CFA Level I Exam · Capital Structure

Modigliani-Miller Propositions I and II Explained

Updated 7 October 2026 · Fact-checked

The Modigliani-Miller (MM) propositions show how capital structure affects value. Without taxes, firm value does not change with leverage (Proposition I), but the cost of equity rises linearly with debt (Proposition II). With taxes, debt adds a tax shield worth t × D, raising firm value. Solve by identifying the case, then applying the matching formula.

Understand Modigliani-Miller Propositions

Start with a simple idea. A firm's assets generate cash flows. Financing only decides how those cash flows are split between debt holders and shareholders. MM asked: does the way you slice the pie change the size of the pie?

In a perfect world with no taxes, no bankruptcy costs and no information asymmetry, the answer is no. This is MM Proposition I (no taxes): V(L) = V(U). Value comes from the assets, not the financing mix. Investors could create their own leverage at home, so they will not pay extra for a levered firm.

MM Proposition II (no taxes) deals with the cost of equity. Debt is cheaper than equity, but swapping equity for debt does not lower the WACC. Why? Equity holders bear more risk as leverage rises, so they demand a higher return. The higher cost of equity exactly offsets the cheaper debt. WACC stays constant and equals the unlevered cost of equity.

Now add corporate taxes. Interest is tax deductible, so debt creates a tax shield. Under MM Proposition I (with taxes), V(L) = V(U) + t × D. Value rises with debt, and in theory the best structure is almost all debt. The cost of equity still rises with leverage, but more slowly, because of the tax shield. WACC falls as debt increases.

Real firms do not use 100% debt, because financial distress costs appear. That leads to the static trade-off theory, which is a separate topic. For the exam, remember what each assumption set implies, and which direction each variable moves.

Key formulas to remember

MM Proposition I, no taxes: V(L) = V(U); Firm value is independent of capital structure. Assumes no taxes, no distress costs, perfect markets.
MM Proposition II, no taxes: r(e) = r(0) + (r(0) − r(d)) × (D ÷ E); r(0) is the cost of capital of the unlevered firm (also the WACC). Cost of equity rises linearly with D/E.
MM Proposition I, with taxes: V(L) = V(U) + t × D; t × D is the present value of the tax shield, assuming permanent debt at a constant tax rate t.
MM Proposition II, with taxes: r(e) = r(0) + (r(0) − r(d)) × (1 − t) × (D ÷ E); Equity cost rises with leverage, but less steeply than without taxes.
WACC: WACC = (D ÷ V) × r(d) × (1 − t) + (E ÷ V) × r(e); Constant with no taxes; declines as D/V rises with taxes.

How to solve Modigliani-Miller Propositions questions

Use this method for any MM question, whether it asks for firm value, cost of equity or the direction of a change.

1Check whether the question assumes taxes. No mention of taxes usually means the no-tax case.
2Write down the given data: V(U), D, E, r(d), r(0) or r(e), and the tax rate t.
3For firm value, choose V(L) = V(U) with no taxes, or V(L) = V(U) + t × D with taxes.
4For cost of equity, compute D/E using market values, then apply the correct Proposition II formula, including (1 − t) only if taxes exist.
5If you need equity value, use E = V(L) − D.
6Sanity check: the cost of equity must be above r(0) when debt is present, and the WACC should be unchanged without taxes or lower with taxes.
7Compute carefully, then pick the option that matches. Options are listed smallest to largest, so check you have the right order of magnitude.

Quickest way: Direction-and-formula shortcut

When to use it: Use when the question asks what happens to value, WACC or cost of equity if leverage changes, or when options differ by a clear amount.

No taxes: value and WACC stay the same, cost of equity rises.
With taxes: value rises by t × D, WACC falls, cost of equity rises but less than without taxes.
For a numerical gain from borrowing, just multiply tax rate by new debt.
For cost of equity, compute the spread r(0) − r(d) first, multiply by D/E (and by 1 − t if taxes), then add r(0).
Eliminate any option that is lower than r(0) when debt is present.

Common mistakes in Modigliani-Miller Propositions

Saying the WACC falls when debt is added in the no-tax case.
Debt is cheaper than equity, so it looks like it should lower the average.
Fix: Remember that the cost of equity rises to offset the cheaper debt exactly. WACC stays at r(0).
Leaving out (1 − t) in Proposition II with taxes.
Students memorise the no-tax formula and apply it everywhere.
Fix: Check the question for a tax rate. If there is one, include (1 − t) in the D/E term.
Using book values for D/E.
Balance sheet numbers are easy to find.
Fix: Use market values of debt and equity, as the theory requires.
Adding the tax shield as t × D when debt is not permanent or a question gives a different shield.
The formula is memorised without its assumption.
Fix: Use t × D for the standard case of permanent debt. Read the question for any other stated assumption.
Thinking MM with taxes means firms should use 100% debt in practice.
The formula shows value always rising with D.
Fix: MM ignores financial distress costs. Practical optimal structure comes from the trade-off theory.
Confusing V(U) with equity value.
Both are called value.
Fix: V(U) is the value of the all-equity firm. Equity of a levered firm is V(L) − D.

Worked examples

Example 1

An all-equity firm is worth $400 million. It issues $150 million of permanent debt and uses the proceeds to repurchase shares. The corporate tax rate is 30%. Under MM with taxes, what is the value of the levered firm? Options: (A) $400 million, (B) $445 million, (C) $550 million.

Show the solution

Taxes exist, so use V(L) = V(U) + t × D.
Tax shield = 0.30 × 150 = $45 million.
V(L) = 400 + 45 = $445 million.
Option A ignores the tax shield. Option C wrongly adds the debt itself.

Answer: (B) $445 million

Example 2

A firm has an unlevered cost of capital r(0) of 10%, a pre-tax cost of debt of 6% and a debt-to-equity ratio of 0.5. There are no taxes. Under MM Proposition II, what is the cost of equity? Options: (A) 10%, (B) 12%, (C) 14%.

Show the solution

No taxes, so r(e) = r(0) + (r(0) − r(d)) × (D ÷ E).
Spread = 10% − 6% = 4%.
Multiply by D/E: 4% × 0.5 = 2%.
r(e) = 10% + 2% = 12%.
Check: the WACC = (1/3)(6%) + (2/3)(12%) = 2% + 8% = 10%, equal to r(0). Correct.

Answer: (B) 12%

Exam tips

Always check if taxes are mentioned. The no-tax and with-tax results are opposite for firm value and WACC.
Know the directions cold: cost of equity rises with leverage in both cases, value only rises with taxes.
Questions with three options often include the no-tax answer as a trap when taxes are given. Eliminate it first.
Use market-value weights and keep percentages in decimals when computing.
Expect conceptual items too, such as which assumption MM relies on. The list is no taxes, no distress costs, no asymmetric information and perfect capital markets.

Practice questions from Capital Structure

Modigliani-Miller Propositions in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Modigliani-Miller Propositions: frequently asked questions

What is the difference between MM Proposition I and II?

Proposition I is about firm value. Proposition II is about the cost of equity. Without taxes, value is unaffected by leverage, while the cost of equity rises linearly with the debt-to-equity ratio.

What is the difference between MM without taxes and with taxes?

Without taxes, capital structure does not matter: value and WACC are constant. With taxes, interest is deductible, so value rises by t × D and the WACC falls as debt increases.

How does leverage affect cost of equity under MM Proposition II?

More debt makes shareholder returns riskier, so the cost of equity rises. With taxes the increase is smaller, because the formula multiplies the D/E term by (1 − t).

Why do real firms not use all debt if MM with taxes says value rises?

MM ignores the costs of financial distress. As debt grows, the chance of bankruptcy and its costs rise, which offsets the tax benefit at some point.