Sensitivity of Performance with the Market

This post relates to a proof I devised in my 2015 corporate finance class. Consider a company - it may be Apple, Nike, META, or even your favourite public company that sells orange juce. One interesting question is the *sensitivity* of that company's performance to the market. This is commonly known as Beta, and notated $\beta$. Unsurprisingly - a company's beta is a function of equity and debt, as they move with the market. One interesting question is how this sensitivity changes in the presence or absence of debt. This relationship was created by Robert Hamada - a former Dean at Chicago's School of Business, who was sent to the Amache internment camp during World War II. Coined as Hamada's Formula - it is presented as: \begin{align*} \beta_{U}=\left[\frac{1}{1+\frac{D}{E}(1-\tau)}\right]\beta_{L}, \end{align*} 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. Observing this in class - I always wondered where it came from. Well, here I present my proof.

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 \begin{align*} 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}}, \end{align*} respectively. Therefore, \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})}\\\\ \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})}. \end{align*} 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: $${U} = L_{U} + E_{U} \implies E_{U} = A_{U} - L_{U}.$$ 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.