Deriving an Interior Optimal Debt Ratio

Why an Interior Leverage Optimum Can Exist

With no taxes or financial distress, changing a firm’s financing mix need not change its total value under the perfect-market assumptions of Modigliani–Miller. Once corporate taxes and expected distress costs are included, however, borrowing can have both a benefit and a cost. The interest tax shield raises value, while expected costs associated with financial distress reduce it. The key question is not simply whether either effect exists, but how each changes as debt rises.

An interior optimum occurs when the value added by a small increase in debt exactly equals the additional expected distress cost from that increase. The following model makes this marginal trade-off explicit. It is a stylized valuation model, not a universal formula: its result depends on the stated tax, distress-cost, and feasibility assumptions.

Definition: Let $V_U$ be the present value of the firm’s operating assets if it is all-equity financed. Define debt intensity $r=D/V_U$, where $D$ is the value of debt. This is a debt ratio relative to unlevered firm value, not the conventional market-value debt ratio $D/V_L$, where $V_L$ is levered firm value. The distinction matters because $V_L$ changes with leverage.

A Simple Firm-Value Model

Suppose the firm maintains a constant amount of perpetual, risk-free debt, has a constant corporate tax rate $T_c$, and can use all interest deductions as they arise. Under these assumptions, the present value of the interest tax shield is $T_cD$: the annual tax saving is the tax rate times interest, and discounting the perpetual shield at the debt rate gives $T_cD$. Assume also that the present value of expected financial distress costs is convex in debt intensity and equals $\kappa V_U r^2$, for a constant $\kappa>0$.

The model excludes other financing effects, such as agency costs, and treats $V_U$, $T_c$, and $\kappa$ as fixed while debt changes. It is intended to apply only over a stated feasible range $0\le r\le r_{\max}$. The firm’s value in that range is therefore:

V_L(r)=V_U+T_cD-\kappa V_Ur^2 =V_U\left(1+T_cr-\kappa r^2\right), \qquad D=rV_U.

The first equality adds the tax-shield value and subtracts expected distress costs from unlevered value. The second uses $D=rV_U$. All terms in the parentheses are dimensionless; multiplying by $V_U$ gives a dollar value. The distress-cost term is a present value in this model, not an annual distress-loss rate.

Model assumptions: The tax shield is fully usable and has present value $T_cD$; $V_U$ is unaffected by the financing choice; expected distress costs have the specified quadratic present value; and the model is evaluated only for feasible $r$ where $V_L(r)>0$. If these assumptions fail, the value function or its parameters must be changed before optimizing.

Deriving the Interior Optimum

An interior optimum must lie strictly between the lower and upper feasible debt intensities. Differentiate firm value with respect to $r$. The first derivative is the marginal change in value, and the second derivative indicates whether a stationary point is a maximum or minimum.

\frac{dV_L}{dr}=V_U(T_c-2\kappa r), \qquad \frac{d^2V_L}{dr^2}=-2\kappa V_U<0.

Since $V_U>0$ and $\kappa>0$, value is strictly concave in $r$. Setting the first derivative equal to zero gives the unconstrained candidate:

T_c-2\kappa r^*=0 \quad\Longrightarrow\quad r^*=\frac{T_c}{2\kappa}.

This condition equates marginal values. An extra unit of debt intensity adds $V_UT_c$ dollars of tax-shield value per unit of $r$; at intensity $r$, it adds $2\kappa V_Ur$ dollars of marginal distress cost. At $r^*$, the two marginal amounts match. The second derivative is negative, so value rises before this point and falls after it.

The candidate is a feasible interior maximum only if it satisfies the constraints:

0<\frac{T_c}{2\kappa}<r_{\max}.

For a positive tax rate and positive $\kappa$, the lower inequality holds. The upper inequality is equivalent to $T_c<2\kappa r_{\max}$. If the unconstrained candidate exceeds the upper bound, value is still increasing throughout the permitted range and the constrained maximum is at $r_{\max}$. If the marginal benefit and cost are equal exactly at the upper bound, that point is a constrained boundary optimum, not an interior one.

Specify the model and range.
State $V_U$, $T_c$, $\kappa$, and the permitted values of $r$. Confirm that the value function is appropriate for the assumptions.

Differentiate value.
Use $dV_L/dr=V_U(T_c-2\kappa r)$ and set it equal to zero to find the unconstrained candidate.

Check curvature and feasibility.
The second derivative is negative, but the candidate must also lie strictly inside the allowed range to be called an interior optimum.

Calculate value and leverage measures.
Find $D=rV_U$, substitute $r$ in the value function, and calculate $D/V_L$ if market-value leverage is requested.

Worked Applications

Worked Example: Finding an Interior Optimum

A hypothetical firm has unlevered value $V_U=\$240$ million. Its corporate tax rate is $T_c=0.30$, and its expected distress-cost parameter is $\kappa=0.24$. Assume the model applies for $0\le r\le1$. Find the optimal intensity, firm value, debt amount, and debt as a share of levered value.

Method. Use the concave value function $V_L(r)=V_U(1+T_cr-\kappa r^2)$, then check the candidate against the feasible interval.

The unconstrained candidate is:

r^*=\frac{0.30}{2(0.24)} =\frac{0.30}{0.48} =0.625.

Because $0<0.625<1$, it is feasible and interior. The second derivative is $-2(0.24)(\$240\text{ million})<0$, confirming a maximum. Debt is:

D=r^*V_U=0.625(\$240\text{ million})=\$150\text{ million}.

The value multiplier is:

$$ 1+(0.30)(0.625)-(0.24)(0.625)^2 =1+0.1875-0.09375 =1.09375. $$

Thus $V_L=1.09375(\$240\text{ million})=\$262.5$ million. The conventional market-value debt ratio is $D/V_L=\$150/\$262.5=0.571428\ldots$, or 57.14% to two decimal places. As a check, the marginal terms match: $T_c=0.30$ and $2\kappa r^*=2(0.24)(0.625)=0.30$.

Common mistake avoided: Reporting $r^*=62.5\%$ as debt divided by levered value. It is $D/V_U$; the market-value ratio is 57.14%.

Worked Example: Comparing the Interior Candidate With the Boundary

A hypothetical insurer’s financing analysis uses $V_U=\$80$ million, $T_c=0.24$, $\kappa=0.30$, and a feasible range $0\le r\le0.50$. Find the optimum and verify that the upper boundary does not have a higher value.

Method. Find the stationary point, verify concavity and feasibility, then compare its value with the value at the upper boundary.

The candidate is $r^*=0.24/[2(0.30)]=0.24/0.60=0.40$. It lies inside the permitted range. Since $V_L''=-2(0.30)(\$80\text{ million})<0$, this is a maximum. At the candidate:

V_L(0.40)=\$80\text{ million} \left[1+(0.24)(0.40)-(0.30)(0.40)^2\right] =\$80\text{ million}(1.048) =\$83.84\text{ million}.

Debt is $D=(0.40)(\$80\text{ million})=\$32$ million, so debt as a share of levered value is $\$32/\$83.84=0.381679\ldots$, or 38.17% to two decimal places. At the upper boundary:

V_L(0.50)=\$80\text{ million} \left[1+(0.24)(0.50)-(0.30)(0.50)^2\right] =\$80\text{ million}(1.045) =\$83.60\text{ million}.

The interior value exceeds the boundary value by $\$83.84-\$83.60=\$0.24$ million. This is consistent with strict concavity and the stationary point being inside the feasible range.

Common mistake avoided: Equating total tax-shield value with total distress cost. At $r=0.40$, those amounts are respectively $(0.24)(0.40)(\$80)=\$7.68$ million and $(0.30)(0.40)^2(\$80)=\$3.84$ million; they are not equal. The optimum equates marginal, not total, effects.

Worked Example: When the Best Feasible Choice Is at the Boundary

A hypothetical company has $V_U=\$150$ million, $T_c=0.36$, and $\kappa=0.20$. Its financing constraints limit $r$ to $0\le r\le0.60$. Determine whether an interior optimum exists and calculate the best feasible value and market-value debt ratio.

Method. Compare the unconstrained candidate with the permitted range. If it lies above the upper bound, use the boundary and verify the direction of the marginal value change.

The unconstrained candidate is $r^*=0.36/[2(0.20)]=0.36/0.40=0.90$, which is infeasible. For every $r\le0.60$, the derivative factor $T_c-2\kappa r$ is at least $0.36-2(0.20)(0.60)=0.12>0$. Value therefore increases across the entire feasible range, so the maximum occurs at $r=0.60$, not at an interior point.

The debt amount is $D=(0.60)(\$150\text{ million})=\$90$ million. The value multiplier at the boundary is:

$$ 1+(0.36)(0.60)-(0.20)(0.60)^2 =1+0.216-0.072 =1.144. $$

Thus $V_L=\$150\text{ million}(1.144)=\$171.6$ million, and the market-value debt ratio is $\$90/\$171.6=0.524475\ldots$, or 52.45% to two decimal places. The derivative is still positive at the boundary, confirming that the constraint, rather than a marginal balance, determines the choice.

Common mistake avoided: Choosing the unconstrained $r=0.90$ even though the stated financing limit is $0.60$. An unconstrained stationary point is not a valid recommendation when it violates a constraint.

Interpreting the Result Carefully

The formula $r^*=T_c/(2\kappa)$ gives comparative insight within this model. Holding $\kappa$ fixed, a higher tax rate raises the value of the marginal tax shield and raises the candidate debt intensity. Holding $T_c$ fixed, a larger $\kappa$ means distress costs rise more steeply, reducing the candidate. Neither conclusion means that the actual debt ratio must move by the same amount: the model’s $r$ uses unlevered value as its denominator, while market-value leverage uses levered value.

The model’s answer is also conditional on its shape. If the distress-cost function is not quadratic, the first-order condition changes. If tax deductions cannot be fully used, the tax-shield benefit may not remain linear in debt. If there are financing constraints, the best feasible choice may be a boundary, as in the third example. These are reasons to check assumptions and constraints rather than apply the formula mechanically.

A frequent exam trap is to compare the total tax-shield benefit with total expected distress cost and select the point where they are equal. Maximizing firm value requires a marginal comparison: differentiate the full value function, set the derivative to zero for an interior candidate, check the second derivative, and then test feasibility. This approach is aligned with the SOA VEE Accounting and Finance syllabus.

Key takeaway: In the specified quadratic-cost model, an interior optimum exists when the stationary point $T_c/(2\kappa)$ lies strictly within the feasible range. Strict concavity makes that point a unique maximum; otherwise, the constrained optimum may be at a boundary.

Check Your Understanding

For each problem, use the stated quadratic-cost model and treat all values as hypothetical. Report market-value debt ratios to two decimal places where requested.

A firm has $V_U=\$120$ million, $T_c=0.25$, $\kappa=0.25$, and $0\le r\le1$. Find the optimal $r$, levered value, debt amount, and market-value debt ratio. (Answer: $r^*=0.50$, $V_L=\$127.5$ million, $D=\$60$ million, and $D/V_L=47.06\%$.)
A firm has $V_U=\$200$ million, $T_c=0.18$, $\kappa=0.30$, and $0\le r\le0.40$. Find the optimum and levered value. (Answer: $r^*=0.30$, an interior maximum, and $V_L=\$205.4$ million.)
A firm has $T_c=0.28$, $\kappa=0.14$, and $0\le r\le0.80$. Does an interior optimum exist? Identify the maximizing feasible $r$. (Answer: No; the unconstrained candidate is $1.00$, so the maximum is at the boundary $r=0.80$.)
For $T_c=0.24$ and $r_{\max}=0.60$, what condition on $\kappa$ makes the unconstrained optimum strictly interior? What happens at equality? (Answer: $\kappa>0.20$ gives an interior optimum; at $\kappa=0.20$, the candidate equals the upper boundary $r=0.60$.)
A firm has $V_U=\$300$ million, $T_c=0.32$, $\kappa=0.25$, and $0\le r\le0.80$. Find the optimal intensity, debt amount, levered value, and market-value debt ratio. (Answer: $r^*=0.64$, $D=\$192$ million, $V_L=\$330.72$ million, and $D/V_L=58.06\%$.)

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