Leverage and the Weighted-Average Cost of Capital for U.S. Banks

Brian Clark

Rensselaer Polytechnic Institute and Office of the Comptroller of the Currency

Jonathan Jones

Office of the Comptroller of the Currency

David Malmquist Office of the Comptroller of the Currency

This Draft: October 2015

Abstract We examine the link between leverage and the weighted-average cost of capital (WACC) for U.S. banks from 1996 to 2012. Our paper differs significantly from prior studies which examine this issue in that we explore this link for different asset-size classes of banks. Not taking this approach results in misestimating the effects of changes in leverage on the WACCs of the largest banks. We estimate that doubling equity capital (halving leverage) would have little effect on the WACCs of the six largest banks, while smaller banks’ WACCs are found to be impacted to a greater extent.

Keywords: systematic risk, equity beta, bank leverage, cost of capital, Modigliani-Miller. JEL Classifications: C33, G21, G28, G32. Brian Clark is at Rensselaer Polytechnic Institute and the Office of the Comptroller of the Currency and can be contacted at Clarkb2@rpi.edu. Jonathan Jones and David Malmquist are at the Office of the Comptroller of the Currency and can be contacted at Jonathan.Jones@occ.treas.gov and David.Malmquist@occ.treas.gov, respectively. The opinions in this paper are those of the authors and do not necessarily reflect those of the Office of the Comptroller of the Currency or the U.S. Treasury Department. An earlier version of this paper was presented at the Financial Intermediation Research Society Conference, held in Reykjavik, Iceland, on May 24-27, 2015. We thank Malcolm Baker, Mike Carhill, Anthony Cassese, John Cochrane, Alireza Ebrahim, Mark Flannery, Lewis Gaul, David Grossman, Ed Kane, David Miles, Joseph Pimbley, Mark Pocock, Natalya Schenck, Til Schuermann, Anjan Thakor, Sheridan Titman, Roger Tufts, Pinar Uysal, seminar participants at Rensselaer Polytechnic Institute, and session participants at the Financial Intermediation Research Society Conference for helpful comments and discussions on earlier drafts. We also thank Rick Moylan for excellent research assistance and Monica Martinez for excellent editorial assistance. We are responsible for any errors in the paper.

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I. Introduction In the aftermath of the global financial crisis of 2007-2009, considerable attention has

focused on quantifying the likely effects of heightened equity capital requirements on banks’

weighted-average cost of capital (WACC). Underpinning much of the debate is the extent to

which Modigliani and Miller’s (1958) (MM) theorem holds for banks. That is, even in the

presence of market frictions,1 to what extent is a bank’s weighted-average cost of capital

independent of its capital structure?

Many misconceptions surround the discussion of the capital regulation of banks,

including the pervasive view that equity capital is expensive and that the cost of equity capital is

invariant to the degree of leverage (see, e.g., Admati et al. (2011), Admati and Hellwig (2013a,

2013b and 2014), and Pfleiderer (2015)). This view is offered as evidence that credit markets

and the real sector would be adversely impacted by heightened capital requirements.2 Despite

the importance of these fundamental questions for prudential financial regulations, such as those

associated with Basel I, II, and III, whose primary regulatory tool is setting capital requirements

for banks,3 there is a surprising lack of empirical work that documents the extent to which the

MM framework may be found to hold for banks (Baker and Wurgler (2013)).

1 These market frictions include the disparate tax treatment of debt and equity, bankruptcy costs, implicit

and explicit government guarantees, agency costs, and asymmetric information. See, e.g., Juks (2010) and Titman (2002) for a detailed discussion of the assumptions underlying the MM irrelevance proposition.

2 See, e.g., Calomiris (2012), Cline (2015), Elliott (2009, 2010, 2013), Institute of International Finance (IIF) (2011), and The Clearing House (2012).

3For Basel III, there is an increased focus on common equity and higher capital buffers. Included among the changes is a non-risk-based leverage ratio, which is measured as Tier 1 capital to a measure of exposure that includes both off-balance-sheet and balance-sheet exposures. In contrast to the microprudential focus of both Basel I and II, Basel III takes a macroprudential approach to setting capital requirements. While a microprudential regulatory approach has a partial equilibrium focus and seeks to prevent the costly failure of individual financial institutions, a macroprudential regulatory approach instead stresses the importance of general equilibrium effects and focuses on safeguarding the financial system as a whole. See Hanson et al. (2011) for a more detailed discussion.

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In this paper, we address the issue by examining the link between leverage and the

weighted-average cost of capital for U.S. bank holding companies (BHCs) from 1996 to 2012,

largely following the empirical framework used in Miles et al. (2012) in their study of U.K.

banks. The framework is based on the Hamada (1972) equation, which relates the beta of a

levered firm to that of its unlevered counterpart, and is derived by combining the Capital Asset

Pricing Model (CAPM) with the first two propositions of the MM (1958) theorem (see, e.g.,

Cohen (2007)). The CAPM and MM are equivalent under the assumption of a zero debt beta,

(βdebt = 0). While banks’ deposit liabilities are close to riskless due to deposit insurance, the

assumption of zero systematic risk for banks’ non-deposit debt may not be as appropriate.

However, it is important to point out that a zero debt beta does not mean that the probability of

default is zero, but instead means that any fluctuation in the value of debt is uncorrelated with

general market movements.4

We first estimate regressions in which a bank’s equity beta is a function of its leverage,

defined as its assets-to-equity ratio. In these regressions, we control for changes in both bank-

specific and systematic riskiness of bank assets.5 Using these results, we examine the impact of

a hypothetical doubling of equity capital on banks’ WACC. If we find evidence that the WACC

is invariant to the increase in equity, we say there is a 100% MM offset. Alternatively, if the

regression has no explanatory power, meaning that the cost of equity is unrelated to leverage,

4 As noted by Reilly and Brown (1997), the evidence on the usefulness of the CAPM as it relates to the

bond market is mixed. For example, Percival (1974) finds that bond betas are more responsive to the intrinsic characteristics of the bond issue (i.e., coupon, maturity, duration, and, call features) than of the issuer. Reilly and Joehnk (1976) find that average bond betas have no significant or consistent relationship with agency ratings, which are set to reflect unique issuer default risk. Reilly, Kao, and Wright (1992) find that interest rate risk has a dominant effect on bond price performance and largely negates the effects of differential, company-specific default risk. In contrast, Weinstein (1983) finds that a bond’s beta is related to firm characteristics (e.g., debt/equity ratios, variance of rate of return on assets) and to bond characteristics (e.g., coupon, term to maturity). Finally, Alexander (1980) finds that the regression results for bond betas are sensitive to the market index and the sample period used in estimating the market model for bonds.

5 The additional variables used in these regressions capture interest rate risk, credit risk, liquidity risk, and various measures of economic activity. Our empirical model is presented in Section IV.

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then the cost of equity remains constant as leverage declines and the WACC increases

accordingly. We term this a 0% MM offset. A partial MM offsets lies between these two

extremes.

Where our paper departs significantly from prior empirical literature based on the

Hamada equation is that we do not take for granted that βdebt = 0 for all banks. Rather, we

indirectly test the reasonableness of that assumption by estimating our model separately for

different asset-size classes of banks. Prior empirical papers (see, e.g., Junge and Kugler (2012),

Baker and Wurgler (2013), and Toader (2013)) that test the relevance of the MM framework for

banks do not do this.6 Assuming that βdebt = 0 in cases where it does not hold can lead to

erroneous conclusions regarding the extent to which the MM theorem holds. For example, we

find that the total MM offset for the 200 BHCs in our sample implied by calculating the

weighted-average MM offsets from separate regressions for different asset-size classes of banks

is almost four times as large as one gets from estimating a single regression for all 200 BHCs in

aggregate. That is, the total weighted-average MM offset across the different size banks is 83.3%

while the MM offset derived from a single regression on all 200 BHCs is 22.8%.7

We find that an empirical model which assumes βdebt = 0 performs well only for the

largest BHCs in our sample, i.e., the too-big-to-fail (TBTF) banks. For these banks—the six

largest U.S. BHCs, which hold roughly two thirds of U.S. banking assets—we find a full (100%)

MM offset for a doubling of equity capital, ignoring the tax effects associated with the banks

using less debt.

We find that the extent to which the MM theorem is observed to hold in our empirical

framework is strongly related to banks’ book asset size. In particular, the six largest U.S. BHCs

6 However, Miles et al. (2012) only look at the six largest U.K. banks. 7 See Table 7 for details.

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in our sample have an MM offset that ranges from 43.6% to 100% in their WACC with a

hypothetical doubling of Tier 1 capital, depending upon the regression specification and not

accounting for tax effects.8 However, our most robust regression specification yields a point

estimate of 100% for the MM offset. While a 100% MM offset may seem unrealistic given real

financial market frictions, it means that we cannot reject the hypothesis that the WACCs of the

six largest banks would not be materially affected by a doubling of equity capital.9

This is an important and new result. That is, contrary to the notion, advanced by some,

that, while the MM framework may be appropriate for banks, it can only weakly apply to the

largest banks due to the implicit TBTF guarantees which prohibit any cost-of-financing

adjustment through the debt channel from occurring.10 Nevertheless, we find the strongest

empirical support for MM among the largest banks. And this is not in spite of, but rather

because of, the TBTF guarantees. This is because the empirical model based on the Hamada

equation, which most empirical studies use, assumes that the debt beta is zero.11 Our results

suggest that this assumption may only be valid for the very largest TBTF banks, and most

especially for the post 2007-2009 crisis period when it became clear how well debt holders at the

very largest BHCs would be protected.

When we conduct our analysis for U.S. BHCs in different asset-size classes, we find that

the MM offset increases with banks’ asset size. In particular, excluding tax effects, banks with

less than $25 billion in assets have an MM offset of 6.1%, banks between $25 and $100 billion

8 For regression specifications that include annual time dummies to control for shifts in the average

riskiness of bank’s assets that are related to changes in economic conditions over time, the size of the MM offset is sensitive to the choice of reference year for the annual time dummies. We address this econometric issue later in the paper.

9 Even our smallest measured MM offset of 43.6%, which is based on a regression specification that uses 2008 as the reference year for the annual time dummies, still suggests that the WACC of the six largest U.S. BHCs’ would only increase by about 31 basis points. See footnote 7 for details.

10 See, e.g., Flannery (2012, pp. 242-243), as an example of this viewpoint. 11 This explains why other studies, which look at samples of hundreds of banks of all asset sizes lumped

together, find only a small MM offset.

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have an MM offset of 32.2%, and banks between $100 and $200 billion have an MM offset of

46.2%. These findings are not surprising, as smaller banks are likely to be more susceptible to

market frictions, such as information asymmetries, and tend to have higher costs of equity capital

(Gandhi and Lustig (2014)). Moreover, it is likely that the Hamada framework does not fully

capture how cost-of-financing adjustments are made for smaller banks because they have debt

that the market perceives as risky. For such institutions, some of the adjustment is likely to be

made through the debt channel, assuming they have non-zero debt betas. Hence, we should not

be surprised to find that smaller banks show a less than full MM offset when using the Hamada

equation as the empirical model for estimation because it forces all cost-of-financing adjustment

through the equity channel.

Preventing the possibility of adjustment through the debt channel doesn’t mean the MM

framework is inoperative; it just means that all of the adjustment must occur through the equity

channel. This translates to more conservative estimates of the extent to which MM is found to

hold for banks. For those TBTF banks whose debt is perceived as risk-free by the market due to

the implicit government guarantee, the Hamada framework, which relies entirely on the equity

channel is probably a good empirical approximation. However, for smaller non-TBTF banks, the

Hamada framework does not appropriately capture how MM actually works, because the market

perceives the debt of these banks to be risky. In this case, it would be necessary to have an

empirical model that has both an equity channel and a debt channel through which cost-of-

financing adjustments can occur.

Tax effects are often touted as a major reason why equity capital is expensive relative to

debt and why the MM framework will not operate for banks. Yet when we calculate this effect

using conservative assumptions regarding effective marginal corporate tax rates, we find the

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effect to be small. Specifically, we find that a hypothetical doubling of equity capital for these

same six U.S. BHCs would increase their WACCs marginally, on the order of 13 basis points

when tax effects are included.12 The main reason that doubling equity capital has such a limited

impact on the tax benefit of debt is because it only decreases leverage—and hence the debt tax

shield—by roughly 6.67% for the six largest BHCs in our sample.13 If equity capital was further

increased, the impact on the WACC would obviously increase as well, because the tax benefits

of debt would further decline.

Finally, we examine the steady-state impact of doubling bank equity on the level of real

GDP. We estimate a permanent 4 basis-points drop in the level of steady-state U.S. real GDP for

all BHCs in our sample.14 Given the significant social costs of large bank failures on the U.S.

economy, this decrease in real GDP seems a small price to pay for a substantially safer financial

system that would move banks closer to engaging in the socially-optimal amount of bank-

lending by eliminating those loans which, ex ante, only appear to be profitable with the lower,

government-subsidized funding rates.15

12 The total effect, including the tax effect is 22 basis points for all 200 BHCs in our sample, where the

weighted average MM offset is 80%. The pure tax effect for all 200 banks is 14.1 basis points. 13 The average total assets-to-equity ratio for the six largest BHCs in our sample is about 16, which

translates to a capital-to-assets ratio of 6.25% and a debt to assets ratio of 93.75%. Doubling equity capital to 12.5% reduces the debt by only about 6.67% to 87.5%. As noted in Admati (2013), Martin Wolf, economics editor of the Financial Times, in 2010 made the following comment with regard to the impact of increased capital requirements under Basel III for U.K. banks which seems quite a propos here: “tripling the previous requirements sounds tough until you realize that tripling almost nothing does not give you very much.” Consistent with this observation, Cecchetti (2014) notes that banks were required to hold virtually no equity capital whatsoever under Basel II. The reasons have to do with the treatment of hybrid instruments and intangibles. Under Basel II, banks were allowed to count as capital a range of hybrid instruments—such as subordinated debt and trust-preferred securities—which were more like debt than equity. Additionally, they were allowed to count in their capital calculations intangible assets such as goodwill, deferred taxes, and mortgage-servicing rights.

14 This estimate, which ignores tax effects (which we treat separately), results from the fact that our estimates for smaller banks show smaller MM offsets. It should also be pointed out that these estimates are based on a Cobb-Douglas production function, which, while it may be a simple approach, is straightforward to apply and produces estimates that can be compared to those of Miles et al. (2012), who use the same production-function approach.

15 Admati et al. (2011) and Admati and Hellwig (2013b, 2014) present a thorough and systematic treatment of these points. See also Cochrane (2013) and Myerson (2014) for an assessment of the arguments made in Admati and Hellwig (2013b).

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We caution against a too-literal interpretation of our findings. Rather, they should be

viewed as lending empirical support in favor of a substantial, if not full, MM offset, excluding

tax effects, especially for the TBTF banks. And while we find no evidence suggesting that

higher equity capital for TBTF banks would have a substantial adverse impact on their pre-tax

WACCs, we do not claim that the pre-tax WACC will always be found to be largely independent

of capital structure for when looked at outside the historical time period of this study. This

caveat notwithstanding, our results show that the MM theorem may be found to hold for banks to

a much greater degree than has been previously believed.

These findings raise the question as to why banks systematically operate with such high

leverage. Nevertheless, it is what they do, and when they do they impose potentially very high

costs on society as a whole.16 Additionally, our findings suggest banks’ motivations for high

leverage must be for reasons other than the failure of the MM framework to be operative.

Our results complement Admati et al. (2011) and Admati and Hellwig (2013b, 2014) by

showing that the potential private and social costs associated with having banks hold more equity

capital are small in comparison to the social benefits of a safer financial system.17 Coupling

their contribution that the MM framework when applied to banks must also include the social

cost of the risk of bank insolvency with our findings of substantial MM offset for the TBTF

banks, the policy response to the question of whether banks should hold more equity capital

becomes that much clearer.

16 Using data for more than 200 banks in 45 countries, Afonso et al. (2014) find higher levels of impaired

loans after an increase in government support. Their findings suggest that banks classified as more likely to receive government support engage in more risk taking.

17 See Thakor (2014), for a comprehensive review of the central issues surrounding the role played by bank capital in financial stability.

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II. Background and Literature Review It is frequently argued that MM cannot hold for banks because of the tax deductibility of

interest on debt, deposit insurance, and implicit TBTF guarantees. However, the tax

deductibility of interest payments is not unique to banks, is rather small according to our

calculations, and is best addressed with a change in tax policy, not capital policy. As such, tax

effects alone cannot be the reason that banks are so highly levered or else one would see such

levels in other industries.18, 19

With regard to the issue of deposit insurance and implicit TBTF guarantees, as noted

previously, we conjecture that our empirical model works for the largest banks because of, not in

spite of, these factors due to the underlying risk-free debt assumption in the Hamada equation.

Furthermore, as Miles et al. (2012, p. 7) observe, the existence of explicit or implicit guarantees

on bank debt does not nullify the underlying mechanism of MM:

The essence of MM is this: higher leverage makes equity more risky, so if leverage is brought down the required return on equity financing is likely to fall. This is true even if debt financing is completely safe— for example because of deposit insurance or other government guarantees. In fact the simplest textbook proofs of the MM theorem often assume that debt is completely safe.

18 Gropp and Heider (2010) present findings that suggest there are considerable similarities between the

capital structures of banks and non-financial firms over the period 1991 to 2004. However, recent data provide evidence that runs counter to their findings. In the retail sector, for example, Macy’s currently has a leverage of approximately 2.24 (retail industry average=1.87). In manufacturing, Intel has a leverage of 1.22 (semiconductor industry average=1.64), General Dynamics has 1.30 (aerospace and defense industry average=1.57), and Du Pont has 1.70 (chemical industry average=1.63). In transportation, Union Pacific has a leverage of 1.51 (road and rail industry average=2.24). These book leverage values are derived by dividing debt/equity by debt/assets, using publicly-available data as of August 2014 from the Fidelity Investments website. While these leverage data show considerable variation, they are significantly lower than we find in the U.S. banking sector, where the largest banks had an average leverage of 13.04 in 2012:Q4 (the most recent quarter in our sample dataset). And while the assets of banks are much more liquid than those of operating firms, in times of financial crisis, banks’ assets can become dramatically less liquid than previously thought possible.

19 Another possible explanation for why banks have higher leverage is the demand-for-liquidity argument offered in DeAngelo and Stulz (2013), according to which debt is a more efficient (i.e., less costly) form of financing for banks, even in the absence of explicit and implicit government guarantees.

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As we show below, it is precisely the presence of these guarantees that allows us to estimate the

extent to which the MM theorem is operative for the largest U.S. BHCs in our sample using an

application of a textbook proof consistent with the Hamada equation.

There are other reasons related to demand-for-liquidity factors and agency costs which

may explain why banks prefer to have high leverage. Most notable is that banks specialize in the

provision of socially-beneficial liquidity claims and are able to operate most efficiently at high

leverage, which provides them with a substantial funding advantage (see, e.g., DeAngelo and

Stulz (2013)).20 However, this argument is based on the notion that banks’ primary source of

funds are demand deposits, which is not the case for large U.S. banks. A bank’s management

may also be better disciplined by the threat of debt funding being withdrawn than by the

existence of shareholders who suffer first losses from any mismanagement of corporate funds

(see, e.g., Calomiris and Kahn (1991)). Admati and Hellwig (2013a), however, question the

plausibility and relevance of the claims made in banking theory that fragility in bank funding

associated with banks’ reliance on short-term debt is useful because it imposes discipline on

bank managers.

Finally, even if the WACCs of the largest banks are largely independent of leverage, the

impact of heightened equity capital requirements on the accounting profitability measures, such

as ROE, could be large. Given this, it is not surprising that banks, especially the largest banks,

oppose any proposed increase in required equity capital. Common arguments put forth by banks

are that loan rates would increase, investment would fall, and economic growth would

20 Flannery (2012) notes, however, that large banks’ liabilities do not differ qualitatively from those of any

other firm at the margin. In addition, see Admati (2013) and Pfleiderer (2014) for a critical assessment of the assumptions underlying the theoretical model developed in DeAngelo and Stulz (2013).

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consequently be adversely affected.21 However, an alternative reason for their resistance to

higher capital requirements might be that lower leverage would mean lower ROEs for

shareholders and managers, whose compensation is largely tied to short-term accounting-

performance metrics.22 While managers may benefit from a high-leverage strategy which drives

up ROE, shareholders may be disadvantaged as the increase in leverage raises risk and the

market’s required ROE, which will adversely affect share prices. As noted by Admati et al.

(2012), a focus on ROE encourages leverage and risk-taking, and exacerbates the inefficiencies

associated with high leverage.23

Buser et al. (1981) point out that the positive incentives for debt—tax deductibility and

fees earned on transactions services—would by themselves produce a zero-equity corner solution

but for regulatory oversight and the costliness of bankruptcy. When financial institutions are

very large, complex, and too big to fail, the tendency toward the corner solution increases.

Miller (1995) makes passing reference to the debate taking place during the 1990s about whether

the deposit insurance subsidy led to a corner solution where banks held as much debt as possible.

He countered this with the argument that it is not necessarily a subsidy. For some banks, at

certain times, the insurance is underpriced, while for other banks, or even the same banks at

different times, it might be overpriced. However, if we hadn’t learned it from Continental

Illinois in 1984, we certainly should have learned from the global financial crisis of 2007-09 that

the government guarantees to the largest financial institutions go well beyond any explicit

21 Other unintended adverse consequences of heightened capital requirements might include decreased

lending activity by banks, migration of financial activity to the shadow banking sector, and potential long-term output loss. See, e.g., The Clearing House (2012) and Hanson et al. (2011) for further discussion.

22 Based on a review of 10-K and 10-Q filings for the six largest U.S. BHCs in our sample, we find evidence suggesting that the underlying metrics used for executive compensation plans are largely based on ROE, or some variation of it.

23 Chu and Ma (2015) find evidence that management compensation packages that include stock options encourage managers of financial institutions to engage in excessive risk-taking.

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deposit insurance. And this additional insurance (i.e., that part that goes beyond explicit deposit

insurance), comes with a zero premium, so it is, by definition, underpriced.

The most extreme examples of government guarantees leading to corner solutions are

Fannie Mae and Freddie Mac. Despite government denials that it guaranteed their debt, the

markets clearly priced their debt as if it was backed by the full faith and credit of the U.S.

government, for Fannie and Freddie were always able to borrow at rates just slightly above that

of U.S. Treasuries.

During the roughly 15-year period leading up to the recent financial crisis, on-balance-

sheet leverage at Fannie Mae and Freddie Mac ranged between 20 and 40. This is more than

double what we would expect for commercial banks. Furthermore, if one includes in the

numerator the credit responsibilities associated with their off-balance-sheet guarantees, the

leverage ratio ranged between 50 and 100.24 This was indeed an extreme corner solution.

Through leverage, Fannie and Freddie were able to transform extremely low ROAs into very

high ROEs – sometimes as high as 30 percent, or even more.25

The problems associated with high bank leverage raise the question of whether such

levels of leverage are necessary. That is, does reducing leverage raise a bank’s WACC? This is

the issue that is explored in the empirical studies of Miles et al. (2012), Junge and Kugler (2012),

Toader (2013), the European Central Bank (ECB 2011), and Cline (2015), as well as in the

present paper. Miles et al. (2012) estimate an MM offset of 45% to 90% for a sample of large

24 White et al. (2013), “Leverage of GSEs.” 25 But, as Miller (1995) points out: “leveraging will indeed raise the expected earnings per share on the

equity, but not by enough to compensate the shareholders for the risk added by the leverage. All this, I might add is just standard M&M Proposition II stuff. An essential message of the M&M propositions as applied to banking, in sum is that you cannot hope to lever up a sow’s ear into a silk purse. You may think you can during the good times; but you’ll give it all back and more when the bad times roll around.”

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U.K. banks from 1997 to 2010.26 Junge and Kugler (2012) find a 36% MM offset for a sample

of Swiss banks from 1999 to 2010. Toader (2013) studies a sample of large European

commercial, universal and investment banks over the period 1997 to 2011 and estimates a 42%

MM offset. The ECB (2011) examines a sample of 54 large international banks from 1995 to

2011 and finds an MM offset between 41% and 73%.27 Finally, Cline (2015) studies a sample of

the 54 largest U.S. banks over the period from 2001 to 2013 and finds an MM offset of less than

50%.

Our paper differs from these studies in that we focus on 200 large U.S. BHCs and

examine the empirical support for the MM theorem across banking institutions of different asset

sizes. Importantly, we show that the impact of heightened equity requirements will have the

smallest effect on the WACC of the largest U.S. BHCs, which are exactly the banking

institutions that pose the greatest systemic risk and have the greatest influence on the U.S.

economy.

Recent studies using alternative methodological approaches to related issues have shed

light on the effect of reducing leverage for U.S. banks, and thus, indirectly, on the extent to

which the MM theorem may or may not apply. In their study of U.S. BHCs over the period 1976

to 2007, Kashyap et al. (2010), for example, find that the long-run, steady-state impact on loan

rates lies in a range of 25 to 45 basis points for a ten percentage-point increase in banks’ capital

requirements from tax effects alone based on the assumption that MM strictly holds.28 They also

26 The six banks in Miles et al. (2012) are: Lloyds TSB (subsequently Lloyds Banking Group), RBS,

Barclays, HSBC, Bank of Scotland, Halifax (and subsequently HBOS). 27 The sample of 54 large international banks used in ECB (2011) includes nine large U.S. banking

institutions: Bank NY Mellon, Bank of America, Citigroup, Goldman Sachs, JP Morgan, Morgan Stanley, PNC Financial, US Bancorp, and Wells Fargo.

28 We should note that our doubling of equity capital for the six largest U.S. BHCs results in only a 6 percentage point reduction in debt financing, while Kashyap et al. (2010) use a 10 percentage point reduction. Therefore, to make a more apples-to-apples comparison, we multiply our pure-tax-effects result of 13.2 basis points

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provide weak empirical support for the MM framework, finding that banks’ equity beta and

stock return volatility are decreasing in the ratio of equity to assets.29 Effectively, they claim to

have empirical support for the fact that MM holds without directly addressing the extent to which

it holds, which is exactly the issue we address in this paper.

Baker and Wurgler (2013) study a sample of 3,952 U.S. BHCs over the period 1971 to

2011. In their paper, they note that the equity capital of better-capitalized banks has lower

systematic risk as measured by the banks’ equity betas—which is consistent with the MM

framework—and that these banks also have lower idiosyncratic risk. Their evidence suggests

that lower-risk banks have higher stock returns, consistent with a low-risk anomaly for banks

that has also been documented for non-banking firms. This finding of a low-risk anomaly is

somewhat of an alternative finding compared to ours in that it suggests that the impact of

systematic risk on the cost of capital might be small and depends on institutional factors and the

behavior of investors. The behavior of investors is potentially important and centers around

whether they will actually require a lower expected return on equity when faced with the lower

risk resulting from a decrease in leverage, as the MM theorem predicts.30 However, the focus of

their paper is not an examination of the extent to which MM applies to banks, but rather to

examine whether the low-risk anomaly, which had been found in an earlier paper for

nonfinancial firms, also obtains for banks.31 Other significant differences between our paper and

by 10%/6%, which yields a 22 basis points increase in WACC resulting from a 10 percentage point reduction in debt financing. This result is close to the low end of what Kashyap et al. found for pure tax effects.

29 Their support for the MM theorem is weak because while their estimated coefficient on leverage is statistically significant, it is also quite small. More importantly, the intercept is quite large and significant suggesting the percent MM offset would be quite small.

30 Hamada (1972) finds evidence that the cost of equity capital increases with leverage, a result which is consistent with the MM cost-of-capital propositions.

31 The low-risk anomaly refers to the empirical pattern that stocks with higher beta or higher idiosyncratic risk tend to earn lower returns on a risk-adjusted or raw basis. See Baker and Wurgler (2013) for a more detailed discussion as well as Cochrane (2014).

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theirs include: we have the 200 largest U.S. BHCs in our sample while they use a sample of

nearly 4,000 U.S. banks and BHCs: and we stratify by banks’ asset size and they do not.

While both Kashyap et al. (2010) and Baker and Wurgler (2013) touch indirectly on what

we address here, neither paper focuses directly on the extent to which the MM theorem applies to

banks. In Kashyap et al. (2010), the findings are that there will be impacts from efforts to

achieve heightened equity capital standards, that these impacts will be greater if banks have to

get there by raising additional equity funding from the capital markets, and that the impacts

might force more banking activities into the shadow-banking sector. Whereas, in Baker and

Wurgler (2013), the primary finding is that the low-risk anomaly applies to banks.

In other related work, Mehran and Thakor (2011) develop a dynamic theoretical model of

bank capital structure in an acquisitions context which predicts that the total target-bank value

(and its components) and the target’s proportion of equity funding are positively related. Using a

sample of all completed M&A deals in the banking industry announced over the period 1989 to

2007, they find that a target bank’s ratio of goodwill to total assets and its ratio of equity capital

to total assets are significantly positively correlated in the cross-section, a result that the authors

argue is inconsistent with the MM leverage indifference theorem. While Mehran and Thakor’s

theoretical model is quite interesting, we do not agree that their empirical model addresses the

issue of MM value irrelevance for banks in the manner they suggest. Instead, we argue that their

results show that acquiring banks are willing to pay more for target banks that are worth more.

We should point out that we do not attempt to prove the validity of the MM theorem.

The empirical question that we address is rather, the degree to which MM’s impact is muted by

the usual frictions plus the presence of government guarantees in U.S. banking. That is, what is

its practical applicability to U.S. banking, and how appropriate is the joint MM/CAPM

16

framework under the zero-debt-beta assumption in measuring the extent to which it is found to

hold for U.S. BHCs? Also, while we examine the steady-state relation between leverage and the

banks’ WACC, we do not estimate the potential costs associated with raising equity capital,

which can be significant because of the flotation and underwriting costs involved (Miller

(1995)).32 Raising equity can also be costly to existing shareholders due to the debt overhang

problem (Myers (1977))33 as well as asymmetric-information issues (Myers and Majluf (1984)).

It is important to point out in this context that compliance with the new Basel III capital

standards has mostly been achieved through increased retained earnings rather than issuing new

equity shares or deleveraging balance sheets.34 Finally, we do not address issues related to the

length of the adjustment period for banks to transition to higher equity capital levels or to what

these higher levels of equity capital should be.35

III. Data

A. Sample Selection We obtain data from several sources. Using quarterly FR Y-9C BHC data available on-

line from the website of the Chicago Federal Reserve Bank and daily stock return data from the

Center for Research in Securities Prices (CRSP), we create an unbalanced panel dataset of

publicly-traded U.S. banks over the period 1996:Q1 to 2012:Q4. For each BHC, we estimate

quarterly equity betas by regressing each bank’s daily stock-market returns on the daily CRSP 32 As noted by Miller (1995, p. 484), “The M&M Propositions are ex ante propositions. They are

concerned with having equity, not with raising equity.” 33 See Admati et al. (2012) for a detailed analysis of debt overhang and its implications for bank capital

regulation. As a result of debt overhang, shareholders have incentives to resist reductions in leverage that make the resulting debt safer. This shareholder resistance occurs even without government subsidies of debt, but it is made worse by government subsidies.

34 As noted by Cecchetti (2014), two thirds of the increase in equity capital levels at banks worldwide associated with the new Basel requirements has been achieved through retained earnings, while one third has been achieved from other sources, primarily net capital issuance. See also Cohen and Scatigna (2014).

35 See Hanson et al. (2011), Birchler and Jackson (2013), and Calomiris (2012), for further discussion of the key issues involved in the debate over the effects of heightened bank-equity capital requirements.

17

equal-weighted market-index returns for each quarter. In order to control for nonsynchronous-

trading effects, we use a Scholes-Williams approach in the estimation of the one-factor market

models for each bank.36

The equity betas for each bank are matched to their corresponding BHC financial data

through a mapping of CRSP ID-PERM numbers and the Federal Reserve System’s RSSID

numbers for BHCs. After this initial matching, we found a number of banks that in some

quarters had equity betas but no financial variables and a few banks that had financial variables

but no equity betas. The first issue involves FR Y-9C reporting thresholds, and the second

involves M&A and/or resolution activity. To address the first issue, we set an asset-size criterion

according to which any BHC which never attains $2 billion in total book assets at any time in its

respective time series is dropped from the database.37 The second issue is addressed by

examining what happens to the BHC in each particular case.38

To produce the final dataset, we apply several additional filters to the data. We drop

those institutions that have: any instance of negative leverage; any instance of leverage that

seemed to be excessively large (greater than 66.67);39 or fewer than five consecutive quarterly

36 See McInish and Wood (1986) for a discussion of analytical and empirical evidence that equity betas of

securities that trade less (more) frequently than the market index used in estimation of the market model are downward (upward) biased.

37 The Federal Reserve has an asset-size threshold, below which BHCs are not required to file FR Y-9C schedules. This size threshold was raised from $150 million to $500 million in the first quarter of 2006. Given this change, we decided to set the minimum-size criterion sufficiently above the Fed’s reporting threshold in order to avoid gaps in the time-series data that would be caused by banks that were too close to the reporting threshold and were required to report one quarter and not the next and so forth. By applying this size criterion, we were able to mitigate the problem of missing FR Y-9C data and also retain a large number of smaller banks in the final dataset.

38 In all, there are five BHCs with data gaps where the institution had financial data but no equity beta. The discrepancy stems from the fact that, when a BHC is merged out of existence, the Federal Reserve may continue to collect BHC data for some period of time (often a year or more) after a BHC ceases to exist as a separate entity from an ownership point of view. This probably occurs because it takes some time for the BHC in question to merge accounting systems and to officially retire the holding-company charter of the former institution. To address this, we determined the merger consummation date for each case and deleted any quarters of FR Y-9C data after that date.

39 The limit on leverage is set at this level because a higher number would mean that the BHC would have less than 1.5% in its Tier I leverage ratio, which would mean that some sort of government intervention should already be underway.

18

observations. This yields an unbalanced panel dataset consisting of 300 U.S. BHCs (before

imposing any other restrictions) over the period 1996:Q1 to 2012:Q4. We subsequently drop

three foreign institutions, impose the restriction that the institution has to be primarily a bank,40

and finally require that the bank must have existed in 2005:Q4 because that is the reference

quarter for dividing our sample of BHCs into groups based on asset size.41 After imposing all of

these requirements, we have a total of 200 U.S. BHCs in our final sample.

B. Definition of Variables The dependent variable in all of our regressions is the Scholes-Williams equity beta that

is estimated using a one-factor market model with the CRSP equal-weighted index as the market

index. The primary independent variable is leverage (i.e., total assets/Tier 1 capital). We also

examine the possible use of three additional bank-specific independent variables: the loan-loss-

reserve ratio, liquidity ratio, and ROA. As suggested by Miles et al. (2012), these three

additional explanatory variables can be used to capture changes in the bank-specific risks of

assets over the sample period. The loan-loss-reserve ratio is measured as total loan-loss reserves

divided by assets; the liquidity ratio is measured as cash and balances due from depository

institutions plus available-for-sale securities divided by total liabilities minus equity; and, the

ROA is measured as net income divided by total assets.

40 There are seven institutions with financial holding company charters that are, for the most part,

something other than a bank. These institutions are: Met Life, Charles Schwab, Goldman-Sachs, Franklin-Templeton Investments, American Express, Morgan-Stanley, and Stifel Financial.

41 We choose the fourth quarter of 2005 as the reference point for determining asset size because it is substantially before the advent of the recent financial crisis. In our analysis, we consider the financial crisis as beginning after the second quarter of 2007. However, that was when the situation began to deteriorate badly in the U.S. financial sector, and some might argue—with some justification—that the early signs of the crisis were beginning to show up in early 2006. Thus, 2005:Q4 seemed a conservative reference point as it comes before the earliest serious indications of a financial crisis.

19

C. Summary Statistics Table 1 presents descriptive statistics for the dependent and independent variables for our

final sample of U.S. BHCs. Panel A reports means, medians, standard deviations, and other

descriptive statistics of these variables for all 200 BHCs. Panel B reports these statistics for the

six largest BHCs with assets greater than $200 billion as of 2005:Q4.42 And Panel C reports

these statistics for the 194 BHCs with assets under $200 billion as of 2005:Q4. The summary

Table 1-A. Descriptive Statistics for All 200 BHCs

Table 1-B. Descriptive Statistics for BHCs with Assets over $200 Billion (6 BHCs)

Table 1-C. Descriptive Statistics for BHCs with Assets under $200 Billion (194 BHCs)

42 The six largest U.S. BHCs are: Citigroup, JP Morgan Chase, Bank of America, Wells Fargo & Co.,

Wachovia, and U.S. Bancorp.

Variable Obs Mean Minimum Maximum Std Dev Lower

Quartile Median Upper

Quartile Scholes-Williams Beta Leverage Loan Loss Reserve Ratio Liquidity Ratio ROA

10577 10577 10577 10577 10577

0.866 12.404 0.010 0.230 0.006

-2.777 4.969 0.000 0.006

-0.172

14.561 48.197 0.061 0.941 0.037

0.713 2.600 0.005 0.119 0.008

0.406 10.635 0.007 0.148 0.003

0.817 12.138 0.009 0.210 0.006

1.252 13.873 0.011 0.287 0.009

Variable Obs Mean Minimum Maximum Std Dev Lower

Quartile Median Upper

Quartile Scholes-Williams Beta Leverage Loan Loss Reserve Ratio Liquidity Ratio ROA

359 359 359 359 359

1.093 15.975 0.011 0.143 0.007

-0.117 10.427 0.005 0.037

-0.044

6.831 25.531 0.024 0.267 0.022

0.750 2.323 0.004 0.045 0.006

0.582 14.607 0.008 0.112 0.003

0.983 16.022 0.010 0.145 0.006

1.466 17.296 0.013 0.171 0.010

Variable Obs Mean Minimum Maximum Std Dev Lower

Quartile Median Upper

Quartile Scholes-Williams Beta Leverage Loan Loss Reserve Ratio Liquidity Ratio ROA

10218 10218 10218 10218 10218

0.858 12.279 0.010 0.233 0.006

-2.777 4.969 0.000 0.006

-0.172

14.561 48.197 0.061 0.941 0.037

0.711 2.519 0.005 0.119 0.008

0.398 10.574 0.007 0.151 0.003

0.811 12.058 0.009 0.214 0.006

1.246 13.730 0.011 0.291 0.009

20

statistics show that the six largest U.S. BHCs have an average leverage of 15.98, while the

smaller BHCs have a much smaller average leverage of 12.28.43 Later in the paper, we present

evidence that shows that average leverage is monotonically increasing in BHCs’ asset size (see

Table 6). The difference in leverage between the smaller and largest BHCs is consistent with the

work of Kashyap et al. (2010), who note that banks with different risk profiles may choose

different capital structures. Also, the minimum and maximum values for the bank-specific

variables show some extreme values—especially with respect to equity beta and leverage—that

we address before performing any estimations.44

IV. The Model Linking Leverage and Equity Beta

A. CAPM and the MM Theorem In this section, we derive an expression that shows the relation between a firm’s asset (or

unlevered) beta and its equity beta.45 This expression serves as the empirical model for our

regression analysis of the link between leverage and the average cost of capital. By definition,

the market value of a levered firm equals the market value of its debt plus the market value of

equity. And, according to Modigliani and Miller (1958), the value of a levered firm can be

written as the value of unlevered firm plus the present value of the tax shields due to debt

financing.

Assuming perpetual debt and ignoring any bankruptcy or distress costs associated with

debt financing, we can equate these two expressions and write the following:

(1)uD E V tD+ = + 43 A t-test shows that the average leverage of the smaller BHCs is significantly less than that of the six

largest BHCs with a p-value = 0.0001. 44 The summary statistics presented in in Table 1 are for the raw data for the 200 BHCs in our final panel

dataset. That is, they have not yet been winsorized to dampen the effect of outliers, as discussed in Section III. 45 The material in this section on the link between the CAPM and MM frameworks is based on Higgins

(2004).

21

where D is interest-bearing debt, E is the market value of equity, Vu is the value of the firm

without any debt, and t is the marginal corporate tax rate. Given that the beta of a portfolio is the

weighted average of the betas of the individual assets that make up the portfolio, we can apply

this insight to both sides of equation (1) and write:

(2)ud e u ITS

u u

VD E tDD E D E V tD V tD

β β β β+ = ++ + + +

where βd is the debt beta, βe is the equity beta, βu is the beta of the unlevered firm, or equivalently,

the firm’s asset beta, and βITS is the beta of the firm’s interest tax shields. If we assume that the

risk of interest tax shields equals the risk of the firm’s unlevered asset cash flows, so that βITS = βu

= βa, we can then simplify and write the following:

( ). 3a e dE D

D E D Eβ β β= +

+ +

Equation (3) shows that the effect of the relative amount of equity and debt a firm has

upon its asset beta works through two channels: the equity channel and the debt channel. If we

then solve for βe, we obtain:

( ). 4e a dD E D

E Eβ β β+

= −

Finally, by imposing the restriction that the systematic risk associated with bank debt is

zero (i.e., dβ = 0), we obtain:

( ), 5e aD E

Eβ β +

=

where D EE+ denotes leverage in the form of the assets-to-equity ratio. Equation (5) shows the

link between the CAPM and the MM theorem and implies that eβ will increase linearly with

leverage, assuming that the systematic risk of debt is zero.

22

The MM theorem primarily consists of two basic propositions with respect to the cost of

capital and capital structure.46 According to Proposition I, the value of a firm is independent of

the firm’s capital structure under the assumption of complete and perfect capital markets.

According to Proposition II (and assuming that Proposition I holds), the cost of equity capital for

a levered firm is a linear increasing function of the debt/equity ratio.47 The CAPM and MM

Proposition II are equivalent under the assumption of a zero debt beta. Thus, it is important to

note that equation (5) does not hold when βd is not equal to zero.48 Equation (5) is a variation of

the well-known Hamada equation.49

Hamada (1969) shows that MM Propositions I and II, for both the no-tax and with-tax

versions, are valid when restated in the CAPM framework (see, e.g., Percival (1973)). While

Haugen and Pappas (1971, 1972) and Imai and Rubenstein (1972) show that the MM

propositions hold in the MM framework with risky debt, the Hamada equation seems to be more

practical for empirical work on banks in a world in which deposit insurance and implicit too-big-

to-fail guarantees come into play.50

46 The MM theorem actually consists of three propositions, although Propositions I and II are the most

well-known. Proposition III is a rule for the optimal investment policy by a firm, assuming that the first two propositions hold. See Modigliani and Miller (1958, pp. 288-296) for a detailed discussion of Proposition III.

47 More specifically, the cost of equity capital for a levered firm is equal to the constant overall cost of capital plus a risk premium, where the risk premium due to debt is equal to the spread between the overall cost of capital and the cost of debt multiplied by the firm’s debt/equity ratio.

48 The zero-debt-beta assumption is referred to in the literature on this subject as the assumption of “risk-free” debt. We attempt to avoid the use of that term—using an assumption of a zero debt beta instead—so as not to cause confusion. It should be noted that the risk-free-debt assumption in this context does not mean what the term literally implies: that the bank’s debt is entirely free of default risk. It simply means that the systematic risk of a bank’s debt is unaffected by general market-wide movements.

49 In corporate finance, Hamada’s equation is used to separate the financial risk of a levered firm from its business risk. The equation combines the MM theorem and CAPM and is used to determine the levered beta and optimal capital structure of firms. The Hamada equation is based on three key assumptions: tax shield values are for constant debt, i.e., the dollar amount of debt is constant over time; the debt beta is zero; and the discount rate used to calculate the tax shield is equal to the cost of debt capital, i.e., the tax shield has the same risk as debt. Based on the first and third assumptions, the tax shield then is proportional to the market value of debt, i.e., the tax shield = t*D, where t is the marginal corporate tax rate. The standard presentation of the Hamada equation includes the marginal corporate tax rate and uses the debt/equity ratio instead of leverage.

50 See Rubenstein (1973) for a detailed discussion of the theoretical work on the MM framework with risky debt.

23

Equation (5) is the basis for our empirical model, which is as follows:

, , 1 , 1 , , ,' ' , (6)i t i t i t t i t i t i i tbLeverage x c z d u uβ a e− −= + + + = +

where for bank i = 1 to N and time period t = 1 to T and where b is the banks’ asset beta, x’i,t-1 is

a vector of regressors that includes additional bank-specific variables, z’t is a vector of regressors

that includes either annual time dummies or macroeconomic variables, and c and d are vectors of

other regression parameters.

Our estimates for the empirical relevance of the MM theorem depend on the results of

this model. In particular, a full MM offset (100%) requires a significant positive coefficient on

lagged leverage (i.e., the estimated asset beta) and an intercept equal to zero. That is, if these

two conditions hold, then we conclude that the WACC is independent of capital structure.

Alternatively, a significant and positive intercept would suggest that there will be a partial MM

offset, meaning that banks’ WACCs will increase as leverage declines.

Equation (6) is the general, linear panel-data regression equation for banks’ estimated

Scholes-Williams equity beta (the dependent variable) at time t. We estimate several different

specifications of this regression model. Either annual time dummies or macroeconomic

variables can be used to attempt to capture those factors that affect the average riskiness of bank

assets from year to year. Additionally, in order to attempt to capture the impact of differences in

the riskiness of portfolios specific across banks and over time, we examine the possible use of

three bank-specific variables in addition to leverage, namely the loan-loss-reserve ratio, liquidity

ratio, and ROA. In order to address the potential issue of endogeneity between the systematic

24

risk of equity and bank-specific risk factors, the Scholes-Williams beta is regressed on one-

quarter lagged values of leverage and the additional bank-specific variables.51

In equation (6), the regression error, ui,t, consists of a stochastic, unobservable bank-

specific effect, αi , and an idiosyncratic disturbance, εi,t. We considered three different panel-data

regression models and estimation approaches: pooled ordinary least-squares (OLS), fixed-effects

(FE), and random effects (RE). In choosing between the FE and RE estimators, the important

econometric issue is whether the bank-specific effects, αi, are correlated with the explanatory

variables. The FE estimator produces consistent parameter estimates even if bank-specific

effects are correlated with lagged leverage and the other regressors in the vector x in equation

(6). The RE estimator yields consistent parameter estimates if the αi are distributed

independently of the regressors, in which case it is the preferred estimator because it is more

efficient than the FE estimator.

In the end, we use an FE regression model, although the RE and OLS regression models

produce qualitatively similar results (available upon request). An F-test indicates that there are

significant unobserved fixed bank-specific effects so the OLS estimates should not be used.

Although a Hausman test shows that RE estimates can be used, we focus on the FE estimates

since they are always consistent.

Before estimating any regressions, all bank-specific variables are winsorized in order to

mitigate the impact of extreme data values on our results. 52 Also, in order to control for the

possibility of cross-sectional or spatial dependence among the banks in our sample, we use

51 The problem of endogeneity could arise if the bank-specific risk variables and a bank’s equity systematic

risk are simultaneously determined by a bank’s management. For example, the bank may set a target risk profile in terms of its equity beta and then choose the leverage that is consistent with the target.

52 Specifically, all the variables are winsorized (or recoded) at the 99.5 and 0.5 percentiles of the variables’ univariate distributions. This addresses a potential for measurement error that could adversely affect regression estimates. See Bollinger and Chandra (2005) for a discussion and analysis of the advantages and disadvantages of using winsorized data in regressions.

25

Driscoll-Kraay (1998) robust standard errors as modified by Hoechle (2007) for unbalanced

panels in calculating the t-statistics. As noted by Driscoll and Kraay (1998), cross-sectional

dependence is a problem for panel datasets in which cross-sectional units are not randomly

sampled. Moreover, robust standard-error estimation techniques that fail to account for the

presence of spatial correlations will yield inconsistent estimates of parameters’ standard errors.53

The first regression equation we estimate—which we refer to as the basic regression

model—involves regressing bank i’s Scholes-Williams beta on lagged leverage and a set of

annual time dummies. The time dummies are meant to capture time-varying heterogeneity in

the average riskiness of bank assets that varies from year to year. Since we include an intercept

in the estimated regressions, it is necessary to drop one of the annual time dummies in order to

avoid the dummy variable trap. We use 2012 as the reference year for the annual time

dummies.54 Given the importance of the intercept in the interpretation of the MM results, we

conduct a careful analysis in the following sections to ensure we have an appropriate estimate of

the intercept that is not sensitive to the choice of reference year for the annual time dummies.

In addition to the basic regression model, we also estimate an extended regression model.

This extended regression model involves regressing bank i’s Scholes-Williams beta on lagged

leverage, a set of annual time dummies, and additional bank-specific variables. Miles et al.

(2012) consider including three additional bank-specific control variables, i.e., the loan-loss-

reserve ratio, liquidity ratio, and ROA, in their regressions. However, they find these variables to

be statistically insignificant for the six largest U.K. banks in their sample and therefore drop

these variables from their final regressions. Recent work by di Biase and D’Apolito (2012)

53 See Driscoll and Kraay (1998, p. 549). Their standard errors are based on the heteroskedasticity and

autocorrelation consistent standard errors of Newey and West (1987) and Andrews (1991) but with an adjustment for spatial dependence.

54 In Miles et al. (2012), the reference year for the annual time dummies in their regressions is 2010.

26

presents evidence that two of these variables, i.e., the loan loss reserve ratio and liquidity ratio,

are significant predictors for bank-equity betas in the Italian banking system.

While previous empirical studies on the determinants of equity beta have included ROA

as an additional bank-specific predictor, the use of ROA as an explanatory variable in our

extended regression model is problematic, since it is highly correlated with leverage. It is well-

known that ROE is ROA adjusted for financial leverage in a firm’s capital structure.55

Accordingly, ROE can be written as ROA multiplied by leverage, an expression that can be

solved for ROA as a function of leverage. Due to the high collinearity between ROA and

leverage, our extended regression model uses only the loan-loss reserve and liquidity ratios as

additional bank-specific predictors.

Table 2 presents FE estimates for the basic and extended regression models for the six

largest U.S. BHCs.56 The results are reported for the full sample period and for two sub-periods,

1996:Q1-2007:Q2 and 2007:Q3-2102:Q4. These two sub-periods correspond to a pre-crisis

period and a combined crisis and post-crisis period.57 The regression results show that the

55 According to the DuPont identity, a popular formula for decomposing ROE into its core components,

ROE = (net income)/(total assets) x (total assets)/(shareholders’ equity), where (net income)/(total assets) is ROA. Therefore, ROA can be expressed as ROE divided by leverage. See Saunders (2000, pp. 26-27) for further discussion of the DuPont identity.

56 We use the xtscc command in Stata to produce the coefficient estimates and Driscoll-Kraay robust standard errors for unbalanced panel datasets. For the results presented in the tables, we use a lag length of 1 for the moving average in producing the robust standard errors. We also use the default lag length in the xtscc procedure as well as a lag length of 4 to calculate Driscoll-Kraay robust standard errors, but the results are largely the same. These results are available upon request.

57 We define the start of the recent financial crisis to be in the third quarter of 2007. A key event in our view was the second-quarter earnings announcement of Countrywide Financial in July 2007—at the beginning of the third quarter. Countrywide reported serious losses overall, especially in its subprime portfolio and stated that it expected to see these losses continue well into the following year. This event rattled the markets. Then, at the end of the third quarter in 2007, Washington Mutual, which had reported record earnings in the second quarter, revealed that it too had had the same experience as Countrywide.

27

estimated coefficient on leverage is much larger in the second sub-period, suggesting a much

stronger link between leverage and equity beta during the recent crisis and afterwards.58

In Table 2, we also examine whether it is important to control for other predictor

variables that capture bank-specific risk characteristics in assessing the empirical link between

equity beta and leverage. The results for the basic regression model are contrasted with those for

the extended regression model that includes the loan-loss-reserve ratio and liquidity ratio to

assess whether these bank-specific factors are significant determinants of the systematic equity

risk of the six largest U.S. BHCs. The coefficient estimate on the loan-loss-reserve ratio is

Table 2. Equity Beta and Leverage FE Basic and Extended Regressions with Annual Time Dummies: Results for Full Time Period and Sub-Periods (Levels) for BHCs > $200B

Basic Regression Model Extended Regression Model 1996:Q1-

2012:Q4 1996:Q1- 2007:Q2

2007:Q3- 2012:Q4

1996:Q1- 2012:Q4

1996:Q1- 2007:Q2

2007:Q3- 2012:Q4

Leveraget-1 0.0644 0.0291 0.1367 0.0622 0.0193 0.1042 (2.48) (1.14) (2.91) (2.40) (0.81) (2.37) Intercept 0.3575 0.1301 -0.5538 0.4135 0.0862 -0.1007 (1.03) (0.30) (-0.90) (1.17) (0.22) (-0.16) LLRt-1 — — — -0.0420 -0.1789 -0.2193 — — — (-0.48) (-2.59) (-1.12) Liquidityt-1 — — — -0.0064 0.0473 -0.0693 — — — (-0.19) (1.51) (-1.15) Within R-Sq. 0.486 0.453 0.456 0.4866 0.466 0.470 ∆WACC — No MM 0.53% — 0.68% 0.31% — 0.52% ∆WACC — w/MM 0.00% — 0.00% 0.00% — 0.00% % MM Offset 100% — 100% 100.0% — 100%

Notes: The dependent variable is the Scholes-Williams equity beta, estimated using the CRSP equal-weighted market index. Leverage is measured as the ratio of total book assets to Tier 1 capital. The intercept in the regressions captures the average unobserved heterogeneity across banks. The results are for fixed-effects panel regressions using one-way (i.e., bank) classification and which include annual time dummies where the reference year is 2012. The numbers in parentheses are t-statistics computed using robust standard errors based on the Driscoll-Kraay method as modified by Hoechle (2007) to adjust for cross-sectional dependence as well as autocorrelation and heteroskedasticity for an unbalanced panel dataset. Our sample includes the six U.S. BHCs which had at least $200 billion in total assets as of 2005:Q4. This list consists of six BHCs, which falls to five near the end of our sample period due to Wells Fargo’s acquisition of Wachovia. The percent MM offset is not reported when neither the intercept nor the leverage coefficient estimate is significant.

58 We use a 5% one-tailed test for the leverage coefficient estimate (critical value = 1.65). We have a

strong prior that this effect must be positive. With regard to the intercept, while our prior is less strong, we are really only interested in the question of whether or not there is a significant positive intercept. Therefore, we use a one-tailed test. The issue of whether there is an MM offset depends on whether there is a significant positive intercept and, if so, how large it is. For this reason, we set a fairly low threshold for significance of the intercept—10%, one-tailed (critical value =1.28). We want to be conservative in assessing the degree to which there is evidence of an MM offset for banks. Setting a high threshold for the significance of the intercept would bias the results toward finding a full MM offset.

28

negative and significant for the pre-crisis sub-period, but insignificant for the full time period

and crisis/post-crisis sub-period. The coefficient estimate on the liquidity ratio is significant only

in the pre-crisis period. While, overall, the results for the extended regression model presented

in Table 2 are mixed, the inclusion of the two additional bank-specific control variables produces

a marginal increase in the within R-squared coefficients for all three time periods.

Comparing our regression results to those presented in the literature, there is a substantial

difference between the estimated asset betas of the six largest U.K. banks reported by Miles et al.

(2012) and what we find for the six largest U.S. BHCs. In particular, Miles et al. find a

statistically significant coefficient of 0.03 (t = 3.49) on lagged leverage (the estimated asset beta

for U.K. banks). Their regression model is most comparable to our basic regression model

presented in the first column of Table 2, where we find a statistically significant leverage

coefficient of 0.06 (t = 2.48) on lagged leverage (the estimated asset beta for U.S. banks). That

is twice as large as that found for the six largest U.K. banks.

Several factors may contribute to the different results for the estimated asset betas.

Banks may choose to hold low-risk assets (at least in their ex ante perception) and then use

leverage to obtain an acceptable return on equity.59 In the case of the U.K. banks, for example,

the estimated asset beta of 0.03 generates an average equity beta that is close to 1 given that the

average leverage of the six largest U.K. banks was around 30 for the time period used by Miles

et al. (2012). In the case of the U.S. banks, the estimated asset beta of 0.06 generates an average

equity beta close to 1 given that the average leverage of the six largest U. S. banks was around

16.7 over our sample period. Another factor that could possibly account for a difference in asset

betas might be that the status of bondholders may have been weaker in the case of the U.K

59 This trade-off is suggested in Choi and Richardson (2012). It is also what Fannie Mae and Freddie Mac

did for many years in the United States and is what Miles et al. (2012) suggest that U.K. banks were doing in their study.

29

financial-crisis resolutions.60 Further, as we will see in Section V for U.S. BHCs’ asset-size

classes below our too-big-to-fail threshold of $200 billion, we find estimated asset betas that are

very similar to those found by Miles et al. (2012) for the six largest U.K. banks.

B. Leverage and the Weighted Average Cost of Capital Next, we calculate the implied impact of changes in leverage on the WACC using the

regression results in Table 2. We ignore the effects of taxes for the time being.61 The WACC

can be expressed as:

(1 ) . (7)e fE EWACC R R

D E D E= + −

+ +

And by the CAPM:

, (8)e f e pR R Rβ= +

where Rf is the nominal risk-free rate and Rp is the equity-market risk premium. Substituting the

coefficient estimates for the intercept and asset beta (i.e., the coefficient estimate on lagged

leverage) from the basic regression model, we obtain the estimated required return on equity

where

ˆˆ( ) . (9)e f pR R a bLeverage R= + +

60 The U.K. government announced its bank rescue package in October of 2008. This was a fairly

complicated operation, and it differed in significant ways from the U.S. bailout program announced the week before—the Emergency Economic Stabilization Act of 2008. One major difference was the fact that the U.K. government took a large equity stake in the banks being reorganized—in particular for the Lloyds-HBOS-RBS combination. As a condition of that equity stake, the government prohibited any payments to bondholders for a specified period of time, and until sufficient equity capital had been built up, leaving some uncertainty as to when such payments might resume. In contrast, in the forced resolutions of U.S. commercial banks, public bondholders were simply paid off. And if that difference were not enough, the Dodd-Frank Act of 2010 appears to formalize too-big-to-fail and the notion of paying off public debt-holders. These differences in the actions taken by the U.S. and U.K. governments to address the recent crisis have implications for whether the Hamada framework that relies only the equity channel (since the debt beta is assumed to be zero) is a good empirical approximation for the largest U.K. banks in depicting how the MM framework works.

61 In our derivation of equation (5), we assume that the asset beta equals the interest-tax-shield beta, which means that the marginal tax rate is not present in equation (5), which serves as our empirical model. As a result, we will need to adjust for tax effects with an add-on later in the paper.

30

We assume a nominal risk-free rate of 5% and a market equity risk premium of 5% and

use our sample mean leverage of 16.56 in 2005:Q4 for our six largest U.S. BHCs.62 Substituting

the FE coefficient estimates for the full-sample period basic regression model from Table 2 into

equation (9), we obtain an estimate of the required return on equity of 10.33% (= 5% + (0 +

0.0644*16.56)*5%). For the extended regression model in column four of Table 2, the required

return on equity is 10.15%. To calculate the WACC for the six largest U.S. BHCs in our sample,

we substitute the estimated required return on equity into equation (7) and get 5.32% for the

basic regression model and 5.31% for the extended regression model.

To estimate the impact of a hypothetical doubling of capital, we first estimate the impact

of halving leverage on the required return on equity in equation (9) and calculate that the

required return on equity would decrease to 7.67% and 7.58% using the basic and extended

regression models, respectively. Furthermore, substituting these hypothetical costs of equity into

the WACC formula in equation (7) yields a new WACC of 5.32% and 5.31% for the basic and

regression extended models, respectively. This means that both our basic and extended

regression models predict that the WACC is unaffected by a hypothetical doubling of capital.

In the empirical literature on the MM framework, it is common to describe these results

in terms of an MM offset. We can determine the extent of the MM offset by comparing changes

in the WACC based on our regression estimates to those based on the assumption of a zero MM

offset. A zero MM offset means that the cost of equity would not decrease as leverage

decreases. While this is an unrealistic assumption, it serves as a useful benchmark from which to

62 The market equity risk premium of 5% is comparable to the survey results reported by Fernandez et al.

(2014) for the United States. Miles et al. (2012) note that a 5% nominal risk-free rate is approximately the average bank rate from 1999 to 2009 for banks in the United Kingdom. Additionally, according to the NYU Stern School’s online database, the average U.S. T-bill rate from 1964-2011 was 5.11%, while the average equity risk premium for the same period was 6.18% when measured against T-bills and 4.32% when measured against 10-year U.S. T-bonds.

31

measure the degree of MM offset. However, in this instance, since the WACC is unchanged

after a doubling of capital, it is obvious that the MM offset is 100%.

Overall, these results show that for the six largest U.S. BHCs, there appears to be a strong

MM offset based on the Hamada empirical framework. In order for a full MM offset to occur,

the intercept in the regressions must be equal to zero. A partial MM offset would occur if the

intercept in the regression is positive and statistically significant. Such a result is not fully

consistent with equation (5), which shows that a bank’s equity beta is proportional to its

leverage, with the asset beta serving as the factor of proportionality. Thus, for the full sample

period, we find a 100% MM offset with both the basic and extended regression models.

C. Sensitivity of Results to Reference Year An important concern with the results presented so far is that the size and statistical

significance of the estimated intercept in the regressions depends on the choice of reference year

for the annual time dummies. To assess the sensitivity of the results to the choice of reference

year, we re-estimate the basic and extended regression models over the full sample period for

each possible reference year from 1996 to 2012. These results are presented in Table 3.63

The basic regression model—with lagged leverage as the only bank-specific—shows a 100%

MM offset in 16 out of 17 cases. The estimated MM offset ranges from 44.8% to 100%, and the

average estimated MM offset is 96.8%. For the extended regression model, with the loan-loss

reserve and liquidity ratios as additional bank-specific control variables, we also find a 100%

offset in 16 out of 17 cases.64 The estimated MM offset for the extended model ranges from

63 All of the empirical studies that use annual time dummies to control for fixed time-effects in regressions

of equity beta on leverage in estimating the size of MM offsets suffer from this issue. This means that the MM offsets reported in these studies could change with a different reference year for the annual time dummies.

64 For both the basic and extended regression models, 2008 is the reference year for the annual time dummies that corresponds to a partial MM offset. For the extended regression model, when either 2002 or 2011 is

32

Table 3. Sensitivity of the Size of MM Offset to Reference Year for BHCs > $200B

Model

Time Period

No. Possible

Ref. Years

No. Ref. Years w/Full Offset

Average Offset

Offset Range

Basic Regression 1996:Q1-2012:Q4 17 16 96.75% 44.81%-100% Extended Regression 1996:Q1-2012:Q4 17 16 96.68% 43.61%-100%

Note: The underlying coefficient estimates used in the calculations are based on one-way, fixed-effects panel regressions with annual time dummies and t-statistics based on Driscoll-Kraay robust standard errors. 43.6% to 100%, and the average estimated MM offset is likewise 96.7%.

Most of the regression estimates indicate a 100% MM offset for the six largest U.S. BHCs.

However, the sensitivity of the results to the choice of reference year is still a potential concern.

We address this issue by controlling for those economic factors that change over the

sample period. Since the annual time dummies are intended to capture non-bank-specific factors

which change over time and that affect the average riskiness of banks’ assets, it seems reasonable

to substitute for the annual time dummies with macroeconomic factors that capture changes in

general economic conditions. To accomplish this, we use a set of macroeconomic factors,

thereby eliminating the need for annual time dummies and avoiding the associated reference-year

problem.65

The regression results using macroeconomic factors instead of annual time dummies are

presented in Table 4. Using this regression specification, we find that there is no longer a

positive and significant intercept regardless of the time period used in the estimation.

used as the reference year, the estimated intercept approaches marginal significance. If we called these intercept coefficients significant, the offsets would be 66% when 2002 is the reference year and 69% when 2011 is the reference year. The same could be said of 2002 as the reference year for the basic model where, if we called the intercept coefficient significant, the offset would be 68%.

65 The macroeconomic factors that we use in our regressions are: the quarterly percentage change in the gross domestic product of the nonfarm business sector (seasonally adjusted); the three-month Treasury rate; the quarterly percentage change in the Implicit Price Deflator (seasonally adjusted); the quarterly percentage change in the Purchasing Managers’ Index (seasonally adjusted); the credit spread (measured as the 10-year Moody’s Baa corporate bond yield minus the 10-year constant maturity Treasury bond yield); the yield spread (measured as the 10-year Treasury yield minus the one-year Treasury yield); the quarterly percentage change in the Case-Shiller Home Price Index (seasonally adjusted); and the quarterly percentage change in output per hour worked in the nonfarm business sector. The three-month Treasury rate, the credit spread, and the yield spread are treated as contemporaneous variables. All other variables are lagged one quarter due to delays in their official publication.

33

Furthermore, with the elimination of the annual time dummies, reference-year dependence is no

longer a concern. While the results are inconclusive for the pre-crisis period, they provide

empirical support to the MM framework for the entire sample period as well as for the

Table 4. Equity Beta and Leverage FE Regressions: Results for Full Time Period and Sub-Periods Using Macroeconomic Factors as Substitutes for Annual Time Dummies (Levels) for BHCs > $200B

Basic Regression Model Extended Regression Model 1996:Q1-

2012:Q4 1996:Q1- 2007:Q2

2007:Q3- 2012:Q4

1996:Q1- 2012:Q4

1996:Q1- 2007:Q2

2007:Q3- 2012:Q4

Leveraget-1 0.0938 0.0325 0.1548 0.0833 0.0352 0.1049 (3.43) (1.44) (6.13) (3.34) (1.43) (3.61) Intercept -0.4019 0.3654 -0.7629 -0.2360 0.3223 -0.0270 (-0.93) (0.99) (-1.96) (-0.60) (0.81) (-0.06) LLRt-1 — — — -0.0398 0.0480 -0.2530 — — — (-0.55) (0.52) (-3.40) Liquidityt-1 — — — -0.0316 0.0070 -0.0976 — — — (-0.76) (0.12) (-1.78) Within R-Sq. 0.415 0.335 0.597 0.418 0.337 0.623 ∆WACC—No MM 0.47% — 0.77% 0.42% — 0.52% ∆WACC—w/MM 0.00% — 0.00% 0.00% — 0.00% % MM Offset 100% — 100% 100% — 100%

Notes: The dependent variable is the Scholes-Williams equity beta, estimated using the CRSP equal-weighted market index. Results are for fixed-effects panel regressions using a one-way (bank ID) classification plus macroeconomic factors to control for changes over time in the economic environment. The numbers in parentheses are t-statistics computed using Driscoll-Kraay robust standard errors. The dependent variable is the Scholes-Williams equity beta, developed using the CRSP equal-weighted market index. The sample used for the estimations includes the six U.S. BHCs with at least $200 billion in total assets as of 2005:Q4 and consists of six BHCs, which falls to five near the end of our sample period due to Wells Fargo’s acquisition of Wachovia. The percent MM offset is not reported when neither the intercept nor the leverage coefficient estimate are significant. All the macroeconomic and bank-specific control variables are standardized to have a zero mean and standard deviation of one, in order to ensure that they have no impact on the estimated intercept.

crisis/post-crisis sub-period. The results for these two periods show a 100% MM offset for both

the basic and extended regression models.66, 67

These results, particularly those for the crisis/post-crisis sub-period, suggest that the

recent financial crisis has had an important impact on the market’s perceived importance of

leverage on the riskiness of bank debt. While the notion of too-big-to-fail banks was discussed

66 We also estimated the regressions presented in Table 4 with betas based on the CRSP value-weighted

index and found the results to be almost identical to the equal-weighted-index results. These results are available upon request.

67 Based on a suggestion by Mark Flannery, we also estimated regression specifications that included interaction terms between leverage and the bank-specific and macroeconomic variables. The results of these regressions were consistent with those regressions that excluded interaction terms. These results are available upon request.

34

before the 2007-2009 financial crisis, prior to that crisis, the only explicit example that could be

referred to was the Continental Illinois bailout back in 1984. And the reality that was eventually

revealed in the recent financial crisis was that if a large commercial bank failed its bondholders

would be made whole.68 This produces the empirical result that leverage takes on more

importance in our regression estimates once the financial crisis unfolds, especially for the too-

big-to-fail banks.69 However, this is not because MM has become more important, but rather

because the assumptions underlying the joint MM/CAPM model with a zero-debt-beta

assumption receive more empirical support when examining the largest U.S. BHCs.

We also consider the issue of non-stationarity in the banks’ Scholes-Williams equity beta

and leverage over the period 1996:Q1 to 2012:Q4.70 In this connection, it is important to address

the potential for unit roots in these two time-series variables, since levels regressions could

produce evidence of statistical significance due to the spurious regression problem (Granger and

Newbold (1974)). In order to address this econometric issue, we conduct Fisher-type unit root

tests for panels of Scholes-Williams equity betas and leverage for the six largest BHCs which are

based on augmented Dickey-Fuller (ADF) tests and can be used with unbalanced panel datasets.

In conducting the unit-root tests, the null hypothesis is that all cross-sectional units in a

panel dataset have unit roots against the alternative hypothesis that at least one cross-sectional

unit in the panel is stationary. Augmented Dickey-Fuller regressions, including one lag (and also

four lags), a drift term, and subtracting the cross-sectional mean from each series, show that we

68 However, in the case of Washington Mutual—a thrift, and not a commercial bank—the institution was

rapidly put through a federal receivership in which the claims of all public debt-holders (around $30 billion) were completely wiped out prior to its being sold to JP Morgan Chase.

69 Haldane (2013) reports that the implicit subsidy to the 29 largest global banks, as measured by the difference between credit rating agencies’ supported and stand-alone bank credit ratings, increased following the recent financial crisis. Prior to the crisis, these banks benefitted from just over one notch of uplift from the ratings agencies due to expectations of government support. In January 2013, the same global banks benefitted from around three notches of implied government support.

70 Plots of both the average equity beta and average leverage showed some signs of non-stationary behavior over the sample period 1996-2012. These plots are available upon request.

35

can reject the null with a very high degree of confidence—all p-values = 0.0000 for both the

equity beta and leverage time series.71 While we can reject the null hypothesis based on the

ADF tests, several of the banks could have unit roots in their equity betas and leverage. In order

to address this issue, we also estimate regressions that include macroeconomic variables instead

of annual time dummies and which use a first-difference specification to check for the robustness

of the regression results for leverage.

The first-difference results presented in Table 5 show that the macroeconomic control

variables are adequate substitutes for the annual time dummies as was also the case for the levels

regressions.72 The estimates are similar to those for the levels regressions shown in Table 4.

The results are strong for the full and crisis/post-crisis periods, but somewhat weaker for the pre-

Table 5. Equity Beta and Leverage FE Regressions with Macroeconomic Factors for BHCs > $200B (First- Differences)

Basic Model Extended Model 1996:Q1-

2012:Q4 1996:Q1- 2007:Q2

2007:Q3- 2012:Q4

1996:Q1- 2012:Q4

1996:Q1- 2007:Q2

2007:Q3- 2012:Q4

∆ Leveraget-1 0.1142 0.0279 0.1644 0.1102 0.0348 0.1702 (2.56) (0.68) (4.00) (2.65) (0.81) (4.48) Intercept 0.0119 0.0002 0.0442 0.0135 0.0039 -0.0451 (0.31) (0.04) (0.97) (0.36) (0.10) (-0.96) ∆ LLRt-1 — — — -0.0296 -0.0063 -0.1535 — — — (-0.63) (-0.13) (-1.80) ∆ Liquidityt-1 — — — -0.0452 -0.0793 0.0174 — — — (-1.30) (-1.77) (0.24) Within R-Sq. 0.185 0.060 0.482 0.191 0.079 0.492

Notes: The dependent variable is the Scholes-Williams equity beta, estimated using the CRSP equal-weighted market index. Results are for fixed-effects panel regressions using one-way (i.e., bank) classification and macroeconomic factors to control for changes over time in the economic environment. Numbers in parentheses are t-statistics computed using robust standard errors. The sample includes the six U.S. BHCs which had at least $200 billion in total assets as of 2005:Q4 and consists of six BHCs, which falls to five near the end of our sample time period due to a merger between Wells Fargo and Wachovia. All the macroeconomic and bank-specific control variables are standardized to have a zero mean and standard deviation of one, in order to ensure that they have no impact on the estimated intercept.

71 We use the Fisher estimation option in Stata’s xtunitroot routine to conduct the augmented Dickey-Fuller

tests. The demean option of this test computes for each time period the mean of the Scholes-Williams beta and leverage series across the six panels and subtracts this mean from each of the series before the ADF test is run. Levin et al. (2002) suggest this approach to lessen the impact of cross-sectional or spatial dependence.

72 Although we include an intercept in the first-difference estimations, the regression model in first differences would not include an intercept since this would imply a linear time trend in the levels regressions. Given this, it is not surprising that the intercept is insignificant in the first-difference regressions. See Harvey (1980) for a discussion of levels versus first-difference regression specifications.

36

crisis period (with respect to the within R-Squared coefficients and the size of the leverage

coefficient estimates). The coefficient estimate on lagged leverage is statistically significant in

the crisis/post-crisis sub-period and for the full-sample period, but not in the pre-crisis period. In

addition to estimating first-difference regressions, we also estimate a double-log regression in

order to obtain a point elasticity estimate with respect to leverage. The results from this

regression model are consistent with a 100% MM offset and are presented in Table A-I of the

Appendix.73

V. The MM Offset and BHCs’ Asset Size

In this section, we examine whether our results on the relevance of the MM framework to

U.S. BHCs are sensitive to banks’ asset size. To do this, we create four mutually-exclusive asset-

size classes: (1) banks with assets below $25 billion (170 BHCs); (2) banks with assets between

$25 billion and $100 billion (18 BHCs); (3) banks with assets between $100 billion and $200

billion (6 BHCs); and (4) banks with assets of $200 billion and above (6 BHCs). Using these

four asset-size classes, we examine the effect of bank asset size with respect to the relative

impact of leverage on equity beta, required return on equity, and WACC. The regression results

and average leverage for banks in the four asset-size classes are presented in Table 6.

Table 6 shows that the effect of leverage on equity beta, as measured by our empirical

model, tends to become stronger for banks of larger asset sizes. In addition, the last column of

the table shows that average leverage is monotonically increasing in asset size. While we obtain

73 A double-log specification was estimated for the basic regression model for the crisis/post-crisis period

using annual time dummies (with a reference year of 2012) to produce an elasticity estimate. The restriction on the sample period was necessary due to the presence of several negative estimated Scholes-Williams equity betas in the pre-crisis period. An elasticity estimate of 1 would indicate a full MM offset. The double-log regression produces a leverage coefficient estimate of 0.9021 with a t-statistic of 2.80. A test of whether the coefficient estimate significantly differs from 1 produces an F-statistic of 0.09 with a p-value of 0.7644. The results of the double-log regressions show no sensitivity to choice of reference year. These results are presented in Table A-I of the Appendix.

37

similar regression results for all BHCs in terms of a significant leverage coefficient estimate, we

also find that the results change in important ways as asset size changes. Specifically, the

leverage coefficient estimate is statistically significant for each asset-size class and

monotonically increases with asset size.

In contrast, the intercept monotonically decreases with asset size and is no longer

statistically significant for those banks with assets of $200 billion and above. This means that,

for the six BHCs in the largest asset-size class, we find a proportional relationship between

equity beta and leverage and, therefore, a full MM offset. This result is consistent with the

Hamada framework shown in equation (5), and is not surprising, given that we are using what is

Table 6. FE Results for Mutually-Exclusive Asset-Size Classes Using the Extended Regression Model with Macroeconomic Factors (Levels)

Intercept Leverage

Size

Coeff

t-Stat

Coeff

t-Stat Within R-Sq.

No. Banks

Average Leverage

<$25 B 0.6980 9.82 0.0123 2.21 0.199 170 12.10 >$25 B < $100 B 0.4762 2.54 0.0399 3.02 0.273 18 13.79

>$100 B < $200 B 0.3989 1.21 0.0434 1.79 0.418 6 14.66 >$200 B -0.2360 -0.60 0.0833 3.34 0.418 6 16.56

Notes: The dependent variable is the Scholes-Williams equity beta, estimated using the CRSP equal-weighted market index. All coefficient estimates are based on one-way, fixed-effects panel regressions using macroeconomic control variables as substitutes for annual time dummies. The estimation period is 1996:Q1-2012:Q4. The t-statistics are computed using Driscoll-Kraay robust standard errors. All the macroeconomic and bank-specific control variables are standardized to have a zero mean and standard deviation of one, in order to ensure that they have no impact on the estimated intercept. Average leverage is the mean leverage as of 2005:Q4. essentially a textbook proof of the MM framework as an empirical model without directly testing

whether the basic underlying assumption of that proof actually holds in the real world.

Therefore, we are testing the applicability of the MM theorem to U.S. BHCs, as well as testing

the validity of the joint MM/CAPM framework with a zero-debt-beta assumption. Specifically,

the Hamada equation may not be applicable for those BHCs for which the zero-debt-beta

assumption does not hold.

The evidence suggests that the MM framework and the zero-debt-beta assumption hold

strongly for BHCs with $200 billion or more in assets. This suggests that it is only for the very

38

largest banking institutions where the bond market appears to be unconcerned about the degree

of financial leverage and where the zero-debt-beta assumption receives empirical support and

appears to be valid.74 For BHCs with assets below $200 billion, the Hamada equation appears to

work less well. By implication, then, it would appear that for BHCs with assets below the $200

billion level, the bond market does have some degree of concern as to whether the relative size of

a bank’s debt might imperil the safety of that debt—i.e., its debt beta is greater than zero. And

this appears to be increasingly true the smaller the BHCs’ asset-size class.

Our findings strongly suggest that for the six largest BHCs (those $200 billion and

larger), the Hamada equation adequately captures the MM framework. The failure to find large

MM offsets for banks with asset sizes below $200 billion may be a consequence of the

underlying zero-debt-beta assumption not being valid below that particular asset-size threshold.

Thus, the results for banks with assets less than $200 billion may not be a rejection of the MM

framework per se, but rather of the appropriateness of the Hamada equation as an empirical

framework for studying the MM offsets for smaller BHCs.75

VI. The Impact of Changes in Banks’ Cost of Funding on Steady State Output

Changes in firms’ borrowing costs should affect equilibrium capital and steady-state

output. Therefore, in this section we estimate the effect of higher capital requirements on U.S.

output using a standard production-function approach. Increases in the estimated borrowing

costs of non-banking firms will change the equilibrium capital stock and produce an estimate of

the long-run effect of strengthened capital requirements for banks on steady-state output. 74 The intercept for the $100-$200 billion asset-size class is only marginally significant (t=1.21). However,

for purposes of our analysis, we consider it to be significant. This is a conservative approach in that it errors on the side of not finding a full MM offset.

75 Similar results on the leverage coefficient estimates are also found for the first-difference results presented in Table A-II of the Appendix.

39

Following Miles et al. (2012), our approach examines the transmission linkages between banks’

cost of capital, non-banking firms’ cost of capital, investment, and real GDP. In order to be

conservative in our estimates of the impact of higher bank funding costs on real GDP (i.e., to

error on the side of finding a larger impact), we assume that the entire increase in banks’ funding

costs is completely passed on to their customers.

We use a standard Cobb-Douglas production function with constant returns to scale and

constant elasticity of substitution, where output (Y) is produced with capital (K) and labor (L).76

It can be shown that the responsiveness of output to the cost of capital is given by:

, (10)k k

k K

P PdY dY K dK P dPdP Y dK Y dP K dP P

=

where P = Pk/Pl, is the relative price of capital to labor.

The right-hand side of equation (10) can be restated as:

1 . (11)1

aσa

=−

In equation (11), α is the elasticity of output with respect to capital and is also the share

of income that flows to capital. The second term, σ, is the responsiveness of capital to P and is

also the elasticity of substitution between capital and labor.77 We set α equal to 0.33 and σ equal

to 0.50, based on data for the United States.78 Based on data from the Federal Reserve Flow of

Funds and Call Report data, we estimate the share of private, non-financial corporations’

(PNFCs) financing that comes from banks to be approximately 25%.

76 In many empirical applications using a production-function approach, a Cobb-Douglas specification is

chosen for the functional form. One of the important advantages of using a Cobb-Douglas production function is that it greatly simplifies estimation of output elasticities, conditional on an assumption of returns to scale. See D’Auria et al. (2010) for further discussion of the use of Cobb-Douglas production functions in estimating potential GDP growth rates and output gaps.

77 See Miles et al. (2012, pp.15-16). 78 The elasticity of substitution assumption is from Young (2010), and the capital share of income is from

Jones (2003).

40

In Table 7, we show the economic cost of higher Tier 1 capital in terms of changes in real

output or GDP. This table presents the required ROE, estimated WACC, and the change in

WACC for a base case and for a case where leverage is halved due to a doubling of Tier 1

Table 7. Expected Impact on Required ROEs, WACCs, and Real GDP from a Halving of Leverage (with and without MM Offset) Using FE Levels Regression Results

Base Case No MM With MM ∆ RGDP (%)

Size

Num.

of Banks

Req. ROE (%)

Est.

WACC (%)

New Req. ROE (%)

New

WACC (%)

∆

WACC (%)

New

WACC (%)

∆

WACC (%)

MM

Offset (%)

PNFC Funding Shares

No MM

With MM

< $25 B

170

09.23

5.35

8.86

5.70

0.35

5.64

0.29

17.57 0.026

-0.023

-0.019

$25B-$100B

18

10.13

5.37

8.76

5.74

0.37

5.54

0.17

53.61 0.032

-0.029

-0.014

$100B-$200B

6

10.18

5.35

08.59

5.71

0.35

5.49

0.14

61.46 0.026

-0.023

-0.009

> $200 B

6

11.90

5.42

8.45

5.83

0.42

5.42

0.00

100 0.166

-0.170

0.000

Weighted Averages and Total Impacts

200

11.21

5.40

8.55

5.79

0.40

5.46

0.07

83.26 0.25

-0.245

-0.041

All 200 Banks

200

9.38%

5.35%

8.88%

5.70%

0.35%

5.62%

0.27%

22.78% 0.25

-0.216

-0.167

Notes: The assumed risk-free rate is 5% and banks’ funding share in the U.S. is 25%. Weighted averages for leverage, ROEs, and WACCs are calculated using the relative shares of total assets in each asset-size class as weights. The PNFC funding shares are determined by multiplying the particular asset-size class’ relative share of total assets for all banks in our sample by the banking system’s share of nonfinancial corporations’ total funding. The results in this table are for the extended regression model with macroeconomic factors estimated over the full sample period. capital. These estimates are reported in the aggregate as well as for each of the four asset-size

classes. The second from the last column of the table shows the share of funding provided to

private, non-financial corporations by BHCs in each of the four asset-size classes.

Equation (11) shows that if private, non-financial corporations’ cost of capital increases

by 1%, then output decreases by σα/(1–α)%. Given that we set α = .33 and σ = .50, this means

that a 1% increase in corporations’ cost of capital would lead to a reduction in output of 0.25%,

or 25 basis points. We use this approach to calculate the economic cost of higher equity capital in

terms of lost output.

As shown in Table 7, our results for the economic cost of higher Tier 1 capital depend on

the particular bank-asset-size class used in the calculation. For example, if we obtain aggregate

41

results for all 200 BHCs in our sample by calculating a weighted sum across the different asset-

size classes, while assuming no MM effect, we find that real GDP falls by 24.5 basis points. But

if we take the MM effect into account, we find that real GDP falls by only about 4 basis points.79

This would represent a permanent fall in the level of real GDP. Further, if we look at the six

largest BHCs, we see that without the MM offset the decline in real GDP would be 17 basis

points. However, with the MM offset there is a 0% decline in the level of real GDP, which is

consistent with the observed full MM offset for the largest BHCs.

In Table 7, we also observe that the size of the MM offset varies directly with the asset-

size class. For example, the MM offset is 17.57% for those banks with assets below $25 billion;

it is 53.61% for banks with assets between $25 billion and $100 billion; it is 61.46% for those

banks with assets between $100 billion and $200 billion; and for those banks with assets of $200

billion and above, the MM offset is 100%. The weighted-average MM offset shown in the last

row for all 200 BHCs is approximately 83%.

Finally, by comparing the results for the total weighted-average MM offset over all four

size classes (next-to-the-last row) with those obtained from a single regression on all 200 BHCs

(last row), we see that the total weighted-average MM offset across the four sizes classes is

83.3% while the MM offset derived from a single regression on all 200 BHCs is only 22.8%.

This illustrates how failing to stratify by asset-size class will not only lead to erroneous results

that overstate the impact of MM for the smallest banks and understate it for the largest ones, but

also seriously understates the overall impact of MM for the banking sector as a whole (as proxied

79In their analysis, Miles et al. (2012) find that real U.K. GDP would fall by 15 basis points for a doubling

of Tier 1 capital for the six largest banks. See Cecchetti (2014) and Cohen and Scatigna (2014) for further discussion of the impact of the increased equity capital requirements of Basel III on economic output in the United States, as well as many other countries.

42

by the 200 BHCs in our sample. All this, we argue, is because the extent to which the zero debt-

beta assumption is valid in the Hamada framework is positively related to banks’ asset size.

VII. The MM Offset with Tax Effects

In all the MM-offset and change-in-real-output calculations up to this point, we have not

explicitly taken into account tax effects. Therefore, in this section, we address the tax effects of a

decrease in the value of the interest tax shield associated with banks using less debt in their

capital structure. In deriving equation (5), we assumed that the risk of interest tax shield equals

the risk of the bank’s unlevered asset cash flows. This assumption resulted in the marginal

corporate tax rate dropping out of equation (5). Given this, it is necessary to account for the

effects with an add-on tax factor.

Table 8 presents results of the effects of doubling Tier 1 capital (halving leverage), and

taking into account the role of corporate income taxes, for the six largest BHCs in our sample. In

this table, we report the change in the banks’ WACC, the change in the PNFCs’ WACC, the fall

in real output, and the present value of lost output. The economic cost of higher capital

requirements is the present value of all lost real GDP taken out to infinity and expressed as a

percentage of current annual real GDP.80

The WACC for the six largest BHCs can be shown to increase by 13.2 basis points due to

tax effects alone.81 For the six largest BHCs, we estimate the cost of equity (the market’s

80 Like Miles et al. (2012), we use a discount rate of 2.5%, which represents the social discount rate, in our

present value calculations. This rate is different from the assumed nominal risk-free rate of 5% that banks offer on their debt.

81 Intuitively, the tax-effect calculations for the six largest BHCs can be done as follows: If leverage is halved, the proportion of banks’ own financing with debt will fall from 15.56/16.56 to 14.56/16.56, for a decline of 1/16.56. Thus, 1/16.56 • 5% (the debt financing rate) yields 0.003019, or 30.2 basis points, which is the amount of financing costs that banks cannot deduct from their taxes. This result multiplied by the combined federal/state statutory tax rate of 43.7% is 0.001319, or 13.2 basis points. This is the pure tax effect, and is reported in the first row in Table 8 which shows the expected change in the six largest banks’ WACC with a 100% MM offset. We use

43

required ROE) to be 11.90% as reported in Table 7 for the base case. If we assume no MM

offset, this cost of equity is the same at leverage 16.56:1 as at 8.28:1, since the required ROE

would not fall with a halving of leverage (or a doubling of Tier 1 capital). We assume a cost of

debt of 5% (gross of tax). This means that the increase in the WACC with no tax effects when

the proportion of debt changes from 15.56/16.56 to 14.56/16.56 is given as 1/16.56 * (11.90% -

5%) = 41.6 basis points.

If we now assume that the gross return on debt is tax deductible at rate t, and also assume

no MM offset, then the calculation for the increase in the WACC from reducing the proportion

of debt from 15.56/16.56 to 14.56/16.56 (i.e., by 1/16.56) becomes 1/16.56 * (11.90% - 5%(1-t))

= 54.8 basis points, where we use a t = 0.437, or a 43.7% marginal corporate tax rate. These

calculations yield a tax-effect increase in the WACC of 13.2 basis points, which is calculated as

54.8 basis points less 41.6 basis points.

To be conservative, we assume that banks pass on increases in their funding costs on a

Table 8. Tax Effects in Basis Points for the Six Largest U.S. BHCs

No Tax Effect

No MM

Tax Effect No MM

No Tax Effect

100% MM

Tax Effect 100% MM

Change in Banks’ WACC 41.6 54.8 0 13.2

Change in PNFCs’ WACC 10.4 13.7 0 3.3

Fall in Long-Run GDP 25.6 33.8 0 8.1

PV of Lost GDP 1,024 1,352 0 324 Notes: The results are based on one-way fixed-effects estimates for all 200 BHCs in our sample having $200 billion or more in total assets. The real discount rate = 2.5%, α = 0.33, σ = 0.50, the risk-free rate and the equity risk premium are both 5%, and banks’ funding share = 25%. The assumed U.S. corporate tax rate is the statutory 35% federal rate plus 8.7% for a representative state (Delaware) tax rate. PNFCs are private non-financial corporations. The results in this table are for the extended regression model with macroeconomic factors estimated over the full sample period.

the U.S. statutory federal corporate tax rate of 35% plus the Delaware statutory corporate income tax rate of 8.7%, which produces a combined rate of 43.7%.

44

one-for-one basis. Since banks account for 25% of the PNFCs’ funding in the United States, an

increase of 41.6 basis points in the WACC for the six largest banks in the no-tax-effect/no-MM-

offset case would translate into a 10.4 basis points increase in the PNFCs’ WACC. If we assume

that the overall cost of capital of private firms is 10% (with a safe rate of 5% and an equity risk

premium of 5% for a firm with a unit equity beta), then the 10.4 basis points would translate into

a 1.04% increase in the cost of capital for private firms in proportional terms. This suggests that

output would fall by 25.6 basis points, which would be a permanent decrease. The present

values of lost GDP reported in the last row of the table are calculated by dividing the fall in long-

run GDP by the real social discount rate of 2.5%. For example, for the case of no tax effect and

no MM offset, the present value of lost output would be given by 0.256%/2.5%, or 1,024 basis

points.

Table 9 presents results of the effects of doubling Tier 1 capital (halving leverage), and

taking into account the role of corporate income taxes, for the 200 BHCs in our sample. The

results are presented for the case with no MM offset both without and with a tax effect and for

Table 9. Tax Effects in Basis Points for All 200 U.S. BHCs

No Tax Effect No MM

Tax Effect No MM

No Tax Effect

80% MM

Tax Effect 80% MM

Change in Banks’ WACC 39.7 53.8 7.9 22.0

Change in PNFCs’ WACC 9.9 13.4 2.0 5.5

Fall in Long-Run GDP 24.5 33.1 4.9 13.5

PV of Lost GDP 980 1,324 196 540 Notes: The results are based on weighted averages from one-way fixed-effects estimates for all 200 BHCs in our sample. The real discount rate = 2.5%, α = 0.33, σ = 0.50, the risk-free rate and the equity risk premium are both 5%, and banks’ funding share = 25%. The assumed U.S. corporate tax rate is the statutory 35% federal rate plus 8.7% for a representative state (Delaware) tax rate. PNFCs are private non-financial corporations. The results in this table are for the extended regression model with macroeconomic factors estimated over the full sample period.

45

the case where there is an 80% weighted-average MM offset for all 200 BHCs.82 As shown in

Table 9, the change in banks’ WACC, the change in PNFCs’ WACC, the fall in long-run GDP,

and the present value of lost GDP are all larger when all 200 BHCs are used in the calculations.

Estimates of the economic cost of higher equity capital in terms of lower output depend

on the market-wide equity risk premium, tax effects, the extent of the MM offset, the share of

private sector funding provided by banks, the social discount rate, and the elasticity of

substitution between capital and labor. Table 10 shows the sensitivity of the output calculations

to changes in several of these underlying assumptions. We examine the sensitivity of the results

to changes in the social discount rate, the share of PNFCs’ funding provided by banks, and the

equity risk premium for all 200 BHCs in our sample. The first column of the table presents the

base case results, with no tax effect and a 65% MM offset. If we double the discount rate, the

present value of lost output is decreased by half. The present value of lost output also falls with

Table 10. Sensitivity of Estimates to Changes in Assumptions in Basis Points with Base Case Measured as No Tax Effect and an 80 % MM Offset for All 200 BHCs

Base Case

No Tax Effect 80% MM Offset

Raise Real Discount to 5%

Reduce Banks’ Share of PNFC Funding to 20%

Increase Equity Risk Premium to

7.5% Change in Banks’

WACC 7.9 7.9 7.9 11.9

Change in PNFCs’ WACC 2.0 2.0 1.6 3.0

Fall in Long-Run GDP 4.9 4.9 3.9 5.9

PV of Lost GDP 196 98 156 236

Notes: The results are based on weighted averages from one-way fixed-effects estimates for all 200 BHCs in our sample. Base-case assumptions are: the real discount rate = 2.5%, α = 0.33, σ = 0.50, the risk-free rate and the equity risk premium are both 5% and banks’ funding share = 25%. The assumed U.S. corporate tax rate is the statutory 35% federal rate plus 8.7% for a representative state (Delaware) tax rate. PNFCs are private non-financial corporations. The results in this table are for the extended regression model with macroeconomic factors estimated over the full sample period.

82 As shown in Table 7, the weighted-average offset for all 200 firms in our sample is 83.26%. For

purposes of this table, we assume an offset of 80% to be conservative.

46

a decrease in the funding share of private firms’ provided by banks from 25% to 20%. When the

equity risk premium is increased from 5% to 7.5%, the present value of lost output rises from

196 basis points to 236 basis points.

VIII. Robustness to Alternative Measures of Equity Capital In this section, we examine the robustness of the estimated MM offsets to the choice of

capital measure used to calculate leverage. Tables 11, 12, and 13 show the sensitivity of the

results to the use of three alternative measures of equity capital—Tier-1 capital, the book value

of equity, and tangible common equity. Under Basel III, the most important form of loss-

absorbing capital is Common Equity Tier 1 (CET1), which is essentially equity. We use tangible

common equity as a proxy for CET1, however, since it is not possible to obtain a time series of

that measure of capital. In addition, we also examine whether the results for the alternative

capital measures vary by asset size. To accomplish this, we estimate separate extended

regressions for two asset-size thresholds, i.e., those banks with assets below $200 billion and

those banks with assets above $200 billion. We estimate these regressions for the full sample

period and for the pre-crisis and crisis/post-crisis sub-periods.

The results presented in Table 11 for the full sample period suggest that the market seems

to consider Tier-1 capital as a more relevant measure of capital than it does either the book value

of equity or tangible common equity for assessing a bank’s riskiness based on some measure of

its assets-to-equity ratio. The results show that leverage based on Tier 1 capital is significant for

both size classes. However, leverage based on the book value of equity is significant only for the

over-$200-billion size class, while leverage based on tangible common equity is not significant

for either size class. Additionally, there are other important differences in the regression results

across the two groups of banks. Specifically, the coefficient estimate on leverage for the six

47

Table 11. FE Regression Results Using Alternative Equity Capital Measures by Asset Size for Full Sample Period (Levels)

$200 Billion and Greater Less Than $200 Billion

Tier 1

Equity Tangible Com Eq.

Tier 1

Equity

Tangible Com Eq.

Leveraget-1: 0.0833 0.0421 0.0089 0.0142 0.0042 0.0008 (3.34) (2.35) (1.22) (2.76) (1.04) (1.18) Intercept: -0.2360 0.5682 0.8828 0.6918 0.8164 0.8536 (-0.60) (2.53) (5.14) (10.38) (15.46) (28.01) No. Banks: 6 6 6 194 194 194 No. of Periods 66 66 66 66 66 66 Within R-Sq.: 0.412 0.396 0.396 0.188 0.186 0.186 MM Offset: 100% 55.10% 0.00% 20.20%83 0.00% 0.00%

Notes: The dependent variable is the Scholes-Williams equity beta, estimated using the CRSP equal-weighted market index. All estimates are based on one-way, fixed-effects panel regressions with bank-specific and macroeconomic control variables. The estimation period is 1996:Q1-2012:Q4. The numbers in parentheses are t-statistics computed using Driscoll-Kraay robust standard errors. All the macroeconomic and bank-specific control variables are standardized to have a zero mean and standard deviation of one, in order to ensure that they have no impact on the estimated intercept.

largest banks is nearly six times bigger than that for the banks with assets under $200 billion. In

addition, while the intercept is insignificant for the largest banks, it is significant for the smaller

banks, which means that the smaller banks will have a partial MM offset.84

In Table 12, we present results for the pre-crisis sub-period. For the banks with assets

over $200 billion, when leverage is calculated using the book value of equity and tangible

common equity, the coefficient estimate on leverage is significant. These results are in sharp

contrast to those for leverage based on Tier 1 capital presented earlier in the paper. For banks

with assets less than $200 billion, the results show that the coefficient estimates on leverage are

insignificant for leverage based on all three measures of capital.

83 The reason that the MM offset for Tier-1 leverage is so small in this table is that the offset is not being

calculated as the weighted average of the results for the three asset-size classes below $200 billion as was done for Table 7. This is because here we are estimating a regression for all banks with assets below $200 billion, instead of separate regressions for each the asset-size classes below this asset-size threshold. The weighted average MM offset for the banks with assets below $200 billion—using the size classes in Table 6—is 44.99%.

84 In addition to the results for alternative measures of leverage presented in this section, we also examine the sensitivity of the results to excluding intangibles from assets in calculating leverage. Removing intangibles has very little impact on the results, producing similar results to those for the regressions using Tier-1capital to calculate leverage. These results are found in the Appendix in Table A-III.

48

Table 12. FE Regression Results Using Alternative Equity Capital Measures by Asset Size for Pre-Crisis Sub-Period (Levels)

$200 Billion and Greater Less Than $200 Billion

Tier 1

Equity Tangible Com Eq.

Tier 1

Equity

Tangible Com Eq.

Leveraget-1: 0.0352 0.0432 0.0563 0.0088 0.0072 -0.0004 (1.43) (2.36) (3.39) (1.63) (1.06) (-0.74) Intercept: 0.3223 0.3442 -0.3353 0.6187 0.6428 0.7361 (0.81) (1.41) (-0.93) (8.33) (7.94) (26.00) No. Banks: 6 6 6 194 194 194 No. of Periods 45 45 45 45 45 45 Within R-Sq.: 0.337 0.342 0.379 0.1812 0.181 0.181 MM Offset: – 67.52% 100% 0.00% 0.00% 0.00%

Notes: The dependent variable is the Scholes-Williams equity beta, estimated using the CRSP equal-weighted market index. All estimates are based on one-way, fixed-effects panel regressions with bank-specific and macroeconomic control variables. The estimation period is 1996:Q1-2007:Q2. The numbers in parentheses are t-statistics computed using Driscoll-Kraay robust standard errors. All the macroeconomic and bank-specific control variables are standardized to have a zero mean and standard deviation of one, in order to ensure that they have no impact on the estimated intercept.

Table 13 presents results for the crisis/post-crisis sub-period. For the six largest banks,

the coefficient estimates on leverage based on Tier 1 and the book value of equity are significant,

while common tangible equity is close to being marginally significant. For the smaller banks,

only the coefficient estimate on leverage based on Tier 1 capital is significant. As was the case

for the results presented in Table 11, the coefficient estimate on leverage based on Tier 1 capital

is substantially bigger for the six largest banks compared to that for the smaller banks.

Table 13. FE Regression Results Using Alternative Equity Capital Measures by Asset Size for Crisis/Post-Crisis Sub-Period (Levels)

$200 Billion and Greater Less Than $200 Billion

Tier 1

Equity Tangible Com Eq.

Tier 1

Equity

Tangible Com Eq.

Leveraget-1: 0.1049 0.0759 0.0082 0.0316 0.0089 0.0004 (3.61) (2.40) (1.41) (3.15) (1.14) (0.44) Intercept: -0.0270 0.6532 1.2993 0.7980 1.0643 1.1537 (-0.06) (1.73) (7.79) (6.88) (12.24) (31.00) No. Banks: 6 6 6 172 172 172 No. of Periods 21 21 21 21 21 21 Within R-Sq.: 0.623 0.600 0.599 0.119 0.112 0.1114 MM Offset: 100% 65.80% 9.02% 32.81% 0.00% 0.00%

Notes: The dependent variable is the Scholes-Williams equity beta, estimated using the CRSP equal weighted market index. All estimates are based on one-way, fixed-effects panel regressions with bank-specific and macroeconomic control variables. The estimation period is 2007:Q3-2012:Q4. The numbers in parentheses are t-statistics computed using Driscoll-Kraay robust standard errors. All the macroeconomic and bank-specific control variables are standardized to have a zero mean and standard deviation of one, in order to ensure that they have no impact on the estimated intercept.

49

IX. Conclusion We show that the extent to which the MM theorem is found to hold for U.S. BHCs is

strongly related to their asset size. We present evidence suggesting that the six largest U.S.

BHCs have an MM offset that ranges from 43.6% to 100% in their WACC with a hypothetical

doubling of Tier 1 capital, depending upon the regression specification used to produce the

estimates and without directly accounting for tax effects. Our most robust regression

specification, however, shows a 100% MM offset before accounting for tax effects. When tax

effects are considered, we find that the six largest U.S. BHCs would see their WACC increase by

roughly 13 basis points from a doubling of Tier 1 capital through tax effects alone.

The joint MM/CAPM model linking equity beta to leverage under the assumption of a

zero debt beta seems to find its strongest empirical support for the MM theorem where the

implicit government guarantees are the strongest–i.e., among the largest, or too-big-to-fail, U.S.

banks. This result suggests that the empirical support for the validity of the Hamada framework

for the six largest U.S. BHCs is actually because of, rather than in spite of, the explicit and

implicit government guarantees. This is because the market appears to assume that the systematic

risk of the largest banks’ debt is unaffected by their degree of leverage.

The relevance of the TBTF guarantees to the zero debt-beta assumption is demonstrated

by comparing our results for the combined impact over the four asset-size classes with those

obtained from a single regression on all 200 BHCs. We found that the MM offsets from the

combined stratified results were nearly four times as large (83.3% versus 22.8%) as those

obtained from the single regression. This result illustrates that the relative strength of the implicit

government guarantees impacts the relative validity of the zero debt-beta assumption. Thus,

50

failing to stratify by asset-size class will seriously understate the overall impact of MM for the

banking sector as a whole.

We do caution against a too-literal interpretation of our regression results in any absolute

sense. Rather, the result of a 100% MM offset should be taken as being supportive of a

substantial, if not full, MM offset. Further, a primary reason that the estimated tax effect is as

small as it is, is that a doubling of equity capital only reduces debt financing by roughly six

percentage points. A larger increase in equity capital (reduction in leverage) would have a more

significant impact. Further, the fact that our results support the notion that banks are not

disadvantaged by having less leverage does not in any way dispute the fact that there might be

significant costs associated with acquiring the additional equity capital necessary to achieve that

lower leverage.

These caveats notwithstanding, we believe that our evidence supports the notion that the

too-big-to-fail banks in the United States are systematically over-leveraged at the expense of the

taxpayer. For example, despite our results showing that there is a substantial MM offset

associated with heightened equity capital requirements for the largest U.S. BHCs, it is telling that

leverage is monotonically increasing with bank asset size. The disparate tax treatment of debt

and management compensation programs based on accounting ROE provide only part of the

explanation for the strong preference of the largest banks for high leverage. However, if we

hadn’t learned it from the Continental Illinois experience in 1984, we certainly should have

learned from the recent global financial crisis that the government guarantees to the largest

financial institutions go well beyond any explicit deposit insurance. And this additional implicit

insurance associated with the too-big-to-fail guarantee comes with a zero premium, and therefore

it is necessarily underpriced.

51

While more equity capital may not be a panacea, it will reduce the moral hazard

associated with the explicit and implicit government guarantees—especially for the TBTF

banks—by creating better incentives to manage risk and avoid excessive risk-taking. In addition,

the agency costs of debt will be reduced. When a bank holds more equity, there is effectively a

larger deductible on the government’s guarantees of its debt. This means that, at the margin at

least, there will be less of a distortive effect of those guarantees, and less shifting of risk from the

bank to the government. In sum, our results are strongly supportive of the direction of recent

developments in setting capital requirements, including Basel III and the enhanced

supplementary leverage ratio, which focus on increasing capital requirements for the very largest

banking institutions.

52

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Appendix

Table A-I. FE Results for Double-Log Specification for Crisis/Post-Crisis Sub-Period (2007:Q3-2012:Q4) for Largest U.S. BHCs

Leverage t-1 0.9021 (2.80) Intercept -2.2018 (-2.66) No. of Banks 6 No. of Periods 21 Within R-Sq 0.3950

Wald 2χ -Test: Coefficient. on Log of Leverage = 1

0.09

Pr> Wald 2χ 0.7644

Notes: The dependent variable is the log of the Scholes-Williams equity beta, estimated using the CRSP equal-weighted market index. Estimation is based on one-way, fixed-effects panel regressions with annual time dummies (reference year=2012). Numbers in parentheses are t-statistics using Driscoll-Kraay robust standard errors. Double-log regressions are not estimated for earlier sample periods due to negative equity betas.

Table A-II. FE Results for Mutually-Exclusive Asset-Size Classes Using Extended Regression Model with Macroeconomic Factors (First Differences)

Intercept Leverage

Size

Coeff

t-Stat

Coeff

t-Stat Within R-Sq.

No. Firms

Average Leverage

<$25 B 0.0119 0.45 0.0441 2.75 0.0253 170 12.10 >$25 B < $100 B 0.0129 0.37 0.0411 1.89 0.113 18 13.79

>$100 B < $200 B 0.0189 0.48 0.0603 1.58 0.149 6 14.66 >$200 B 0.0114 0.30 0.1042 2.91 0.188 6 16.56

Notes: The dependent variable is the Scholes-Williams equity beta, estimated using the CRSP equal-weighted market index. Estimation is based on one-way, fixed-effects panel regressions using macroeconomic control variables as substitutes for annual time dummies. Estimation period is 1996:Q1-2012:Q4 .The t-statistics are computed using Driscoll-Kraay robust standard errors. All the macroeconomic and bank-specific control variables are standardized to have a zero mean and standard deviation of one, in order to ensure that they have no impact on the estimated intercept. Table A-III. FE Results for Bank Book Assets Adjusted for Intangibles by Asset Size (Levels)

Notes: The dependent variable is the Scholes-Williams equity beta, estimated using the CRSP equal-weighted market index. All coefficients based on one-way, fixed-effects panel regressions with firm-specific and macroeconomic control variables. The estimation period is 1996:Q1-2012:Q4. Numbers in parentheses are t-statistics computed using Driscoll-Kraay robust standard errors. All the macroeconomic and bank-specific control variables are standardized to have a zero mean and standard deviation of one, in order to ensure that they have no impact on the estimated intercept.

$200 Billion and Greater Less Than $200 Billion Stand.

Spec. Adj.

Assets Stand. Spec.

Adj. Assets

Leveraget-1 0.0833 0.0814 0.0142 0.0120 (3.34) (3.33) (2.76) (2.33) Intercept -0.2360 -0.1712 0.6918 0.7212 (-0.60) (-0.46) (10.38) (10.92) No. of Banks 6 6 194 194 No. of Periods 66 66 66 66 Within R-Sq 0.418 0.417 0.188 0.187 MM Offset 100% 100% 20.20% 17.02%