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








Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.