MM Proposition II (without taxes)

Leverage raises expected EPS but not share price, because a higher EPS is offset by a higher discount rate.

\[ r_A=\frac{D}{D+E}r_D+\frac{E}{D+E}r_E, \qquad \text{Price per share}=\frac{\text{EPS}}{r_A}\;\text{(all-equity benchmark)}. \]

Cost of equity with leverage:

\[ r_E=r_A+(r_A-r_D)\frac{D}{E}. \]

Before borrowing (\(D/E=0\)):

\[ r_E=r_A=\frac{1{,}500}{10{,}000}=0.15, \qquad \text{Price}=\frac{1.5}{0.15}=10. \]

After borrowing (\(D=E=5{,}000\)):

\[ r_E=0.15+(0.15-0.10)\frac{5{,}000}{5{,}000}=0.20, \qquad \text{Price}=\frac{2}{0.2}=10. \]

Shift in capital structure example

  • Initial: \(D=33.3\), \(E=66.7\), \(r_D=7.25\%\), \(r_E=15.5\%\), \(V=100\).
  • Asset return:

\[ r_A=\frac{33.3}{100}7.25\%+\frac{66.7}{100}15.5\%=12.75\%. \]

  • After recap: \(D=50\), \(E=50\), \(r_D=8\%\), keep \(r_A=12.75\%\).
  • Solve for equity return:

\[ r_E=12.75\%+(12.75\%-8\%)\frac{50}{50}=17.5\%. \]

Weighted-average return on debt and equity stays constant (no-tax MM), while higher leverage raises equity risk and required equity return.

If debt were fully risk-free, \(r_D\) would be constant and \(r_E\) would rise linearly with \(D/E\); in a more realistic setting, \(r_D\) also rises with leverage and \(r_E\) rises more slowly.