Operating Income, Free Cash Flow, and Value Decomposition

Definitions

\(\text{Rev}\)

Revenues

\(\text{VC}\)

Variable costs of operations

\(\text{FCC}\)

Fixed cash costs (administrative costs, real estate taxes, etc.)

\(\text{dep}\)

Noncash charges (depreciation and deferred taxes)

\(\text{EBIT}\)

Earnings before interest and taxes

\(k_d D\)

Interest on debt (interest rate times principal \(D\))

\(\text{EBT}\)

Earnings before taxes

\(T\)

Taxes \(= t_c(\text{EBT})\)

\(\text{NI}\)

Net income

From Operating Income to Free Cash Flow

\[ \widetilde{EBIT} = \widetilde{Rev} - \widetilde{VC} - FCC - dep \]

\[ \widetilde{EBIT} - t_c\widetilde{EBIT} = (\widetilde{Rev} - \widetilde{VC} - FCC - dep)(1-t_c) \]

\[ \widetilde{FCF} = (\widetilde{Rev} - \widetilde{VC} - FCC - dep)(1-t_c) + dep - I \]

In a no-growth firm, \(dep = I\), so:

\[ \widetilde{FCF}=(\widetilde{Rev}-\widetilde{VC}-FCC-dep)(1-t_c)=\widetilde{EBIT}(1-t_c). \]

For a perpetual no-growth firm with constant free cash flow:

\[ V_U = \frac{E(\widetilde{FCF})}{\rho} = \frac{E(\widetilde{EBIT})(1-t_c)}{\rho}. \]

Interpretation: the value of an unlevered firm is the present value of after-tax operating income.

If the firm issues debt, after-tax cash flows are split between claimholders:

• shareholders receive net cash flows after interest, taxes, and replacement investments;
• debtholders receive interest on debt.

\[ \widetilde{NI}+\widetilde{dep}-I+k_dD = (\widetilde{Rev}-\widetilde{VC}-FCC-dep-k_dD)(1-t_c)+k_dD. \]

With \(dep=I\):

\[ \widetilde{NI}+k_dD=(\widetilde{Rev}-\widetilde{VC}-FCC-dep)(1-t_c)+k_dDt_c. \]

So cash flow separates into:

• unlevered operating after-tax cash flow, discounted at \(\rho\);
• interest tax shield, discounted at \(k_b\) (risk-free debt assumption).

\[ V^L=\frac{E(\widetilde{EBIT})(1-t_c)}{\rho}+\frac{k_dDt_c}{k_b}=V_U+t_cB. \]

Thus, value added by leverage is the present value of the interest tax shield.