📖 MM 定理：有税情况

Corporate Finance

I. Lesson Focus

II. The Problem

Consider an all-equity firm that generates a stable EBIT of CNY 1,000,000 per year with a corporate tax rate of 25%. Management is evaluating whether to issue debt and use the proceeds to repurchase shares. Introducing debt creates an annual interest tax shield, but it also increases the risk to equity holders, raising the cost of equity. The fundamental question is: Does leverage increase the overall value of the firm? If so, by how much? How does WACC change? This is a core topic in CFA Level I Corporate Finance and the starting point for understanding capital structure decisions.

III. Review of MM Propositions: No Taxes vs. With Taxes

The Modigliani-Miller (MM) propositions form the foundation of modern corporate finance.

• MM Proposition I without taxes: Firm value is independent of capital structure. $V_L = V_U$.
• MM Proposition II without taxes: Leverage increases the cost of equity exactly enough to offset the lower cost of debt: $r_e = r_0 + (r_0 - r_d)\frac{D}{E}$, where $r_0$ is the required return on assets (unlevered cost of equity).

When corporate taxes are introduced, the conclusion changes dramatically. Interest expense is tax-deductible, creating an “interest tax shield” that increases firm value.

IV. MM Proposition I with Taxes

Core Conclusion: The value of a levered firm equals the value of an unlevered firm plus the present value of the interest tax shield.

$$V_L = V_U + t_c D$$

where: - $V_L$ = value of levered firm - $V_U$ = value of unlevered firm - $t_c$ = corporate tax rate - $D$ = market value of debt (assumed perpetual and risk-matched to the firm)

This formula assumes debt is constant and the risk of the tax shield matches the risk of the debt, so it is discounted at $r_d$. For perpetual debt, the present value of the tax shield simplifies to $t_c D$.

Practical Implication: In a world with only corporate taxes (no personal taxes, no bankruptcy costs), the optimal capital structure would be 100% debt. This is unrealistic, which is why later readings introduce financial distress and agency costs.

V. MM Proposition II with Taxes

The cost of equity rises with leverage, but the increase is smaller than in the no-tax case because the tax shield partially offsets the added financial risk.

$$r_e = r_0 + (r_0 - r_d)(1 - t_c)\frac{D}{E}$$

where $r_0 = \frac{\text{EBIT}(1-t_c)}{V_U}$, the required return for the unlevered firm (which equals its WACC).

VI. Behavior of Weighted Average Cost of Capital (WACC)

In a world with taxes, WACC declines as debt increases:

$$WACC = \frac{E}{V_L} r_e + \frac{D}{V_L} r_d (1 - t_c)$$

Substituting the expression for $r_e$ shows that WACC falls monotonically with leverage. This is consistent with MM Proposition I: because $V_L$ rises, the same after-tax EBIT divided by a larger value implies a lower overall cost of capital.

VII. Alternative Expressions for the Tax Shield

When EBIT is perpetual and debt is maintained at a constant level, the annual tax shield is $t_c \times r_d \times D$. Its present value is:

$$\text{PV(Tax Shield)} = \frac{t_c r_d D}{r_d} = t_c D$$

If debt is rebalanced proportionally to firm value, the tax shield risk approximates the unlevered asset risk and should be discounted at $r_0$. However, CFA Level I primarily tests the constant-debt assumption leading to the simple $t_c D$ result.

Worked Cases

Case 1: Basic Valuation

An unlevered firm has EBIT = CNY 1,000,000, $r_0 = 12\%$, $t_c = 25\%$. The firm plans to issue perpetual debt of CNY 2,000,000 at $r_d = 6\%$.

1. Unlevered value: $V_U = \frac{1,000,000 \times (1-0.25)}{0.12} = 6,250,000$
2. Levered value: $V_L = 6,250,000 + 0.25 \times 2,000,000 = 6,750,000$
3. Tax shield value = CNY 500,000
4. Equity value $E = V_L - D = 4,750,000$

Case 2: Cost of Equity and WACC Calculation

Using data from Case 1:

• $D/E = 2,000,000 / 4,750,000 \approx 0.421$
• $r_e = 0.12 + (0.12 - 0.06)(1-0.25) \times 0.421 = 0.12 + 0.01895 \approx 13.895\%$
• WACC = $(4,750,000/6,750,000)\times13.895\% + (2,000,000/6,750,000)\times6\%\times(1-0.25) \approx 9.722\% + 1.333\% = 11.055\%$

Note that WACC falls from 12% to 11.055%, confirming that leverage reduces the cost of capital.

Case 3: Sensitivity to Tax Rate

Increase the tax rate to 40%, holding other inputs constant.

• $V_L = 6,250,000 + 0.40 \times 2,000,000 = 7,050,000$
• $E = 5,050,000$
• $D/E \approx 0.396$
• $r_e = 0.12 + (0.12-0.06)(1-0.4)\times0.396 \approx 0.12 + 0.01426 = 13.426\%$
• WACC $\approx (5.05/7.05)\times13.426\% + (2/7.05)\times6\%\times0.6 \approx 9.62\% + 1.02\% = 10.64\%$

The higher the tax rate, the greater the tax shield value and the larger the decline in WACC.

Traps

Key Formulas

• $V_L = V_U + t_c D$
• $r_e = r_0 + (r_0 - r_d)(1 - t_c)\frac{D}{E}$
• $WACC = \frac{E}{V_L} r_e + \frac{D}{V_L} r_d (1 - t_c)$
• $V_U = \frac{\text{EBIT}(1-t_c)}{r_0}$
• Present value of tax shield (perpetual debt) $= t_c D$
• With taxes, WACC declines as leverage increases (unlike the constant WACC without taxes)

Practice Questions

Q1. According to MM Proposition I with corporate taxes, which statement is correct?
A. Firm value decreases as debt increases
B. $V_L = V_U + t_c D$
C. Tax shield value increases with personal tax rates
D. Optimal capital structure is 50% debt

Q2. An all-equity firm has $V_U = \$10$ million and $t_c = 30\%$. If it issues $\$4$ million of perpetual debt, firm value becomes:
A. \$10 million
B. \$11.2 million
C. \$12.4 million
D. \$14 million

Q3. The reason the cost-of-equity formula with taxes multiplies by $(1-t_c)$ is that:
A. Debt cost increases
B. The tax shield reduces the incremental risk borne by equity
C. Firm value decreases
D. It avoids double taxation

Q4. Which of the following would invalidate the MM conclusions with taxes?
A. Existence of corporate income tax
B. Existence of financial distress costs
C. Fixed interest rate on debt
D. Constant perpetual EBIT

Q5. Given $r_0=15\%$, $r_d=8\%$, $t_c=25\%$, and $D/E=1$, the cost of equity $r_e$ equals:
A. 15.0%
B. 16.5%
C. 17.25%
D. 18.75%

Q6. In a world with taxes, the relationship between WACC and leverage is:
A. U-shaped
B. Continuously declining
C. Constant
D. Inverted U-shaped

Q7. A firm has EBIT = \$800,000, $t_c=25\%$, $r_0=10\%$. After issuing \$3 million of perpetual debt, the present value of the tax shield is:
A. \$300,000
B. \$750,000
C. \$1,200,000
D. Cannot be determined (missing $r_d$)

Q8. Compared with a no-tax world, at the same leverage ratio the cost of equity in a world with taxes is:
A. Higher
B. Lower
C. The same
D. Cannot be compared

Answers

Takeaways

• In a world with taxes, leverage increases firm value via the formula $V_L = V_U + t_c D$
• The slope of the cost-of-equity line is flattened by the factor $(1-t_c)$, reflecting risk sharing by the tax shield
• WACC declines with leverage; this is the mathematical consequence of higher firm value
• Always calculate $V_U$ using after-tax EBIT: $\text{EBIT}(1-t_c)/r_0$
• CFA Level I focuses on the simplified perpetual-debt tax shield $t_c D$
• Master the algebraic difference between the no-tax and with-tax versions of Proposition II, especially the $(1-t_c)$ term