Lev and Unlev Beta

Unlevered betas and the cost of equity capital: An empirical approach

Draft Copy

By

Mehdi Sadeghi* and Julio S. Sabogal

Unlevered betas and the cost of equity capital: An empirical approach

Abstract Current paper investigates the significance of financial leverage in computation of

systematic risk within the context of CAPM model. First, we attempt to test whether variables such as growth rate, target leverage, and the proper rate of discounting tax shields affect Unlevered Betas (βu). Second, investigate whether utilizing the basic idea behind βu would help to overcome the information shortfalls in calculating the cost of capital for non-traded firms. We develop a model that allows us to empirically test these assumptions. Our results suggest that the use of the Levered Proxy Betas in solving the lack of market information problem for both non-traded firms and individual business units is not misleading, even when we use book value of debt instead of market-values. We also find more support in statistical performance term for Modigliani-Miller (1958, 1963) assumptions in in calculating (βu) compared with Miles and Ezzel (1985) model.

Corresponding author: Mehdi Sadeghi

Department of Applied Finance & Actuarial Studies,

Macquarie University,

Sydney, Australia, 2109,

Tel: (61) 2-98508527,

E-mail: [email protected].

2

1 Introduction

Since the publication of seminal paper by Hamada (1972) on the role of financial leverage

in the computation of systematic risk and development of unlevered Betas (βu), this

concept has been the subject of investigation by the academic researchers and practitioners

in two different ways. While researchers have been more engaged in the discussion of how

variables such as growth rate, target leverage, or and an appropriate discount tax shields

may affect βu. , practitioners have been concerned about how to utilise the basic idea

behind βu to resolve the lack of information problem in calculation of the cost of capital

for non-traded firms. However, little effort has been made to empirically examine how

relevant or effective these approaches might be.

Current study is an attempt to fill this research gap and contribute to the literature in two

different ways. First, we empirically test a theoretical model for unlevered beta. However,

instead of checking the Unlevered Betas directly, we develop a theoretical target that

allows us to test their performance. Second, we evaluate the performance of unlevered

beta to confirm the robustness of practioners, methodology as we believe that the real

importance of the unlevered betas is its usefulness as a proxy method to calculate the cost

of equity for non-traded firms.

Need one or two more introductory sentence at the beginning to tell the reader what will

follow. Previous theoretical discussion has been mainly focussed on the impact of

corporate taxes (τ) on βu. Some studies1 agree with two underlying assumptions in βu

calculation. First, the absolute value of debt does not change over the time as suggested

by Modigliani-Miller (1958, 1963) (MM hereafter). Second, the correct rate to discount

Tax Shields is Cost of Debt (Ke). On the other hand, following Miles and Ezzell (1985) (ME

1 (Fernandez, 2004, 2005, 2007; Massari, Roncaglio & Zanetti, 2008)

3

hereafter), another group of authors2 consider that the absolute value of debt changes

periodically to maintain a target Leverage Ratio (D/EL) and the correct rate to discount Tax

Shields is Cost of Unlevered Equity (Ku). Our results suggest that both definitions (MM and

ME) have a strong statistically significant relationship with the market beta (βm).

Nevertheless, both tend to increase the average of the (re-calculated) proxy levered betas

above the expected value of one. Our empirical evidence doesn’t support ME warning that

inclusion of taxes in βu tends to overestimate the cost of capital. Instead, our results

suggest that MM approach has two advantages in our study: First, the mean of the

levered proxy betas using MM approach are closer to the predicted value of one in our

dataset. Second, estimated coefficients according to that approach are higher compared

with the ME approach. Our findings also diverge from the empirical results obtained by

Marston & Perry (1996). However, the immediate concern is about the capacity of

leverage to explain changes in βm, rather than an attempt to show the direct relationship

between βm and βu.

In testing the application of βu to determine the cost of equity capital for unlisted

companies,, our paper is related to the empirical findings of Bowman (1980). Brown

relaxed the assumption on the requirement that D and EL should be as stated in their

respective market values and asserted that by using the accounting measure (book value

of debt) as a proxy for the market value of debt, the variations in the results should

become insignificant. We document a strong positive relationship between βm and proxy

levered betas calculated with the book values of debt and equity. These results essentially

confirm the findings obtained by Bowman & Bush (2006), but differ in the sense that we

do not control for size as they did, because for us the theoretical foundation between size

and systematic risk of the firm in the CAPM context is not clear.

The remainder of this paper is as follows: In Section 2 we present the unlevered betas and

the method to calculate a proxy of the Market-based Beta using such metrics. In Section 3

2 See for example: (Arzac & Glosten, 2005; Cooper & Nyborg, 2006; Fieten et al., 2005; Harris & Pringle, 1985; Myers, 1974; Ruback, 2002; Taggart, 1991; Tham & Vélez-Pareja, 2004)

4

we develop our testing model. In section 4.1 we describe our dataset and methodology.

Section 5 presents the empirical results. Section 6 examines the robustness of our results,

and Section 7 concludes.

2 Unleveredbetasandproxymethods

According to the MM theory, the Firm Value (VL) (in absence of Bankruptcy costs) is a

function of its own hypothetical value when is non-debt financing (VU) and the present

value of the effect of tax shields (VTS) produced by the financial debt. Thus,

L UV V VTS (0)

As an implication of the MM theory, Hamada (1972) define the relationship between the

systematic risk and leverage: / /m u EL Eu . Where EL is the Market Value of Levered

Common Equity and Eu is the Expected Market Value of the Unlevered Common Equity.

Rubinstein (1973) extended the Hamada model by incorporating the impacts of corporate

taxes (τ) and the market value of debt (D) on beta according to the following model:

1 / 1MMLu m D E (0)

Miles and Ezzell (1985) argue that equation (0) implies constant D. Additionally, this

equation assumes that the correct discount rate for VTS in (0) is the Cost of Debt (Kd)

(Fieten et al., 2005; Tham & Vélez-Pareja, 2004). An alternative model by Bowman

suggests that3:

1MELu m D E (0)

Although the only difference between (0) and (0) is the use of tax shields, the extensive

discussion regarding which one is the correct model to calculate the Unlevered Beta is

inconclusive. On one hand, Fernandez (2004, 2005, 2007) and Massari, Roncaglio &

Zanetti (2008) among others, support the underlying assumptions in (0), while Arzac &

Glosten (2005); Cooper & Nyborg (2006); Fieten et al. (2005), Harris & Pringle (1985),

Myers (1974) Ruback (2002); Taggart (1991) and Tham & Vélez-Pareja (2004) argue that

3 The Bowman (1980) definition of βu can be also derived from Miles and Ezzell (1985), assuming constant growth.

5

(0) is the correct derivation of βu. These studies have been largely limited to the

theoretical issues, with little reference to the empirical implications of this conundrum. A

couple of empirical research that is also available on this subject use either (2) [Bowman

(1980)] or (3) [Marston & Perry (1996)](0), giving little attention to the theoretical issues.

On the other hand, practitioners calculate the cost of equity capital (Ke) using either (0) or

(0) as a proxy method for non-traded firms or individual business units. It is a common

technique to calculate an average industry sector unlevered beta u and recalculate a

new Proxy Levered Beta l using the leverage of the private company.

1L u (0)

Where Ψ represents the correct relationship between the Leveraged Ratio (L) and the

Systematic Risk. Thus, Ψ= L for ME and Ψ=L*(1-τ) for MM.

3 CorporatetaxesandSystematicrisk

The tax rate has a diminishing effect in the decomposition of systematic risk. Without

corporate taxes there is not conflict between MM and ME approaches (equations (0) and

(0)). When tax rate is more than zero, the beta estimated according to MM model are

higher than the results from ME model (Miles & Ezzell, 1985). However, the argument

regarding which model is more accurate may not be solved through an empirical test due

to the absence of a theoretical target. Define the theoretical target in one sentence.

Therefore, differences between the two models are confined to the theoretical discussion

about their underlying assumptions. However, the use of the levered proxies allows us to

compare the performance of the two models by comparing empirical results with

predicted values. This predicted value (in the absence of disturbances) is the same as

Market-Based Beta, as the average of all Levered Proxies should be equal to one. To

discuss this idea, consider the average of the unlevered betas u for a group of firms in

the same risk-class during a given period of time:

6

1

(1 / )*1

ni

ii

mu n

(0)

where Ψ = L for ME and Ψ=L*(1-τ) for MM. In the absence of any market imperfections, all

unlevered betas in the same risk-class group should be equal (Hamada, 1972), in such

case (0) becomes: 1i i iu m u for all i in the same risk-class. Certainly, an

industry sector might be considered as a proxy for a risk-class. However, it is not realistic

to assume that all unlevered betas in the same industry sector are equal. Let λi be an error

term that encloses all market disturbances and risk-class’ misspecifications are defined as:

i iu u (0)

Using λi, the relationship (0) can be redefined as:

1 *U i i im (0)

Using the practitioners’ approach (equation (0)) we calculate a general expression for the

Proxy Levered Beta (βL) of any firm in a specific industry sector in a given period of time:

* 1

1 * * 1

*

i i i

i i i i

i i

l u

m

m

(0)

From (0) we can establish that the expected value of the Proxy Levered Beta is the same

as the Market-Base Beta in absence of both market disturbances and risk-class’

misspecifications (λi=1). Add a concluding sentence before following to the next section

7

4 DataandMethodology

4.1 Dataset

In one sentence or so, define the nature of the panel data you are intending to use

For instance, “In this section we discuss the details of the panel data set we used in our

study.”

Our panel data contains the annual financial information from Compustat and market

prices and dividends from CRSP of 159 companies listed in US Markets in the period from

1972 to 2007. We exclude those firms that do not have all of the accounting information,

as well as those that are not traded at the beginning or at the end of the period under

scrutiny. We also exclude firms with market capitalization of less than 10 million.

Furthermore, we dropped potentially bankrupt companies with negative D/E ratios, or D/E

ratios over 10 from our sample4. Firms in the Global Industry Classification (GIC) sector #

50 are also excluded due to the small number of firms in the sample.

4.2 Methodology

The methodology is applied in four steps: First, we compute the Market-based Beta using

an equally weighted portfolio conformed for all companies in the sample. Second, we

calculate the Unlevered Beta for each economic sector. Third, we compute the Levered

Proxy Beta of each firm using its unlevered sector beta. Finally, we run the panel

regressions between Market-based Betas and Levered Proxy Betas.

The descriptive statistics of the sample are presented in Table 1. Panel A shows Financial

Fundamentals and Panel B present Market Return.

See (Marston & Perry, 1996) for a further discussion.

8

Table 1 Descriptive statistics of the sample data

Panel A. Descriptive statistics of Firm financial fundamentals Global Industry Classification (GIC) Sector

(10) (15) (20) (25) (30) (35) (45) (55) Total

Firms 14 16 39 17 11 10 6 46 159 Years 31 31 31 31 31 31 31 31 31 Observations 434 496 1,209 527 341 310 186 1,426 4,929

Market Value of Equity

Average 17,661 5,789 8,441 4,734 18,248 33,352 20,276 3,195 9,764Stnd Dev 47,320 10,128 34,121 7,238 32,881 51,160 40,061 4,296 29,482Min 78 33 10 12 16 119 46 36 10Max 432,187 64,034 455,607 54,850 183,159 284,853 202,342 40,251 455,607

Book Value of Equity


Debt


Debt over Market Equity

Average 0.57 0.36 0.57 0.31 0.18 0.12 0.15 1.24 0.64Stnd Dev 0.60 0.30 0.83 0.35 0.17 0.12 0.14 0.72 0.74Min 0.02 0 0 0 0 0.01 0 0.14 0Max 7.16 2.97 7.95 2.49 0.87 1.29 0.93 8.11 8.11

Debt over Book Equity

Average 0.73 0.58 0.88 0.55 0.67 0.39 0.34 1.40 0.89Stnd Dev 0.56 0.46 1.11 0.45 0.95 0.26 0.34 0.63 0.83Min 0.03 0 0 0 0 0.03 0 0.48 0Max 4.13 4.43 7.71 2.36 9.81 1.57 1.53 8.17 9.81

Taxes Average 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40

Panel B. Market return Global Industry Classification (GIC) Sector

(10) (15) (20) (25) (30) (35) (45) (55) Total

Firms 14 16 39 17 11 10 6 46 159 Months 432 432 432 432 432 432 432 432 432 Observations 6,048 6,912 16,848 7,344 4,752 4,320 2,592 19,872 68,688

Market Return Average 1.46 1.28 1.34 1.19 1.32 1.25 1.29 1.10 1.25 Stnd Dev 9.10 8.52 8.90 8.41 7.40 7.78 11.51 6.17 8.07

Table 1 presents the descriptive statistics of the sample. The sample is composed by nonfinancial and nontelecomunication firms listed in US markets during 1972 to 2007. Accounting information is obtained from COMPUSTAT yearly file and market-based information comes from CRSP monthly file. Market Value is calculated as the Closing Price at the end of the year (Compstat item#199) multiplied by the number of outstanding shares (#25) at the end of each year. Book Value of Equity is Common Equity (#60). Debt is defined as Debt in Current Liabilities (Compustat item # 34) plus Long-Term Debt (#9). Taxes are defined as the top statutory corporate taxes and are available at Tax Policy Center. The firms are divided using Global Industry Classifications (GIC) sectors: Energy (10), Materials (15),Industrials (20), Consumer discretionary (25), Consumer staples (30), Health Care (35), Information Technology (45) and Utilities (55). Total column corresponds to the sum of all GIC sectors in rows Firms, Years, Months and Observations.

9

4.2.1 MarketBetas

We use the Firm’s Market Return (Rit), calculated in the CRSP database as

it it it itR P D P 1/ , where Pit is the stock price at month t, Dit is the Cash Dividend Paid,

and Pit-1 is the stock price at the end of the previous month. The result was filtered to the

range (-50, 100) to avoid leverage caused by extreme observations (Campbell, Polk &

Vuolteenaho, 2010). We obtain the monthly Market Beta for each firm using both the

individual market returns from both the 60 60m and 12 previous moths 12m , using

the standard model: 2,i m mm R R R . Where Rm is the average return of an

equally weighted portfolio composed by all the firms contained in the dataset5, ,i mR R

is the covariance of the stock return and market return and 2mR is the market

variance.

We have a number of reasons to justify the use of two different measures of βm: First, the

estimation of 60m is standard in the literature, so we need it to compare our results

with previous findings. Second, we are concerned about whether the use of shared

information can affect our results. This problem arises when we calculate our coefficients

with a month-firm base, using information from 60 previous months. For example, the

market return obtained in 1980/01 is an observation needed to compute the βm from the

same period to 1984/12. Finally, the use of a βm based upon 60 months implies that we

have five different leverage ratios (using yearly financial statements), then, is not clear for

us whether the use of an average of such points can affect our results. Consequently, we

compute βm based upon 12 previous months 12m as an alternative that does not

share information with the monthly betas calculated in overlapping periods.

4.2.2 Unleveragedprocessandleveredproxybetas

5 Previous research (Beaver, Kettler & Scholes, 1970; Beaver & Manegold, 1975; Marston & Perry, 1996) shows that using a value weighted portfolio rather than the equally weighted portfolio in the context of comparing accounting and market based betas does not significantly alter the results.

10

We obtain a matrix with monthly βm for each company in the sample between 1977 and

2007. Note that observations from 1972 to 1976 are needed to compute 60m . We match

the monthly market information with the annual financial information of each firm

dropping all months but December each year.

The unlevered betas are computed using the two calculated versions of market-based

betas 12 60 and m m for the corresponding periods, using also both relationships (0)

and (0) for each one of the two versions of the market-based betas. Although theory

advises for the use of the market price of debt and equity as the correct measure for the

leverage (D/E), we use the book value of debt as a proxy of its market value. The use of

Market Value of debt implies an enormous amount of additional work beyond the focus of

this paper and the book value is a good substitute at the firm level (Bowman, 1980;

Mulford, 1985). We define Debt (D) as Debt in Current Liabilities (Compustat item # 34)

plus Long-Term Debt (#9). Equity (E) is the market value of the firm at the end of the year

calculated as the Closing Price at the end of the year (#199) multiplied by the number of

outstanding shares (#25) it may be OK for your thesis to refer to Compustat numbers.

However, you have to remove them from a submitted copy to a journal for publication

Corporate taxes are defined as the top statutory corporate taxes for each year6. We use

two definitions of Leverage: D/E and D/BE. The latter corresponds to the use of the Book

Value of Equity (BE)7 instead of its market value as used by practitioners for non-traded

firms. We average across… the average firm’s leverage of the five previous years when we

compute Eu and BEu based upon 60m .

We obtain four alternative definitions of the Levered Proxy Beta: MMEl and MM

BEl

represent the unleveraged and leveraged process, according to MM approach (equation

(0), using the market-value and the book value of equity, respectively. MEEl and ME

BE

6 US top statutory corporate taxes were 48% between 1972 and 1978; 46% from 1979 to 1986; 40% in 1987; 34% between 1988 and 1992 and 35% thereafter.

7 BE is Common Equity (Compustat item #60).

11

correspond to the unleveraged- if this is a minus sign, then use “minus” leveraged process

in equation (0), using again both market-value and book value of equity. We run the

process twice using our two market market-based betas 12 60 and m m .

To summarize, we calculate two versions of market-base beta for each firm-year, then we

unleverage the beta using the specific leverage (D/E and D/BE) of the firm. We average

the unleveraged betas for each sector-year and finally we recalculate a Proxy Levered Beta

for each firm-year using the specific leverage of the firm. Therefore we obtain four

versions of the levered proxy beta for each version of the market-based beta:

1 / 1MM MMBE BEl l u D BE (0)

1 / 1MM MME El l u D E (0)

1 /ME MEBE BEl l u D BE (0)

1 /ME MEBE Bl l u D E (0)

Where Eu and BEu are the cross-sectional average of the industry sector unlevered

beta for each firm using the market and the book value of equity respectively. D/E and

D/BE corresponds to the average leverage using five previous years for 60m .

12

Table 2 Summary statistics of calculated variables

Variable Mean Std. Dev. Min Max

Panel A. Calculations based upon 60 previous months

60m 1.00 0.42 -0.29 3.48

MMu

0.78 0.39 -0.23 3.25

MEu

0.70 0.38 -0.23 3.16

MMBEu

0.71 0.36 -0.23 2.91

MEBEu

0.62 0.34 -0.23 2.69

MMBEl 1.03 0.37 0.38 4.31

MEBEl 1.05 0.47 0.33 6.30

MMBEl 1.04 0.41 0.36 4.90

MEBEl 1.07 0.52 0.33 7.18

MME

1.13 1.66 -74.05 45.62

MEE

1.15 1.67 -72.42 46.91

MMBE

1.15 1.79 -78.17 50.34

MEBE

1.18 1.88 -80.00 54.01

Panel B.

Calculations based upon 12 previous months

12m 1.00 0.74 -2.28 5.30

MMu

0.79 0.64 -1.74 5.07

MEu

0.71 0.61 -1.56 4.99

MMBEu

0.71 0.58 -1.62 4.99

MEBEu

0.62 0.54 -1.46 4.99

MMBEl 1.03 0.48 -0.13 6.22

MEBEl 1.06 0.58 -0.14 9.40

MMBEl 1.05 0.52 -0.10 5.32

MEBEl 1.09 0.64 -0.09 7.78

MME

1.87 23.11 -171.05 934.04

MEE

1.93 23.76 -177.25 928.84

MMBE

2.00 26.21 -180.61 999.86

MEBE

2.24 27.79 -199.86 1,022.40

13

Table 2 Shows the summary statistics of the calculated coefficients. The Market Beta (βm) for each firm using both the individual market returns from both the 60 and 12 previous moths (Panel A and Panel B respectively). βu is Unlevered Beta calculated usingequations (0) or (0). βl is proxy levered beta (equation (0)). λ is the error term in the unleveraged-leveraged process (equation (0)). Subscript E indicates calculation using the market value of debt and subscript BE specify the Book Value of Equity. Superscripts ME and MM correspond to the unleveraged-leveraged process suggested by ME.

Table 2 presents the summary statistics for two versions of Market-based Betas using 60

previous months (Panel A) and 12 previous months (Panel B) and the Levered Proxy Betas.

The averages of the proxy levered betas are higher than the theoretical expectation of 1 in

both panels. The results using Book Values are greater than those where the Market

Values are employed. This is not surprising since the average of the Market Value of the

equity is greater than the Book value of Equity (see Table 1). Therefore, the factor (1+ D/E)

in equations (0) and (0) tend to be greater than (1+ D/BE) in equations (0) and (0).

Although we obtained the result MM

U > ME

U predicted by ME, this relationship is contrary

in the case of Levered Proxy Betas. The reasons why those results are arisen will be

explained latter. The specification error λ not only exceeds the expected average of one in

all cases, but also contains a great amount of noise.

5 Empiricalresults

Our empirical model is extracted from equation (0). We report Generalized Least Squares

(GLS) panel regression estimates, robust to heteroskedasticity and cross-sectional

correlation. They are also adjusted for serial correlation in the time series dimension of

the panel using Prais-Winsten correction:

1 2 3it it it itL m (0)

, 1it i t it

| | 1

Table 3 summarizes our findings. Panel A reports the results using market-based beta,

estimated with data for 60 previous months. Panel B shows the estimates using market-

based beta calculated from 12 previous months’ data. For both panels the relationship

14

between all Levered Proxy Betas and he Market-Based Beta incomplete sentence. All

proxy beta definitions are related with the market beta at the one precent level sentence

is not clear. This is an interesting result due to the great amount of noise incorporated

during the unleveraged-leveraged process. The specification error λ are also related with

Market-Based Beta in seven out of eight specifications, indicating that λ encloses

information that is missing by Levered Proxy Beta. This result reiterates that industry

sector is just a proxy of a class risk and there are factors other than industry and leverage

that affect systematic risk.

Table 3 also let us making some interesting comparisons: First, contrasting both panels,

our results suggest that the relationship between βm and βu is stronger when using 60

months to calculate betas. This is a symptom of the use of sharing information somehow

may alter the information dynamics and tend to artificially increase the results. Second,

Proxy Levered Betas using book-values of debt show weaker relationships than those

computing market-values, accordingly with previous results. Finally, the relationships

using MM definition present stronger relationship than ME counterparts. Perhaps this last

is our most important results because it adds an empirical light to the discussion regarding

which one of the theoretical models are most accurate.

Table 3 Parameters estimates of regressions of Levered Proxy Betas on Market‐Based

Betas

MMBEl l ME

BEl l MMEl l ME

El l

Panel A.

1 2 3(60)it it it itl m

m 0.270*** 0.275*** 0.287*** 0.300*** (17.26)

(15.66)

(18.97)

(17.79)

it 0.00509**

0.008***

0.00350**

0.00591***

(2.84)

(3.30)

(3.00)

(4.27)

Observations 4,929 4,929 4,929 4,929

Panel B.

15


m 0.187*** 0.185*** 0.193*** 0.193*** (15.17)

(14.32)

(15.85)

(14.91)

it 0.00026*

0.00026

0.00035*

0.00037*

(1.77)

(1.64)

(2.46)

(2.27) Observations 4,929 4,929 4,929 4,929 Table 3 reports the results of the Generalized Least Squares panel regression model adjusted using Prais-Winsten correction. Panel A reports the results using market-based beta (βm) calculation using 60 previous months. Panel B shows the estimates using results using market-based beta calculation using 12 previous months. βl is the Levered Proxy Beta equations (0) to (0). λ is the error term in the unleveraged-leveraged process (equation (0)). Subscript E indicates calculation using the market value of debt and subscript BE specify the Book Value of Equity. Superscripts ME and MM correspond to the unleveraged-leveraged process suggested by ME. First row of each panel reports the coefficients and second row in each panel reports the corresponding t statistics (in parentheses). * p<0.05, ** p<0.01, *** p<0.001.

5.1 ComparingtheempiricalperformanceofMEvs.MMmodel

We are interested in confirm whether the larger mean when we compute the proxy using

the ME model in Table 2 is a signal of larger results than those obtained using MM model.

If ME proposal tend to overestimate the risk, this may indicate that the cost of equity

capital using such method is also overestimated. We run Wilcoxon sign-rank test and

Somers’ D test on the pooled sample. These two tests are nonparametric test that allow

us testing the null hypothesis MM ME . The results in Table 4 indicate that the null

hypothesis is soundly rejected in all possible interpretations of the accounting-based

betas. As signalled by ME, the unlevered betas using MM model are higher, however there

are not a theoretical predicted value to use as a target, therefore is not possible to reach

any conclusion regarding this finding. The results are contrary in all cases of the Levered

Proxy Betas. The sign of the z-statistic indicates that ME proxies tend to be higher and may

indicate that using ME tend to overestimate the systematic risk due to the results are

above the expected value of one.

Table 4 Tests of difference between the results using MM and ME models

Panel A. Calculations based upon 60 previous months

Panel B. Calculations based upon 12 previous months

16

Pair comparation

Wilcoxon z-statistic

Somers’ D z-statistic Wilcoxon

z-statistic Somers’ D z-statistic

MM MEu u 60.80*** 9.31*** 58.08*** 16.69***

MM MEE El l -6.51*** -11.76*** -7.06*** -11.76***

MM MEBE BEl l -11.60*** -11.34*** -12.99*** -11.34***

Table 4 presents the results (z-statistics) of sign-rank test and Somers’ D for paired observations on the pooled sample. Panel A reports the results using market-based beta calculation using 60 previous months. Panel B shows the results using market-based beta calculation using 12 previous months. The null hypothesis is βMM=βME. βu is Unlevered Beta calculated using equations (0) or (0). βl is the Levered Proxy Beta equations (0) to (0). Subscript E indicates calculation using the market value of debt and subscript BE specify the Book Value of Equity. Superscripts ME and MM correspond to the unleveraged-leveraged process suggested by ME. * p<0.05, ** p<0.01, *** p<0.001.

6 Robustnesschecks

The results obtained from Table 3 allow us to make a general conclusion regarding the

empirical validity of the unleveraged-leveraged process for our sample. Nevertheless, the

performance of the process may be different for each economic sector and the general

result might be influenced by the model specification. We check the results in two ways:

Analysing the validity of unleveraged-leveraged process on each specific sector (section

6.1) and testing the regression in different periods of time (section 6.2).

6.1 Resultsbyeconomicsector

The first concern we have is whether our results might be affected by unlevered betas

coming from some specific sectors. Especially for those that have fewer companies in our

sample, due to in those cases the specific weight of each firm in the sector average is high.

Table 5 reports the results regression model by sector. The results indicate that all

versions of the levered proxy betas are highly correlated with the market-based betas in

seven out of eight sectors. A striking result from Table 5 is that not for all specifications

the estimates the levered betas using market values are bigger than those using book

values. This finding contradicts the expected results.

17

Table 5 Parameters estimates regressions by sector

Panel A.


GIC Sector

(10) (15) (20) (25) (30) (35) (45) (55) MEEl l 0.8*** 0.22*** 0.25*** 0.14*** 0.23*** 0.3*** 0.37*** 0.37***

(21.42) (7.3) (5.95) (5.63) (7.03) (7.74) (9.41) (10.16)

MEBEl l 0.83*** 0.24*** 0.33*** 0.19*** 0.19*** 0.31*** 0.41*** 0.35***

(22.55) (6.81) (8.18) (7.21) (5.45) (7.67) (8.99) (9.24)

MMEl l 0.71*** 0.21*** 0.16*** 0.12*** 0.23*** 0.29*** 0.36*** 0.36***

(17.28) (7.16) (5.98) (6.01) (7.16) (7.69) (9.27) (10.09)

MMBEl l 0.74*** 0.22*** 0.22*** 0.17*** 0.21*** 0.3*** 0.38*** 0.34***

(18.85) (6.8) (8.0) (8.06) (6.25) (7.55) (9.19) (9.3)

Panel B.


GIC Sector

(10) (15) (20) (25) (30) (35) (45) (55) MEEl l 0.31*** 0.17*** 0.05** 0.06*** 0.17*** 0.32*** 0.28*** 0.27***

(8.96) (6.71) (2.61) (4.24) (6.45) (9.51) (7.63) (9.62)

MEBEl l 0.33*** 0.16*** 0.05** 0.07*** 0.15*** 0.31*** 0.28*** 0.21***

(9.27) (5.19) (3.04) (4.49) (4.79) (9.47) (6.72) (6.91)

MMEl l 0.31*** 0.17*** 0.05*** 0.06*** 0.17*** 0.31*** 0.27*** 0.25***

(9.3) (7.02) (3.67) (5.29) (6.57) (9.48) (7.55) (9.33)

MMBEl l 0.32*** 0.16*** 0.05*** 0.07*** 0.16*** 0.31*** 0.27*** 0.2***

(9.45) (5.79) (4.06) (5.34) (5.47) (9.59) (6.95) (7.03)

Firms 14 16 39 17 11 10 6 46

Years 31 31 31 31 31 31 31 31

Observations 434 496 1209 527 341 310 186 1426 Table 5 reports the results of the Generalized Least Squares panel regression model adjusted using Prais-Winsten correction. Panel A reports the results using market-based beta (βm) calculation using 60 previous months. Panel B shows the estimates using results using market-based beta calculation using 12 previous months. The firms are divided using Global Industry Classifications (GIC) sectors: Energy (10), Materials (15), Industrials (20), Consumer discretionary (25), Consumer staples (30), Health Care (35), Information Technology (45) and Utilities (55). βl is the Levered Proxy Beta equations (0) to (0). λ is the error term in the unleveraged-leveraged process (estimates not reported). Subscript E indicates calculation using the market value of debt and subscript BE specify the Book Value of Equity. Superscripts ME and MM correspond to the unleveraged-leveraged process suggested by ME.

18

First row of each panel reports the coefficients and second row in each panel reports the corresponding t statistics (in parentheses). * p<0.05, ** p<0.01, *** p<0.001.

6.2 Regressionresultsinsubsamples

Another concern is whether the selected period in our sample affects the results. Especially

because our results does not confirm the previous findings from Marston & Perry (1996). Table 6

presents the results when the sample is divided in two subsamples: From 1977 to 1991 and 1992

to 1997. Although the estimates are lower than those obtained from the full sample, the statistical

significance is very high for both sub periods.

Table 6 Parameters estimates regressions using subsamples

MMBEl l ME

BEl l MMEl l ME

El l

Panel A.


1977 1991m 0.822*** 0.907*** 0.835*** 0.952***(24.58) (21.7) (24.39) (22.42)

N 2385 2385 2385 2385

1992 2007m 0.262*** 0.271*** 0.280*** 0.297***(14.11) (13.23) (16.2) (16.35)

N 2544 2544 2544 2544

Panel B.


1977 1991m 0.184*** 0.182*** 0.189*** 0.189***(9.39) (8.41) (9.2) (7.88)

N 2385 2385 2385 2385

1992 2007m 0.193*** 0.192*** 0.203*** 0.206***(12.33) (11.73) (13.59) (13.43)

N 2544 2544 2544 2544Table 6 reports the results of the Generalized Least Squares panel regression model adjusted using Prais-Winsten correction. The sample is divided in two subsamples: 1977 to 1991 and 1992 to 2007. Panel A reports the results using market-based beta (βm) calculation using 60 previous months. Panel B shows the estimates using results using market-based beta calculation using 12 previous months. βl is the Levered Proxy Beta equations (0) to (0). λ is the error term in the unleveraged-leveraged process (estimates not reported). Subscript E indicates calculation using the market value of debt and subscript BE specify the Book Value of Equity. Superscripts ME and MM correspond to the unleveraged-leveraged process suggested by ME. First row of each panel reports the coefficients and second row in each panel reports the corresponding t statistics (in parentheses). * p<0.05, ** p<0.01, *** p<0.001.

19

6.3 PanelRegressionIssues

Recent literature has been concerned about possibility of biased regression estimates

in the panel data (due to the nature of this type of datasets which may present

dependence in both cross-sectional and time series dimensions.) Petersen (2009) and Gow

(2010) suggests using cluster regressions as a robust method. Nevertheless, we decide

using a GLS with Prais-Winsten correction regression that is also robust also for

heteroskedasticity. Is the later method as good or better? From table Table 7 we can

conclude that our results do not change using the commonly used Fama-MacBeth

regression or a cluster regression.

Table 7 Parameters estimates of OLS regressions

MMBEl l ME

BEl l MMEl l ME

El l

Panel A.

1 2 3(60)it it it itl m βm Fama-MacBeth 0.744*** 0.820*** 0.728*** 0.807*** (16.18)

(17.45)

(14.49)

(14.73)

βm Cluster Regression 0.588***

0.521***

0.521***

0.595***

(10.25)

(9.00)

(10.47)

(8.69)

Observations 4,929 4,929 4,929 4,929

Panel B.


βm Fama-MacBeth 0.299*** 0.327*** 0.308*** 0.338*** (10.69)

(10.25)

(11.6)

(11.04)

βm Cluster Regression 0.273***

0.287***

0.282***

0.300*** (11.33)

(10.66)

(12.01)

(11.26)

Observations 4,929 4,929 4,929 4,929 Table 7 reports the results of Fama-MacBeth regression or a cluster regression. Panel A reports the results using market-based beta (βm) calculation using 60 previous months. Panel B shows the estimates using results using market-based beta calculation using 12 previous months. βl is the Levered Proxy Beta equations (0) to (0). λ is the error term in the unleveraged-leveraged process (estimates not reported). Subscript E indicates calculation using the market value of debt and subscript BE specify the Book Value of Equity. Superscripts ME and MM correspond to the unleveraged-leveraged process suggested by ME. First row of each panel reports the coefficients and second row in each panel reports the corresponding t statistics (in parentheses). * p<0.05, ** p<0.01, *** p<0.001.

20

6.4 Valuesofλandendogeneityissues

Equation (0) allows us to compare the performance of all versions of the Proxy Levered

Beta. However, the same equation encloses a possible endogeneity in our estimations

when λ is equal to one. From Table 2 we can conclude that the large standard deviation in

all versions of λ as well as the huge difference between the minimum and maximum of the

same variable are indicators of its real value. Nevertheless, we conduct a formal test to

check if the median of λ is equal to one (Ho: λ=1). The results from Table 8 suggest that

we can reject the null hypothesis in all specifications of λ.

Table 8 Wilcoxon test on median of λ equal to one



MM MM MMBE BE BEu 10.602*** 3.935*** ME ME MEBE BE BEu u 11.680*** 5.180*** MM MM MME E Eu u 10.502*** 3.354*** ME ME MEE E Eu u 11.703*** 4.501***

Table 9 presents the results (z-statistics) of Somers’ D test on the pooled sample. Panel A reports the results using market-based beta calculation using 60 previous months. Panel B shows the results using market-based beta calculation using 12 previous months. The null hypothesis is λ=0. β is Unlevered Beta calculated using equations (0) or (0). βl is the Levered Proxy Beta equations (0) to (0). Subscript E indicates calculation using the market value of debt and subscript BE specifying the Book Value of Equity. Superscripts ME and MM correspond to the unleveraged-leveraged process suggested by ME. * p<0.05, ** p<0.01, *** p<0.001.

7 Conclusions

Current paper investigates the significance of financial leverage in computation of

systematicriskwithinthecontextofCAPMmodel.Ourresultssuggestthattheuseof

theLeveredProxyBetas insolving the lackofmarket informationproblemforboth

non‐tradedfirmsand individualbusinessunits isnotmisleading,evenwhenweuse

bookvalueofdebtinsteadofmarket‐values.Wealsofindmoresupportinstatistical

21

performance term forModigliani‐Miller (1958, 1963) assumptions in in calculating

(βu)comparedwithMilesandEzzel(1985)model.

The strong correlation between the Market-based Beta and its levered proxy counterpart

we found in Table 3 is consistent with the empirical findings of the majority of previous

studies. This suggests that the use of the Levered Proxy Betas to solve the lack of market is

a sound procedure. Although the relation is stronger for market values, book values of

equity also show statistical significant relationship. This is good news for practitioners who

have been using the unleveraged approach for years without much empirical support

about the validity of their procedures.

Our results show that the MM assumption regarding the inclusion of tax effect in the

unlevered beta calculation has a more statistical performance than the Miles and Ezzel

(1985) with the market-based beta and also their results are closer to the theoretical

expected result of one.

22

References

Arzac, E. R., & Glosten, L. R. 2005. A Reconsideration of Tax Shield Valuation. European Financial Management, 11(4): 453-461.

Beaver, W., Kettler, P., & Scholes, M. 1970. The Association between Market Determined and Accounting Determined Risk Measures. The Accounting Review, 45(4): 654-682.

Beaver, W., & Manegold, J. 1975. The Association Between Market-Determined and Accounting-Determined Measures of Systematic Risk: Some Further Evidence. The Journal of Financial and Quantitative Analysis, 10(2): 231-284.

Bowman, R. G. 1980. The Importance of a Market-Value Measurement of Debt in Assessing Leverage. Journal of Accounting Research, 18(1): 242-254.

Bowman, R. G., & Bush, S. R. 2006. Using Comparable Companies to Estimate the Betas of Private Companies. Journal of Applied Finance, 16(2): 71-81.

Campbell, J. Y., Polk, C., & Vuolteenaho, T. 2010. Growth or Glamour? Fundamentals and Systematic Risk in Stock Returns. Review of Financial Studies, 23(1): 305-344.

Cooper, I. A., & Nyborg, K. G. 2006. The value of Tax Shields IS Equal to the Present Value of Tax Shields. Journal of Financial Economics, 81(1): 215-225.

Fernandez, P. 2004. The Value of Tax Shields is NOT Equal to the Present Value of Tax Shields. Journal of Financial Economics, 73(1): 145-165.

Fernandez, P. 2005. Reply to "Comment on the Value of Tax Shields is NOT Equal to the Present Value of Tax Shields". The Quarterly Review of Economics and Finance, 45(1): 188-192.

Fernandez, P. 2007. A More Realistic Valuation: Adjusted Present Value and WACC with Constant Book Leverage Ratio. Journal of Applied Finance, 17(2): 13-20.

Fieten, P., Kruschwitz, L., Laitenberger, J., Löffler, A., Tham, J., Vélez-Pareja, I., & Wonder, N. 2005. Comment on "The Value of Tax Shields is NOT Equal to the Present Value of Tax Shields". The Quarterly Review of Economics and Finance, 45(1): 184-187.

Gow, I. D. O. G. D. J. 2010. Correcting for Cross-Sectional and Time-Series Dependence in Accounting Research. Accounting Review, 85(2): 483-512.

Hamada, R. S. 1972. The Effect of the Firm's Capital Structure on the Systematic Risk of Common Stocks. The Journal of Finance, 27(2): 435-452.

Harris, R. S., & Pringle, J. J. 1985. Risk-Adjusted Discount Rates-Extensions from the Average-Risk Case. Journal of Financial Research, 8(3): 237.

Marston, F., & Perry, S. 1996. Implied Penalties for Financial Leverage: Theory Versus Empirical Evidence. Quarterly Journal of Business & Economics, 35(2): 77-97.

Massari, M., Roncaglio, F., & Zanetti, L. 2008. On the Equivalence between the APV and the wacc Approach in a Growing Leveraged Firm. European Financial Management, 14(1): 152-162.

Miles, J. A., & Ezzell, J. R. 1985. Reformulating Tax Shield Valuation: A Note. Journal of Finance, 40(5): 1485-1492.

Modigliani, F., & Miller, M. 1958. The Cost of Capital, Corporation Finance and the Theory of Investment. The American Economic Review, 48(3): 261-297.

Modigliani, F., & Miller, M. 1963. Corporate Income Taxes and the Cost of Capital: A Correction. The American Economic Review, 53(3): 433-443.

Mulford, C. W. 1985. The Importance of a Market Value Measurement of Debt in Leverage Ratios: Replication and Extensions. Journal of Accounting Research, 23(2): 897-906.

Myers, S. C. 1974. Interactions of Corporate Financing and Investment Decisions-Implications for Capital Budgeting. The Journal of Finance, 29(1): 1-25.

23

Petersen, M. A. 2009. Estimating Standard Errors in Finance Panel Data Sets: Comparing Approaches. Review of Financial Studies, 22(1): 435-480.

Ruback, R. S. 2002. Capital Cash Flows: A Simple Approach to Valuing Risky Cash Flows. Financial Management, 31(2): 85-103.

Rubinstein, M. E. 1973. A Mean-Variance Synthesis of Corporate Financial Theory. The Journal of Finance, 28(1): 167-181.

Taggart, R. A., Jr. 1991. Consistent Valuation and Cost of Capital Expressions with Corporate and Personal Taxes. Financial Management, 20(3): 8-20.

Tham, J., & Vélez-Pareja, I. 2004. Principles of Cash Flow Valuation London: Elsevier Academic Press.

Lev and Unlev Beta

Documents

Transcript of Lev and Unlev Beta