Why the No-Tax WACC Stays Constant
Debt usually has a lower required return than equity. It may therefore seem that a firm could reduce its overall financing cost simply by replacing equity with debt. In the no-tax Modigliani–Miller model, that conclusion is incorrect: as leverage rises, equity becomes riskier and shareholders require a higher return. The changing component costs offset the changing weights, leaving the firm’s overall cost of capital unchanged.
This result depends on the model’s assumptions, not on a claim that debt and equity are equally risky or have equal required returns. The firm’s business risk and investment policy are held fixed, and the capital markets are assumed to be perfect. In particular, there are no corporate taxes, transaction costs, or costs of financial distress, and investors can borrow or lend on the same terms as firms. These assumptions isolate the effect of changing financing mix.
The weights are market-value weights, not book-value weights. All returns in a calculation must be measured on a consistent basis, such as annual required returns. The result concerns the cost of financing the firm’s assets as a whole; it does not say that the required return on each individual security is constant.
How Leverage Changes the Component Returns
Modigliani–Miller Proposition II without taxes states that, when the firm’s business risk is unchanged, the required return on equity increases with the market debt-to-equity ratio:
The amount by which \(r_E\) exceeds \(r_A\) depends on both leverage and the spread between the asset return and the debt return. For example, if \(r_A>r_D\), increasing \(D/E\) raises \(r_E\). This relation does not assert that \(r_D\) must remain constant as leverage changes. If debt itself becomes riskier at higher leverage, its required return may change too; the corresponding equity return must then be determined using the applicable \(r_D\).
Deriving the Constant-WACC Result
The result follows directly by substituting the Proposition II return on equity into the weighted-average formula. Because \(V=D+E\), the debt and equity weights add to one. The full algebra shows how the apparent advantage of a lower debt return is offset by the higher return required by equity holders.
By definition, \(\operatorname{WACC}=(D/V)r_D+(E/V)r_E\), where \(V=D+E\).
Proposition II gives \(r_E=r_A+(D/E)(r_A-r_D)\). Therefore, $$ \operatorname{WACC} =\frac{D}{V}r_D+\frac{E}{V}\left[r_A+\frac{D}{E}(r_A-r_D)\right]. $$
Since \((E/V)(D/E)=D/V\), $$ \operatorname{WACC} =\frac{D}{V}r_D+\frac{E}{V}r_A+\frac{D}{V}(r_A-r_D). $$
The two terms involving \(r_D\) cancel, leaving $$ \operatorname{WACC} =\frac{E}{V}r_A+\frac{D}{V}r_A =\frac{D+E}{V}r_A =r_A. $$ Thus the WACC equals the asset required return.
This is an algebraic expression of the economic mechanism. Debt’s weight may rise, but the return required by equity holders adjusts to reflect the extra financial risk they bear. Under the model assumptions, changing the financing split reallocates risk between security holders; it does not change the risk or required return of the firm’s assets.
Calculating WACC Across Capital Structures
Worked Example: Equal Debt and Equity Values
A hypothetical firm has market-value debt of $60 million and market-value equity of $60 million. Its debt has a required return of 6% and its asset cost of capital is 12%. Under the no-tax model, calculate the equity required return and verify the WACC. Use annual rates and keep the calculations in millions of dollars.
Method. First use Proposition II to find the equity return; then apply market-value weights. This avoids the common mistake of assuming that a lower-cost debt component automatically lowers WACC while the equity return stays unchanged.
The market debt-to-equity ratio is $$ \frac{D}{E}=\frac{60}{60}=1. $$ Thus, $$ r_E=12\%+1(12\%-6\%)=12\%+6\%=18\%. $$ Total financing value is $$ V=60+60=120\text{ million}. $$ The debt and equity weights are both \(60/120=0.5\). Therefore, $$ \operatorname{WACC}=0.5(6\%)+0.5(18\%)=3\%+9\%=12\%. $$ The result equals the given asset cost of capital, as required by the model.
Worked Example: Converting a Debt-to-Equity Ratio into Weights
A hypothetical business has market-value debt of $80 million and equity of $100 million. Its asset cost of capital is 10.5%, and its debt required return is 5.5%. Find the equity required return and verify the no-tax WACC. Report rates to three decimal places.
Method. Use the stated dollar values to obtain both \(D/E\) and the WACC weights. The trap is to treat \(D/E\) as the debt share of total value; the debt weight is \(D/(D+E)\), not \(D/E\).
First, $$ \frac{D}{E}=\frac{80}{100}=0.8. $$ Proposition II gives $$ r_E=10.5\%+0.8(10.5\%-5.5\%) =10.5\%+0.8(5.0\%) =10.5\%+4.0\% =14.5\%. $$ The total market value is \(V=80+100=180\) million, so the weights are $$ \frac{D}{V}=\frac{80}{180}=\frac{4}{9}, \qquad \frac{E}{V}=\frac{100}{180}=\frac{5}{9}. $$ The weighted average is $$ \operatorname{WACC} =\frac{4}{9}(5.5\%)+\frac{5}{9}(14.5\%) =\frac{22\%+72.5\%}{9} =10.5\%. $$ To three decimal places, the equity return is 14.500% and WACC is 10.500%. The WACC check agrees with the asset cost of capital.
Worked Example: Comparing Two Leverage Levels
Two hypothetical financing plans fund the same assets, with an asset cost of capital of 11%. Under Plan A, \(D/E=0.5\) and debt’s required return is 4%. Under Plan B, \(D/E=1.5\) and debt’s required return is 7%. For each plan, calculate the equity return, the debt and equity weights, and WACC. The difference in debt returns represents a change in debt risk between the plans.
Method. Apply Proposition II separately using each plan’s own debt return, then calculate WACC using the weights implied by \(D/E\). The key trap is to carry Plan A’s debt return into Plan B even though the problem specifies a different required return.
For Plan A, \(D/E=0.5\), so $$ r_E=11\%+0.5(11\%-4\%)=11\%+3.5\%=14.5\%. $$ When \(D/E=0.5\), take \(E=1\) unit and \(D=0.5\) unit. Then \(V=1.5\) units, giving debt weight \(0.5/1.5=1/3\) and equity weight \(1/1.5=2/3\). Hence $$ \operatorname{WACC}_A =\frac{1}{3}(4\%)+\frac{2}{3}(14.5\%) =\frac{4\%+29\%}{3} =11\%. $$ For Plan B, $$ r_E=11\%+1.5(11\%-7\%)=11\%+6\%=17\%. $$ Take \(E=1\) unit and \(D=1.5\) units. Total value is \(2.5\) units, so the debt weight is \(1.5/2.5=0.6\) and the equity weight is \(1/2.5=0.4\). Therefore, $$ \operatorname{WACC}_B=0.6(7\%)+0.4(17\%)=4.2\%+6.8\%=11\%. $$ Both plans produce the same WACC of 11%, although the debt return and equity return differ between plans. The check is consistent with the fixed asset cost of capital.
What the Result Does—and Does Not—Mean
A constant no-tax WACC does not mean that every financing choice is equivalent in every real-world setting. It is a conclusion within a defined model. If a firm faces corporate taxes, financial distress costs, transaction costs, or other frictions, the assumptions used in the derivation may fail. Those effects are outside the no-tax result.
It is also important to distinguish the firm’s asset cost of capital from the required return on a particular security. As leverage changes, equity holders’ claim becomes more sensitive to the firm’s operating outcomes because debt has a prior contractual claim on cash flows. In the no-tax model, \(r_E\) adjusts accordingly. A lower \(r_D\) alone is not enough to conclude that the firm’s financing cost has fallen.
For an actuarial or corporate-finance analysis, this distinction matters when estimating the rate used to evaluate the firm’s assets or investment opportunities. A change in financing proportions should not, by itself, be used to justify changing the no-tax asset discount rate under the model. The candidate must first identify whether the assumptions hold and whether the question asks for the asset cost, the equity cost, or WACC.
Common Exam Traps
- Using book values for weights. The no-tax WACC definition here uses market values \(D\) and \(E\). If the problem supplies both book and market values, do not substitute book amounts without an explicit instruction to do so.
- Confusing \(D/E\) with \(D/V\). Proposition II uses \(D/E\), while the WACC formula uses \(D/V\) and \(E/V\). Convert between them using \(V=D+E\).
- Holding the equity return fixed as leverage changes. That omits the risk adjustment in Proposition II and generally produces a WACC below \(r_A\) when \(r_D<r_A\).
- Assuming the debt return can never change. The constant-WACC result does not require \(r_D\) to be identical at every leverage level. Use the debt return appropriate to the capital structure being analyzed.
- Applying the no-tax conclusion when taxes or distress costs are present. Those conditions change the model. Do not import a no-tax result into an after-tax calculation.
Check Your Understanding
For each problem, use market values and the no-tax assumptions stated or implied. Check that your result is consistent with the asset cost of capital.
- A hypothetical firm has debt worth $45 million and equity worth $105 million. Its asset cost of capital is 9%, and its debt required return is 5%. Find \(D/E\), the equity required return, and WACC. (Answer: \(D/E=0.4286\); \(r_E=10.714\%\); WACC \(=9.000\%\).)
- A firm’s debt-to-equity ratio is 1.25, its asset cost of capital is 13%, and its debt required return is 8%. Find \(r_E\), then calculate the debt and equity weights and verify WACC. (Answer: \(r_E=19.250\%\); debt weight \(=5/9\), equity weight \(=4/9\); WACC \(=13.000\%\).)
- A hypothetical company has market-value debt of $72 million and equity of $128 million. The asset cost is 10.8%, while debt requires 6.8%. Calculate the equity required return and WACC. Explain why the equity return exceeds the asset cost. (Answer: \(D/E=0.5625\); \(r_E=13.050\%\); WACC \(=10.800\%\); equity bears additional financial risk.)
- Two plans fund the same assets with \(r_A=12\%\). Plan X has \(D/E=0.4\) and \(r_D=5\%\); Plan Y has \(D/E=1.6\) and \(r_D=8\%\). Find each equity return and WACC, using each plan’s own debt return. (Answer: Plan X \(r_E=14.800\%\), WACC \(=12.000\%\); Plan Y \(r_E=18.400\%\), WACC \(=12.000\%\).)
- An analyst uses debt of $90 million and equity of $60 million, with \(r_D=4\%\) and \(r_E=16\%\). The analyst mistakenly treats \(D/E=1.5\) as the debt weight and reports WACC as \(1.5(4\%)+16\%\), or 22%. Find the correct weights and WACC, and identify the error. (Answer: debt weight \(=0.6\), equity weight \(=0.4\); WACC \(=8.800\%\); \(D/E\) was used in place of \(D/V\), and the stated returns imply \(r_A=8.800\%\) under the no-tax relation.)