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

Question

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.

Thanks to all in advanced.

Answer 1

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.

Answer 2

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/