Modigliani–Miller Proposition II Without Taxes

Introduction

Debt may have a lower required return than equity, but replacing equity with debt does not eliminate the risk of the firm’s operating assets. It changes how that risk is divided between investors. With a fixed debt claim ahead of equity, shareholders receive what remains after interest is paid, so their returns become more sensitive to operating outcomes as leverage rises.

Modigliani–Miller Proposition II without taxes quantifies this effect. Under its idealized assumptions, the required return on equity increases linearly with the market-value debt-to-equity ratio. The proposition is aligned with the SOA VEE Accounting and Finance syllabus.

Definition: Let $D$ and $E$ be the market values of a firm’s debt and equity, and let $V=D+E$ be the total market value of its financing claims. Let $r_D$ be the required return on debt, $r_E$ the required return on equity, and $r_A$ the required return on the firm’s operating assets if financed without debt. The ratio $D/E$ is the market-value debt-to-equity ratio.

Model and Proposition

The proposition is a result of the idealized, no-tax capital structure model. It assumes perfect capital markets: investors and firms can trade securities without transaction costs; information is available to market participants; and investors can borrow or lend on terms consistent with the model. There are no corporate or personal taxes, financial distress costs, or agency effects. Financing does not change the firm’s operating decisions or the risk of its assets.

The firm’s operating cash flows have the same business risk under each financing choice. Debt promises a contractual payment before equity receives anything, and the firm can meet that payment in the model. For the familiar straight-line form of the proposition, take $r_D$ as constant as leverage changes. These assumptions are restrictive: the proposition isolates how financing reallocates risk rather than describing every real firm.

Theorem: Under the no-tax Modigliani–Miller assumptions, the required return on equity is $$ r_E = r_A + \frac{D}{E}(r_A-r_D). $$ Here $D/E$ uses market values, not book values. If $r_A>r_D$ and $r_D$ remains constant, increasing $D/E$ increases $r_E$ linearly. The spread $r_A-r_D$ is the increase in the equity return per unit increase in the debt-to-equity ratio.

The formula does not say that debt is risk-free in every possible capital structure. If debt becomes riskier as leverage increases, its required return $r_D$ may also rise. The return identity underlying the proposition still applies, but a constant $r_D$ should not be assumed in the straight-line version when debt pricing changes.

Why Leverage Raises the Required Return on Equity

The derivation begins with how the operating return is divided between debt and equity. In a one-period description, the expected return on the assets is the total expected return attributable to the firm’s operations divided by the market value $V$. Debt holders receive their required return on $D$; equity holders receive the residual return on $E$. Thus the asset return, weighted by the values of the claims, equals the returns paid to the two classes of investors.

$$ r_A V = r_D D + r_E E $$

Because the firm’s total financing value is $V=D+E$, substitute that identity and rearrange:

\begin{aligned} r_A(D+E) &= r_DD+r_EE,\\ r_A D+r_A E &= r_DD+r_EE,\\ r_AE &= r_EE+D(r_D-r_A),\\ r_EE &= r_AE+D(r_A-r_D),\\ r_E &= r_A+\frac{D}{E}(r_A-r_D). \end{aligned}

The subtraction in the final line is important: the extra equity return depends on $r_A-r_D$, not on $r_D-r_A$. When debt has a lower required return than the assets, equity must earn more than the asset return to compensate for bearing the residual risk. If $D=0$, the formula gives $r_E=r_A$, as expected for an all-equity firm.

A perpetuity provides another way to see the same result. Suppose expected operating income is constant forever, there is no growth, and the income is not reduced by taxes. Then the asset value is operating income capitalized at $r_A$. The debt claim receives interest, and the equity claim receives the remaining income. Dividing those claims’ expected income by their market values leads to the same return identity. This perpetuity illustration is not required for the general return allocation, but it makes the residual claim especially clear.

Applying the Relation

Before substituting numbers, check the leverage measure. Proposition II uses $D/E$. If a problem gives the debt fraction $D/V$, convert it using $V=D+E$:

\frac{D}{E} = \frac{D/V}{E/V} = \frac{D/V}{1-D/V}.

A debt-to-value ratio and a debt-to-equity ratio are not interchangeable. For example, if debt is 40% of total value, equity is 60%, so $D/E=0.40/0.60$, not 0.40.

Worked Example: Calculating the Equity Return From Market Values

A hypothetical firm has constant expected annual operating income of $9.6 million, no taxes, and no growth. Its operating assets have a required return $r_A=12\%$. The firm has perpetual debt with market value $D=\$32$ million and required return $r_D=5\%$. Find its market value of equity and required return on equity. Use the no-tax perpetuity model and report the final rate to two decimal places.

Method and assumptions. Under the stated no-growth perpetuity assumption, the total asset value is operating income divided by the required return on assets. This assumes the annual operating income continues indefinitely and is level; it is not a general valuation rule for growing or changing cash flows.

Step 1: Find total value. $$ V=\frac{\$9.6\text{ million}}{0.12}=\$80\text{ million}. $$

Step 2: Find equity value and the relevant leverage ratio. $$ E=V-D=\$80-\$32=\$48\text{ million}, \qquad \frac{D}{E}=\frac{32}{48}=\frac{2}{3}. $$

Step 3: Apply Proposition II. $$ r_E=12\%+\frac{2}{3}(12\%-5\%) =12\%+\frac{2}{3}(7\%) =12\%+4.6667\% =16.6667\%. $$ The required return on equity is 16.67%, rounded to two decimal places.

Check. Annual debt interest is $0.05(\$32\text{ million})=\$1.6$ million. The remaining operating income is $\$9.6-\$1.6=\$8.0$ million. Dividing by equity value gives $\$8.0/\$48=0.166667$, or 16.67%. The residual-income calculation agrees with the proposition. The common mistake this method avoids is treating $D/V=32/80=0.40$ as if it were $D/E$.

Worked Example: Converting a Debt-to-Value Ratio

A hypothetical company has a market value of total financing $V=\$100$ million, of which debt is 40%. Its asset required return is $r_A=11\%$, and its debt required return is $r_D=5\%$. Assume no taxes, constant operating risk, and debt priced at the stated rate. Find the required return on equity. Report the final rate to two decimal places.

Method. Convert the stated debt-to-value ratio to $D/E$ before using Proposition II. The given debt percentage is not the ratio in the formula.

Step 1: Find the market values. $$ D=0.40(\$100\text{ million})=\$40\text{ million}, \qquad E=\$100-\$40=\$60\text{ million}. $$

Step 2: Calculate the debt-to-equity ratio. $$ \frac{D}{E}=\frac{40}{60}=\frac{2}{3}. $$

Step 3: Calculate the equity return. $$ r_E=11\%+\frac{2}{3}(11\%-5\%) =11\%+\frac{2}{3}(6\%) =11\%+4\% =15.00\%. $$

Check. A 40% debt share leaves a 60% equity share, so $D/E=40/60=2/3$. The return spread is 6 percentage points, and two-thirds of that spread is 4 percentage points. Adding this to 11% gives 15.00%. The common mistake avoided is using 0.40 directly as the leverage ratio, which would incorrectly produce 13.40%.

Worked Example: Finding Leverage From a Target Equity Return

An analyst evaluates a hypothetical no-tax firm with $r_A=10\%$ and debt required return $r_D=4\%$. The firm’s target required return on equity is 16%. Assuming debt continues to have the stated required return and the firm’s asset risk is unchanged, find the target $D/E$, and state the corresponding debt fraction $D/V$. Give the ratios to four decimal places and the percentage to two decimal places.

Method. Rearrange Proposition II to solve for $D/E$, then convert that ratio to a fraction of total value.

Step 1: Isolate leverage. $$ r_E-r_A=\frac{D}{E}(r_A-r_D), \qquad \frac{D}{E}=\frac{r_E-r_A}{r_A-r_D}. $$

Step 2: Substitute the required returns. $$ \frac{D}{E}=\frac{16\%-10\%}{10\%-4\%} =\frac{6\%}{6\%}=1.0000. $$

Step 3: Convert to debt as a share of total value. If $D/E=1$, debt and equity have equal market values. Therefore, $$ \frac{D}{V}=\frac{D}{D+E}=\frac{1}{1+1}=0.5000=50.00\%. $$

Check. Substituting $D/E=1$ into the original relation gives $r_E=10\%+1(10\%-4\%)=16\%$, the stated target. The common mistake avoided is assuming that $D/E=1$ means debt is all of the firm’s value; it means debt equals equity, so debt is half of total value.

Interpretation and Limits

The formula separates two sources of required return. The asset return $r_A$ reflects the risk of the firm’s operations. The second term, $(D/E)(r_A-r_D)$, reflects the additional return shareholders require because debt has a prior claim. Holding $r_A$ and $r_D$ fixed, each increase in $D/E$ magnifies that financial-risk adjustment.

This explains why cheaper debt does not make equity safer. The debt holders receive their contracted amount first, leaving equity holders with a more variable residual. In the model, the apparent financing advantage of using a lower-return security is offset by the higher return required by equity investors. The proposition itself describes the equity-return adjustment; it should not be used to claim that operating risk has changed.

Key takeaway: In the no-tax model, use market-value $D/E$ and the relation $r_E=r_A+(D/E)(r_A-r_D)$. Leverage raises the required return on equity when $r_A>r_D$, because shareholders bear more financial risk. The result depends on the model assumptions, especially unchanged asset risk and the stated debt return.

A practical limitation is that real firms may face taxes, distress costs, transaction costs, changing investment decisions, or agency conflicts. Debt may also become riskier as leverage grows, causing $r_D$ to change. Those effects are outside the no-tax, constant-debt-return version developed here. Always identify the model assumptions before applying its linear relation.

Check Your Understanding

For each problem, use the no-tax Proposition II relation and market-value leverage. Report percentage answers to two decimal places unless the problem requests a ratio.

A hypothetical firm has $r_A=9\%$, $r_D=4\%$, and $D/E=0.80$. Find $r_E$. (Answer: 13.00%)
A company has market-value debt of $18 million and equity of $42 million. If $r_A=9.5\%$ and $r_D=5.5\%$, find $r_E$. (Answer: 11.21%)
A no-tax perpetual firm has operating income of $9 million, $r_A=12\%$, debt market value of $30 million, and $r_D=7.5\%$. Find total value, equity value, and $r_E$. (Answer: Total value $75 million; equity value $45 million; $r_E=15.00\%$)
A hypothetical company has debt equal to 30% of total market value, $r_A=10\%$, and $r_D=4\%$. Convert the debt share to $D/E$, then find $r_E$. (Answer: $D/E=0.4286$; $r_E=12.57\%$)
A firm targets $r_E=16\%$, with $r_A=10\%$ and $r_D=4\%$. Find the required $D/E$ and the corresponding $D/V$. (Answer: $D/E=1.0000$; $D/V=50.00\%$)

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