# Proof of Hamada's Formula (Relationship between levered and unlevered beta)

Hamada's formula is presented as follows:

$$\beta_{U}=\left[\frac{1}{1+\frac{D}{E}(1-\tau)}\right]\beta_{L},$$

where $\beta_{U}$ and $\beta_{L}$ are the unlevered and levered betas of a firm respectively. $D$ is the market value of debt. $E$ is the market value of equity and $\tau$ is the tax rate.

May anyone please provide a proof for this formula? I have found some sources on the internet, though they are not convincing.

• I might but it's good to know what you found unconvincing about the other sources you found. Can you elaborate? – Bob Jansen Apr 5 '15 at 7:05
• @BobJansen. Thank you for your time. I thought about it a little more and was able to figure it out. I posted my proof as an answer. I used the wikipedia source found here: en.wikipedia.org/wiki/Hamada%27s_equation – Gustavo Louis G. Montańo Apr 5 '15 at 12:47
• That's the best :) Welcome to Quant.SE! – Bob Jansen Apr 5 '15 at 18:11
• The Hamada equation contains some crippling assumptions that make it unusable in the real world. Please see researchgate.net/publication/… – chaamjamal Mar 27 '17 at 3:35

Proof: Recall that

$$\beta_{i} = \frac{\mathrm{Cov}(r_{i},r_{m})}{\mathrm{Var}(r_{m})}.$$

Now, the returns on unlevered and levered equity are given by

$$r_{U} = \frac{\mathrm{EBIT}(1-\tau) - \mathrm{CAPEX} + \mathrm{Depreciation}}{E_{U}}$$ $$r_{L} = \frac{\mathrm{EBIT}(1-\tau) - \mathrm{CAPEX} + \mathrm{Depreciation} + \mathrm{Net\ Debt} - \mathrm{Interest}}{E_{L}},$$

respectively.

Therefore,

$$\beta_{U} = \frac{\mathrm{Cov}\left(\frac{\mathrm{EBIT}(1-\tau) - \mathrm{CAPEX} + \mathrm{Depreciation}}{E_{U}},r_{m}\right)}{\mathrm{Var}(r_{m})}$$ $$\beta_{L} = \frac{\mathrm{Cov}\left(\frac{\mathrm{EBIT}(1-\tau) - \mathrm{CAPEX} + \mathrm{Depreciation + \mathrm{Net\ Debt} - \mathrm{Interest}}}{E_{L}},r_{m}\right)}{\mathrm{Var}(r_{m})}.$$

Working with the $$\beta_{U}$$ equation,

\begin{align} \beta_{U} &= \frac{\mathrm{Cov}\left(\frac{\mathrm{EBIT}(1-\tau) - \mathrm{CAPEX} + \mathrm{Depreciation}}{E_{U}},r_{m}\right)}{\mathrm{Var}(r_{m})} \\ &= \frac{\mathrm{Cov}\left(\frac{\mathrm{EBIT}(1-\tau)}{E_{U}} - \frac{\mathrm{CAPEX}}{E_{U}} + \frac{\mathrm{Depreciation}}{E_{U}},r_{m}\right)}{\mathrm{Var}(r_{m})} \\ &= \frac{\mathrm{Cov}\left(\frac{\mathrm{EBIT}(1-\tau)}{E_{U}}, r_{m}\right) + \mathrm{Cov}\left(\frac{-\mathrm{CAPEX}}{E_{U}}, r_{m}\right) + \mathrm{Cov}\left(\frac{\mathrm{Depreciation}}{E_{U}},r_{m}\right)}{\mathrm{Var}(r_{m})} \\ &= \frac{\mathrm{Cov}\left(\frac{\mathrm{EBIT}(1-\tau)}{E_{U}}, r_{m}\right)}{\mathrm{Var}(r_{m})} + \frac{\mathrm{Cov}\left(\frac{-\mathrm{CAPEX}}{E_{U}}, r_{m}\right)}{\mathrm{Var}(r_{m})} + \frac{\mathrm{Cov}\left(\frac{\mathrm{Depreciation}}{E_{U}},r_{m}\right)}{\mathrm{Var}(r_{m})}.\\ \end{align}

Since $$E_{U}$$ is the value of unlevered equity from the last financial year, it is constant. Hence,

\begin{align} \beta_{U} &= \frac{\mathrm{Cov}\left(\frac{\mathrm{EBIT}(1-\tau)}{E_{U}}, r_{m}\right)}{\mathrm{Var}(r_{m})} + \frac{\mathrm{Cov}\left(\frac{-\mathrm{CAPEX}}{E_{U}}, r_{m}\right)}{\mathrm{Var}(r_{m})} + \frac{\mathrm{Cov}\left(\frac{\mathrm{Depreciation}}{E_{U}},r_{m}\right)}{\mathrm{Var}(r_{m})} \\ &= \frac{1}{E_{U}} \left[ \frac{\mathrm{Cov}\left(\mathrm{EBIT}(1-\tau), r_{m}\right)}{\mathrm{Var}(r_{m})} - \frac{\mathrm{Cov}\left(\mathrm{CAPEX}, r_{m}\right)}{\mathrm{Var}(r_{m})} + \frac{\mathrm{Cov}\left(\mathrm{Depreciation}, r_{m}\right)}{{\mathrm{Var}(r_{m})}} \right] \\ &= \frac{1}{E_{U}} \left[\beta_{\mathrm{EBIT}(1-\tau)} - \beta_{\mathrm{CAPEX}} + \beta_{\mathrm{Depreciation}}\right]. \end{align}

We next assume that that the correlation between the market, CAPEX, depreciation, net debt and interest is $$0$$. That is, we assume that $$\beta_{\mathrm{CAPEX}} = \beta_{\mathrm{Depreciation}} = \beta_{\mathrm{Net\ Borrowing}} = \beta_{\mathrm{Interest}} = 0.$$ Thus the equation for unlevered, and by similar computation, levered beta are given by the following formulas:

\begin{align} \beta_{U} &= \frac{1}{E_{U}} \left[\beta_{\mathrm{EBIT}(1-\tau)}\right] \implies E_{U}\beta_{U} = \left[\beta_{\mathrm{EBIT}(1-\tau)}\right] \\ \beta_{L} &= \frac{1}{E_{L}} \left[\beta_{\mathrm{EBIT}(1-\tau)}\right] \implies E_{L}\beta_{L} = \left[\beta_{\mathrm{EBIT}(1-\tau)}\right] \\ \end{align}

Equating the equations yields

\begin{align} E_{U}\beta_{U} &= E_{L}\beta_{L} \\ \implies \beta_{U} &= \frac{E_{L}}{E_{U}} \beta_{L}.\\ \end{align}

Now, for an unlevered firm it is known that:

$$A_{U} = L_{U} + E_{U} \implies E_{U} = A_{U} - L_{U}.$$

Say that the assets and liabilities of this firm are fixed with the exception of new debt capital issued. That is, the firm is levered and therefore, \begin{align} E_{L} &= \left(A_{U} + \mathrm{Tax\ Shield}\right) - \left(L_{U} + D\right) \\ &= \left(A_{U} - L_{U}\right) - D + \mathrm{Tax\ Shield} \\ \implies E_{L} &= E_{U} - D + \mathrm{Tax\ Shield}.\\ \end{align}

Thus we are left to calculate the tax shield. Assume that the pre tax cost of debt is $$k_{d}$$. Therefore,

\begin{align} \mathrm{Tax\ Shield} &= \sum_{i = 1}^{\infty} \frac{k_{d}D\tau}{(1+k_{d})^{i}} \\ &= k_{d}D\tau \sum_{i=1}^{\infty} \frac{1}{(1+k_{d})^{i}} \\ &= k_{d}D\tau \cdot \frac{1}{k_{d}} \\ \implies \mathrm{Tax\ Shield} &= D\tau.\\ \end{align}

Therefore

\begin{align} E_{L} &= E_{U} - D + D\tau \\ \implies E_{L} &= E_{U} - D(1-\tau) \\ \implies E_{U} &= E_{L} + D(1-\tau).\\ \end{align}

Recalling that $$\beta_{U} = \frac{E_{L}}{E_{U}} \beta_{L}$$ and substituting our newly created equation for $$E_{U}$$ yields

\begin{align} \beta_{U} &= \frac{E_{L}}{E_{U}} \beta_{L} \\ &= \frac{E_{L}}{E_{L} + D(1-\tau)}\beta_{L} \\ \implies \beta_{U} &= \left[\frac{1}{1 + \frac{D}{E}(1-\tau)}\right]\beta_{L}, \\ \end{align}

as required. Thanks.

This formula is a direct result of the Modigliani-Miller theorem. After some searching I found a fairly simple proof of this here:

https://quantcoyote.com/2017/04/09/modigliani-miller-unlevered-betas/