grad-mm-theorem（梯度 mm 定理）

Modigliani-Miller Theorem

Overview

The Modigliani-Miller theorem (1958) establishes that in perfect capital markets, firm value is independent of capital structure. This irrelevance result serves as the benchmark — every real-world reason capital structure matters is a violation of MM's assumptions.

When to Use

• Evaluating whether a financing decision creates or destroys value
• Identifying which market imperfections make capital structure relevant
• Calculating the value of the tax shield from debt
• Teaching or analyzing the logical foundations of capital structure theory

When NOT to Use

• As a literal prescription — real markets are never frictionless
• When the analysis requires explicit bankruptcy cost modeling (use tradeoff theory)
• For financial institutions where capital structure is regulated

Assumptions

IRON LAW: MM irrelevance holds ONLY in perfect markets — every
real-world deviation (taxes, bankruptcy costs, agency costs) makes
capital structure matter. MM is the null hypothesis, not the answer.

Key assumptions (for irrelevance):

1. No taxes (corporate or personal)
2. No bankruptcy costs or financial distress costs
3. No agency costs — managers act in shareholders' interest
4. Symmetric information — insiders and outsiders know the same things
5. Individuals and firms borrow at the same rate

Methodology

Step 1 — State MM Propositions

• Proposition I (no tax): VL = VU — firm value is independent of leverage
• Proposition II (no tax): Re = R0 + (D/E)(R0 - Rd) — cost of equity rises linearly with leverage
• Proposition I (with tax): VL = VU + Tc x D — debt creates a tax shield

Step 2 — Identify Market Imperfections

For each deviation, assess its magnitude:

• Corporate taxes: create incentive for debt (tax shield)
• Bankruptcy costs: create incentive against excessive debt
• Agency costs: debt disciplines managers (Jensen, 1986) but may cause asset substitution
• Information asymmetry: leads to pecking order behavior

Step 3 — Apply Tradeoff Framework

Optimal capital structure balances marginal tax shield benefit against marginal bankruptcy and agency costs.

Step 4 — Compute WACC Impact

WACC = (E/V)Re + (D/V)Rd(1-Tc). Optimal structure minimizes WACC.

Output Format

## Capital Structure Analysis: [Firm]

### Current Structure
| Metric | Value |
|--------|-------|
| Debt (D) | $X |
| Equity (E) | $X |
| D/E Ratio | x.xx |
| WACC | x% |

### MM Imperfections Present
| Imperfection | Magnitude | Direction |
|-------------|-----------|-----------|
| Tax shield | [high/medium/low] | Favors debt |
| Bankruptcy costs | [high/medium/low] | Favors equity |
| Agency costs | [high/medium/low] | [depends] |

### Recommendation
- [Optimal direction of adjustment with reasoning]

Gotchas

• MM Proposition II is frequently misunderstood: WACC stays constant (no tax) because cheaper debt is exactly offset by rising equity cost
• Tax shield value Tc x D assumes perpetual debt — temporary debt requires PV calculation
• Personal taxes (Miller, 1977) can offset corporate tax advantage of debt
• Empirical leverage ratios vary wildly by industry, suggesting no single "optimal" structure
• Financial distress costs are hard to measure but can be 10-20% of firm value
• MM assumes operating cash flows are independent of financing — this fails when leverage affects investment decisions

References

• Modigliani, F. & Miller, M. (1958). The cost of capital, corporation finance and the theory of investment. *American Economic Review*, 48(3), 261-297.
• Modigliani, F. & Miller, M. (1963). Corporate income taxes and the cost of capital: a correction. *American Economic Review*, 53(3), 433-443.
• Jensen, M. (1986). Agency costs of free cash flow. *American Economic Review*, 76(2), 323-329.