• interest on debt is tax deductible
• when firm adds debts, then it reduces taxes, all else equal
• reduction of taxes means an increase of the cash flows
• reduction in taxes reduces the net income

\[R_a = WACC = \frac{E}{E+D}\times R_e + \frac{D}{E+D}\times R_d \times (1-\tau_c)\]

\(WACC\) decreases with the Debt-to-Equity ratio. But Cost of equity \(R_e\) rises, because

\[R_e = R_a + (R_a-R_d) \times \frac{D}{E}(1-\tau_c)\]

Assuming constant flow of the tax shield (\(D r_D\tau_c\)), the following holds

\[PV = \frac{D R_d \tau_c}{(1+R_d)} + \frac{D R_d \tau_c}{(1+R_d)^2} + \dots = D \tau_c\]

\[EV = \frac{FCFE}{(1+R_a)} + \frac{FCFE}{(1+R_a)^2} + \dots = \frac{FCFE}{R_a}\]

\[FV = \frac{FCFE}{R_a} + D \tau_c\]

1. Direct costs

• legal and administrative costs
• bondholders incur additional losses

1. Indirect bancruptcy costs

• Larger than direct costs but difficult to measure
• Losses of non-effective management
• lost sales, interrupted operations, low morale…

\[\frac{BC}{E}\]

where \(BC = const\) is absolute value of the bancruptcy costs in total E = equity \[\lim_{E\rightarrow 0, D \rightarrow E}\frac{BC}{E} = +\infty \quad \frac{\partial(BC/E)}{\partial D} < 0 \quad E+D = const.; BC = const.\]

We have a set of different tax rates:

• \(\tau_p\) personal tax rate of the income
• \(\tau_{CG}\) personal tax rate of the capital gains
• \(\tau_{div}\) personal tax rate of the dividends
• \(\tau_E\) personal equity tax rate

\[\tau_E = \frac{div.}{NetIncome}\tau_{div}+\frac{NI-div.}{NI}\tau_{CG}; \quad \text{NI = Net Income}\]

• • \(\tau_p\) personal tax rate of the dividends

Each year an individual receives \[(1-\tau_C)(1-\tau_e)(x - rD) + (1-\tau_p)rD = \] \[(1-\tau_C)(1-\tau_e)x + ((1-\tau_p)-(1-\tau_C)(1-\tau_e))rD = \]

where \[V_L = V_U + PV ((1-\tau_p)-(1-\tau_C)(1-\tau_e)rD )\] \(PV\) is a present value of the present and future ## quotation 3 interests.

An investment in knowledge pays the best interest. – Benjamin Franklin

“Always Do What You Are Afraid To Do” – Ralph Waldo Emerson

The roots of education are bitter, but the fruit is sweet. – Aristotle