Why Taxes Change the Capital Structure Result
With no corporate taxes, changing how a firm divides financing between debt and equity does not change its total value under the perfect-market assumptions. Corporate income taxes alter that result because interest expense is deductible: for a firm that can use the deduction, interest reduces taxable income and therefore reduces taxes paid. The resulting tax saving is called the interest tax shield.
Modigliani–Miller Proposition I with corporate taxes expresses the value effect of that shield. The proposition is not a claim that borrowing improves a firm’s operations. It compares firms with the same operating assets and cash flows but different financing, under specific assumptions about taxes, debt, and the tax savings.
Assumptions Behind the Proposition
The formula is useful only when its assumptions are understood. In the basic model, the firms being compared have identical operating assets and operating cash flows, and the only difference is financing. The corporate tax rate is constant, and interest is deductible. The levered firm maintains a fixed amount of debt in perpetuity, with a constant required return \(r_D\) on that debt.
The firm has enough taxable income to use the interest deduction as it arises, so each period’s tax saving is realized immediately. The tax shield has the same risk as the debt and is discounted at \(r_D\). The model also excludes personal taxes, financial distress costs, transaction costs, and other financing effects. These assumptions isolate the tax benefit rather than describing every real firm.
The value in the proposition is total firm value: the market value of debt plus the market value of equity. It is not the value of equity alone. Once \(V_L\) is determined, equity value is \(E=V_L-D\), provided \(D\) denotes the market value of debt.
Deriving the Perpetual Tax-Shield Value
The derivation starts with the tax saving in one period. Suppose the firm has fixed debt \(D\) and pays interest at rate \(r_D\). Interest expense per year is \(r_DD\). If the firm can use the deduction and its corporate tax rate is \(T_c\), the reduction in tax paid each year is \(T_c\) times the deductible interest.
The shield is received at the end of each year forever. Under the stated assumptions, it is a level perpetuity discounted at \(r_D\). Its present value is the annual saving divided by the discount rate:
The division cancels \(r_D\), which is why the basic proposition gives the tax-shield value as \(T_cD\). The levered firm has the value of the same operating business as the unlevered firm, plus this tax-shield value:
This derivation depends on a perpetual, level shield discounted at the debt rate. It does not establish that every tax shield, regardless of duration or risk, is worth \(T_cD\). For example, if deductions cannot be used when generated, the timing or amount of the savings can differ. Those cases require a different valuation rather than an automatic application of the formula.
Use \(V_U\) for the value of the firm’s operating assets without debt.
For fixed perpetual debt, use \(T_cD\), after confirming that the deductions are usable and that the standard risk assumption applies.
Calculate \(V_L=V_U+T_cD\). Do not subtract debt at this step; debt is already included in total levered firm value.
If equity value is requested, calculate \(E=V_L-D\) and verify that \(E+D=V_L\).
Applying Proposition I
The following examples use the proposition only after specifying the assumptions and quantities needed. All dollar amounts are hypothetical. Where values are stated in thousands, calculations remain in thousands until rounding.
Worked Example: Valuing a Firm With Fixed Perpetual Debt
A hypothetical firm has an unlevered value of $12 million. It issues $4 million of fixed debt that remains outstanding forever. The debt’s required return is 6% per year, and the corporate tax rate is 25%. Assume the firm has enough taxable income to use every interest deduction, the tax shield is discounted at the debt rate, and there are no personal taxes, distress costs, or transaction costs. Find the annual tax saving, its present value, and the levered firm value.
Method. First find the interest and annual tax saving, then value the level perpetuity. This avoids treating the annual tax saving as though it were itself the total value added.
Annual interest is \(0.06\times \$4{,}000{,}000=\$240{,}000\). The tax saving each year is \(0.25\times \$240{,}000=\$60{,}000\).
The tax shield is a level perpetuity discounted at 6%, so its present value is \(\$60{,}000/0.06=\$1{,}000{,}000\). Equivalently, \(T_cD=0.25\times \$4{,}000{,}000=\$1{,}000{,}000\).
Therefore, \(V_L=\$12{,}000{,}000+\$1{,}000{,}000=\$13{,}000{,}000\). As a check, the shield value from the perpetuity calculation equals \(T_cD\), and the levered value exceeds the unlevered value by exactly that amount.
Common mistake avoided: adding the annual $60,000 tax saving directly to firm value. That is a yearly cash flow, not a present value; the perpetual stream is worth $1,000,000 under the assumptions.
Worked Example: Solving for Debt From a Debt-to-Value Ratio
A hypothetical firm has an unlevered value of $8,000 thousand. It plans to issue perpetual fixed debt, and its corporate tax rate is 30%. The target debt-to-total-firm-value ratio is \(D/V_L=30\%\). Under the standard assumptions for Proposition I, find the required debt and the resulting levered firm value, in thousands.
Method. The target ratio uses levered firm value in its denominator. Substitute \(V_L=V_U+T_cD\) into the ratio before solving. Using \(V_U\) as the denominator would answer a different question.
The ratio condition is \(D=0.30V_L\). Since \(V_L=8{,}000+0.30D\), in thousands,
Subtract \(0.09D\) from both sides: \(0.91D=2{,}400\). Therefore, \(D=2{,}400/0.91=2{,}637.362637\) thousand.
The levered firm value is \(V_L=8{,}000+0.30(2{,}637.362637) =8{,}791.208791\) thousand. To three decimal places in thousands, \(D=2{,}637.363\) and \(V_L=8{,}791.209\).
Check. \(D/V_L=2{,}637.362637/8{,}791.208791=0.300000\), as required. Also, \(V_L-V_U=791.208791=0.30D\).
Common mistake avoided: setting \(D=0.30V_U\). The stated target is 30% of levered firm value, which includes the tax-shield increase; the debt amount must therefore be solved algebraically.
Worked Example: Comparing Two Debt Levels
A hypothetical firm has unlevered value \(V_U=\$20\) million. Consider two financing plans: Plan A has fixed perpetual debt of $3 million, and Plan B has fixed perpetual debt of $7 million. The corporate tax rate is 25%. Assume sufficient taxable income and all other standard assumptions. Calculate each levered firm value and the change in value from Plan A to Plan B.
Method. Apply \(V_L=V_U+T_cD\) separately to each debt level, keeping the operating benchmark and tax rate unchanged.
Under Plan A, the tax-shield value is \(0.25\times \$3{,}000{,}000=\$750{,}000\). Thus \(V_{L,A}=\$20{,}000{,}000+\$750{,}000=\$20{,}750{,}000\).
Under Plan B, the tax-shield value is \(0.25\times \$7{,}000{,}000=\$1{,}750{,}000\). Thus \(V_{L,B}=\$20{,}000{,}000+\$1{,}750{,}000=\$21{,}750{,}000\).
The debt increase is \(\$7{,}000{,}000-\$3{,}000{,}000=\$4{,}000{,}000\). The firm-value increase is \(\$21{,}750{,}000-\$20{,}750{,}000=\$1{,}000{,}000\), which also equals \(0.25\times \$4{,}000{,}000\).
Check. Both plans use the same \(V_U\), so the difference in total value is exactly the difference in shield values. This comparison does not claim that debt changes operating cash flows.
Common mistake avoided: applying the 25% tax rate to total firm value or adding the entire increase in debt to firm value. The proposition adds \(T_c\) times debt, not all debt.
Interpretation and Limits
The proposition isolates one financing benefit: interest deductions can create value when they produce usable corporate tax savings. Under its assumptions, each additional dollar of perpetual debt adds \(T_c\) dollars of firm value through the shield. This result does not mean that the firm’s operating assets become more productive, or that the value added is equal to the debt proceeds themselves.
In practice, the result can fail to describe the full effect of financing. A firm with losses may be unable to use deductions immediately. Debt may not remain fixed forever, and the tax shield’s risk may not match the debt’s risk. Taxes, distress costs, and other financing effects can also matter. In those cases, the simple \(T_cD\) expression should not be used without checking its assumptions.
- Keep value categories separate. \(V_L\) is debt plus equity value; equity alone is \(E=V_L-D\).
- Use the correct ratio denominator. If debt is stated as a proportion of \(V_L\), solve using \(V_L=V_U+T_cD\).
- Distinguish a cash flow from its value. The annual tax saving is \(T_c r_DD\); its perpetual present value under the model is \(T_cD\).
- Check tax capacity and duration. A deduction that cannot be used, or a shield that is not perpetual, does not automatically have the value in the basic proposition.
Check Your Understanding
For each problem, assume the standard Proposition I conditions unless the question states otherwise. Amounts are hypothetical.
- A firm has \(V_U=\$9\) million, fixed perpetual debt \(D=\$2\) million, and \(T_c=20\%\). Find \(V_L\). (Answer: \(\$9.4\) million.)
- A firm has \(V_U=\$10\) million, perpetual debt of $6 million, a 4% debt rate, and \(T_c=25\%\). Find the annual tax saving and levered value. (Answer: annual tax saving \(=\$60{,}000\); \(V_L=\$11.5\) million.)
- A firm has \(V_U=\$15\) million, \(T_c=25\%\), and a target debt-to-total-value ratio \(D/V_L=40\%\). Find \(D\) and \(V_L\). (Answer: \(D=\$6.667\) million and \(V_L=\$16.667\) million, to three decimals in millions.)
- A firm’s levered value is $13.2 million, its unlevered value is $12 million, and \(T_c=30\%\). Under the model, infer its perpetual debt. (Answer: \(D=\$4\) million.)
- A hypothetical firm has \(V_U=\$18\) million, \(D=\$5\) million, and \(T_c=24\%\). Find its total levered value and equity value. (Answer: \(V_L=\$19.2\) million and \(E=\$14.2\) million.)