Is the Beta Anomaly Real? A Correction in Existing Theories of Cost of Capital and Asset Pricing

Abstract

Researchers argue about the beta anomaly and related anomalies in the capital market based on existing theories of asset pricing. This article shows that the observed beta anomaly is added due to the mathematical errors, inconsistencies, and limitations in existing theories. We propose a general theory for central concepts in asset pricing, including beta and cost of capital, that holds for growth, taxes, and risky debt. Our theory addresses observed beta-related anomalies and other phenomena, and provides a clearer taxonomy for ongoing research and a step toward resolving several issues. The findings are highly significant for researchers and firms.

JEL Codes: G32, G12, G11, G35

Keywords

Beta anomaly capital asset pricing model capital structure cost of capital risky debt tax shield

1. Introduction

Estimating the cost of capital and value of capital structure based on financial theories is a central component in pricing risky assets, making capital structure decisions, and maximizing firm value. Financial theories play a good role in academic research, policy-making, and regulatory decisions. A good asset pricing theory is a necessity for the development of the financial market, economic growth, policy decisions, and future theoretical developments.

Though there are several theories for pricing risky assets and capital structure, however, since many years the researchers and practitioners have been reporting various anomalies and puzzles in the financial market when making calculations based on existing theories. Ross et al. (2019, p. 545) mention that no formula in capital structure theories supports actual practice by the firms. Miller (1977) argued that despite the accepted theories of capital structure that debt increases firm value in presence of taxes and there is an optimum capital structure for a firm, the market trend says that well-managed companies decrease debt. Chauhan (2016) found that firms generally do not chase some target debt ratio, as suggested by some capital structure theories. Various researchers have reported that high-beta stocks are undervalued in the market and low beta stocks are overvalued (Baker et al., 2014; Black et al., 1972). Baker et al. (2014) further mentioned that the asset beta of a firm decreases with increasing debt. Bali et al. (2017) argued that this beta anomaly is “one of the most persistent and widely studied anomalies in empirical research of security returns” (p. 2369). Baker et al. (2020) stated this beta anomaly and related anomalies as evidence of mispricing rather than a misspecified risk model. They argued that “Beta risk is overvalued in equity securities, but not in debt securities. For firms with high-beta assets, this increase is high even at low levels of leverage.… For firms with low-beta assets, this increase remains low until leverage is high” (p. 2).

In addition to the anomalies, the existing basic theories for estimating cost of capital fail to reconcile other models of valuation. Damodaran (2012, pp. 338–342) proposed two approaches for valuing a firm with cash holdings and debt—gross debt and net debt approaches, and two approaches for valuing equity in a firm with cash—consolidated estimation and separate estimation. These models are mathematically equivalent and give the same value for the firm or equity if the corporate tax rate x_c = 0. However, if x_c > 0 then the values calculated by the different approaches differ. Moreover, we get different asset prices under different theories, and these differences raise questions.

While a good theory helps in several ways, an erroneous model creates several problems. It may give erroneous results and fails to explain several market phenomena. Erroneous theories create puzzles/anomalies that may not exist, diverting valuable academic resources to the study of these puzzles, and also infects future theoretical developments that use these theories. New theories and explanations built on previous theories only augmented existing conflicts. No doubt there are market phenomena and puzzles that need explanation and sometimes a new theory, but if a puzzle is created by an erroneous theory or if that puzzle persists even after it is noticed, then the theory may require revisiting.

Regarding existing theories in capital structure and asset pricing, financial theorists have argued for the benefit of debt in the capital structure in the form of tax savings and reduced cost of capital. Over the last few decades, several financial theories have been proposed for pricing risky assets and capital structures, such as the Modigliani and Miller (MM) theory (Modigliani & Miller, 1958, 1963), the capital asset pricing model (CAPM) independently developed by Treynor (1962), Sharpe (1964), Lintner (1965), and Mossin (1966). However, several theories conflict with each other in terms of their assumptions and the valuations they provide for firms. This has led to multiple assumptions and treatments for the same fundamentals. For example, the weighted average cost (WACC) in the constant growth model for the valuation of a firm assumes a constant debt-equity ratio (dollar debt increases with firm growth), but the adjusted present value (APV) model proposed by Myers (1974) assumes constant dollar debt (decreasing debt-equity ratio in growing firms). In some cases, the assumptions made for an input in one part of the theory conflict with the assumptions made about the same input in other parts of the same theory. In the face of multiple theories for the valuation of the tax shield and capital structure, assumptions (including implicit ones) are frequently conflicting and internally inconsistent. Further, every model has limitations in its application due to its assumptions.

In the MM theory, the tax rate on debt income (τ_d) and the rate of growth of income (g) of the firm are zero (implicit assumptions). The Hamada (1962) model also assumes g = 0 and that the cost of debt (R_d) is equal to the risk-free rate (R_f). Both models assume R_d to be the discount rate for tax saving (R_TS) to value the tax shield (V_TS) and calculate V_TS = τ_c D for a firm, where the market value of the debt is D, but for different reasons. The V_TS in the MM theory reduces if τ_d increases, and V_TS in the Hamada model reduces if R_d – R_f increases. The Daves and Ehrhardt (2002) model introduces a non-zero growth rate (g ≠ 0), but argues that R_TS should not be less than the cost of unlevered equity (R_eU). This is to avoid an unacceptable situation in which the levered cost of equity (R_eL) becomes less than R_eU. The Daves and Ehrhardt model, therefore, proposes a different formula for the tax shield. Conine (1980) argued that the tax shield must be τ_c D even if R_d > R_f, and a debt beta (b_d) component should be added to the equation for R_eL in the CAPM. The Hamada model, Conine model, and Daves and Ehrhardt model assume a tax shield (though each model proposes a different formula) even if τ_c = τ_d and so there is no increase in free cash flow to the levered firm (FCFF_L).

Further, there is no general theory that holds for all cases of growth g, τ_d, τ_c, R_d, and R_f. Therefore, the demand for better models to resolve these conflicts is unmet. The errors, inconsistencies, limitations, and conflicts in existing models have created artificial puzzles and unduly affect practitioners, academicians, and researchers. This gap in good theory also poses several challenges for decision-makers in the business and policy-making.

This article contributes to filling this gap. The MM model assumes x_d = 0, g = 0, and R_TS = R_d; the Hamada model implicitly assumes x_d = 0 in the beta and cost of equity equation, but τ_d = τ_c in WACC (internal inconsistency), g = 0, and R_d = R_f; the Daves and Ehrhardt model implicitly assumes x_d = 0 in the beta and cost of equity equation (internal inconsistency), but x_d = x_c in the WACC. Here, we have developed a general model without any assumptions about τ_d, τ_c, g, R_TS, or R_d, and without internal inconsistencies.

This article is organized as follows. Section 2 analyzes the basic theories in corporate structure and asset pricing, and revisits the estimation of inputs provided by these models to the CAPM and other theories to identify the source of conflict in previous theories. In Section 3, we mathematically derive our general formula for the cost of equity and firm valuation using a consistent approach, and determine inputs for the CAPM. In Section 4, we analyze our model and demonstrate that our correction in the theory explains and corrects various anomalies, puzzles, and other observations in the financial market. In this section, we also show that the limitations, errors, and inconsistencies in the previous theories have led or added to such anomalies and puzzles that may not exist. Section 5 provides concluding remarks and future research directions.

2. Basic Theories in Corporate Finance

The MM theory argues that, if the growth rate g = 0, x_d = 0, the cost of debt is r, the market value of debt is D, then the FCFF_L in the firm will increase by rx_cD. If the value of the unlevered firm is V_U, and the value of the levered firm is V_L such that

V_{L} = V_{U} + τ_{c} D

(1)

where the tax shield τ_cD is the present value of the perpetual cash flow rx_cD discounted by the r. With V_TS = x_cD, the MM theory shows that, if the value of the equity in the levered firm is S, then the cost of capital of the levered firm (WACC_L)¹ and R_eL are as follows:

{WACC}_{L} = R_{e U} - (R_{e U} - r) \frac{τ_{c} D}{V_{L}}

(2)

R_{e L} = R_{e U} + (R_{e U} - r) (1 - τ_{c}) \frac{D}{S}

(3)

From Equations (2) and (3), it can be seen that MM uses

WACC = \frac{D}{V} R_{d} + \frac{S}{V} R_{e L}

(4)

Equation (4) shows that x_d = 0 is assumed. Therefore, in the MM theory, r is the post-tax cost of debt. If x_d = x_c, there is no tax saving and hence no tax shield. The MM theory argues that R_TS must be equal to r, and therefore V_TS = x_cD. With the help of two portfolios each made by a $1 investment in the levered and the unlevered firm, as shown in Table 1, the MM model argues that V_TS = x_cD is a necessary condition for no arbitrage.

Table 1.

Arbitrage Portfolio in Levered Firm.

Case	Portfolio A (Equity E and Debt D in Levered Firm)		Portfolio B (Unlevered Equity V_U)		Remarks About No-arbitrage Condition
	Fraction of Investment		Fraction of Investment
	Equity	Debt	Equity	Debt
V_L < V_U + x_cD	$\frac{S}{V_{L} - τ_{c} D}$	$\frac{(1 - τ_{c}) D}{V_{L} - τ_{c} D}$	1	0	Portfolio A gives excess risk-adjusted return.
V_L > V_U + x_cD	1	0	$1 + \frac{(1 - τ_{c}) D}{S}$	$- \frac{(1 - τ_{c}) D}{S}$	Portfolio B gives excess risk-adjusted return.

An important point for subsequent discussion is that Equations (2) and (3) are derived from V_TS = x_cD. Therefore, if we assume Equation (2) for WACC_L or Equation (3) for R_eL in advance, we will invariably get V_TS = x_cD, even if other assumptions of MM theory are ignored (i.e., that x_d = 0, or r is post-tax cost of debt). Some researchers have knowingly or unknowingly fallen into this trap, by either

applying Equation (2) or (3), ex ante, to show V_TS = x_cD,

or assuming either V_TS = x_cD or tax saving = rx_cD, even after assuming x_d = x_c.

We later show that these inconsistencies have led to conflict in the values obtained under these models and even an inability in those models to handle growth rates.

MM theory defines R_eL as a function of R_eU, but does not provide a model to estimate R_eU. This gap was filled by the CAPM, based on the efficient frontier model of Markowitz (1952). According to the CAPM,

E [R_{e L}] = R_{f} + β_{L} \times {E [R_{m}] - R_{f}}

(5)

where

E[Variable] is the expectation operator,

b is the measure of systematic risk in equity security, b_L is the levered beta, and b_U is the unlevered beta,

R_m is the return in a market portfolio and R_f is the risk-free rate.

E[R_m] – R_f is referred to as the market risk premium (R_mp ). The CAPM, however, does not provide a model to estimate b_L from b_U and D/E Subsequently, Hamada (1969) proposed a mathematical model for b_L. The Hamada model defined the cost of equity for levered and unlevered firms as follows:

E (R_{e U}) = R_{f} + λ c o v (R_{e U} ​, R_{m})

(6a)

E (R_{e L}) = R_{f} + λ c o v (R_{e L} ​, R_{m})

(6b)

where m is a parameter. Hamada further defined

E (R_{e U}) = \frac{E [X_{A} (1 - τ_{c})]}{V_{u}}

(7a)

E (R_{e L}) = \frac{E [(X_{A} - R_{f} D) (1 - τ_{c})]}{S}

(7b)

where X_A is the one-period pre-tax income of the unlevered firm. X_A (1 – τ_c) and (X_A – R_fD) (1 – τ_c) are the free cash flow to equity (FCFE), if g = 0. From Equations (6a) and (6b), we see that g = 0, and R_d = R_f are implicit assumptions.

In addition, using Equations (6a) and (6b), Hamada wrote

cov (R_{e U} ​, R_{m}) = \frac{1 - τ_{c}}{V_{U}} cov (X_{A} ​, R_{m})

(8a)

cov (R_{e L} ​, R_{m}) = \frac{1 - τ_{c}}{S} cov (X_{A} ​, R_{m})

(8b)

With the help of Equations (7a) to (7b), where all these equations are ex ante assumed, Hamada showed that

S + D = V_{U} + τ_{c} D

(8c)

V_{L} = V_{U} + τ_{c} D

(8d)

E (R_{e L}) = E (R_{e U}) + \frac{D (1 - τ_{c})}{S} [E (R_{e U}) - R_{f}]

(8e)

Thereafter, using the MM model, the Hamada model provides the following equations:

β_{L} = β_{U} [1 + \frac{D (1 - τ_{c})}{S}]

(9a)

Combining Equations (5) and (9a), we get

E (R_{e L}) = R_{f} + β_{U} [1 + \frac{D (1 - τ_{c})}{S}] [E (R_{m}) - R_{f}]

(9b)

Using E(R_eU) = R_f + b_U[E(R_m) – R_f], we can see that Equation (9b) is the same as

E (R_{e L}) = E (R_{e U}) + \frac{D (1 - τ_{c})}{S} [E (R_{e U}) - R_{f}]

(9c)

which is the same as Equation (8e). Equation (8e) is implicitly built into the assumptions and definitions (Equations (6a)–(7b)) in the Hamada model (see Appendix A(d)). Further, Equation (8e) is similar to Equation (3), and this automatically leads to V_TS = x_cD. Since the Hamada model is derived using MM equations, it is consistent with the MM model (τ_c = 0), if the additional restriction of R_d = R_f holds.

However, the Hamada model deviates from the MM model if x_d > 0 and R_d > R_f. The difference between R_eL in the MM model (Equation (3)) and Hamada model (Equation (9c)) is as follows

R_{e L} (M M) - R_{e L} (Hamada) = [\frac{D (1 - τ_{c})}{S}] [R_{f} - R_{d} (1 - τ_{d})]

If R_d > R_f and x_d > 0, then R_eL(MM) < R_eL(Hamada), and the tax shield in the MM model is still τ_cD, but the tax shield in the Hamada model, using Equation (9b), becomes less than τ_cD. This is because the MM equation (3) measures equity risk premium against R_d, but the Hamada equation (9b) measures equity risk premium against R_f (recall that Equation (9b) is not derived for R_d > R_f). The higher cost of equity in the Hamada equation leads to a lower.

If x_d > 0 and R_d = R_f, then R_eL(MM) < R_eL (Hamada), and the Hamada model deviates from the MM model in the opposite direction. Being a free cash flow to the firm (FCFF) based model, the MM model estimates V_TS < x_cD due to decreased tax savings. However, in the Hamada equation (9b), V_TS is independent of τ_d, and therefore, V_TS = x_cD (recall that Equation (9b) is not derived for x_d > 0).

Thus, an increase in τ_d decreases the value of the tax shield in the MM model, but not in the Hamada model. Similarly, an increase in R_d decreases the value of the tax shield in the Hamada model but does not have an effect in the MM model. The departure of τ_d and R_d from the Hamada assumption impacts V_TS in the opposite direction and reduces the net error. Therefore, the valuation error using the Hamada model is observed more in firms when τ_c ≈ τ_d and R_d ≈ R_f. This is a testable hypothesis.

In addition, Equation (9b) suggests that an increase in τ_c will increase firm value, ceteris paribus, which is counter-intuitive and differs from the market reality.

2.1. Issue of Inconsistency

Further, the inconsistency in the Hamada model becomes clear when we compare Equation (9b) with Equation (3). In Equation (3), the cost of debt (r) is the post-tax cost, but in Equation (9b) this is not the case. If τ_d ≥ τ_c (as assumed in the formula $WACC = \frac{D}{V} R_{d} (1 - τ_{c}) + \frac{S}{V} R_{e L}$ ), then R_f is the pre-tax cost of debt. Therefore, in Equations (6a) and (6b) of the Hamada model, where the left-hand side is the post-tax cost, the right-hand side is a mix of pre-tax and post-tax numbers (if τ_d > 0), which is an inconsistency. In fact, the only source of the tax shield and reduced WACC in the Hamada model with debt is the use of R_f for R_f (1 – τ_d), even though τ_d ≥ τ_c is assumed in WACC. By using R_f (1 – τ_d) in Equations (6a) and (6b), we can show that V_L = V_u (see Appendix A(a)). In addition,

With τ_d ≥ τ_c (i.e., $\frac{D}{V} R_{d} (1 - τ_{d}) + \frac{S}{V} R_{e L} = \frac{D}{V} R_{d} (1 - τ_{c}) + \frac{S}{V} R_{e L}$ ), the formula $WACC = \frac{D}{V} R_{d} (1 - τ_{c}) + \frac{S}{V} R_{e L}$ is correct, but Equations (9a) and (9b) are not correct. With τ_d ≥ τ_c, the levered beta b_L derived in the Hamada equations, after putting τ_d ≥ τ_c in Equations (6a) and (6b), should be independent of τ_c (see Appendix A(b)). Additionally, in such cases the arguments of a U-shaped curve for WACC and optimum D/S ratio will not hold (see Appendix A(c)).

With τ_d ≥ 0, Equations (9a)–(9c) are correct, but WACC = $\frac{D}{V} R_{d} (1 - τ_{c}) + \frac{S}{V} R_{e L}$ is incorrect, because $\frac{D}{V} R_{d} (1 - τ_{d}) + \frac{S}{V} R_{e L} \neq$ $\frac{D}{V} R_{d} (1 - τ_{c}) + \frac{S}{V} R_{e L}$ . In such cases, the WACC in the Hamada model should be similar to Equation (2) in the MM model (see Appendix A(c)). That means, assuming the Hamada equation (9b) to be true, $WACC = \frac{D}{V} R_{d} (1 - τ_{c}) + \frac{S}{V} R_{e L}$ is not consistent.

With τ_d ≠ τ_c, and using the correct WACC formula, WACC = $\frac{D}{V} R_{d} (1 - τ_{d}) + \frac{S}{V} R_{e L}$ , with Equation (9b), the firm value under the WACC model is different from (S + D), that is, $\frac{F F L^{τ}_{1}}{W A C C_{L} - g} \neq (\frac{F E L^{τ}_{1}}{R_{e L} - g} + D)$ . This indicates errors in the Hamada equations. Additionally, the tax shield can sometimes be negative in (V_L < V_u), for example, when τ_d ≈ 0 and R_d – R_f is high.

Importantly, $\frac{F F L^{τ}_{1}}{W A C C_{L} - g} = (\frac{F E L^{τ}_{1}}{R_{e L} - g} + D)$ with Equation (9b), irrespective of τ_d, only if one uses $WACC = \frac{D}{V} R_{d} (1 - τ_{c}) + \frac{S}{V} R_{e L}$ (instead of WACC = $\frac{D}{V} R_{d} (1 - τ_{d}) + \frac{S}{V} R_{e L}$ , used by MM) (see Appendix D). Here, both values are equal, but not necessarily correct. Since, $WACC = \frac{D}{V} R_{d} (1 - τ_{c}) + \frac{S}{V} R_{e L}$ assumes τ_d ≥ τ_c, in such cases, Equation (9b) is incorrect (see point 1 above).

Therefore, with any assumption about τ_d, the Hamada equations for R_eL and beta are not consistent with the WACC formula used in CAPM. Additionally, the Hamada equations are very restrictive in their assumptions (x_d = 0, g = 0, and R_d = R_f), which makes them incompatible for real firms. The tax shield V_TS ≤ τ_cD, and WACC_L < WACC_U observed in the CAPM framework (even though change in FCFF due to debt financing is zero) are only due to using the Hamada equations (8a)–(9b), without correcting these equations for x_d > 0 and ignoring that these equations may not be consistent for real firms and the CAPM.

2.2. Issue of Growth in the Firm

Another issue in the valuation of corporate structure is that the Hamada model and MM theory are zero-growth models. Thus, Daves and Ehrhardt (2002) introduced growth to the MM approach $(V_{T S} = \frac{r D τ_{c}}{r})$ and showed that discounting tax saving rDx_c by the cost of debt $(V_{T S} = \frac{r D τ_{c}}{r - g})$ , as argued in the MM model, can lead to an unacceptable situation of R_eL < R_eU. To avoid this anomaly in the MM model, the model suggests that R_TS should not be less than R_eU, that is, R_TS ≥ R_eU. Using R_TS ≥ R_eU, the Daves and Ehrhardt model proposes the following formulas:

V_{T S} = \frac{r D τ_{c}}{(R_{e U} - g)}

(9a)

R_{e L} = R_{e U} + (R_{e U} - r) \frac{D_{0}}{S}

(9b)

β_{L} = β_{U} + (β_{U} - β_{D}) \frac{D_{0}}{S}

(9c)

Thus, if g > 0 then V_TS > τ_cD. Applying the Hamada model in a growing firm (which is derived under assumptions g = 0, R_d = R_f, and x_d = 0, but used in other situations) can also provide V_TS > Dx_c (when R_d ≈ R_f) irrespective of τ_d. However, Equation (9b) does not allow R_eL < R_eU. The Hamada model does not value the tax saving separately but values total cash flow using R_eL or WACC, and, therefore, the denominator in the value of the firm formula does not come close to 0. Additionally, V_TS =Dx_c in the Hamada model is a conditional outcome and departs from the MM model when conditions differ.

The fault with the Daves and Ehrhardt model is that the model ex ante assumes that the tax saving is rDx_c (i.e., x_d = 0, and r is post-tax interest rate), even when this model uses τ_d ≥ τ_c in the WACC formula $(\frac{D}{V} R_{d} (1 - τ_{c}) + \frac{S}{V} R_{e L}) .$ This is an internally inconsistent approach. In addition, the tax shield in the Daves and Ehrhardt model does not reduce to V_TS =Dx_c for g = 0 and R_TS =r; however, the Daves and Ehrhardt model does not answer the arbitrage argument in the MM model that R_TS must be r.

2.3. Issue of Risky Debt

An additional challenge in the valuation of the corporate structure pertains to incorporating risky debt. The MM model argues that riskiness in debt does not matter. The MM model states that if R_d > R_f, then R_eL will increase with the $\frac{D}{E}$ ratio at a slower rate and may even decrease to keep WACC as per Equation (2). CAPM (with the Hamada equations) incorporates risky debt through the term R_d (1 – τ_c) in the WACC formula and by measuring R_mp against R_f (pre-tax) instead of the post-tax R_d (due to using Equations (9a) and (9b)).

Conine (1980) argued that the tax shield V_TS must be equal to Dx_c, even with risky debt (R_d > R_f) and irrespective of the value of τ_d. Because we get V_TS < Dx_c in the Hamada model when R_d > R_f, Conine suggested adding a term $[- β_{D} (\frac{D (1 - τ_{c})}{S}) R_{m p}]$ to the Hamada equation (8b) for R_eL, to apply this assumption. That is,

E (R_{e L}) = R_{f} + β_{U} [1 + \frac{D}{S} (1 - τ_{c})] E (R_{p m}) - β_{D} [\frac{D}{S} (1 - τ_{c})] E (R_{p m})

(11a)

E (R_{e L}) = E (R_{e U}) + [E (R_{e U}) - R_{f}] [\frac{(1 - τ_{c}) D}{S}] - β_{D} [\frac{D}{S} (1 - τ_{c})] E (R_{p m})

(11b)

where debt beta (b_D) is defined by Conine as

E (R_{d}) = R_{f} + β_{D} E (R_{p m})

(11c)

By combining Equations (11b) and (11c), it can be seen that Conine proposes converting Equation (9b) of the Hamada model (where R_d = R_f) into Equation (3) of the MM model (where R_d ≥ R_f), where x_d = 0 is assumed in both models. However, Conine ex ante assumes Equation (3), irrespective of τ_d, (an inconsistency with the MM model), and then Equation (11a) automatically leads to V_TS =Dx_c. Not only do Conine’s arguments lack theoretical support and are inconsistent with the MM model, but the model borrows inconsistencies from the Hamada model as well, including the failure to include g. Additionally, if x_d = 0, then Equation (11c) is inconsistent as R_d becomes a pre-tax rate.

The Daves and Ehrhardt model incorporates risky debt by using r, instead of R_f, as cost of debt, but uses internally inconsistent assumptions about τ_d (explicitly assumes x_d = x_c in WACC, but implicitly assumes x_d = 0 in increase in FCFF = R_d x_cD₀). Therefore, this model also proposes erroneous formulas that fail to reconcile with previous models.

There are various other conflicts as well; for example, the WACC model assumes constant D/E ratio (dollar debt is increasing with firm growth), but APV model assumes constant dollar debt (decreasing D/E ratio in growth firms).

We argue that the conflicts in the value of the tax shield and the firm are due to inconsistencies in the assumptions in previous models. Further, real firms grow and take on risky debt, and debt holders pay tax on interest income. However, the previous theories or models have not addressed all these inputs together and they do not provide a consistent input to the CAPM framework (Daves and Ehrhardt proposed a general model but, due to inconsistency in the use of τ_d, it does not cover all the inputs). This article revisits the other theories and models, addresses the inconsistencies, and proposes a general theory that holds for growth rate, tax on debt, and riskiness of debt, and provides consistent inputs for the CAPM.

3. Proposed Theory: Cost of Capital in the Presence of Growth and Risky Debt

The following assumptions from previous theories are followed in our model:

If a change in debt does not change the future FCFF and its distribution, then the firm’s value should not change (MM theory).

If τ_c ≥ τ_d, then the FCFF will not change with a change in capital structure.

Post-tax cash flow should be discounted by the post-tax cost of capital (Modigliani & Miller, 1963). We assume R_f as the pre-tax risk-free rate to avoid confusion.

Let us define the expected post-tax operating income $(E [{\bar{X}}^{τ}_{t}])$ and pre-tax operating income $(E [{\bar{X}}_{t}])$ of the unlevered firm in year t as

E [{\bar{X}}_{t}] = E [{\bar{X}}_{0}] {(1 + g)}^{t}

(12)

E [{\bar{X}}^{τ}_{t}] = E [{\bar{X}}_{t}] (1 - τ_{c})

(13)

where the operating income of the firm is growing at a constant rate g. Therefore, the expected free cash flow to the firm (FFU^x_t) will be

E [F F U^{τ}_{t}] = E [{\bar{X}}_{t}] (1 - τ_{c}) (1 - b_{t})

(14)

where b_t is the fundamental reinvestment rate for the firm in year t given by the relationship

b_{t - 1} = \frac{g_{t}}{R O C_{t}}

(15)

where ROC_t is return on capital in year t, defined as

R O C_{t} = \frac{E [{\bar{X}}_{t}] (1 - τ_{c})}{n e t o p e r a t i n g a s s e t a t t - 1}

Assuming the ROC_t as constant for the firm, we have

R O C = \frac{E [{\bar{X}}_{1}] (1 - τ_{c})}{n e t o p e r a t i n g a s s e t a t t = 0}

Now, if the value of the unlevered firm is V_u₀ at year 0, and cost of unlevered equity is R_eU, then, we have

E [R_{e U}] = \frac{E [{\bar{X}}_{1}] (1 - τ_{c}) (1 - b)}{V_{u 0}} + g

(16)

3.1. Levered Firm With Growth

Let us assume that in year 0 the firm raises debt D₀ to introduce a debt ratio X. Owing to the debt, the value of the levered firm becomes V_L₀ and the value of equity become S₀. Further, the expected value of the levered firm E[V_Lt] in the year t is

E [V_{L t}] = V_{L 0} {(1 + g)}^{t}

(17)

With a constant debt ratio X, the market value of the debt and the equity in year t is

D_{t} = D_{0} {(1 + g)}^{t}

(18a)

S_{t} = S_{0} {(1 + g)}^{t}

(18b)

Ω = \frac{D_{t}}{V_{L t}}

(19)

The increase in the value of the firm (DV₁) and the new debt raised (DD₁) in year 1 are

Δ V_{1} = g * V_{L 0}

(20)

Δ D_{1} = g D_{0} = Ω * Δ V_{1}

(21)

The expected post-tax operating income of the levered firm $({\bar{Y}}^{τ}_{t})$ is

E [{\bar{Y}}^{τ}_{t}] = (E [{\bar{X}}_{t}] - R_{d} D) (1 - τ_{c}) + R_{d} D_{t - 1} (1 - τ_{d})

(22)

E [{\bar{Y}}^{τ}_{t}] = E [{\bar{X}}_{t}] (1 - τ_{c}) + R_{d} D_{t - 1} (τ_{c} - τ_{d})

(23)

Therefore, the expected free cash flow to the levered firm (FFL^x_t) is

E [F F L^{τ}_{t}] = E [{\bar{X}}_{t}] (1 - τ_{c}) (1 - b) + R_{d} D_{t - 1} (τ_{c} - τ_{d})

(24)

From Equations (23) and (24), we see that the increase in firm cash flow due to tax saving is $R_{d} D_{t - 1} (τ_{c} - τ_{d})$ and not $R_{d} D_{t - 1} τ_{c}$ . If the discount rate for this tax saving is R_TS then the tax shield V_TS is

V_{T S} = \frac{R_{d} D_{0} (τ_{c} - τ_{d})}{R_{T S} - g}

(25)

Since the free cash flow to equity (FFU^x_t) for the levered firm is

\begin{array}{l} E [F E L^{τ}_{t}] = E [F F L^{τ}_{t}] - p o s t t a x i n t e r e s t p a y m e n t + \\ n e t a d d i t i o n t o t h e d e b t \end{array}

we have

E [F E L^{τ}_{t}] = E [{\bar{X}}_{t}] (1 - τ_{c}) (1 - b) + R_{d} D_{t - 1} (τ_{c} - τ_{d}) - R_{d} D_{t - 1} (1 - τ_{d}) + g D_{t - 1}

(26a)

where gD_t_{– 1} is new debt raised in period t. Equation (26a) also suggests that, for $E [F E L^{τ}_{t}] \leq E [F F L^{τ}_{t}]$ on a continuous basis, we should have $g \leq R_{d} (1 - τ_{d})$ . That is, the terminal growth rate of a firm should have an upper limit $R_{d} (1 - τ_{d})$ . Damodaran (2012) suggested that $g \leq R_{f}$ .

Free cash flow to debt (FCFD_t) in the year t, therefore, is

F C F D_{t} = E [F F L^{τ}_{t}] - E [F E L^{τ}_{t}] = D_{t - 1} [R_{d} (1 - τ_{d}) - g]

(26b)

where the (– gD_t_{– 1})term in FCFD_t is required to maintain a constant D/S ratio in the capital structure in a growing firm. This also ensures that the market value of a portfolio invested in the firm (debt and equity component in the D₀/S₀ ratio) continues to grow at the rate g; that is, the expected cash flow in the period t is $E [F E L^{τ}_{t} + F C F D_{t} + V_{L (t - 1)} (1 + g)]$ .

For constant growth, the cost of equity of the levered firm (R_eL) is

E [R_{e L}] = \frac{E [F E L^{τ}_{1}]}{S_{0}} + g

(27)

E [R_{e L}] = \frac{E [{\bar{X}}_{1}] (1 - τ_{c}) (1 - b) + R_{d} D_{0} (τ_{c} - τ_{d}) - R_{d} D_{0} (1 - τ_{d}) + g D_{0}}{S_{0}} + g

where S₀ is the value of equity in the levered firm such that

V_{L 0} = S_{0} + D_{0}

(28)

Using $E [{\bar{X}}_{1}] (1 - τ_{c}) (1 - b) = (E [R_{e U}] - g) V_{U 0}$ from Equation (16), we can write

E [R_{e L}] = [\frac{(E [R_{e U}] - g) V_{U 0} + R_{d} D_{0} (τ_{c} - τ_{d}) - R_{d} D_{0} (1 - τ_{d}) + g D_{0}}{S_{0}}] + g

(29)

From Equation (29), we can show that (see Appendix B)

E [R_{e L}] = E [R_{e U}] + \frac{D_{0}}{S_{0}} [E [R_{e U}] - R_{d} (1 - τ_{d}) + R_{d} (τ_{c} - τ_{d}) \frac{R_{T S} - E [R_{e U}]}{R_{T S} - g}]

(30)

Equation (30) is a general solution derived with a few assumptions. After dropping the expectation operator (for simplicity), we can write

R_{e L} = R_{e U} + \frac{D_{0}}{S_{0}} [R_{e U} - R_{d} (1 - τ_{d}) + R_{d} (τ_{c} - τ_{d}) \frac{R_{T S} - R_{e U}}{R_{T S} - g}]

(31)

3.2. Inputs for the CAPM

The mathematical equations for beta and other inputs in the CAPM are developed using the definition of equity risk premium from the CAPM (Equation 6). For the consistency, it is required that the risk-free rate used in the cost of equity should be the post-tax risk-free rate. Equation (30) also suggests that cost of debt used in the cost of equity equation should be the post-tax cost. If we denote pre-tax risk-free rate as R_f, we can write the CAPM equations as

E [R_{e L}] - R_{f} (1 - τ_{d}) = β_{L} E [R_{p m}]

(32a)

E [R_{e U}] - R_{f} (1 - τ_{d}) = β_{U} E [R_{p m}]

(32b)

where $E [R_{e U}] - R_{f} (1 - τ_{d})$ and $E [R_{e L}] - R_{f} (1 - τ_{d})$ are the equity risk premium for the unlevered firm (R_pU) and levered firm (R_pL), respectively. Therefore, the market risk premium (R_pm) and unlevered equity risk premium (R_pU) are

E [R_{p m}] = E [R_{m}] - R_{f} (1 - τ_{d})

(33a)

E [R_{p U}] = E [R_{e U}] - R_{f} (1 - τ_{d})

(33b)

Equation (33b) can be written as

E [R_{e U}] = R_{f} (1 - τ_{d}) + E [R_{p U}]

(34)

Putting the value of $E [R_{e U}]$ from Equation (34) into Equation (30), we have

E [R_{e L}] = E [R_{e U}] + \frac{D_{0}}{S_{0}} [E [R_{p U}] - (R_{d} - R_{f}) (1 - τ_{d}) + R_{d} (τ_{c} - τ_{d}) \frac{R_{T S} - E [R_{e U}]}{R_{T S} - g}]

(35)

Equation (35) suggests that the slope of the cost of equity line for a firm with risky debt is less than the slope of the line if the debt is risk-free.

Further, by putting $E [R_{e U}] = R_{f} (1 - τ_{d}) + E [R_{p U}]$ from Equation (34) into Equation (35) (dropping the time 0 subscript for D and S), we get

E [R_{e L}] = R_{f} (1 - τ_{d}) + E [R_{p U}] (1 + \frac{D}{S}) - \frac{D}{S} [(R_{d} - R_{f}) (1 - τ_{d}) - R_{d} (τ_{c} - τ_{d}) \frac{R_{T S} - E [R_{e U}]}{R_{T S} - g}]

(36)

Based on Equation (36), the levered equity risk premium R_pL is

E [R_{p L}] = E [R_{p U}] (1 + \frac{D}{S}) - \frac{D}{S} [(R_{d} - R_{f}) (1 - τ_{d}) - R_{d} (τ_{c} - τ_{d}) \frac{R_{T S} - E [R_{e U}]}{R_{T S} - g}]

(37a)

E [R_{p L}] = E [R_{p U}] + \frac{D}{S} [E [R_{p U}] - (R_{d} - R_{f}) (1 - τ_{d}) + R_{d} (τ_{c} - τ_{d}) \frac{R_{T S} - E [R_{e U}]}{R_{T S} - g}]

(37b)

From the CAPM equations (32a) and (32b), we have

E [R_{p U}] = β_{U} E [R_{p m}]

(38a)

E [R_{p L}] = β_{L} E [R_{p m}]

(38b)

Putting these values from Equations (38a) and (38b) into Equation (37b), we have

\begin{array}{l} β_{L} E [R_{p m}] = β_{U} E [R_{p m}] + \frac{D}{S} [β_{U} E [R_{p m}] - (R_{d} - R_{f}) (1 - τ_{d}) + R_{d} (τ_{c} - τ_{d}) \frac{R_{T S} - E [R_{e U}]}{R_{T S} - g}] \\ β_{L} = β_{U} (1 + \frac{D}{S}) - [\frac{{(R_{d} - R_{f}) (1 - τ_{d}) - R_{d} (τ_{c} - τ_{d}) \frac{R_{T S} - E [R_{e U}]}{R_{T S} - g}} (\frac{D}{S})}{E [R_{p m}]}] \end{array}

(39)

E [R_{e L}] = R_{f} (1 - τ_{d}) + β_{U} (1 + \frac{D}{S}) E [R_{p m}] - (\frac{D}{S}) [(R_{d} - R_{f}) (1 - τ_{d}) - R_{d} (τ_{c} - τ_{d}) \frac{R_{T S} - E [R_{e U}]}{R_{T S} - g}]

(40)

Therefore, an increase in beta in a firm with risky debt depends on the $\frac{D}{S}$ , expected market risk premium (R_pm), and riskiness of the debt (R_d – R_f). Since (R_d – R_f) itself may depend on $\frac{D}{S}$ , the relationship between b_L and $\frac{D}{S}$ is non-linear.

The cost of capital in the levered firm WACC_L is (by dropping the expectation operator)

\begin{array}{l} W A C C_{L} = \frac{{\overset{�}{X}}_{t} (1 - τ_{c}) (1 - b) + R_{d} D_{t - 1} (τ_{c} - τ_{d})}{V_{L}} + g \\ W A C C_{L} = \frac{{\overset{�}{X}}_{t} (1 - τ_{c}) (1 - b) + R_{d} D_{t - 1} (τ_{c} - τ_{d})}{V_{U} + V_{T S}} + g \end{array}

(41)

which can be simplified (Appendix A(e)) as

W A C C_{L} = W A C C_{U} + \frac{D}{V} R_{d} (τ_{c} - τ_{d}) \frac{R_{T S} - R_{e U}}{(R_{T S} - g)}

(42)

Equation (42) can also be obtained by using the equation $W A C C_{L} = \frac{S}{V} R_{e L} +$ $\frac{D}{V} R_{d} (1 - τ_{d})$ .

\begin{array}{l} W A C C_{L} = \frac{S}{V} [R_{e U} + \frac{D}{S} [R_{e U} - R_{d} (1 - τ_{d}) + R_{d} (τ_{c} - τ_{d}) \frac{R_{T S} - R_{e U}}{R_{T S} - g}]] + \frac{D}{V} R_{d} (1 - τ_{d}) \\ W A C C_{L} = W A C C_{U} + \frac{D}{V} R_{d} (τ_{c} - τ_{d}) \frac{R_{T S} - R_{e U}}{(R_{T S} - g)} \end{array}

(43)

Thus, WACC does not reduce with debt if R_TS > R_eU or τ_c > x_d. In addition,

V_{L} = V_{U} + V_{T S} = V_{U} + \frac{R_{d} D_{0} (τ_{c} - τ_{d})}{(R_{T S} - g)}

(44)

These findings are summarized in Table 2 (the expectation operator, and the subscripts for debt and equity are dropped).

The b_U and R_m in Table 2 are the beta and expected market returns, respectively, in the CAPM. The derived formulas in Table 2 reduce to formulas proposed in previous models if we use the implicit and explicit assumption in those models.

Table 2.

Summary of the Proposed General Model With R_T_S ≥ R_eU.

Inputs	Proposed Formula
Risk-free rate	$R_{f} (1 - τ_{d})$
R_pm	$R_{m} - R_{f} (1 - τ_{d})$
R_eU	$R_{f} (1 - τ_{d}) + β_{U} R_{p m}$
R_pU	$β_{U} R_{m p} = R_{e U} - R_{f} (1 - τ_{d})$
R_eL	• $R_{f} (1 - τ_{d}) + β_{L} R_{p U}$ • $R_{e U} + \frac{D}{S} [R_{e U} - R_{d} (1 - τ_{d}) + R_{d} (τ_{c} - τ_{d}) \frac{R_{T S} - R_{e U}}{(R_{T S} - g)}]$ • $R_{f} (1 - τ_{d}) + β_{U} (1 + \frac{D}{S}) R_{p m} - (\frac{D}{S}) [(R_{d} - R_{f}) (1 - τ_{d}) - R_{d} (τ_{c} - τ_{d}) \frac{R_{T S} - R_{e U}}{R_{T S} - g}]$
b_L	$β_{U} (1 + \frac{D}{S}) - [\frac{{(R_{d} - R_{f}) (1 - τ_{d}) - R_{d} (τ_{c} - τ_{d}) \frac{R_{T S} - R_{e U}}{R_{T S} - g}} (\frac{D}{S})}{R_{p m}}]$
WACC_L	$R_{e U} + \frac{D}{V} R_{d} (τ_{c} - τ_{d}) \frac{R_{T S} - R_{e U}}{(R_{T S} - g)}$
V_L	$V_{U} + \frac{R_{d} D_{0} (τ_{c} - τ_{d})}{(R_{T S} - g)}$
V_TS	$\frac{R_{d} D_{0} (τ_{c} - τ_{d})}{(R_{T S} - g)}$

MM model: To obtain the R_eL or WACC equation, substitute τ_d = , g = 0, r = R_d, and R_TS = r.

Hamada model: To obtain the R_eL equation, substitute τ_d = 0, g = 0, R_TS = R_f , and R_d = R_f , even if R_d > R_f . To calculate WACC, use this R_eL, but for the debt component use τ_d =τ_c and for the cost of debt use the actual cost of debt (R_d).

Daves and Ehrhardt model: To obtain the R_eL, substitute R_TS = R_d, τ_d =0, and g = 0. For the WACC, use this R_eL, but use τ_d =τ_c and the actual R_d for the debt component.

To get b_L we can also use $R_{d} (1 - τ_{d}) = R_{f} (1 - τ_{d}) + β_{d} R_{p m}$

APV model: To obtain the V_L equation, substitute τ_d = , g = 0, and R_TS = R_f (ignoring the distress cost).

Nevertheless, we propose to use R_TS = R_eU in the above formulas for the following reasons:

We do not follow the reasoning provided by the MM theory, because:

The MM theory argument that R_TS ≡ R_D fails in firms with high growth rates if R_d is very close to R_f.

The arbitrage argument in the MM model has issues. The MM model derived his arbitrage condition by equating the cash flows in the two portfolios, but the cost of equity in the two portfolios, required to apply the arbitrage argument, is not considered. Of course, cash flow in the two MM portfolios with equal investment only will be equal if V_TS =τ_cD. But this occurs only with the definition of R_eL in Equation (3), and Equation (3) itself is a product of V_TS =τ_cD. This makes it a circular argument.

In fact, for any other V_TS we can show that: the no-arbitrage condition requires that value of tax shield must be equal to V_TS, provided that the R_eL is derived already from such an assumed V_TS. Assuming a portfolio of $1 investment in S_L₀ and D₀, (called a levered portfolio) where the fraction of investment in S_L₀ and D₀ are $\frac{R_{e U} - R_{d} (1 - τ_{d})}{R_{e L} - R_{d} (1 - τ_{d})}$ and $\frac{R_{e L} - R_{e U}}{R_{e L} - R_{d} (1 - τ_{d})}$ , respectively, we can show that

$\frac{R_{e U} - R_{d} (1 - τ_{d})}{R_{e L} - R_{d} (1 - τ_{d})} R_{e L} + \frac{R_{e L} - R_{e U}}{R_{e L} - R_{d} (1 - τ_{d})} [R_{d} (1 - τ_{d})] = R_{e U}$

The total cash flow from this levered portfolio is equal to the total cash flow from same investment in the unlevered portfolio (see Appendix C).

The no-arbitrage argument in the MM model is similar. For the MM model, $\frac{R_{e U} - R_{d} (1 - τ_{d})}{R_{e L} - R_{d} (1 - τ_{d})} = \frac{S_{0}}{V_{L 0} - τ_{c} D_{0}}$ . A similar argument for the no-arbitrage condition also holds for R_TS = R_eU, provided that the R_eL is given by Equation (30).

Therefore, $R_{T S} = R_{d} (1 - τ_{d})$ is not a binding condition for no arbitrage, as argued in the MM model. The R_TS should depend on the cash flow distribution of tax saving for the firm. With a constant D_t/S_t ratio assumption, the volatility in “tax saving” should follow the volatility in S_t. If D_t is assumed as constant dollar and S_t is stochastic, then the D_t/S_t ratio changes with S_t, and R_eL is stochastic. That means, R_TS should be more than R_d.

In our model, R_TS = R_eU is associated with our assumption of constant D_t/S_t.

This article agrees with the reasoning provided by the Daves and Ehrhardt model for growing firms that R_TS ≥ R_eU. Damodaran (2012) also argued that a firm with a higher g should have a higher cost of equity, and this condition is satisfied in Equation (30) only if R_TS ≥ R_eU.

Additionally, since R_eU captures the extra risk premium for the higher risk in the higher growth firm, we propose the use of R_TS = R_eU. This proposal is as the MM model (1958), and Daves and Ehrhardt (2002).

Accordingly, by substituting R_TS = R_eU in Equation (35), the results obtained are presented in Table 3 (expectation operator, and the subscripts for D and S are dropped).

4. Further Discussion

Some of the findings that can be highlighted from the equations presented in Table 3 are as follows.

Table 3.

Summary of the Proposed Model, if R_T_S =R_eU.

Inputs	Proposed Formula
R_eL	• $R_{e U} + \frac{D}{S} [R_{e U} - R_{d} (1 - τ_{d})]$ • $R_{f} (1 - τ_{d}) + β_{U} (1 + \frac{D}{S}) R_{p m} - (\frac{D}{S}) [(R_{d} - R_{f}) (1 - τ_{d})]$
R_pL	$R_{p U} + \frac{D}{S} [R_{p U} - (R_{d} - R_{f}) (1 - τ_{d})]$
b_L	$β_{U} (1 + \frac{D}{S}) - [\frac{(R_{d} - R_{f}) (1 - τ_{d}) (\frac{D}{S})}{R_{p m}}]$
b_U	$\frac{[β_{L} + \frac{(R_{d} - R_{f}) (1 - τ_{d}) (\frac{D}{S})}{R_{p m}}]}{(1 + \frac{D}{S})}$
WACC_L	$R_{e U} = W A C C_{U}$
V_L	• $V_{U} + \frac{R_{d} D_{0} (τ_{c} - τ_{d})}{(R_{e U} - g)}$ • $V_{U} i f τ_{d} = τ_{c}$
V_TS	• $\frac{R_{d} D_{0} (τ_{c} - τ_{d})}{(R_{e U} - g)}$ • $0 i f τ_{d} = τ_{c}$

By combining the MM equation (3) with CAPM equations (32a) and (32b), one can derive the following equations for b_L and R_eL (Appendix A(f)):

R_{e L} = R_{f} + β_{U} (1 + (1 - τ_{c}) \frac{D}{S}) R_{p m} - (R_{d} - R_{f}) (1 - τ_{c}) \frac{D}{S}

and

β_{L} = β_{U} (1 + \frac{D (1 - τ_{c})}{S}) - [\frac{(R_{d} - R_{f}) (1 - τ_{c}) (\frac{D}{S})}{R_{p m}}]

These equations have similar structures to our equations in Table 3, providing support to our formulations (the few differences are due to the conditions τ_d =0 and R_TS = R_d in the MM model).

This derived b_L (the MM model) further reduces to $β_{L} = β_{U} (1 + \frac{D (1 - τ_{c})}{S})$ for R_d = R_f, which is the Hamada equation. This again shows that the b_L in the Hamada model is true only if τ_d =0 and R_d = R_f.

Our equation for R_eL has a negative term $- (\frac{D}{S}) [(R_{d} - R_{f}) (1 - τ_{d})]$ for the riskiness of debt. One may relate it with the MM model as the MM equation (3) can be written as

R_{e L} = R_{e U} + (R_{e U} - R_{f}) (1 - τ_{c}) \frac{D}{S} - (R_{d} - R_{f}) (1 - τ_{c}) \frac{D}{S}

This also has a negative term, $- (R_{d} - R_{f}) (1 - τ_{c}) \frac{D}{S}$ , when the risk premium is measured against the risk-free rate, as in our equation for R_eL. This provides support to our model (the differences are due to the conditions τ_d =0 and R_TS = R_d in the MM model). One can also argue this as support to the Conine model (which includes a premium $- (\frac{D}{S}) β_{D} E (R_{p m})$ in R_eL for debt beta b_D, where b_D is defined by $R_{d} (1 - τ_{d}) = R_{f} (1 - τ_{d}) + β_{D} E (R_{p m})) .$

However, we offer a different explanation for the term $- (\frac{D}{S}) [(R_{d} - R_{f}) (1 - τ_{d})]$ . In the MM model, the equity risk premium is measured against R_d ( 1 – τ_d) (actual cost of debt, not the risk-free rate). The equity risk premium in the MM model is the additional premium over and above the debt risk premium because a decrease in debt risk premium is in lieu of business risk transferred from equity holders to debt holders as per the MM model (this is also intuitive). Therefore, the debt risk premium and equity risk premium in the MM model do not overlap. However, in our formulations, we have used R_f ( 1 – τ_d) as a common basis to define all the risk premiums (to make debt risk premiums and equity risk premiums across firms with different costs of debt comparable). Therefore, the risk premium (R_d – R_f)( 1 – τ_d) in our model (or, in any model that uses R_f as its base) is counted in the cost of debt, as well as in the equity risk premium. This double counting must be removed. Therefore, the term $- (\frac{D}{S}) [(R_{d} - R_{f}) (1 - τ_{d})]$ is an adjustment to eliminate the double counting in the cost of equity and WACC for debt-equity ratio $\frac{D}{S}$ .

The Hamada model has the double-counting problem. In the case of high beta firms, this double counting partially neutralizes the underestimation of the cost of equity. In contrast, in low beta firms, the double counting of the debt risk premium adds to the already overestimated cost of equity, leading to undervaluation of low beta firms.

Campbell (2000) explained that levered beta (b_L) in a distressed firm increases when the market risk premium increases. Our model explains this phenomenon within the CAPM framework; in addition, it shows how beta fits into the cost of equity.

Baker et al. (2020) and various other researchers mentioned that in a firm with low-risk debt, the beta model overvalues high beta firms and undervalues low beta firms but it has no effect on debt securities. Comparing the equation $R_{e L} = R_{e U} + \frac{D}{S} [R_{e U} - R_{d} (1 - τ_{d})]$ in our model with the Hamada equation $R_{e L} = R_{e U} + \frac{D (1 - τ_{c})}{S} [R_{e U} - R_{f}],$ we can see that for R_d ≈ R_f, the cost of equity measured in the Hamada model is higher for high beta firms and lower for low beta firms compared to our model, and there is no difference in the case of zero beta firms.

Our model shows that the problem of beta anomalies in valuation (overvaluation in high beta firms that increases with increasing D/S ratio, and the undervaluation in low beta firm that reduces with increasing D/S ratio) is a consequence of using the Hamada model. In a time of low risk aversion, the problems increase further. The Hamada model can erroneously justify and motivate levered buyouts of low beta firms with low debt. This also reflects problems in valuing debt securities with CAPM using the Hamada equations.

The WACC remains constant if R_TS = R_eU but never decreases due to debt. The WACC increases if b_u or τ_d increases because R_eU increases with these inputs. The argument of optimum capital structure is incorrect. In the MM model, the WACC reduces because of assumption R_TS < R_eU, and in the Hamada model, the WACC decreases because of the internally inconsistent assumptions about τ_d. Baker et al. (2020) argued that the asset beta of a firm reduces with increasing debt. This happens because, in the Hamada model, the WACC reduces with debt (U-shaped WACC curve). Since the WACC does not reduce with debt in our model, so the asset beta of firm will also reduce.

The relationship between b_L with $\frac{D}{S}$ is non-linear because (R_d – R_f) may depend on $\frac{D}{S}$ . Additionally, the beta increases at a slower rate because some portion of the business risk is assumed by risky debt owners.

The corporate tax rate τ_c does not play a role in b_L (for R_TS = R_eU). A similar finding is provided by the Daves and Ehrhardt model, but due to internal inconsistencies, the model overestimates in the presence of risky debt (and because this model overestimates b_L, is also overestimated DFCFF, V_TS). Further, we get a similar finding in the Hamada model if we use R_f ( 1 – τ_c) in the Hamada equations (6a) and (6b).

The growth rate g does not appear in any components of the cost of capital (for R_TS = R_eU). The difficulty faced in previous models in incorporating growth rate was mainly due to the inconsistency in assumptions in those models. Since the equation for beta and cost of capital is independent of g, the formula holds even if g changes. An uneven projected growth rate can be considered as a constant growth rate in the same manner as bond yield is a substitute for an upward/downward sloping yield curve.

The slope of the equation for R_eL can be negative if R_d ( 1 – τ_d) > R_eU (highly distressed firms); that is, the cost of equity may decrease with increasing $\frac{D}{S}$ ratio after a certain level of $\frac{D}{S}$ . A similar argument is provided by the MM model.

5. Concluding Remarks, Policy Implications, and Future Research Direction

We propose a general theory for the cost of capital that also provides inputs in the CAPM framework, and holds for tax rates on debt, growth, and riskiness of debt, and hence applies to real-life firms. Our model corrects the inconsistencies found in previous theories and the associated valuation errors. Our theory is internally consistent and contributes to the integration of financial theories in corporate finance and asset pricing. It also resolves problems of different valuations of same firm in mathematically equivalent theories, such as consolidation valuation versus separate estimation approaches, and the gross debt approach versus net debt approach, as proposed by Damodaran (2012).

Our theory also shows that several financial market puzzles and anomalies are produced by the errors in existing theories. Our theory corrects observed puzzles created by errors in previous theories. Our proposition to use the post-tax risk-free rate as the measure for the risk-free rate also has implications for other fields of finance, such as pricing of derivative products and related anomalies. The theory answers various questions we face while applying the existing capital structure and asset pricing theories.

The article shows that the reduction in the WACC due to debt and the argument of optimum capital structure are not true. We argue that debt can add value to the firm through increasing the investment and growth rate and not by changing the capital structure per se. A firm can increase value by attracting investors who have tax advantages or exemptions on interest income. The findings also suggest that a firm with a high credit rating and τ_c ≈ τ_d, with moderate growth opportunity, may not find debt valuable.

The proposed theory has several policy implications. Our correction can make significant differences in project evaluations and new investments where erroneous valuations can lead to costly errors. Considering that most new ventures and the technology industry are high-beta projects, anomalies in existing theory caused negative impacts on project selection, productivity of capital, and economic development. Our theory corrects this problem and will have significant impact in future. The findings in this article will also have valuable implication for portfolio management, merger and acquisitions.

Our findings can initiate fresh research on financial market issues such as capital structure, beta anomaly, risk premium puzzle, and tax shield with a new perspective. The findings in this article may contribute to resolving various issues in asset pricing. By integrating previous theories, our theory will help advance theories in the future. A mathematically correct and consistent model will also contribute to the development of better algorithm-based trading or financial technology.

Supplemental Material

Supplemental material for this article is available online.

Footnotes

Declaration of Conflicting Interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author have received no financial support for the research, authorship, and/or publication of this article.

Note

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