Modigliani–Miller Proposition I Without Taxes

Why Compare Total Firm Value?

A firm can finance the same operating assets with different combinations of debt and equity. Debt gives creditors a prior claim to specified payments, while shareholders receive what remains. The central question in Modigliani–Miller Proposition I without taxes is whether that change in the division of cash flows changes the value of the firm as a whole.

The result is deliberately narrow: it holds when financing changes only the division of operating cash flows among investors, not the cash flows generated by the assets or the market conditions under which those cash flows are valued. It does not say that debt and equity are equally risky, or that their individual market values stay unchanged.

Definition: Let $V_U$ denote the market value of an unlevered firm, financed entirely by equity. Let $V_L$ denote the total market value of a levered firm, financed by debt and equity. Let $D$ be the market value of the levered firm’s debt and $E_L$ the market value of its equity. Proposition I without taxes states that, for otherwise identical firms under perfect-market assumptions, $V_L=V_U$, with $V_L=D+E_L$.

Here, $D$ means the debt’s market value, not its face amount or promised repayment. A risky debt claim may be worth less than its promised payment because it does not receive that payment in every state. Likewise, $E_L$ is the market value of the residual claim held by shareholders.

Assumptions and Cash-Flow Logic

The perfect-market assumptions are essential, not decorative. In this no-tax version, assume there are no corporate taxes, transaction costs, or costs of financial distress; investors and firms can trade the relevant claims on comparable terms; and the firm’s financing choice does not alter its operating decisions or operating cash flows. The comparison holds the operating assets, time horizon, and operating outcomes fixed. The earlier material on perfect-market assumptions explains why relaxing these conditions can change the conclusion.

At the horizon being valued, suppose the operating assets generate cash flow $C_s$ in state $s$. If the firm has debt, creditors receive $B_s$ and shareholders receive the residual $R_s$. In the absence of value-destroying costs or other claims, the cash-flow division satisfies $B_s+R_s=C_s$ in every state. It is the equality state by state—not a claim that debt and equity have the same risk—that supports the result.

For a transparent valuation calculation, suppose a market has state prices $q_s$: the current price of a claim paying one dollar in state $s$, and zero in other states. In a complete market, the value of a claim with state-contingent payments $X_s$ is $\sum_s q_sX_s$. The $q_s$ already incorporate timing and risk valuation; they are not probabilities and should not be treated as discount rates. If state prices are not supplied, the same argument can be expressed with market values of the claims, provided the market values are additive for the same cash-flow portfolio.

V_U=\sum_s q_s C_s \qquad\text{and}\qquad V_L=\sum_s q_s(B_s+R_s)

The model is one-period here for convenience: all listed payments occur at the same future horizon. There is no mortality assumption or probability distribution to specify; each example lists the possible states and provides the state prices directly. In a multi-period setting, the same comparison applies to the full dated cash-flow stream, not just to one payment date.

Deriving Proposition I

The proof follows from the cash-flow identity and the linear valuation of claims. It does not require debt and equity to have equal risk or to be valued using the same discount rate.

Fix the operating assets.
In each state $s$, the unlevered firm’s investors collectively receive $C_s$. The levered firm’s debt and equity investors collectively receive $B_s+R_s$.

Use the financing identity.
With no taxes, distress losses, or other deductions, $B_s+R_s=C_s$ in every state. Thus the combined debt-and-equity claim has exactly the same state-contingent payoff as the unlevered equity claim.

Value the identical payoff.
For state prices $q_s$, additivity gives $V_L=\sum_s q_s(B_s+R_s)=\sum_s q_sC_s=V_U$. Therefore $D+E_L=V_U$.

This derivation also shows the limit of the conclusion. It establishes that the sum of debt and equity values is unchanged when their combined payoff remains the same. It does not establish that either claim’s value is fixed: changing the debt contract changes which investors receive cash in each state, and can change the value of each claim separately.

Worked Example: Valuing a Firm with Risky Debt

A one-period firm has operating cash flows of $240,000, $140,000, or $60,000 in three possible states. Its debt promises $100,000 at the horizon, subject to the cash available; shareholders receive any remaining cash. The state prices are $0.10$, $0.20$, and $0.40$ dollars per dollar of payoff in the corresponding states. Find the unlevered value, debt value, equity value, and total levered value.

Method and assumptions. Use the supplied state prices to value each claim. The three states and their state prices are fully specified; no probability distribution or separate interest rate is needed. Assume the firm has no taxes, distress costs, or other deductions, so debt and equity divide the operating cash flow.

Step 1: Find each claim’s payoff. Debt receives the lesser of the cash available and its $100,000 promise. Its payments are therefore $100,000, $100,000, and $60,000. Shareholders receive residual payments of $140,000, $40,000, and $0. In each state, the two payments sum to the operating cash flow.

Step 2: Value the unlevered firm. If entirely equity-financed, the firm’s investors receive the operating cash flows directly:

V_U=0.10(240{,}000)+0.20(140{,}000)+0.40(60{,}000) =24{,}000+28{,}000+24{,}000 =\$76{,}000.

Step 3: Value debt and equity separately.

D=0.10(100{,}000)+0.20(100{,}000)+0.40(60{,}000) =10{,}000+20{,}000+24{,}000 =\$54{,}000, $$ $$ E_L=0.10(140{,}000)+0.20(40{,}000)+0.40(0) =14{,}000+8{,}000+0 =\$22{,}000.

Step 4: Check the result. The market values add to $D+E_L=\$54{,}000+\$22{,}000=\$76{,}000=V_U$. The debt is risky because it receives only $60,000 in the lowest-cash-flow state; using its $100,000 face amount as its market value would be incorrect.

Common mistake avoided: Do not value debt at its promised face amount merely because that amount appears in the contract. Value the debt’s actual state-contingent payments.

Different Financing, Same Total Value

The debt-equity split can change substantially even when total value does not. In the next example, the operating cash flows and state prices remain fixed, while the debt promise changes. As debt takes a larger share of the cash flows in some states, the residual equity claim changes as well.

Worked Example: Comparing Two Debt Levels

A one-period firm’s operating assets pay $300,000, $180,000, or $90,000 in three states. The corresponding state prices are $0.08$, $0.16$, and $0.30$. Compare a debt contract promising $100,000 with one promising $180,000. In each case debt receives no more than the available operating cash flow, and equity receives the residual.

Method and assumptions. Apply state-price valuation separately to the operating assets, debt, and equity. The three states and state prices are given, with no separate probability distribution or interest rate required. Both contracts leave the operating assets unchanged, and there are no taxes or distress costs.

Step 1: Value the operating assets. This is the unlevered value under either financing plan:

V_U=0.08(300{,}000)+0.16(180{,}000)+0.30(90{,}000) =24{,}000+28{,}800+27{,}000 =\$79{,}800.

Step 2: Evaluate the $100,000 debt promise. Debt pays $100,000, $100,000, and $90,000. Equity receives $200,000, $80,000, and $0.

D=0.08(100{,}000)+0.16(100{,}000)+0.30(90{,}000) =8{,}000+16{,}000+27{,}000 =\$51{,}000, $$ $$ E_L=0.08(200{,}000)+0.16(80{,}000)+0.30(0) =16{,}000+12{,}800+0 =\$28{,}800.

The check is $D+E_L=\$51{,}000+\$28{,}800=\$79{,}800$.

Step 3: Evaluate the $180,000 debt promise. Debt pays $180,000, $180,000, and $90,000. Equity receives $120,000, $0, and $0.

D=0.08(180{,}000)+0.16(180{,}000)+0.30(90{,}000) =14{,}400+28{,}800+27{,}000 =\$70{,}200, $$ $$ E_L=0.08(120{,}000)+0.16(0)+0.30(0) =9{,}600 =\$9{,}600.

The second check is $D+E_L=\$70{,}200+\$9{,}600=\$79{,}800$. The higher debt promise increases debt’s value and reduces equity’s value, but their sum remains the value of the unchanged operating assets.

Common mistake avoided: Comparing debt values alone can suggest that the financing change created value. Proposition I compares the total market value of all claims, not the value of one claim in isolation.

A State-by-State Check of the Proposition

The equality is easiest to audit by checking each state before doing any valuation. For example, if the operating assets pay $250,000, $120,000, and $40,000, a debt contract that pays $50,000, $50,000, and $40,000 leaves equity with $200,000, $70,000, and $0. In every state, debt plus equity equals the asset payoff. A different debt contract can rearrange those amounts, but the same check applies.

This is why the result is not a statement that leverage has no effect on investors. It changes the contingent payments made to creditors and shareholders, and therefore the risk and value of each claim. Proposition I says that, under its assumptions, this rearrangement does not change the combined market value.

Common Errors and Scope

Using face value instead of market value. Face value is the contractual amount due if paid in full. Market value reflects the debt’s actual payments across states.
Adding promised payments instead of actual claims. If cash is insufficient in a state, debt and equity payments must still add to the cash actually available, not to an impossible total promise.
Discounting debt and equity at one arbitrary rate. Their risks can differ. Value each claim using its own market price or appropriate state-contingent valuation, then compare the sum with the value of the operating assets.
Claiming each security’s value is unchanged. The proposition concerns $D+E_L$, not $D$ or $E_L$ separately.
Ignoring the assumptions. Taxes, distress costs, transaction costs, or financing-induced changes in operating decisions can break the assumptions used in the derivation. Those cases are outside this no-tax result.

Key takeaway: With unchanged operating cash flows and perfect-market assumptions without taxes, debt and equity divide the same total payoff that unlevered equity would receive. Because that combined payoff is unchanged, $V_L=D+E_L=V_U$; the individual claim values may still differ.

Check Your Understanding

For each problem, use the stated state prices to value the claims. Treat each as a one-period model with no taxes, distress costs, or other deductions. State prices are supplied directly, so no separate probability distribution or interest rate is required.

A firm’s operating cash flows across three states are $200,000, $100,000, and $50,000. State prices are $0.10$, $0.25$, and $0.40$. Debt promises $80,000 and receives no more than available cash. Find the unlevered value, debt value, and equity value. (Answer: $V_U=\$65{,}000$, $D=\$48{,}000$, and $E_L=\$17{,}000$; $D+E_L=\$65{,}000$.)
Operating cash flows are $240,000, $120,000, and $40,000, with state prices $0.15$, $0.30$, and $0.50$. Debt promises $150,000 and is paid only from available cash. Calculate the unlevered value and the market values of debt and equity. (Answer: $V_U=\$92{,}000$, $D=\$78{,}500$, and $E_L=\$13{,}500$; $D+E_L=\$92{,}000$.)
Operating cash flows are $300,000, $150,000, and $60,000, with state prices $0.20$, $0.35$, and $0.45$. Compare debt promises of $120,000 and $200,000. For each plan, calculate debt value and equity value, then check the common total. (Answer: $V_U=\$139{,}500$; with the $120,000 promise, $D=\$93{,}000$ and $E_L=\$46{,}500$; with the $200,000 promise, $D=\$119{,}500$ and $E_L=\$20{,}000$. Both plans total $139,500.)
Operating cash flows are $250,000, $120,000, and $40,000, and state prices are $0.12$, $0.20$, and $0.45$. Plan A has debt payments of $50,000, $50,000, and $40,000, with equity receiving the residual. Plan B has debt payments of $100,000, $100,000, and $40,000, also with equity receiving the residual. Find the value of each claim under both plans and verify Proposition I. (Answer: $V_U=\$72{,}000$; Plan A has $D=\$34{,}000$, $E_L=\$38{,}000$, and $V_L=\$72{,}000$; Plan B has $D=\$50{,}000$, $E_L=\$22{,}000$, and $V_L=\$72{,}000$.)

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