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





Note: Karnen in Italics


I can’t see under Tax Shield (TS) discount rate = Ku, then Ku = pre-tax WACC, or D/(D+E) Kd + E/(E+D) Ke?

I found a side note in my Stephen Ross Corporate Finance textbook (7th Ed., the book I read during my master study) showing Ku = WACC without corporate tax.

(1) WACC = D/ (D + E) * Kd + E/ (D+E) Ke

(2) Ke = (Earnings before Interest EBI – Kd* D)/E

Incorporating (2) into (1), we will have

WACC = (kd*D + EBI – Kd*D)/(D + E)

Since under unlevered situation D= zero, then


EBI/E here is Ku!

However you gave me one statement that the above is correct only under TS discount rate is Ku….here I don’t get it..???

What is the relationship between TS discount rate for WACC pretax with Ku?

I guess your statement probably is unintentionally wrongly said?

Ignacio Velez-Pareja:

Yes, it is valid for any value of discount rate for TS.

My comment on Ross derivation is this: one thing is to design WACC without taxes and another thing is that a situation without taxes means D=0.

Let’s see

Let’s see this


Adjusted WACC applied to the FCF

Let WACCAdji be the adjusted WACC that is applied to the FCF in year i. Then we have

VLi-1×WACCAdji  = Di-1×Kdi – TS+ ELi-1×Kei                                (24)

VLi-1×WACCAdji  = VUni-1×Kui + VTSi-1×yi – TS                            (25)

VLi-1×WACCAdji  = (VLi-1 – VTSi-1)×Kui + VTSi-1Xyi – TS              (26)

VLi-1×WACCAdji  = VLi-1×Kui – (Kui – yi)×VTSi-1 – TS                   (27)

In equation 27, if not taxes, we obtain,

WACCAdji  = Kui                     (28)

However, when there is no taxes, FCF and FCF are identical. Remember


When no taxes, TS =0 and


And WACC for FCF = WACC for CCF = Ku.

From (24)

VLi-1×WACCAdji  = Di-1×Kdi – TS+ ELi-1×Kei

BUT, if no taxes , then

VLi-1×WACCAdji  = Di-1×Kdi + ELi-1×Kei

and WACCAdj =  KdiD%i-1 + Ei-1%XKe = Ku

I would not say that D=0. What we want is to define WACC BEFORE Taxes, BUT it doesn’t mean that it is WITHOUT debt.

In short, I think that Ross approach depart from the assumption that no taxes implies no debt and that is wrong.



Yes, I guess you are correct. WACC pretax can’t be read as no Debt.

Today I got time to go thru again Corporate Finance textbook by Jonathan Berk and Peter Demarzo (B&DM). I believe their teaching on valuation is correct, which mean they are not departed from what you have in your books. However, they further said that TS discount rate will be Ku if only the Debt Equity ratio is being kept constant, meaning that Debt and Equity will be kept adjusted at t=1, t=2, etc following the target D/E ratio. If it is permanent debt, then Kd will be appropriate for TS discount rate. Bottom line, Berk and DeMarzo (B&DM) recognized and even put that in their book that the relationship between Ku and Ke will be determined by which assumption we put for TS discount rate.

Ignacio Velez-Pareja:

Yes, I agree

HOWEVER, the idea of constant debt or D%, is borrowed from the original idea of perpetuities.

But, let’s accept it. How do you implement that in practice? Let’s see.


Karnen: yes, I agree with you 100%. This constant D/E ratio is not observable in the reality. most companies are very careful in using debt, though technically, its tax savings and financial leverage is enticing. From finance book we know, this financial leverage increases the risk of the cash flows. So long the ROA, or EBIT level could support the cost of debt, theoretically, EPS will be levered much higher than that without debt. 


If you use a model like the one I have sent to you, it is very easy to implement the idea of constant debt. You just set LT + ST debt constant and equity will contribute to any LT investment/deficit up to the value needed. The procedure would be to discount the CFs with the proper formulas for Ke and WACC.

If D% is constant the solution is a little bit more complicated: it yields another source of circularity because D will be D% times total value and you have to somehow, apportion ST and LT.



 I guess, in reality, people confuses Debt Constant (=permanent debt, the one that M&M uses) and Debt % constant. The latter creates circularity.


Personally, I am choosing Debt with Scheduled Payment. In this cases, separately valuing the FCF and then added on that the value of TS will make more efficient to handle the TS. Again this will necessitate us to put explicitly the discount rate for TS. From B&DM, sounds to me the authors will support Kd as the discount rate for TS in the case scheduled payment of Debt could be detailed. only in the case in which the firm adjusts its debt continuously to maintain a target debt to value ratio, then it is reasonable to expect the risk of the interest tax shield will equal that of the firm FCF.


On the other hand, I wonder if B&DM will offer the proper formulae for Ke and WACC for each world: Kd and Ku as discount rate for TS. Do them?


Karnen: yes, though the way they present it not always easy to follow. Your book is better. Yet, I guess, each finance scholar would like to have his/her own way of presenting the concept. Similar to M&M proposition, I noted that each corporate finance textbook is not always having the same idea in explaining it away.

By the way, you should buy B&DM book, it is a good book indeed, the best of all corporate finance textbooks in the market so far. I always use both your book/papers and B&DM book as my anchor in case I am confused with something related to valuation.

Would you like to to play with the model and try to work under the two assumptions as I mentioned above? [Karnen: sure…have tried it anyway…I like your approach. B&DM approach is assuming constant D%, which I believe in reality, it’s not easy to apply.]


By the way, I just came to realize that it is why you keep saying that there is a strong assumption behind traditional WACC formula, that is EBIT will be sufficient to cover the interest expense, meaning that EBIT > interest expense for the company to enjoy full Tax Savings = Kd (1-T). In situation, which EBIT < Interest expense, then traditional WACC is not working, and that’s well known formula can’t be used. Is there anyway that if that traditional WACC with its Kd(1-T) could handle the situation in which EBIT < interest expense? Or is it really a very special case in which EBIT > Interest expense that we could only use this Traditional WACC?


I am re-reading your paper now: Returns to Basics: Are You Properly Calculating Tax Shields.

I guess, not many people/readers really appreciate the contents of your papers in which you keep saying traditional WACC has a very strong assumptions. Even the Berk&Peter Demarzo do not say anything about this assumption and keep using the example in which the EBIT > Interest Expense.

Ignacio Velez-Pareja:

See this:

In next table I show how to handle the TS.

OI = Other income, FE Financial expenses, TS = tax savings


No Debt With debt TS= Change in taxes
0 FE
Impuesto = T×(EBIT+OI) Impuesto = T×(EBIT+OI – FE) T×FE
Tax = T×(EBIT + OI) Tax = 0 T×(EBIT+OI)
Case 3 EBIT+OI < 0 EBT=EBIT+OI< 0 EBT <EBIT + OI – FE <0
Tax = 0 Tax = 0 0


This situation can be expressed as


This means

TS = Maxim(T ´ Minimum(EBIT+OI, FE), 0).

In Excel: =Max(T*Min(EBIT+OI;FE);0)

In this way you can model the TS.


Thanks for the table. Yet, can we still use traditional WACC in situation in which EBIT less than Interest expense?



Ignacio Velez-Pareja:

Well, that is the problem with traditional WACC. It works only for the case when EBIT >FE!!!

The case you are mentioning IS NOT case 1.

When psi=Ku the formula is

WACC = Ku – TS_t/V_t-1

See that when you are in this formula with case 1, you end up with traditional WACC. SEE: WACC = Ku – TxKdxD_t-1/V_t-1, but Ku =KdxD%_t-1 + KexE%_t-1


WACC_t= KdxD%_t-1 + KexE%_t-1 – TxKdxD_t-1/V_t-1,

WACC_t= KdxD%_t-1 + KexE%_t-1 – TxKdxD%_t-1

Identical to traditional WACC.

The new formula above is much better than the traditional one because you can cover ALL 3 cases plus include ANY other sources of TS. The best example of this is the losses in exchange when you have a loan in foreign exchange and the above mentioned cases.

It is a more general formula. Follow?


Other sources of TS not related to Kd might be bank commissions paid at the issue of the loan and for only one time and similars. There are banks that charge a commission using the loan or a fine for not using a loan and so on. There is an interesting case in Brazil where they have part of dividends paid as a deductible expense, hence you earn tax shields on that.


Thank you for the clarification. I see your points…

Unfortunately many corporate finance textbooks, instead of giving us general WACC formula, they teach a very special case…The problem with this teaching, we, as the reader (and new baby in finance), swallow it and use it to the general case….The right way, it is supposed to be the book teaching the general approach and bring it to specific situation, instead of the other way around.

Jakarta, Sept-Oct 2018








Readers, this article is pretty good, reminding us that pre-tax and post-tax discount rate is not the same.

Ignacio Velez-Pareja:

I agree with the paper. The reason is simple.  Debt creates value through tax savings. See the cashflows and value equations.
V_unlevered + VTS = D + E
If you disregard taxes, you lose VTS
FCF = Free Cash Flows
TS = Cash Flow from Tax Shield
CFE = Cash Flow to/from the Equityholders
CFD = Cash Flow to/from the Debtholders
VTS  = Value of Tax Shield
D = Debt (Book value)
E = Equity (Market value)


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)?