. If you are able to define when the risk of the unlevered project changes, you tell me.

In addition, consider that you don’t work with Ku using unlevered beta from today and keepimg it constant over the period, but you use Ke. Would you be more informed for changing the levered beta in the future? If you are able to change levered beta more than the change due to leverage, you tell me.

In summary, I use Bu all over the N periods. IF I were to use BL, I would adjust that levered beta, BL, only by leverage.

As CCF is not affected by leverage, I don’t adjust Bu, but keep it constant and adjust Ku by inflation. In other words, I deflate the initial Ku and keep constant ku (with all small letters)  and this real ku, I inflate to get Ku for every year.

Let me know if this is clear enough and if it makes sense to you.




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 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.) (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 Debt Betas and Beta Unlevering Formulas