Modigliani-Miller theorem

A business produces a stream of operating cash flows. Before any financing decision, that stream has a size, a growth rate and a risk. The financing decision does not touch any of the three. What it does is decide who holds a claim on the stream, in what order, with what rights on default.

Picture the stream as a pie of fixed diameter. Lenders take a fixed slice, cut first, with a legal right to force a sale if their slice is short. Owners take everything left. Every proposition in this file is an answer to one question: does re-cutting the pie change its diameter?

Does the financing mix change a firm's value?

The honest answer is that in a frictionless world it cannot, and that the world has exactly four frictions worth naming. Taxes make the interest slice deductible and the owners' slice not, which enlarges the after-tax pie as debt rises. Financial distress shrinks the pie in the states where the fixed slice cannot be cut, because customers leave, suppliers demand cash, capable staff resign and the capital budget is cancelled at the worst moment. Information asymmetry makes issuing a new claim a message about what management privately believes. And splitting the pie between two groups with different payoffs creates incentives to change the diameter itself.

Each friction gets a section. Together they produce a curve with an interior maximum, and the last thing these pages will do is say how much to trust it.

MM Proposition I without taxes, and the arbitrage that proves it

Modigliani and Miller's 1958 result says that under the frictionless assumptions, the total market value of a firm's securities does not depend on how those securities are divided between debt and equity. Written out, V_L = V_U: the levered firm and the otherwise identical unlevered firm are worth the same.

The proof is a trade you could actually place. Take Brandywine Valve Corp (synthetic), which earns a perpetual $100 million of operating profit. There are no taxes. Its unlevered cost of equity is 12%, so the unlevered firm is worth 100 / 0.12 = $833.333333 million. A levered twin carries $500 million of perpetual debt at 6%.

Suppose the market prices the levered twin at $900 million, so its equity is worth $400 million. Own 1% of that equity and you receive 0.01 × (100 − 30) = $0.700000 million a year, and your stake is worth $4.000000 million. Now unwind it. Sell the equity for $4.000000 million and borrow $5.000000 million personally at 6%, which is 1% of the firm's debt. You hold $9.000000 million of cash. Buy 1% of the unlevered firm for $8.333333 million and you receive 0.01 × 100 = $1.000000 million a year, out of which you pay 0.06 × 5 = $0.300000 million of personal interest, leaving $0.700000 million. Identical income, identical risk, and $0.666667 million left in your pocket. That residue is exactly 1% of the $66.666667 million by which the levered twin was overpriced.

The trade runs the other way too. If the unlevered firm were the one priced at $900 million, sell 1% of it for $9.000000 million, buy 1% of the levered firm's equity and 1% of its debt for a combined $8.333333 million, and collect 0.700000 + 0.300000 = $1.000000 million, the same income you gave up, with $0.666667 million left over. Buying both halves of a levered firm is homemade unlevering, the mirror of homemade levering.

Investors can therefore manufacture any capital structure they want inside their own accounts. A firm that changes its capital structure supplies something investors could already make for themselves, and nobody pays for that. The proof consumes four assumptions, each of which fails somewhere in the real world: individuals borrow on the same terms as corporations, there are no taxes, there are no costs of bankruptcy, and everyone sees the same cash-flow distribution.

MM Proposition II without taxes: the cost of equity does the work

If total value is flat in leverage and the weighted average cost of capital is NOPAT / V, then the weighted average cost of capital must also be flat in leverage. Something has to give, and it is the cost of equity.

Proposition II states it: r_E = r_0 + (r_0 − r_D) × D/E. Equity holders in a levered firm bear the same business risk plus the financial risk that the fixed claim was cut first, and they are paid for both. The relationship is linear in the debt-to-equity ratio, with slope equal to the spread between the unlevered cost of equity and the cost of debt.

Brandywine, at three leverage levels, with V = 833.333333 throughout:

Every cost of equity in that table can be produced twice, which is the check that matters. The formula gives 0.12 + 0.06 × 1.5 = 21.000000% at the highest level. Dividing net income by equity gives 70 / 333.333333 = 21.000000%. They agree because they are the same statement.

The cost of equity nearly doubles across the table, rising nine percentage points from 12% to 21%, while the cost of capital does not move at all. A treasurer who says "our debt costs 6% and our equity costs 12%, so more debt is cheaper" has read the first column and stopped. Cheap debt raises the price of equity by precisely enough to leave the blend alone.

Corporate taxes, and the reconciliation to the beta arithmetic

Interest is deductible and dividends are not. That single asymmetry breaks the irrelevance result in the government's direction.

With a corporate tax rate t and perpetual debt D, the levered value is V_L = V_U + tD, and the cost of equity becomes r_E = r_0 + (r_0 − r_D)(1 − t) × D/E. Brandywine again, now taxed at 25%, so V_U = 100 × 0.75 / 0.12 = $625.000000 million:

Three independent checks close on the middle row. Net income over equity gives 63.75 / 437.5 = 14.571429%. The shortcut r_0 (1 − tD/V_L) gives 0.12 × (1 − 0.25 × 250/687.5) = 10.909091%. And capitalising after-tax operating profit at that rate gives 75 / 0.10909091 = 687.500000, the levered value you started from. A capital-structure model that fails any of those three is broken, and the failure is usually a book-value weight where a market-value weight belongs.

Does the Hamada relevering formula overstate the cost of equity?

Now the reconciliation that most people skip. The valuation phase relevers beta with β_L = β_U [1 + (1 − t) D/E], which is the Hamada relation, and Hamada assumes the debt carries no market risk. Brandywine's numbers let you test that. With a risk-free rate of 4% and an equity risk premium of 5%, an unlevered cost of equity of 12% implies β_U = 1.6, and a 6% cost of debt implies a debt beta of (6 − 4)/5 = 0.4, which is not zero.

Run both versions at D/E = 0.571429. Hamada with a zero debt beta gives β_L = 1.6 × (1 + 0.75 × 0.571429) = 2.285714, so r_E = 4 + 2.285714 × 5 = 15.428571%. Carrying the debt beta through, β_L = β_U + (β_U − β_D)(1 − t) D/E = 1.6 + 1.2 × 0.428571 = 2.114286, so r_E = 14.571429%, which is exactly what Proposition II produced. The overstatement is 0.857143 percentage points, and it has a closed form: β_D (1 − t)(D/E) × ERP = 0.4 × 0.75 × 0.571429 × 5% = 0.857143%. At D/E = 2.0 the same error is 3.000000 percentage points.

The practical reading is that the standard relever is conservative in the direction of a higher cost of equity, and the error grows linearly in both leverage and the credit spread. On an investment-grade issuer it is a rounding error. On a high-yield issuer at three turns of leverage it is worth hundreds of basis points, and a discounted cash flow built on it will systematically undervalue the equity.

What are the limits of the tax shield formula?

Two limits, stated honestly. The perpetual-debt assumption behind V_L = V_U + tD is generous, because real debt matures and is sometimes not replaced, so the shield has a life rather than a perpetuity. And it is worth nothing in a year with no taxable income, which is exactly when a levered firm most often has none.