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JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS

COPYRIGHT 2019, MICHAEL G. FOSTER SCHOOL OF BUSINESS, UNIVERSITY OF WASHINGTON, SEATTLE, WA 98195 doi:10.1017/S0022109019000322

Leverage and the Beta Anomaly

MalcolmBaker, Mathias F.Hoeyer, and JeffreyWurgler*

Abstract

The well-known weak empirical relationship between beta risk and the cost of equity (the beta anomaly) generates a simple tradeoff theory: As firms lever up, the overall cost of cap- ital falls as leverage increases equity beta, but as debt becomes riskier the marginal benefit of increasing equity beta declines. As a simple theoretical framework predicts, we find that leverage is inversely related to asset beta, including upside asset beta, which is hard to ex- plain by the traditional leverage tradeoff with financial distress that emphasizes downside risk. The results are robust to a variety of specification choices and control variables. I.

Intr oduction

Millions of students have been taught corporate finance under the assump- tion of the capital asset pricing model (CAPM) and integrated equity and debt markets. Yet, it is well known that the link between textbook measures of risk and realized returns in the stock market is weak, or even backward. For example, a dollar invested in a low beta portfolio of U.S. stocks in 1968 grows to $70.50 by 2011, while a dollar in a high beta portfolio grows to just $7.61 (see Baker,

Bradley, and Taliaferro (

2014
)). The evidence on the anomalously low returns to high-beta stocks begins as early as Black, Jensen, and Scholes ( 1972
). In a re- cent contribution, Bali, Brown, Murray, and Tang ( 2017
) identify it as "one of the most persistent and widely studied anomalies in empirical research of secu- rity returns" (p. 2369). A large literature has come to view the beta anomaly, and related anomalies based on total or idiosyncratic risk, as evidence of mis-

pricing as opposed to a misspecified risk model. In this paper, we consider these*Baker, mbaker@hbs.edu, Harvard Business School and NBER; Hoeyer, mfhoeyer@gmail.com,

University of Oxford; and Wurgler (corresponding author), jwurgler@stern.nyu.edu, NYU Stern School of Business and NBER. For helpful comments we thank Heitor Almeida (the referee), Hui Chen, Robin Greenwood, Sam Hanson, Lasse Pedersen, Thomas Philippon, Ivo Welch, and seminar participants at the American Finance Association, Capital Group, George Mason University, George- town University, the Ohio State University, NYU, the New York Fed, Southern Methodist University, University of Amsterdam, University of Cambridge, University of Tokyo, University of Toronto, Uni- versity of Miami, University of Utah, USC, and University of Warwick. Baker and Wurgler also serve as consultants to Acadian Asset Management. Baker gratefully acknowledges financial support from the Division of Research of the Harvard Business School. 1

2 Journal of Financial and Quantitative Analysis

explanations sufficiently plausible to consider their implications for leverage. In doing so, we bring ideas from the anomalies literature into corporate finance, which often assumes efficient and integrated securities markets to highlight other frictions. We show that a beta anomaly in equity markets, with the absence of any other deviations from the Modigliani-Miller setup, leads to a simple tradeoff. Intuitively, under the anomaly, beta risk is overvalued in equity securities, but not indebtsecurities.Ideally,then,tominimizethecostofcapital,riskisconcentrated in equity. A firm will always want to issue as much riskless debt as it can. This lowers the cost of equity by increasing its beta without any "inefficient" transfer of risk from equity to debt. But, as debt becomes risky, further increases in leverage have a cost. Shifting overvalued risk in equity securities to fairly valued risk in debtsecuritiesincreasestheoverallcostofcapital.Forfirmswithhigh-betaassets, this increase is high even at low levels of leverage. For firms with low-beta assets, this increase remains low until leverage is high. Using the Merton ( 1974
) model to characterize the functional form of debt betas and the underlying transfer of risk from equity to debt, we show that there is an interior optimum leverage ratio that is inversely related to asset beta. The theory"s immediate prediction is that leverage is negatively related to asset beta. Empirically, we confirm that there is indeed a robust negative rela- tionship between leverage and asset beta. The relationship remains strong when controlling for overall asset variance (a control for distress risk), using alternative measures of leverage and industry measures of asset beta and overall risk, and including various other control variables. While it is reassuring that the beta anomaly tradeoff generates a leverage- asset beta prediction that is borne out in the data, an obvious alternative expla- nation is based on the standard tradeoff theory, which also predicts an inverse relationship between leverage and risk (most intuitively, total asset risk, not asset beta). For example, Long and Malitz ( 1985
) include systematic asset risk, not just overall risk, in leverage regressions with the traditional tradeoff between taxes and financial distress costs in mind. We do not wish to cast doubt on the relevance of the standard tradeoff theory. Instead, we derive and test additional predictions of the beta anomaly tradeoff to show that it has incremental explanatory power. In particular, we prove that, under a beta anomaly, both "upside" and "down- side" betas are inversely related to leverage. Financial distress costs, on the other hand, clearly emphasize downside risk. Moreover, consistent with the beta anomaly tradeoff, we find that leverage is indeed inversely related to upside as- set beta as well. In some specifications, the upside beta relationship is actually stronger than the downside beta relationship. While our results do not rule out the traditional tradeoff theory, given that downside beta still matters, this prediction and empirical relationship is most easily explained by a beta anomaly tradeoff. We tentatively suggest that the beta anomaly tradeoff may also help to ex- plain extremely high and low leverage levels that challenge the standard tradeoff.

For instance, Graham (

2000
) and others have pointed out that hundreds of prof- itable firms, with high marginal tax rates, maintain essentially zero leverage. This is often called the low-leverage puzzle. Conversely, a number of other profitable firms maintain high leverage despite no tax benefit. The beta anomaly tradeoff

Baker, Hoeyer, and Wurgler 3

could help to explain these patterns. If low leverage firms have determined that the tax benefit of debt is less than the opportunity cost of transferring risk to lower-cost equity, low leverage may be optimal even in the presence of additional frictions; a minor, realistic transaction cost of issuance could drive some firms to zero leverage. Meanwhile, low asset beta firms with no tax benefits of debt still resist equity because of its high risk-adjusted cost at low levels of leverage, and instead use a very large fraction of debt finance. In addition to providing a novel theory of leverage based on a single devi- ation from the Modigliani-Miller setup, the beta anomaly approach has an at- tractive and unifying conceptual feature. The traditional tradeoff theory, under rational asset pricing, cannot fit both the leverage and asset pricing evidence on the pricing of beta. If betaistruly a measure of risk relevant for capital structure, then presumably it would also help to explain the cross section of asset returns, which it does not. Investors, recognizing the associated investment opportunities, would demand higher returns on assets exposed to the systematic risk of fire sales, with high risk-adjusted costs of financial distress. If betais nota measure of risk, as the large literature that follows Fama and French ( 1992
1993
) has claimed, then asset beta should not be a constraint on leverage, after controlling for total asset risk. Although the beta anomaly is far from the only force at work in real-life capital markets (following the standard approach in corporate finance theory, we focus on a single "friction" (the beta anomaly) for simplicity), it is worth noting that it offers an internally consistent explanation for both of these patterns.

Section

I briefly re viewsthe beta anomaly and deri vesoptimal le verageun- der the anomaly. Section II contai nsempirical h ypothesesand tests. Section III concludes. II.

The Bet aAnomal yT radeoff

A.

Motiv ationand Setup

Over the long run, riskier assetclasseshave earned higher returns, but the historical risk-return tradeoffwithinthe stock market is flat at best. The CAPM predicts that the expected return on a security is proportional to its beta, but in fact stocks with higher beta have tended to earn lower returns, particularly on a risk-adjusted basis. As mentioned earlier, Bali et al. ( 2017
) highlight the beta anomaly as one of the mostprominentof the many dozen anomalies studied in the literature. We refer the reader to that paper, as well as Baker, Bradley, and

Wurgler (

2011
) or Hong and Sraer ( 2016
) for an overview of the large literature that shows the beta anomaly"s robustness across sample periods and international markets and reviews some of its theoretical foundations. The simple linear specification for the beta anomaly that we employ here is (1)rD(1)

CrfCrp,

whererfis the risk-free rate,rpis the market risk premium, and <0 measures the size of the anomaly. The beta anomaly is that a stock has "alpha" and the alpha falls with its beta. A few comments on this simple functional form. In terms of the "neutral point"whereequityriskisfairlypriced,equation( 1 )assumesthistakesplaceatan

4 Journal of Financial and Quantitative Analysis

equity beta of 1.0. The equity premium puzzle suggests that, historically, location might be at a higher level. But, even if equity were undervalued or overvalued on average, the mean leverage behavior would change, but predictions about the cross section of leverage, our focus here, would not. If the beta anomaly extended in equal force into debt, Modigliani-Miller ir- relevancewouldstillhold.Neithertheliteraturenorourowntestsindicateamean- ingful beta anomaly in debt, however. In Frazzini and Pedersen ( 2014
), short- maturity corporate bonds of low risk firms have marginally higher bond-market- beta risk-adjusted returns, but bond index betas have little relationship to stock index betas, the basis of the beta anomaly. In fact, Fama and French ( 1993
) show that stock market betas are nearly identical for bond portfolios of various ratings, and Baele, Bekaert, and Inghelbrecht ( 2010
) find that even the sign of the corre- lation/beta between government bond and stock indexes is unstable. Nonetheless, just to be sure, we directly compared the returns on beta-sorted stock portfolios with beta-risk-adjusted corporate bond portfolios and can easily reject an inte- grated beta anomaly. These results are omitted for brevity but available on request.

See Harford, Martos-Vila, and Rhodes-Kropf (

2015
) for perspectives on financing under (non-beta-based) debt mispricing. With a beta anomaly in equity, the overall cost of capital depends not only on asset beta but on leverage:

WACC(e)DerC(1e)rd(2)

DrfCarpC(a1)y(1e)(d1)y,

whereais asset beta andeis capital structure as measured by the ratio of equity to firm value. The second to last term (the asset beta minus 1 times ) is the uncontrollable reduction in the cost of capital that comes from having high-risk assets. The last term is the controllable cost of having too little leverage while debt beta remains low. An advantage of using the CAPM to develop comparative statics is to see the familiar textbook transfers of beta risk from equity to debt as leverage increases. However, any asset pricing model that features a stronger beta anomaly in equity will lead to the same qualitative conclusions. B.

Optimal C apitalStr ucture

Next we outline a simple, static model of optimal capital structure with no frictions other than a beta anomaly. There are no taxes, transaction costs, issuance costs, incentive or information effects of leverage, or costs of financial distress. It is interesting that unlike other tradeoff approaches, which require one friction to limit leverage on the low side and another to limit it on the high side, this single mechanism can drive an interior optimum. The optimal capital structure minimizes the last term of equation ( 2 ) by satis- fyingthefirst-orderconditionfore.Withthefurtherassumptionofadifferentiable debt beta, for a given level of asset beta the optimal capital ratioesatisfies (3)y

1dTe(a)UCT1e(a)U@dTe(a),aU@e

D0,

Baker, Hoeyer, and Wurgler 5

or, in terms of optimal debt beta, dTe(a),aU D1CT1e(a)U@dTe(a),aU@e: The optimum leverage does not depend on the size of the beta anomaly, but this is a bit of a technicality. If there were any other frictions associated with leverage, suchastaxesorfinancialdistresscosts,theanomaly"ssizewouldbecomerelevant. Observation 1.Firms will issue some debt, and as much debt as possible as long as it remains risk-free. The first-order condition cannot be satisfied as long as the debt beta is 0. At a zero debt beta, the left side of equation ( 3 ) is positive (recall <0). In other words, issuing more equity at the margin will raise the cost of capital. With zero debt, the asset beta is equal to the equity beta and the WACC reduces to: (4) WACC .1/D(a1)yCryCarp: A first-order Taylor approximation aroundeD1 shows that even marginal debt will decrease the cost of capital: (5) WACC(e)WACC(1)C(1e) ). One might ask why nonfinancial firms do not increase their leverage ratios further to take advantage of the beta anomaly: It is initially unclear how the low leverage ratios of nonfinancial firms represent an optimal tradeoff between the tax benefits of interest and the costs of financial distress, much less an extra benefit of debt arising from the mispricing of low risk stocks.

The answer from equation (

3 ) is that many low leverage firms (e.g., the stereotypical unprofitable technology firm) already start with a high asset beta or overall asset risk. Their assets are already quite risky at zero debt. Even at modest levels of debt, meaningful risk starts to be transferred to debt. While equation ( 3 cannot on its own explain why a firm would have exactly zero debt, it can explain why some firms have low levels of debt, despite the tax benefits and modest costs of financial distress.

Observation 2.Leverage has an interior optimum.

Zero equity is also not optimal. With all debt finance, the debt beta equals the asset beta and equation ( 2 ) reduces to the traditional WACC formula without the beta anomaly. This establishes that optimum leverage must be interior. The intuition is that, with the assumption of fairly priced debt, the firm will be fairly priced if it is funded entirely with debt (i.e. 100% leveraged). Can it increase value by reducing its leverage ratio? Yes. This new equity, an out-of-the-money call option, will be high risk, and hence overvalued. As a consequence, neither

0% nor 100% leverage are optimal, so there must be an interior optimum.

To further our understanding of optimal debt levels, we must characterize the dynamics underlying the transfer of risk from equity to debt with increas- ing levels of leverage, and in particular the dependence of debt beta on leverage.

6 Journal of Financial and Quantitative Analysis

A natural candidate for the functional form of debt betas is the Merton ( 1974
model. Merton uses the isomorphic relationship between levered equity, a Euro- pean call option, and the accounting identityDDVEto derive the value of a single, homogeneous debt claim, such that (6)D(d,T)DBerft8Tx2(d,T)UCVf18Tx1(d,T)Ug, whereVis firm value with volatility,Dis the value of the debt with maturity in and face valueB. LetTD2be the firm variance over time, anddBerfV the debt ratio, where debt is valued at the risk-free rate, thusdis an upward biased estimateoftheactualmarketbaseddebtratio(Merton( 1974
),pp.454-455).Here,

8(x) is the cumulative standard normal distribution andx1andx2are the familiar

terms from the Black-Scholes formula.

Following the approach of Black and Scholes (

1973
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