Size Anomalies in U.S. Bank Stock Returns

• Author: PRIYANK GANDHI,HANNO LUSTIG
• Published date: 01 April 2015
• Date: 01 April 2015
• DOI: http://doi.org/10.1111/jofi.12235

THE JOURNAL OF FINANCE •VOL. LXX, NO. 2 •APRIL 2015

Size Anomalies in U.S. Bank Stock Returns

PRIYANK GANDHI and HANNO LUSTIG∗

ABSTRACT

The largest commercial bank stocks, ranked by total size of the balance sheet, have

significantly lower risk-adjusted returns than small- and medium-sized bank stocks,

even though large banks are significantly more levered. Weuncover a size factor in the

component of bank returns that is orthogonal to the standard risk factors, including

small minus big, which has the right covariance with bank returns to explain the

average risk-adjusted returns. This factor measures size-dependent exposure to bank-

specific tail risk. These findings are consistent with government guarantees that

protect shareholders of large banks, but not small banks, in disaster states.

BANKS ARE DIFFERENT FROM NONFINANCIAL FIRMS in many ways. One of the most

salient distinctions is that banks are subject to bank runs during banking

panics and crises, not just by depositors, but also by other creditors (see Gor-

ton and Metrick (2012)andDuffie(2010)). Because financial crises are high

marginal utility states for the average investor, the expected return on bank

stocks should be especially sensitive to variation in the anticipated financial

disaster recovery rates of bank shareholders related to bank size, the regu-

latory regime, implicit government guarantees, and other characteristics. For

example, if a bank is deemed too big to fail, the expected return on its stock is

lower in equilibrium than that of smaller banks holding the exact same assets

in their portfolio because the government absorbs some of the large bank’s tail

risk. We find evidence that the pricing of bank-specific tail risk in the stock

market depends on all of these bank characteristics.

To explore the asset pricing implications of financial disasters, our paper

studies historical bank stock returns in the United States. We find that there

∗Gandhi is with Mendoza College of Business, University of Notre Dame and Lustig is with

Anderson School of Management, University of California at Los Angeles. The authors thank the

Editors, Cam Harvey and Ken Singleton, as well as the Associate Editor and several referees for

excellent suggestions. The authors also thank Viral Acharya, Martin Bodenstein, Markus Brun-

nermeier, John Campbell, John Cochrane, Ron Feldman, Cesare Fracassi, Etienne Gagnon, Mark

Garmaise, Amit Goyal, Mark Grinblatt, Jennifer Huang, John Krainer, Arvind Krishnamurthy,

David Laibson, Jonathan Parker, Lasse Pedersen, Andrea Raffo, Richard Roll, Jan Schneider,

Martin Schneider, Clemens Sialm, Skander Vandenheuvel, Rob Vishny, and Baolian Wang for

many for detailed and helpful comments, as well as seminar participants at University of Texas

at Austin, Chicago Booth, New York University Stern, Harvard Economics, the Federal Reserve

Board of Governors, the Federal Reserve Bank of San Francisco, the Society for Economic Dynam-

ics 2010 Annual Meeting in Montreal, and the 2010 Stanford Institute for Theoretical Economics

for detailed and helpful comments.

DOI: 10.1111/jofi.12235

733

734 The Journal of Finance R

is a size effect in bank stock returns that is different from the market capital-

ization effects that have been documented in nonfinancial stock returns (see

Banz (1981) and many others). All else equal, a 100% increase in a bank’s book

value lowers its annual return by 2.23% per annum. For nonfinancial stocks,

there is no similar relation between book value and returns (Berk (1997)).

These return differences cannot be imputed to differences in standard risk

exposure. A long position in the stock portfolio of the largest commercial banks,

measured by deciles of total book value, and a short position in the stock

portfolio of the smallest banks underperforms an equally risky portfolio of all

(nonbank) stocks and bonds by more than 7% per annum. The average alphas

are small but positive for commercial banks in the first five deciles and then

decrease for the largest banks in the top three deciles.

Small banks differ from large banks in many ways, but these differences

should not lead to differences in average risk-adjusted returns on bank port-

folios unless there is bank-specific tail risk that is priced but not spanned by

the traded returns on other stocks in the sample. We find evidence of such a

risk factor in bank stock returns: the second principal component of the risk-

adjusted returns on size-sorted portfolios of commercial banks is a size factor

that has exactly the right covariance with the portfolio returns to account for

most of this pricing anomaly. By construction, this size factor is orthogonal to

the stock and bond risk factors.

This highly levered size portfolio, determined by the second principal com-

ponent, which goes long in small bank stocks and short in large bank stocks,

loses an average of 41 cents during National Bureau of Economic Research

(NBER) recessions per dollar invested at the start, after hedging out exposure

to standard stock and bond risk. We attribute the cyclical banking size factor

in the data to size-dependent differences in the perceived shareholder recovery

rates on these bank portfolios during financial disasters.

In a version of the Barro (2006), Rietz (1988), and Longstaff and Piazzesi

(2004) asset pricing model with a time-varying probability of rare events, de-

veloped by Gabaix (2012), Wachter (2013), and Gourio (2008), financial disas-

ters that disproportionately impact bank cash flows contribute an additional

bank-specific risk factor. These rare events are priced into the expected returns

on portfolios of banks, but are not fully spanned by the returns on other assets

in a small sample. A general equilibrium version of our model can match the

average alphas in a sample without disasters if the financial disaster recovery

rate is 35 cents higher for large banks, in line with the failure rate of banks in

the lowest decile during the latest crisis.

Historically, the probability of a financial disaster increases during reces-

sions. Because of the size-contingent nature of the recovery rate for bank stock-

holders in the case of a financial disaster, the variation in the probability of

a financial disaster generates a common business cycle factor in the normal

risk-adjusted returns of size-sorted bank stock portfolios; the loadings of bank

stock portfolio returns on this size factor are determined by the recovery rates

and hence by size. Small banks have positive loadings while large banks have

Size Anomalies in U.S. Bank Stock Returns 735

negative loadings. As the probability of a financial disaster increases, the ex-

pected return gap between small and large banks grows.

In the United States, shareholder recovery rates for banks depend on size.

During financial disasters, large banks fare much better, even though they are

more levered than their smaller counterparts. A total of 30% of publicly traded

commercial banks in the first size decile were delisted in 2009 but there were

no delistings in the last decile.

Why study the effect of bailouts on bank equity? The anticipation of future

bailouts of bondholders and other creditors always benefits shareholders (see

Kareken and Wallace (1978)) ex ante. Furthermore, during a crisis, there may

be massive uncertainty about the resolution regime, especially for large finan-

cial institutions. As a result, government guarantees will inevitably tend to

benefit shareholders ex post as well. Clearly, the U.S. government and regu-

lators are willing to let small banks fail, but not large banks. Acharya and

Yorulmazer (2008) point out that bailouts may be ex post efficient if a suffi-

ciently large fraction of banks is impacted. Of course, ex ante, one could have

expected that the government would wipe out shareholders of large financial

institutions in the case of a bailout. Our evidence suggests that this is not what

market participants expect.

Government guarantees essentially grant stockholders of large banks a menu

of path-dependent put options that can only be exercised after large declines

in a broad index of stocks. This essentially reduces the negative co-skewness

of large bank stock returns, but not of small banks. In our sample, large bank

stock returns are indeed less negatively skewed and feature less co-skewness,

even though the Harvey and Siddique (2000) skewness factors constructed from

nonfinancial stocks cannot fully account for the variation in average returns

on size-sorted bank portfolios.

Toback out the implicit financial tail risk premium or discount charged by the

shareholders of commercial banks, we multiply the loadings on the size factor

by its risk price. The implicit insurance provided against financial disaster

risk lowers the expected equity return for the largest U.S. commercial banks

by 1.97%, but the additional exposure to bank-specific tail risk increases the

expected return on the smallest bank stocks by 2.85%, compared to a portfolio

of nonbank stocks and bonds with the same standard risk characteristics. The

largest banks have an average market capitalization of $140 billion in 2005

dollars. For the largest commercial banks, this amounts to an annual savings of

$2.76 billion per bank. The market imposes large financial tail risk “subsidies”

(“taxes”) on large (small) bank stocks compared to a portfolio of stocks and

bonds with the same observed risk profile. There is direct evidence from option

markets to support this conclusion: Kelly, Lustig, and Nieuwerburgh (2011)

find that out-of-the-money put options on large banks were cheap during the

crisis.

The pricing of financial tail risk depends not only on bank size. We relate

the financial disaster premium of banks to the regulatory regime. Commercial

banks, which have access to the discount window and benefit from deposit

insurance, and government-sponsored enterprises (GSEs), which benefit from