^{1}

^{*}

^{2}

We define debt ratio as the market value of a firm’s debt divided by the market value of the firm. In a perfect market with corporate taxes, given that the cost of debt is increasing and concave up and that the firm rebalances its debt, the cost of equity is an increasing and concave up function of the debt ratio if and only if the third derivative of the cost of debt is non-negative; otherwise, the cost of equity is increasing but its exact shape cannot be ascertained. In all cases, however, the cost of equity must be concave up initially. Also in this world, the weighted average cost of capital of the firm, WACC, is decreasing and concave down. In an imperfect market, the WACC may not have an absolute minimum between zero and 100 percent debt. Even if it does, the minimum may not occur at the debt ratio that maximizes firm value. The “pure-play” method to determine a new project’s discount rate is correct only if the opportunity cost of capital of the comparable firm remains constant with respect to the debt ratio or if the debt ratio of the comparable firm is equal to the target debt ratio of the firm evaluating the project. Strictly speaking, even if the two debt ratios are the same, the opportunity cost of capital of the comparable firm is not necessarily equal to that of the project unless the two costs of capital are identical functions of the debt ratio. Therefore, this method may not be valid.

[

Later authors examine how market imperfections such as bankruptcy costs and agency costs affect firm value and the choice between debt and equity. When bankruptcy imposes significant monetary penalties on the firm, corporate financing decisions become a trade-off between interest tax savings and bankruptcy costs, resulting in an optimal capital structure somewhere between zero and 100% debt. These results can be seen in [

[

Combining the considerations on interest tax savings, financial distress costs, and agency costs emerges the trade-off capital structure theory, in which the value of the firm is maximized when the optimal trade-off between the savings and the costs is reached. Less well explored, however, is how the inclusion of these costs affects the firm’s costs of capital. Nevertheless, the general view is that the cost of debt is an increasing and concave up function of the debt ratio, the cost of equity is an increasing function of the debt ratio, and that there exists a unique capital structure between zero and 100% debt that simultaneously maximizes the firm value and minimizes the weighted average cost of capital. However, no explanation can be found in the literature on why the weighted average cost of capital should simultaneously reach the minimum. Most likely, researchers simply assume that the firm’s expected cash flow stream is invariant to changes in the debt ratio. With this assumption, the present value of the expected cash flow stream is maximized when the discount rate, namely, the weighted average cost of capital, is at its lowest. However, is this assumption valid when market imperfections such as financial distress and agency costs exist?

As mentioned above, the cost of debt, cost of equity, and weighted average cost of capital are expressed as functions of the debt ratio in general, but the debt ratio is defined either as the debt to firm value ratio or the debt to equity value ratio. This inconsistency causes unnecessary confusion. In this paper, we use a single definition of debt ratio to reconstitute capital structure and cost of capital theories. In the process, we clarify the relationships among capital structure, costs of capital, and firm value in perfect and imperfect markets. In a perfect market, if the debt is risky and the firm rebalances its debt, the cost of equity is an increasing and concave up function of the debt ratio if and only if the third derivative of the cost of debt is non-negative; otherwise, the cost of equity is increasing but its exact shape cannot be ascertained. In all cases, however, the cost of equity must be concave up initially. Also in this world, the weighted average cost of capital is decreasing and concave down. In an imperfect market, the weighted average cost of capital may not have an absolute minimum between zero and 100% debt. Even if such minimum exists, the maximum firm value and the minimum weighted average cost of capital may not occur at the same debt ratio.

In the next section, we review the literature, develop our arguments, and provide the clarifications. Based on the findings in this section, we show in Section 3 that the validity of the popular “pure-play” method to estimate a project’s cost of capital is uncertain. Concluding remarks are provided in Section 4.

In the Modigliani and Miller theorems (1958, 1963), the cost of debt is equal to the risk-free rate R_{f}, and the cost of equity K_{s} is equal to

ρ = the discount rate for an all-equity firm;

B = the market value of debt;

S = the market value of equity;

T = the corporate income tax rate.

Let V be the market value of the firm, thus

Substituting _{S} in the above equation and simplifying the result, we obtain

The firm’s opportunity cost of capital, K_{a}, is equal to

Again, substituting _{S} and simplifying the result, we obtain

If

Note that both K_{a} and WACC are functions of_{s} is a function of

market value of the firm when it is levered, V_{L}, comes from the value of operations and the value of interest tax shields:

NOI is the expected net operating income, V_{U} is the value of the firm when it is unlevered and TB is the value of interest tax shields. If T = 0, V_{L} = V_{U}.2 For the purposes of this paper, we change the notations for K_{S}, WACC, and K_{a} and express all of them as

functions of the debt ratio

s(x) in Equation (2) is thus defined on the interval [0, 1). From Equations (3) and (4), if_{U} remains the same, TB becomes a larger proportion of V_{L}. Since TB is risk- free, it lowers the firm risk and thus r(x). Equation (3) shows that w(x) is also a linear function; however, it has a more negative slope than r(x) since

and

Therefore, s(x) is an increasing and concave up function of x. In summary, if debt is risk-free, there is no debt rebalancing, and

Let f(x) denote the cost of debt. It is generally accepted that for risky debt, f(x) is an increasing and concave up function of x; that is,

we can express s(x) from the above equation as follows:

What is the behavior of s(x)? It is commonly believed that the cost of equity is an increasing function of the debt ratio. [

cially if the debt ratio is defined as x instead of

and concave up and r(x) is constant, s(x) is strictly increasing. The proof is given in the following.

Proposition 1:

Suppose f(x) is twice continuously differentiable on [0, 1], increasing, and concave up. Let

Proof:

We first note that

To see that

on

Since

Introducing the notation

that

First note that

Next, we claim that

By the Mean Value Theorem for derivatives, there exists a d with

However, using the fact that

which is a clear contradiction, showing that no c with

If the third derivative of f(x),

Proposition 2:

Suppose

Proof:

It will be sufficient to show that

and since

This last inequality is equivalent to

On the other hand, since

for some

Since by assumption

where in the last inequality we used the facts that

Thus we can conclude that

and since

Proposition 3:

Suppose f(x) is twice continuously differentiable on

Proof:

Using the formula for

It is now sufficient to see that the above expression is positive. Thus we aim to prove that

By the Mean Value Theorem for derivatives, there exists

and since

as needed.

Example 1 in Appendix shows the possibility of an increasing and eventually concave down s(x).6 In this example, f(0) = 0.02 and s(0) = 0.07. The reasons to use these numbers are given later in this Section. What is the behavior of w(x)?

Based on this equation,

and

Therefore, w(x) is a decreasing and concave down function of x. In summary, if debt is risky,

If [

Again, the market value of the firm when it is levered, V_{L}, comes from the value of operations and the value of interest tax shields:

In this equation, V_{TS} is the value of the tax shields. Since ρ is greater than f(x),7 V_{TS} is less than TB in Equation (1).8 Also,

Equation (6) is commonly known as the WACC method. Equations (5) and (6) are in fact equivalent. Multiplying both sides of Equation (6) by

Substituting

Equation (5). We will assess firm value in an imperfect market based on Equation (6) in the following Section; therefore, this proof is necessary to ensure the continuity of the theory. Moreover, this proof implies that the WACC method requires constant debt rebalancing.

In an imperfect market, the trade-off capital structure theory posits that the value of the firm is a trade-off among the interest tax shields, agency costs of equity, financial distress costs, and agency costs of debt. The optimal trade-off is reached at a certain debt ratio, denoted by x^{*}, between zero and 100%, at which the value of the firm is maximized. Based on this theory, researchers posit that corresponding to the rise and fall of the firm value, w(x) would first decrease, reach a minimum at x^{*}, and then rise. In other words, the firm value would reach the peak and w(x) would reach the minimum at x^{*}; therefore, x^{*} is the optimal debt ratio. For convenience, let’s call this result the “maximum-minimum proposition.” The relationships among s(x), f(x), and w(x) can be depicted in the following

Note that s(0) is equal to f(1); meaning that the cost of debt when the debt ratio is 100% equals the cost of unlevered equity. There is no discussion in the literature of whether f(1) should be equal to, higher than, or lower than s(0).

Furthermore, no justification can be found in the literature for the “maximum- minimum proposition”. In other words, no clear explanation has been offered for why w(x) should be U-shaped with an absolute minimum between zero and 100% debt and why the maximum firm value and the minimum of w(x) should occur at the same debt ratio. Most likely this proposition is based on the assumption that the firm’s expected cash flow stream is invariant to changes in debt ratio; therefore, the value of the firm is maximized when the discount rate, w(x), is minimized. However, if this result stems

from the trade-off capital structure theory, can we make such an assumption? [

According to the trade-off theory, the financial distress costs and the agency costs of debt are expected to increase with debt ratio; while the agency costs of equity are expected to decrease with the debt ratio. Thus, if market imperfections exist, NOI(1 − T) in Equation (6) may vary across debt ratios, depending on the likelihood for these costs to occur and the magnitude of these costs.9 In short, we may not hold expected cash flow stream constant across debt ratios.

In Equation (6), w(x) should reflect systematic risks. According to the literature, financial distress and debt-related agency problems are more likely to occur during economic downturns.1^{0} Therefore, the costs incurred by these problems may present systematic risks, and r(x) and w(x) should reflect these risks. Again according to the trade-off theory, these problems become more severe with the increase of debt ratio; therefore, the systematic risks reflected in r(x) and w(x) should also increase. In addition, according to the literature, financial distress and debt-related agency problems give management the incentive to change the firm’s assets. For example, financial distress costs include the loss incurred by fire-selling assets, and debt-related agency problems include switching to more risky assets and “cashing-out.”1^{1} Changing the assets is likely to change the asset risk (and thus the systematic risk). Again, since a higher debt ratio increases the probability for financial distress and debt-related agency problems to occur, the resulting impact on asset risk should also be reflected in r(x) and w(x). Based on these arguments, r(x) and w(x) may not have the same functions as depicted in

To use Equation (6), we need to determine how the financial distress and agency problems affect both the numerator and denominator of the equation for a given debt ratio. This appears to be an impossible task, given the current state of literature. Therefore, we can’t say for sure how V_{L} and w(x) behave as functions of the debt ratio in an imperfect market. Taking the first derivative of w(x) with respect to x,

Here, in addition to the assumptions made about f(x), we assume that s(x) is continuously differentiable and increasing on^{2} w(x) would even increase initially (that is, in a neighborhood of x = 0) if

Given T, f(0), and s(0), the value of ^{3} Since this firm is average-risk, let’s use 7% as s(0). For the corporate tax rate T, let’s use the average of 35%. Based on these numbers,

The behavior of r(x) in an imperfect market has not been examined in the literature. Both r(x) and w(x) reflect the firm’s asset risk; therefore, both must have a certain relationship. In the following, we prove that the first derivative of r(x) is greater than that of w(x).

Proposition 4:

Suppose

^{4}

Proof:

First observe that

and thus

since

Based on Proposition 4, when w(x) is decreasing, r(x) could still be rising or could be also decreasing but at a slower rate, and when w(x) is rising, r(x) must also be rising but at a faster rate. Thus, r(x) cannot be a constant if w(x) is U-shaped, implying a changing asset risk across the debt ratios.1^{5}

The proponents of the “maximum-minimum proposition” would need to explain why expected cash flow stream can be assumed constant across debt ratios in the world of the trade-off capital structure theory, if this is an assumption for the proposition. They would also need to explain how and why the asset risk is perceived changing across debt ratios in the way implied by the shape of r(x) discussed in the last paragraph. Also, as mentioned earlier, financial distress costs and agency costs involve possible changes in the firm’s assets and their occurrences are functions of the debt ratio; therefore, there is no guarantee that s(0) = f(1) = ρ. f(1) may be higher than, equal to, or lower than s(0). This potentially adds complications to the shapes of r(x) and w(x). In Appendix, Examples 2, 3, and 4 show possible shapes of r(x) and w(x) when f(1) = s(0), f(1) > s(0) and f(1) < s(0) respectively. In each example w(x) does not have an absolute minimum between zero and 100% debt; also, since the slope of s(x) is greater than 0.057, w(x) is initially increasing. Although s(x) and f(x) in each example are specified arbitrarily, they satisfy the conditions that

If expected cash flows may not be held constant across debt ratios and if the function of w(x) cannot be determined, what can we say about a firm’s optimal capital structure in an imperfect world? In a nutshell, the optimal capital structure, if exists, can be anywhere from zero to 100% debt depending on the expected cash flows and w(x) at each debt ratio. w(x) may not be U-shaped as conventionally believed. Even if w(x) has an absolute minimum between zero and 100% debt, there is no guarantee that the debt ratio that minimizes w(x) also maximizes the value of the firm. This result does not prescribe a specific shape of w(x) and hence r(x), and does not specify whether f(1) is equal to, higher than, or lower than s(0).

Based on the aforementioned analysis, the firm’s opportunity cost of capital r(x) may not be a constant across the debt ratio in an imperfect world. A non-constant r(x) poses a major difficulty in evaluating a new project. It is necessary to assess a project’s risk to determine the project’s opportunity cost of capital r(x)_{j} and hence the project’s weighted average cost of capital w(x)_{j} to calculate the net present value. Since the risk of a new project is usually difficult to determine, the “pure play” method is often used. In this method, r(x) of a comparable firm whose entire business is similar to the project is calculated based on the firm’s f(x), s(x), and x.1^{6} Since the comparable firm and the project are of the same business, r(x) is used as r(x)_{j}.

In this equation, x_{j} is the debt ratio for the project. Based on current literature, it should be equal to the target debt ratio of the firm that is considering the project (henceforth “the project firm”). f(x)_{j} is the cost of debt and s(x)_{j} is the cost of equity for the project; both are determined by x_{j} and by the project’s risk. The debt ratio for the comparable firm may not be the same as the target debt ratio for the project firm; that is, x may not be equal to x_{j}. If they are not equal, f(x) is not equal to f(x)_{j} and s(x) is not equal to s(x)_{j}, although the project risk is deemed equal to that of the comparable firm. A typical procedure is to first estimate f(x)_{j} based on the project’s risk and x_{j} and then solve for s(x)_{j} based on the now known r(x)_{j}, f(x)_{j}, and x_{j}. The project’s weighted average cost of capital, w(x)_{j}, can then be calculated:

However, this method is valid only if the opportunity cost of capital is constant with respect to debt ratio or if x happens to be equal to x_{j}. If the opportunity cost of capital is not constant, r(x) is not necessarily equal to r(x)_{j} unless x is equal to x_{j}. Strictly speaking, even if the two debt ratios are equal, r(x) is not necessarily equal to r(x)_{j} unless r(x) and r(x)_{j} are identical functions of the debt ratio. We can’t say for sure that they are. If r(x) cannot be used to proxy r(x)_{j}, the “pure-play” method breaks down because the cost of capital for the project cannot be determined.

Theories in capital structure and costs of capital are important because they lay the foundation for making financing and investment decisions in a firm. However, inconsistent and questionable arguments exist concerning the relationships among capital structure, costs of capital, and firm value. We critically re-examine these relationships and provide several clarifications.

We define the debt ratio as the market value of the debt divided by the market value of the firm, and express the cost of debt, cost of equity, the firm’s opportunity cost of capital, and the weighted average cost of capital as functions of this debt ratio. In a perfect market with corporate taxes, given that the cost of debt is increasing and concave up and that the firm rebalances its debt, the cost of equity is increasing and concave up if and only if the third derivative of the cost of debt is non-negative; otherwise, the cost of equity is increasing but its exact shape cannot be ascertained. In all cases, however, the cost of equity must be concave up initially. Also in this world, the weighted average cost of capital is decreasing and concave down.

In an imperfect market, the optimal capital structure can be anywhere from zero to 100% debt, depending on the expected cash flows and the weighted average cost of capital at each debt ratio. We can’t say for sure that the weighted average cost of capital has an absolute minimum between zero and 100% debt. Even if such minimum exists, the debt ratio that results in this minimum may not be the one that maximizes the firm value. These results are contrary to the conventional view that there exists a unique debt ratio between zero and 100% that simultaneously minimizes the weighted average cost of capital and maximizes the firm value. The proponents of this view would need to explain how the expected cash flow stream can be assumed constant across debt ratios in the context of the trade-off theory, if this is an assumption. We prove that if the weighted average cost of capital is U-shaped, the firm opportunity cost of capital cannot be a constant across debt ratios. Since the opportunity cost of capital reflects asset risk, the proponents would also need to explain how and why the asset risk is perceived changing across debt ratios in the way implied by the shape of the opportunity cost of capital.

The “pure-play” method to determine a new project’s discount rate is correct only if the opportunity cost of capital remains constant with respect to debt ratio or if the debt ratio of the comparable firm is equal to the target debt ratio of the firm evaluating the project. Strictly speaking, even if the two debt ratios are the same, the opportunity cost of capital for the comparable firm is not necessarily equal to that for the project unless the two costs of capital are identical functions of the debt ratio. Therefore, this method may not be valid.

Hsieh, C. and Szarvas, T. (2016) Critical Re-Examinations on the Relationships among Capital Structure, Costs of Capital, and Firm Value. Journal of Mathematical Finance, 6, 792- 809. http://dx.doi.org/10.4236/jmf.2016.65055

Example 1. This example shows that if

Example 2. Possible shapes of r(x) and w(x) when

Example 3. Possible shapes of r(x) and w(x) when

Example 4. Possible shapes of r(x) and w(x) when