Leverage, Tax Shields, and After-Tax WACC

Why Debt Changes the Weighted Cost of Capital

Debt and equity are both sources of financing, but their costs enter a firm’s financing calculation differently when corporate interest is tax-deductible. The interest deduction reduces the effective cost of debt to the firm. In the idealized Modigliani–Miller (MM) model with corporate taxes, this tax benefit makes the firm’s after-tax weighted average cost of capital (WACC) fall as debt replaces equity.

That conclusion does not mean that borrowing makes equity less risky or lowers its required return. As leverage increases, equity holders bear more financial risk, and the cost of equity rises. The result for WACC follows because the after-tax debt component receives a tax adjustment while the increase in the cost of equity does not fully offset that benefit under the model’s assumptions.

Definition: Let $E$ and $D$ be the market values of equity and debt, $V=E+D$ the total market value of financing, $r_E$ the required return on equity, $r_D$ the required return on debt, and $T_c$ the corporate tax rate. The after-tax WACC is

\operatorname{WACC} = \frac{E}{V}r_E + \frac{D}{V}r_D(1-T_c).

The weights use market values, not book values. The debt term uses $r_D(1-T_c)$ because interest is deductible in the model: each dollar of interest reduces taxable income by one dollar and reduces taxes by $T_c$ dollars. This adjustment presumes that the firm can use the deduction. The formula is a model result, not a general claim that debt is always preferable.

Model Assumptions and the MM Relation

The tax-adjusted WACC implication is derived within the MM model with corporate taxes. For this derivation, assume a constant corporate tax rate, interest that is tax-deductible, and sufficient taxable income to use the interest deduction. Assume also that operating cash flows and business risk are unchanged by financing, and that the idealized model’s financing and valuation assumptions apply. In particular, the relation below treats $r_U$, the required return on the unlevered firm’s assets, as fixed as leverage changes.

The MM Proposition II relation with corporate taxes expresses how the cost of equity responds to leverage:

r_E = r_U+\frac{D}{E}(1-T_c)(r_U-r_D).

Here, $D/E$ is the debt-to-equity ratio, not the debt share of total financing. The relation says that leverage raises the cost of equity when $r_U>r_D$, because equity becomes a riskier residual claim. The tax term affects how that risk increase relates to leverage.

Deriving the Tax-Adjusted WACC Formula

Substitute the MM Proposition II relation into the definition of after-tax WACC. Writing $E/V$ and $D/V$ as the financing weights, and using $E/V+D/V=1$, gives:

\begin{aligned} \operatorname{WACC} &= \frac{E}{V}\left[r_U+\frac{D}{E}(1-T_c)(r_U-r_D)\right] +\frac{D}{V}r_D(1-T_c)\\ &= \frac{E}{V}r_U +\frac{D}{V}(1-T_c)(r_U-r_D) +\frac{D}{V}r_D(1-T_c)\\ &= \frac{E}{V}r_U +\frac{D}{V}(1-T_c)r_U\\ &= r_U\left[\frac{E}{V}+\frac{D}{V}(1-T_c)\right]\\ &= r_U\left(1-T_c\frac{D}{V}\right). \end{aligned}

In the second line, the factor $E/V$ cancels the $E$ in $D/E$, leaving $D/V$. In the third line, the terms involving $r_D$ cancel: the increase in the weighted cost of equity associated with $r_U-r_D$ combines with the after-tax debt term. Finally, because $E/V+D/V=1$, the result can be written compactly as:

Theorem: Under the stated MM corporate-tax assumptions,

\operatorname{WACC}=r_U\left(1-T_c\frac{D}{V}\right).

This expression shows the predicted direction clearly. If $r_U>0$, $T_c>0$, and the debt share $D/V$ increases, then $1-T_c(D/V)$ decreases, so WACC decreases. The conclusion is conditional on the idealized model. It does not by itself identify an optimal debt ratio or establish that a firm should borrow without limit.

Notice that this formula uses $D/V$, whereas the MM Proposition II formula uses $D/E$. They are related but not interchangeable:

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

For example, a debt-to-equity ratio of $1$ means debt and equity have equal market values, so debt is half of total financing: $D/V=1/2$. It does not mean that debt is all of the firm’s financing.

Worked Examples

Worked Example: Calculating WACC from Leverage and the MM Relation

A hypothetical firm has market-value debt of $40$ million dollars and equity of $60$ million dollars. Its unlevered cost of capital is $10\%$, its cost of debt is $5\%$, and its corporate tax rate is $25\%$. Use the MM Proposition II relation with corporate taxes to calculate the cost of equity and after-tax WACC. Check the result using the tax-adjusted WACC formula.

Method and common mistake avoided. First calculate the debt-to-equity ratio for the equity-cost formula. Then calculate the market-value weights for WACC. Using $D/V$ in place of $D/E$ when finding $r_E$ would give the wrong result.

Step 1: Find total value and both leverage measures. Total financing is $V=40+60=100$ million dollars. Therefore $D/E=40/60=2/3$, while $D/V=40/100=0.40$. The equity weight is $E/V=0.60$.

Step 2: Calculate the cost of equity. The difference $r_U-r_D$ is $0.10-0.05=0.05$, and $1-T_c=0.75$. Thus

r_E =0.10+\frac{2}{3}(0.75)(0.05) =0.10+0.025 =0.125 =12.5\%.

Step 3: Calculate after-tax WACC. The after-tax cost of debt is $0.05(0.75)=0.0375$, or $3.75\%$. Hence

\operatorname{WACC} =(0.60)(0.125)+(0.40)(0.0375) =0.075+0.015 =0.090 =9.0\%.

Check. The compact formula gives $0.10[1-(0.25)(0.40)]=0.10(0.90)=0.090$, also $9.0\%$. The two calculations agree.

Worked Example: Comparing Two Debt Shares

A hypothetical company has $r_U=12\%$, $r_D=6\%$, and $T_c=30\%$. Compare its after-tax WACC when $D/V=25\%$ with its WACC when $D/V=50\%$. For each case, calculate the debt-to-equity ratio and cost of equity as well as WACC.

Method and common mistake avoided. Convert each debt share into $D/E$ before applying MM Proposition II. A higher debt share raises $r_E$; do not conclude that the equity cost stays fixed while computing the weighted average.

Case 1: $D/V=0.25$. The equity share is $E/V=0.75$, so $D/E=0.25/0.75=1/3$. Since $1-T_c=0.70$ and $r_U-r_D=0.06$,

r_E=0.12+\frac{1}{3}(0.70)(0.06)=0.12+0.014=0.134=13.4\%.

The after-tax cost of debt is $0.06(0.70)=0.042$, or $4.2\%$. Therefore

\operatorname{WACC} =(0.75)(0.134)+(0.25)(0.042) =0.1005+0.0105 =0.111 =11.1\%.

Case 2: $D/V=0.50$. The equity share is also $0.50$, so $D/E=1$. The cost of equity is

r_E=0.12+(1)(0.70)(0.06)=0.162=16.2\%.

Thus

\operatorname{WACC} =(0.50)(0.162)+(0.50)(0.042) =0.081+0.021 =0.102 =10.2\%.

Check. The compact formula gives $0.12[1-(0.30)(0.25)]=0.111$ in Case 1 and $0.12[1-(0.30)(0.50)]=0.102$ in Case 2. WACC falls by $0.9$ percentage points as the debt share rises from $25\%$ to $50\%$, even though the cost of equity rises from $13.4\%$ to $16.2\%$.

Worked Example: Checking WACC Against a Perpetual Tax-Shield Valuation

A hypothetical firm is expected to generate perpetual EBIT of $12$ million dollars per year. Its corporate tax rate is $25\%$, its unlevered cost of capital is $10\%$, and it has $30$ million dollars of perpetual debt. Assume the debt tax shield has the risk and valuation treatment specified by the idealized MM tax model. The debt rate is $5\%$. Calculate the unlevered value, levered value, equity value, cost of equity, and after-tax WACC. Then check that after-tax operating cash flow discounted at WACC gives the levered value.

Method and common mistake avoided. First value the after-tax operating cash flow of the unlevered firm, then add the tax shield value $T_cD$. Do not treat the tax shield as if it were an extra deduction from operating cash flow.

Step 1: Calculate annual after-tax operating cash flow and unlevered value. With no growth and perpetual EBIT, after-tax operating cash flow is $12(1-0.25)=9$ million dollars each year. At $r_U=10\%$,

V_U=\frac{9}{0.10}=90\text{ million dollars}.

Step 2: Value the tax shield and the levered firm. Under the stated idealized assumptions, the present value of the tax shield on permanent debt is $T_cD=(0.25)(30)=7.5$ million dollars. Thus $V_L=90+7.5=97.5$ million dollars, and equity value is $E=V_L-D=97.5-30=67.5$ million dollars.

Step 3: Calculate the cost of equity. Here $D/E=30/67.5=4/9$. Applying the MM relation,

r_E =0.10+\frac{4}{9}(0.75)(0.10-0.05) =0.10+\frac{1}{60} =0.116666\ldots \approx 11.67\%.

Step 4: Calculate and check WACC. The weights are $E/V=67.5/97.5=9/13$ and $D/V=30/97.5=4/13$. The after-tax debt cost is $0.05(0.75)=0.0375$. Therefore

\begin{aligned} \operatorname{WACC} &=\frac{9}{13}(0.116666\ldots)+\frac{4}{13}(0.0375)\\ &=0.080769\ldots+0.011538\ldots\\ &=0.092307\ldots\approx 9.23\%. \end{aligned}

The compact formula gives $0.10[1-(0.25)(4/13)]=0.092307\ldots$, the same result. Finally, $9/0.092307\ldots=97.5$ million dollars, which matches the levered value. Rates are rounded to two decimal places only at the final step.

Interpreting the Result and Avoiding Common Errors

The WACC decline is the net result of two opposing movements. The debt component becomes less costly after tax, while financial leverage raises the required return on equity. The derivation shows exactly how they combine in this model: the debt-related $r_D$ terms cancel, leaving a reduction determined by $T_cD/V$. Without corporate taxes, $T_c=0$, so this expression reduces to $\operatorname{WACC}=r_U$; changing the financing mix does not change WACC under the corresponding no-tax MM assumptions.

A common error is to use the tax-adjusted formula with the debt-to-equity ratio. For instance, if $D/E=1$, then $D/V=1/2$, not $1$. With $r_U=10\%$ and $T_c=25\%$, the incorrect substitution gives $10\%[1-(0.25)(1)]=7.5\%$. The correction is to convert the ratio: $D/V=1/(1+1)=0.5$, so the correct WACC is $10\%[1-(0.25)(0.5)]=8.75\%$.

A second frequent mistake is to apply the tax adjustment to equity cost. Interest is deductible in the model; dividends to equity holders are not. Thus the debt term in WACC is $r_D(1-T_c)$, while the equity term remains $r_E$. Another is to report the decrease as a guaranteed real-world benefit of borrowing. The formula is an implication of stated idealized assumptions, not a complete financing recommendation. It is useful for understanding the tax-shield effect and for calculations that explicitly invoke the MM tax model.

Key takeaway: Under the MM corporate-tax assumptions, $\operatorname{WACC}=r_U(1-T_cD/V)$. Debt raises the required return on equity, but the tax benefit on interest causes after-tax WACC to fall as the debt share increases in this idealized model.

Check Your Understanding

For each problem, state the leverage ratio used and show enough intermediate steps to distinguish the debt share from the debt-to-equity ratio.

A hypothetical firm has $r_U=9\%$, $r_D=4\%$, $T_c=20\%$, and $D/V=25\%$. Calculate $D/E$, $r_E$, and after-tax WACC. (Answer: $D/E=1/3$, $r_E=10.33\%$, WACC $=8.55\%$.)
A company has $r_U=11\%$, $r_D=5\%$, $T_c=30\%$, and $D/E=0.5$. Calculate $D/V$, $r_E$, and WACC. (Answer: $D/V=1/3$, $r_E=13.10\%$, WACC $=9.90\%$.)
For $r_U=10\%$, $r_D=4\%$, and $T_c=25\%$, calculate WACC at debt shares of $20\%$ and $60\%$. Also calculate the corresponding cost of equity in each case. (Answer: At $D/V=20\%$, $D/E=0.25$, $r_E=11.125\%$, WACC $=9.50\%$; at $D/V=60\%$, $D/E=1.5$, $r_E=16.75\%$, WACC $=8.50\%$.)
A hypothetical firm has perpetual EBIT of $8$ million dollars, $T_c=25\%$, $r_U=8\%$, perpetual debt of $20$ million dollars, and $r_D=4\%$. Under the idealized MM tax model, calculate $V_U$, $V_L$, equity value, $r_E$, and WACC. Check the value using after-tax operating cash flow divided by WACC. (Answer: $V_U=\$75$ million, $V_L=\$80$ million, $E=\$60$ million, $r_E=9.00\%$, WACC $=7.50\%$; $\$6\text{ million}/0.075=\$80\text{ million}$.)
A calculation uses $D/E=0.4$, $r_U=10\%$, $r_D=6\%$, and $T_c=25\%$, and reports WACC as $10\%[1-(0.25)(0.4)]=9.00\%$. Identify the error and give the corrected WACC and cost of equity. (Answer: $0.4$ is $D/E$, so $D/V=0.4/1.4=2/7$; $r_E=11.20\%$ and WACC $=9.29\%$.)

What You'll Learn