Why Leverage Changes the Cost of Equity
Debt can change how a firm’s operating risk is divided between its investors. Bondholders have a contractual claim, so equity holders receive what remains after interest is paid. When the firm’s operating results vary, that residual can vary more sharply than the firm’s total operating results. Corporate interest deductions modify the size of this leverage effect: under the Modigliani–Miller model with taxes, the increase in the cost of equity is scaled by one minus the corporate tax rate.
The result is useful for analyzing how debt financing affects the return required by shareholders. It is not a general rule that can be applied regardless of the tax position, debt policy, or risk of a company. The model’s assumptions are essential to the formula.
The Proposition and Its Assumptions
In the standard Modigliani–Miller model with corporate taxes, the relation between leverage and the cost of equity is
The leverage measure in this equation is debt-to-equity, \(D/E\), not debt as a fraction of total firm value, \(D/(D+E)\). The cost of equity generally rises as \(D/E\) rises when \(r_U>r_D\). The factor \(1-T_C\) reflects the value of the interest deduction under the model.
The standard relation relies on a set of restrictive assumptions. The firm and its unlevered counterpart have the same operating risk; investors and firms can borrow and lend under the model’s market conditions; and there are no transaction costs or costs of financial distress. The corporate tax rate is constant, and interest deductions are fully usable at that rate. Debt is treated as permanent, with its market value and required return stable, and the tax shield is valued consistently with that debt policy. The basic derivation below also assumes level perpetual operating earnings, no growth, and sufficient taxable income to use the deduction every period.
These conditions matter. For example, if a firm cannot use its interest deductions, it does not receive the full modeled tax benefit. If debt is changing substantially over time, or if financial distress affects value or required returns, the simple relation may not describe the firm’s financing choice. Later analysis can address such limitations; this proposition isolates the tax effect under its stated assumptions.
Why the Formula Holds
A short derivation shows how the tax adjustment enters. Let expected annual EBIT be level and perpetual. With no debt, after-tax operating earnings are \(EBIT(1-T_C)\), so the value of the unlevered firm is \(V_U=EBIT(1-T_C)/r_U\). With permanent debt and fully usable interest deductions, the standard tax-shield value relation is \(V_L=V_U+T_CD\). Since \(V_L=D+E\), rearranging gives \(V_U=E+(1-T_C)D\).
For the levered firm, annual interest is \(r_DD\). Its after-tax cash flow to equity is therefore \((EBIT-r_DD)(1-T_C)\). The required return on equity means that the expected cash flow to equity equals \(r_EE\). Also, the unlevered firm’s after-tax operating earnings equal \(r_UV_U\). Combining these statements gives:
Dividing both sides by \(E\) yields the proposition:
The derivation makes the economic point precise: shareholders bear the residual after debt claims, but interest deductibility reduces the modeled after-tax leverage increment. If \(T_C=0\), the relation becomes the no-tax version, \(r_E=r_U+(D/E)(r_U-r_D)\). If \(D=0\), it gives \(r_E=r_U\), as it should.
Applying the Relation
Before calculating, identify the ratio supplied by the problem. If debt is given as a fraction of total market value, convert it to debt-to-equity. Writing \(V=D+E\), a debt fraction \(D/V\) implies:
This conversion prevents a frequent error: substituting \(D/V\) directly into a formula that requires \(D/E\).
Worked Example: Calculating the Levered Cost of Equity
A hypothetical firm has \(r_U=10.00\%\), \(r_D=4.00\%\), \(T_C=25\%\), and market debt-to-equity of \(0.50\). Assuming permanent debt, a constant tax rate, fully usable interest deductions, and the other standard Modigliani–Miller assumptions, calculate \(r_E\).
Method. Use the tax-adjusted Proposition II relation with the debt-to-equity ratio as given. This avoids treating \(D/E\) as a fraction of total value.
First, find the difference between the unlevered cost of equity and the cost of debt:
Next, calculate the tax-adjusted leverage factor:
The leverage increment is \(0.375(6.00\%)=2.25\%\). Therefore:
Check. Because \(r_U>r_D\) and the firm has positive debt, the levered cost of equity should exceed \(r_U\). The calculated \(12.25\%\) satisfies this check.
Worked Example: Converting a Debt-to-Value Fraction
A hypothetical firm has \(r_U=11.00\%\), \(r_D=5.00\%\), and \(T_C=30\%\). Debt equals \(40\%\) of total market value. Assuming the standard model conditions, calculate the firm’s cost of equity.
Method. Convert the debt fraction of total value to \(D/E\), then use Proposition II. The common mistake is to insert \(0.40\) directly as though it were \(D/E\).
If \(D/V=0.40\), equity is the remaining \(0.60\) of total value. Thus:
The return spread is \(11.00\%-5.00\%=6.00\%\), and the tax-adjusted leverage factor is:
The leverage increment is \(0.466666\ldots(6.00\%)=2.80\%\). Therefore:
Check. Substituting \(D/E=2/3\) gives \(0.70(2/3)(0.06)=0.028\), or a \(2.80\%\) increment. The final cost of equity is \(13.80\%\), not \(12.68\%\), which would result from the incorrect substitution \(D/E=0.40\).
Worked Example: Inferring the Debt-to-Equity Ratio
A hypothetical company’s levered cost of equity is \(15.00\%\). Its unlevered cost of equity is \(9.00\%\), its cost of debt is \(5.00\%\), and its corporate tax rate is \(25\%\). Under the standard Proposition II assumptions, infer \(D/E\), then state what fraction of total market value is debt.
Method. Rearrange the proposition to isolate \(D/E\). This avoids confusing the leverage ratio being solved for with the debt share of total value.
The observed increase over the unlevered cost is \(15.00\%-9.00\%=6.00\%\). The return spread is \(9.00\%-5.00\%=4.00\%\), and the tax adjustment is \(1-0.25=0.75\). Therefore:
Debt is twice equity. To convert that ratio to a debt fraction of total market value, use \(D/(D+E)\). Setting \(E=1\) gives \(D=2\), so:
Check. Substituting \(D/E=2\) into Proposition II gives \(9.00\%+0.75(2)(4.00\%)=15.00\%\), matching the stated cost of equity.
Interpreting the Result and Avoiding Traps
Under the model, the relation is linear in \(D/E\). For fixed \(r_U\), \(r_D\), and \(T_C\), each additional unit of \(D/E\) increases \(r_E\) by \((1-T_C)(r_U-r_D)\). This is a comparative result under fixed assumptions; it does not say that a firm’s actual cost of debt stays constant as borrowing rises, or that the firm can increase debt without changing risk or market conditions.
A useful exam check is the direction of the result. When \(r_U>r_D\), \(T_C<1\), and \(D/E>0\), the formula implies \(r_E>r_U\). When there is no debt, the two costs are equal. If the calculation violates these implications, revisit the ratio, the tax adjustment, and the return spread.
- Wrong ratio: using \(D/V\) instead of \(D/E\) understates the leverage term when debt is a fraction of total value. Convert using \(D/E=(D/V)/(1-D/V)\).
- Missing tax adjustment: multiplying by \(D/E\) alone applies the no-tax relation, not the corporate-tax relation.
- Reversed return spread: the term is \(r_U-r_D\), not \(r_D-r_U\). Reversing it can incorrectly make more leverage appear to reduce the cost of equity.
- Ignoring the assumptions: the formula does not capture distress costs, unusable tax deductions, or changes in the firm’s underlying operating risk.
This proposition isolates how leverage and corporate interest deductions affect the required return on equity. It should not be read as a standalone recommendation to borrow more. Its prediction depends on the model’s idealized tax and financing conditions, and it addresses the equity return rather than every consequence of a financing decision.
Check Your Understanding
Use the stated assumptions in each problem. For calculations, retain full precision until the final answer.
- A hypothetical firm has \(r_U=12.00\%\), \(r_D=6.00\%\), \(T_C=25\%\), and \(D/E=0.75\). Find \(r_E\) under the standard model. (Answer: \(15.375\%\))
- A firm’s debt is \(30\%\) of total market value. Its \(r_U\) is \(10.00\%\), \(r_D\) is \(4.00\%\), and \(T_C\) is \(25\%\). Convert the leverage ratio and calculate \(r_E\). (Answer: \(D/E=3/7\); \(r_E\approx11.93\%\))
- Under the standard assumptions, \(r_E=14.00\%\), \(r_U=8.00\%\), \(r_D=3.00\%\), and \(T_C=40\%\). Find \(D/E\), then find debt as a fraction of total market value. (Answer: \(D/E=2\); debt is \(2/3\), or \(66.67\%\), of total value)
- A firm has \(r_E=16.00\%\), \(r_D=6.00\%\), \(T_C=25\%\), and \(D/E=1.50\). Assuming the model applies, solve for \(r_U\). (Answer: \(r_U\approx10.71\%\))
- A hypothetical firm has \(r_U=9.00\%\), \(r_D=5.00\%\), \(T_C=30\%\), and \(D/E=1\). Its annual EBIT is \(40\) monetary units, while annual interest is \(60\); it has no tax refunds or carryforwards, so it cannot use the interest deduction. What does mechanical substitution into Proposition II produce, and is that result valid for this firm? (Answer: Mechanical substitution gives \(11.80\%\), but it is not a valid estimate under Proposition II because the interest deduction is not usable.)