# NACVA ARTICLE : ESTIMATING DEBT BETAS AND BETA UNLEVERING FORMULAS

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.)

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
• Psi = Discount rate for Tax Shield
• 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

FCF = CFD + CFD – TS

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]

Then

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],

Or

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

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