the hamada equation reconsidered

THE HAMADA EQUATION RECONSIDERED JAMAL MUNSHI ABSTRACT: A relationship between levered β and unlevered β that is consistent with the Miller-Modigliani

Theorem (MM) is presented as a Modified Hamada Equation (MHE). It differs from the Original Hamada Equation (OHE). A scenario analysis reveals that the OHE results are anomalous when the firm's cost of debt is not equal to the risk-free rate and when its operating returns are different from market returns. The MHE is proposed as a more stable and reliable alternative1.

1. INTRODUCTION An unsettled question in finance is the existence of an optimal capital structure (Myers, 1989) at which the cost of capital is minimized and the value of the firm is maximized ceteris paribus simply by changing the ratio of debt to equity financing (Wikipedia, 2014). The Miller-Modigliani Irrelevance Theorem, or MM (Modigliani, 1958), proposes that the value of the firm depends on operating parameters only and therefore cannot be changed by financing decisions (Wikipedia, 2014) but this result is thought by many to be an artifact of the assumptions or of an under-specified model. Variables such as agency cost (M. Jensen, 1976), bankruptcy cost (Wikipedia, 2013), and tax shields (DeAngelo and Masulis 1980)(Kane, Marcus, and Alan 1984)2 have been proposed to explain the assumed existence of an optimal capital structure. The relationship between capital structure and the cost of capital was further developed by Hamada (Hamada, 1972) who combined the capital asset pricing model or CAPM (Sharp, 1964) (Wikipedia, 2014) and the MM to derive an equation that established a relationship between leverage and the β market risk of the firm (Wikipedia, 2013). Although the Hamada equation became popular with analysts, many deficiencies in it have been noted and remedies proposed (Conine, Divisional cost of capital estimation, 1985) (Conine, 1980) (Cohen R. D., 2007). Issues raised include the reliance on the risk-free rate of return as the cost of debt to corporations, whether it is consistent with MM, and with regard to anomalous results under certain conditions (Gonzales-Litzenberger-Rolfo, 1977). New parameters have been proposed to address these deficiencies (Conine, Debt capacity and the capital budgeting decision, 1980). In this paper we take a new approach to a critical evaluation the Hamada equation. Instead of using it as the starting point and then proposing modifications that would make it behave the way we think it should, we start with basic financial principles and derive an equation relating levered beta to unlevered beta anew to arrive at what we call the Modified Hamada Equation (MHE) and compare it with the Original Hamada Equation (OHE).

1

Date: Presented at the American Finance Association annual meeting 1993, Revised March, 2014. Some typographical errors were corrected on March 23, 2014. Key words and phrases: Hamada, levered beta, financial leverage, WACC, CAPM, capital asset pricing model, MillerModigliani, capital structure, optimal capital structure, cost of capital, effect of debt on beta, numerical methods Author affiliation: Professor Emeritus, Sonoma State University, Rohnert Park, CA, 94928, [email protected] 2 See (Teall, 2014) for a summary.

THE HAMADA EQUATION RECONSIDERED, JAMAL MUNSHI, 2014

2

2. THEORY

2.1` Notation. Consider a firm whose assets are financed only with equity and debt and whose relevant accounting and financial market parameters are defined as follows: Total assets Assets financed with debt Net operating income (EBIT) The risk free rate of return Ratio of interest on debt to the risk free rate Market risk premium The keep rate = 1-tax rate Rate of return to common shareholders for the unlevered firm Rate of return to common shareholders for the levered firm Market beta risk of the unlevered firm Market beta risk of the levered firm

1 δ ρ φ α μ τ κU κL βU βL

We can now write the following relationships using accounting and CAPM principles Equity financing Debt/Equity ratio Interest rate on debt CAPM rate of return for the unlevered firm CAPM rate of return of the levered firm Net operating income Net income after taxes Accounting returns to shareholders of the unlevered firm Accounting returns to shareholders of the levered firm Multiply through by τ

1- δ δ/(1- δ) αφ κU = μβU + φ κL = μβL + φ EBIT = NOI = ROI = ρ NIAT = τ (ρ - αфδ) κU = τρ κL = τ (ρ - αφδ)/( 1- δ) κL = (τρ - ταφδ)/( 1- δ)

2.2 The relationship between βL and βU . The relationship between levered β and unlevered β is developed by equating accounting returns to CAPM returns as shown below. Equate CAPM returns to accounting returns for the unlevered firm: ρτ = μβU + φ Solving for βU βU = (ρτ - φ)/ μ To solve for βL we write ρτ in in terms of βU in the accounting returns for the levered firm: κL = (μβU + φ - ταφδ)/( 1- δ) Equate CAPM returns and accounting returns for the levered firm:


μβL + φ = (μβU + φ - ταφδ)/( 1- δ) Subtract φ from both sides of the equation: μβL = (μβU + φ - ταφδ)/( 1- δ) - φ Divide by μ βL = (βU + φ/μ - ταφδ /μ)/( 1- δ) - φ/μ Define θ as the ratio of the risk free rate over the market premium. θ = φ/μ Rewrite the equation for βL = f(βU) in terms of θ: Equation 1

MHE

βL = (βU + θ - ταδθ)/( 1- δ) - θ

This is our Modified Hamada Equation (MHE). It differs from the Original Hamada Equation (OHE) which may be represented using our notation as: Equation 2

OHE

βL = (βU + τδ/(1-δ) )

Structurally, the two equations describing βL =f(βU ) differ in several important ways. The θ term and division by ( 1- δ) of the βU term in the MHE are missing in the OHE. We also note that the OHE assumption that the firm can borrow at the risk free rate ф is avoided in the MHE by adding the multiplier α. These structural differences imply that the OHE and MHE will predict different values of β L and therefore of the weighted average cost of capital or WACC. Numerical analysis is used to display these differences and to describe their implications Note that in both equations the value of βU is computed as Equation 3

βU = ρτ/μ - θ

This equation implies that for the all equity firm, the cost of equity and therefore the WACC is μβ U + ф or μ(ρτ/μ - θ) + ф = μ(ρτ/μ - ф/μ) + ф = ρτ. Therefore, according to the MM principle, the effect of leverage on β will be such as to exactly offset the effect of the cost of debt on WACC so as to keep the cost of capital constant at WACC = ρτ. Only ρ or τ can materially affect the cost of capital and the value of the firm. As we see in the following numerical examples, The MHE predicts exactly this relationship but the OHE behaves very differently. The equation for WACC in general when the firm is allowed to used debt may be written as WACC = δαфτ + (1-δ)( μβL + ф). The value of α is used to accommodate any level of interest rates. In the case that the cost of debt to corporations rises with the debt fraction (Conine, Debt capacity and the capital budgeting decision, 1980), we write α=f(δ) as a power function as α = 1 + δε . The value of the exponent ε may be estimated from empirical data. Cohen's data (Cohen R. , 2004) implies a value of ε=2.4. For scenarios where we use risky debt - that is, where increasing the debt fraction increases the risk of default, we use α=1+δ2.4 to estimate the cost of borrowing.

3


3. DATA ANALYSIS AND RESULTS

We present a comparison of the two equations using numerical methods. The Microsoft Excel file used for these computations is available in the data archive for this paper (Munshi, 2014). Graphical analysis is used to compare β and WACC predicted by the OHE and the MHE under four different combinations of taxation and the cost of debt listed below. 1. The base case: The firm can borrow at the risk-free rate and the tax rate is set to zero. 2. The effect of taxation: The firm can borrow at the risk-free rate but faces a tax rate of 40%. This tax rate is used for the remaining two conditions. 3. The effect of a higher interest rate (with taxation): The firm can borrow at a fixed rate of interest that is higher than the risk-free rate. 4. Default risk (with taxation): The firm faces higher interest rates according to its debt ratio defined by the empirical power function α=1+δ2.4 . Each of these four cases is examined under three different relationships between the return on investment ρ and the market rate of return μ+φ as shown below. 1. ρ > the market rate of return μ+φ 2. ρ = the market rate of return μ+φ 3. ρ < the market rate of return μ+φ The twelve combinations of these conditions are listed below with assigned scenario number for ease of identification. Scenario 1.1: Scenario 1.2: Scenario 1.3:

The base case with ρ > the market rate of return μ+φ The base case with ρ = the market rate of return μ+φ The base case with ρ < the market rate of return μ+φ

Scenario 2.1: Scenario 2.2: Scenario 2.3:

Effect of tax with ρ > the market rate of return μ+φ Effect of tax with ρ = the market rate of return μ+φ Effect of tax with ρ < the market rate of return μ+φ


Higher interest rate with ρ > the market rate of return μ+φ Higher interest rate with ρ = the market rate of return μ+φ Higher interest rate with ρ < the market rate of return μ+φ


Default risk with ρ > the market rate of return μ+φ Default risk with ρ = the market rate of return μ+φ Default risk with ρ < the market rate of return μ+φ

4


5

Scenario 1.1: The base case with ρ > the market rate of return μ+φ

3.1

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

β

β

OHE 1.411 1.550 1.729 1.967 2.300 2.800 3.633 5.300 10.300

MHE 1.444 1.625 1.857 2.167 2.600 3.250 4.333 6.500 13.000

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

wacc

wacc

OHE 0.177 0.174 0.171 0.168 0.165 0.162 0.159 0.156 0.153

MHE 0.180 0.180 0.180 0.180 0.180 0.180 0.180 0.180 0.180

Scenario 1.1 Tax rate ρ μ+ф Interest ф Is ρ = μ+ф ? Interest μ

0 0.18 0.15 0.05 0.05 Greater Rskfree rate 0.1

Table 1 Scenario 1.1 parameters and results

Scenario 1.1 13.000 11.000 β

9.000 7.000

OHE

5.000

MHE

3.000 1.000 0

0.2

0.4

0.6

0.8

1

δ Figure 1 Scenario 1.1 β values

3

Scenario 1.1 0.180 WAC

0.160 0.140 0.120

OHE

0.100

MHE

0.080 0

0.2

0.4

0.6

0.8

1

δ Figure 2 Scenario 1.1 weighted average cost of capital 3

The graphs make use of color to distinguish between MHE and OHE values. If the paper is printed out in black and white, manual notation may be necessary to make this distinction in the printed copy of the paper.


6

Scenario 1.2: The base case with ρ = the market rate of return μ+φ

3.2

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

β

β

OHE 1.111 1.250 1.429 1.667 2.000 2.500 3.333 5.000 10.000

MHE 1.111 1.250 1.429 1.667 2.000 2.500 3.333 5.000 10.000

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

wacc

wacc

OHE 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150

MHE 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150


0 0.15 0.15 0.05 0.05 Equal Rskfree rate 0.1


Scenario 1.2 8.500 β

6.500 4.500

OHE

2.500

MHE

0.500 0

0.2

0.4

0.6

0.8

1


4


0.160 0.140 0.120

OHE

0.100

MHE

0.080 0

0.2

0.4

0.6

0.8

1


When only one curve is visible in the graph it indicates that the curves for the OHE and the MHE are coincident.


7

Scenario 1.3: The base case with ρ < the market rate of return μ+φ

3.3

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

β

β

OHE 0.811 0.950 1.129 1.367 1.700 2.200 3.033 4.700 9.700

MHE 0.778 0.875 1.000 1.167 1.400 1.750 2.333 3.500 7.000

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

wacc

wacc

OHE 0.123 0.126 0.129 0.132 0.135 0.138 0.141 0.144 0.147

MHE 0.120 0.120 0.120 0.120 0.120 0.120 0.120 0.120 0.120


0 0.12 0.15 0.05 0.05 Less Rskfree rate 0.1



6.500 4.500

OHE

2.500

MHE

0.500 0

0.2

0.4

0.6

0.8

1



0.160 0.140 0.120

OHE

0.100

MHE

0.080 0

0.2

0.4

0.6 δ

Figure 6 Scenario 1.3 weighted average cost of capital

0.8

1


8

Scenario 2.1: Effect of tax with ρ > the market rate of return μ+φ

3.4

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

β

β

OHE 0.838 0.935 1.060 1.227 1.460 1.810 2.393 3.560 7.060

MHE 0.861 0.988 1.150 1.367 1.670 2.125 2.883 4.400 8.950

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

wacc

wacc

OHE 0.124 0.122 0.120 0.118 0.116 0.113 0.111 0.109 0.107

MHE 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126


0.3 0.18 0.15 0.05 0.05 Greater Rskfree rate 0.1



6.500 4.500

OHE

2.500

MHE

0.500 0

0.2

0.4

0.6

0.8

1


Scenario 2.1

WAC

0.140 0.120 OHE 0.100

MHE

0.080 0

0.2

0.4

0.6 δ


0.8

1


9

Scenario 2.2: Effect of tax with ρ = the market rate of return μ+φ

3.5

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

β

β

OHE 0.628 0.725 0.850 1.017 1.250 1.600 2.183 3.350 6.850

MHE 0.628 0.725 0.850 1.017 1.250 1.600 2.183 3.350 6.850

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

wacc

wacc

OHE 0.105 0.105 0.105 0.105 0.105 0.105 0.105 0.105 0.105

MHE 0.105 0.105 0.105 0.105 0.105 0.105 0.105 0.105 0.105


0.3 0.15 0.15 0.05 0.05 Equal Rskfree rate 0.1



6.500 4.500

OHE

2.500

MHE

0.500 0

0.2

0.4

0.6

0.8

1


5

Scenario 2.2

WAC

0.140 0.120 OHE 0.100

MHE

0.080 0

0.2

0.4

0.6

0.8


When only one curve is visible it indicates that the two curves are co-incident

1


10

Scenario 2.3: Effect of tax with ρ < the market rate of return μ+φ

3.6

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

β

β

OHE 0.418 0.515 0.640 0.807 1.040 1.390 1.973 3.140 6.640

MHE 0.394 0.463 0.550 0.667 0.830 1.075 1.483 2.300 4.750

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

wacc

wacc

OHE 0.086 0.088 0.090 0.092 0.095 0.097 0.099 0.101 0.103

MHE 0.084 0.084 0.084 0.084 0.084 0.084 0.084 0.084 0.084


0.3 0.12 0.15 0.05 0.05 Less Rskfree rate 0.1



6.500 4.500

OHE

2.500

MHE

0.500 0

0.2

0.4

0.6

0.8

1


Scenario 2.3

WAC

0.140 0.120 OHE 0.100

MHE

0.080 0

0.2

0.4

0.6 δ


0.8

1


11

Scenario 3.1: Higher interest rate with ρ > the market rate of return μ+φ

3.7

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

β

β

OHE 0.838 0.935 1.060 1.227 1.460 1.810 2.393 3.560 7.060

MHE 0.849 0.961 1.105 1.297 1.565 1.968 2.638 3.980 8.005

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

wacc

wacc

OHE 0.125 0.124 0.123 0.122 0.121 0.120 0.119 0.118 0.117

MHE 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126


0.3 0.18 0.15 0.065 0.05 Greater Higher 0.1



6.500 4.500

OHE

2.500

MHE

0.500 0

0.2

0.4

0.6

0.8

1


Scenario 3.1

WAC

0.140 0.120 OHE 0.100

MHE

0.080 0

0.2

0.4

0.6 δ


0.8

1


12

Scenario 3.2: Higher interest rate with ρ = the market rate of return μ+φ

3.8

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

β

β

OHE 0.628 0.725 0.850 1.017 1.250 1.600 2.183 3.350 6.850

MHE 0.616 0.699 0.805 0.947 1.145 1.443 1.938 2.930 5.905

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

wacc

wacc

OHE 0.106 0.107 0.108 0.109 0.110 0.111 0.112 0.113 0.114

MHE 0.105 0.105 0.105 0.105 0.105 0.105 0.105 0.105 0.105


0.3 0.15 0.15 0.065 0.05 Equal Higher 0.1



6.500 4.500

OHE

2.500

MHE

0.500 0

0.2

0.4

0.6

0.8

1


Scenario 3.2

WAC

0.140 0.120 OHE 0.100

MHE

0.080 0

0.2

0.4

0.6 δ


0.8

1


13

Scenario 3.3: Higher interest rate with ρ < the market rate of return μ+φ

3.9

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

β

β

OHE 0.418 0.515 0.640 0.807 1.040 1.390 1.973 3.140 6.640

MHE 0.383 0.436 0.505 0.597 0.725 0.918 1.238 1.880 3.805

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

wacc

wacc

OHE 0.087 0.090 0.093 0.097 0.100 0.103 0.106 0.109 0.112

MHE 0.084 0.084 0.084 0.084 0.084 0.084 0.084 0.084 0.084


0.3 0.12 0.15 0.065 0.05 Less Higher 0.1



6.500 4.500

OHE

2.500

MHE

0.500 0

0.2

0.4

0.6

0.8

1


Scenario3.3

WAC

0.140 0.120 OHE 0.100

MHE

0.080 0

0.2

0.4

0.6 δ


0.8

1


3.10

14

Scenario 4.1: Default risk with ρ > the market rate of return μ+φ

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

β

β

OHE 0.838 0.935 1.060 1.227 1.460 1.810 2.393 3.560 7.060

MHE 0.861 0.986 1.142 1.341 1.604 1.971 2.536 3.581 6.504

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

wacc

wacc

OHE 0.124 0.122 0.120 0.119 0.119 0.120 0.122 0.126 0.132

MHE 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126


0.3 0.18 0.15 Risky 0.05 Greater Higher 0.1



6.500 4.500

OHE

2.500

MHE

0.500 0

0.2

0.4

0.6

0.8

1


Scenario 4.1 0.150 0.140 WAC

0.130 0.120

OHE

0.110

MHE

0.100 0.090 0

0.2

0.4

0.6 δ


0.8

1


3.11

15

Scenario 4.2: Default risk with ρ = the market rate of return μ+φ

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

β

β

OHE 0.628 0.725 0.850 1.017 1.250 1.600 2.183 3.350 6.850

MHE 0.628 0.723 0.842 0.991 1.184 1.446 1.836 2.531 4.404

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

wacc

wacc

OHE 0.105 0.105 0.106 0.107 0.108 0.111 0.115 0.121 0.129

MHE 0.105 0.105 0.105 0.105 0.105 0.105 0.105 0.105 0.105


0.3 0.15 0.15 Risky 0.05 Equal Higher 0.1



6.500 4.500

OHE

2.500

MHE

0.500 0

0.2

0.4

0.6

0.8

1


Scenario 4.2 0.150 0.140 WAC

0.130 0.120

OHE

0.110

MHE

0.100 0.090 0

0.2

0.4

0.6 δ


0.8

1


3.12

16

Scenario 4.3: Default risk with ρ < the market rate of return μ+φ

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

β

β

OHE 0.418 0.515 0.640 0.807 1.040 1.390 1.973 3.140 6.640

MHE 0.394 0.461 0.542 0.641 0.764 0.921 1.136 1.481 2.304

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

wacc

wacc

OHE 0.086 0.088 0.091 0.094 0.098 0.103 0.109 0.117 0.127

MHE 0.084 0.084 0.084 0.084 0.084 0.084 0.084 0.084 0.084


0.3 0.12 0.15 Risky 0.05 Less Higher 0.1


Scenario 4.3

8.250 β

6.250 4.250

OHE

2.250

MHE

0.250 0

0.2

0.4

0.6

0.8

1


Scenario 4.3

WAC

0.140 0.120 OHE 0.100

MHE

0.080 0

0.2

0.4

0.6 δ


0.8

1


4. DISCUSSION OF RESULTS 4.1 OHE assumptions not violated. In Scenarios 1.2 and 2.2 we find that the OHE, the MHE and the MM are in full agreement. The WACC is invariant with debt holding its value at WACC = ρτ defined by operating parameters and invariant with capital structure. The apparent advantage of debt along with its assumed "tax shield"6 is exactly offset by the effect of leverage on the cost of equity by way of higher values of β. What these two scenarios have in common are that (1) the firm can borrow at the risk free rate at any value of leverage and (2) the firm's operating returns ρ is exactly equal to the market returns μ+ф. Complete agreement among the OHE, MHE, and MM is observed only under these conditions and in no other scenario. Accordingly, we propose that the validity of the OHE rests on these assumptions and that its use under conditions that violate these assumptions may be expected to generate spurious results as is clearly demonstrated in many of the scenarios presented. Taxation is not an issue as the OHE results are in full conformity with MHE and MM with and without taxes as can be seen by comparing Scenario 1.2 (no tax) with Scenario 2.2 (with tax). The effect of taxation is only that the WACC is reduced by the factor τ according to the equation WACC = ρτ, which we may think of as the "intrinsic cost of capital" for the firm completely defined by operating parameters and the tax rate. The assumption implicit in the OHE that the firm can borrow at the risk free rate at any value of leverage is well known but its importance may not be fully appreciated. The importance of the second assumption of the OHE, that ρ=μ+φ is neither well known nor appreciated. As a result, it is common to find that spurious and anomalous results of the OHE when it is applied under conditions that violate these assumptions are interpreted either as a refutation of the OHE (Gonzales-Litzenberger-Rolfo, 1977) or as having implications for financial theory (Cohen R. , 2004). 4.2 One of the OHE assumptions is violated. The rest of the scenarios serve as examples of the kind of anomalies one may expect from the OHE when its twin assumptions are violated. Scenarios 1.1, 1.3, 2.1, and 2.3 conform to the first assumption that the firm can borrow any amount at the risk free rate but violate the assumption that ρ=μ+φ. What we find in these scenarios is that when ρ>μ+φ, the OHE underestimates the effect of debt on β and predicts a gradually declining cost of capital as debt is added to the capital structure; and when ρ