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)]

where:

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.

**Karnen:**

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

**IVP:**

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= Ku_{t} + (Ku_{t} – Kd_{t})D_{t-1}/E_{t-1} or without some sub indexes, Ke = Ku + (Ku-Kd)D_{t-1}/E_{t-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.

**Karnen:**

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)

**u 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.

**IVP:**

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

**Sukarnen:**

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?

**IVP:**

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.