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DIVERSIFICATION, SIZE, AND RISK AT BANK HOLDING COMPANIES
by
Rebecca S. Demsetz and Philip E. Strahan
Federal Reserve Bank of New York
Research Paper No. 9506
April 1995
This paper is being circulated for purposes of discussion and comment only.
The contents should be regarded as preliminary and not for citation or quotation without
permission of the author. The views expressed are those of the author and do not necessarily
reflect those of the Federal Reserve Bank of New York or the Federal Reserve System.
Single copies are available on request to:
Public Information Department
Federal Reserve Bank of New York
New York, NY 10045
Abstract
This paper shows that large BHCs are better diversified than small BHCs based on market measures
of diversification. We find, however, that better diversification does not translate into reductions in
overall risk. The risk reducing potential of diversification at large BHCs is offset by their lower capital
ratios, larger C&I loan portfolios, and greater use of derivatives. Our results suggest that asset growth
should enhance diversification but that the effects on risk will depend on the extent to which growth is
accompanied by changes in portfolio attributes. Using data from 1980 to 1993, we find that BHC asset
growth has, in fact, been accompanied by economically important reductions in risk.
The authors wish to thank Fred Herriman, Beverly Hirtle, Mark Levonian, Stavros Peristiani and
Lawrence Radecki for helpful comments. Richard Duke, James Weston and August Moret provided
outstanding research assistance.
I. Introduction
Bank diversification takes on a vatiety of forms, most of which are commonly associated
with
asset size. Compared to small institutions, large banks have access to a wider variety of
borrowers and
a broader deposit base. Their loan originations and funding sources are likely to span larger
geographical markets. The commercial and industrial (C&I) loan portfolios of large banks
will generally
represent a greater mix of industries than those of small banks, both because large banks
are likely to
have expertise in a wider variety of borrower industries and because they are more active
than small
institutions in the secondary market for C&I loans. Furthermore, large banks will in general
have more
sophisticated risk management systems than small institutions. It seems noncontroversial
then that large
banks have better diversification opportunities than small banks. But diversification is not
the only
determinant of risk, and certain activities and characteristics typically associated with large
banking
institutions may be inherently risky.
This paper focuses on the empirical relationships between bank size, diversification, and
risk.
We begin by constructing a conceptually sound measure of overall bank holding company
(BHC)
diversification and establishing a strong positive correlation between that measure and asset
size. We
measure diversification using the R' statistic from an equity return-generating regression,
or "market
model." This application is not novel to the finance literature but has been overlooked in
the banking
literature. In our opinion, it is superior to diversification measures derived from the balance
sheet
because the balance sheet lacks information on key aspects of diversification such as industria
l or
regional loan concentrations. Moreover, a measure based on equity returns reflects all types
of
diversification within the holding company, including diversification associated with each
of the bank and
nonbank subsidiaries and diversification across those subsidiaries.
Using total stock return variability to measure risk, we illustrate that the positive correlatio
n
between size and diversification does not translate into a negative correlation between size
and risk. We
explore this result further by decomposing total risk into its two components: systematic
risk and firm-
specific risk. Our analysis indicates that asset size is negatively related to firm-spe
cific risk but
positively related to systematic risk. This result is puzzling since a fundamental
implication of portfolio
theory is that only non-systematic risks are affected by the scale of a portfolio
. If a large BHC can be
characterized simply as a collection of smaller BHCs, portfolio theory predicts
that larger BHCs will
have less firm-specific risk than smaller BHCs but similar systematic risk. Our
findings confirm that
BHC size and firm-specific risk are negatively related but contradict the predicti
on that BHC size and
systematic risk are unrelated. These results highlight the importance of differen
ces in the activities of
larger and smaller BHCs.
The final section of this paper explores the relationship between size, systema
tic risk, and firmspecific risk controlling for activities and characteristics related to BHC size.
We find that certain
characteristics and activities, particularly capitalization and C&I lending, can
help explain the
unconditional positive correlation between BHC size and systematic risk. In
contrast, the negative
relationship between BHC size and firm-specific risk becomes stronger when
we control for activities
and characteristics related to BHC size.
Our analysis has several interesting policy implications. The first relates to the
safety and
soundness of the banking system. While BHC size is positively related to diversif
ication, that
relationship comes about because larger institutions have lower firm-specific
risk and because larger
institutions have greater systematic risk. From the standpoint of the safety and
soundness of the banking
system, this is somewhat disconcerting. Reductions in firm-specific risk are
certainly advantageous, but
their advantages are primarily realized by individual institutions. The safety
and soundness of the
banking system depends more on levels of systematic risk, wh1ch we find to
be positively correlated with
BHC size. Our results also have implications for merger policy. The negative
relationship between
BHC size and firm-specific risk lends empirical support to the notion that growth
enhances
diversification. The associated risk reduction will depend on the extent to which
that growth is
accompanied by changes in an institution's activities and portfolio characteristics.
We find that over the
2
past decade, BHC asset growth has been accompanied by economically important reductions in risk.
The next section motivates the use of the market model R2 as a measure of BHC diversification
2
and presents R statistics for a sample of BHCs using three alternative return-generating models. In
section III, we explore the unconditional relationships between BHC size, d;versification, and risk.
Section IV presents a multiple regression analysis relating risk and diversification to BHC size,
controlling for activities related to size. We also estimate a fixed effects model that addresses changes in
BHC size over time. Section V concludes.
II. Measuring Diversification
Barnea and Logue (1973) first used the market model R2 to measure the industrial diversification
of conglomerate corporations. 1 They adopted the market model R2 statistic as a measure of industrial
diversification because it reflects the degree to which a given conglomerate "mirrors the diversity of the
economy and the relative importance attaching to each segment of the firm's activities within the context
of the whole economy." We argue that the concept of R2 as a measure of diversification has broader
applicability. In (l'articular, it provides a convenient summary measure of overall diversification for
BHCs that does not suffer from the deficiencies of measures derived from the balance sheet, which lack
information on key aspects of diversification such as industrial or regional loan concentrations. 2
The market model for a BHC is an ordinary least squares (OLS) regression relating the BHC's
stock return to the return on a stock market index:
(I)
In equation (1), the market index serves as a proxy for systematic factors common to the returns on most
stocks. Since the residual vector is uncorrelated with the vector of market returns, the following
1
Barnea and Logue's approach has been adopted in several subsequent works calling for measures
of conglomerate diversification. See, for example, Amihud and Lev (1981) and Amihud, Kamin, and
Ronen (1983).
2
Roll (1988) explores the relationship between R2 and size for firms in a wide range of industries.
3
variance decomposjtion holds:
(2)
The left-hand side of equation (2) represents the return variability, or
"total risk," associated with the
I\
given BHC's equity. Equation (2) decomposes that return variability
into two parts: /f'a2(R,,,) represents
the estimated return variability stemming from systematic factors; u2( A
e) represents the estimated return
variability specific to the given BHC. Following conventional termino
logy we will refer to the two risk
components as "systematic risk" and "firm-specific risk," respectively.
We use the R' statistic (the ratio
of systematic risk to total risk) as a measure of overall BHC diversification
.
Alternative Return-Generating Models
The one-factor model depicted in (I) uses a stock market index as a
proxy for systematic factors
common to most stock returns. With BHCs, we may consider some
additional systematic factors not
adequately captured by the stock market index. For instance, variabl
es related to interest rate risk and
credit risk may provide additional explanatory power over that provide
d by stock market returns.
We estimate a second return-generating model by adding three regress
ors. Yield is the change in
the three-month Treasury Bill rate, where that rate is defined as the
bond equivalent yield. Positive
values of yield represent increases in short term rates. 3 Term is the
change in the spread between 30year and three-month Treasury rates with positive values representing
a steepening of the yield curve.4
Credit quality spread is the change in the spread between rates on Moody
's Baa-rated corporate bonds
and 30-year Treasury Bonds. Positive values represent an increase in
the premium charged for credit
risk.' Our multi-factor model is thus defined as:
' Specifically, Yield,= three-month rate, - three-month rate,.,.
4
Specifically, Ter111i=(30-year rate - three-month rate), - (30-year rate
- three-month rate)t-1'
' Specifically, credit quality spread,
= (Baa rate-30-year rate), - (Baa
4
rate-30-year rate),.,.
+ ~ 4Credit
Quality Spread, +
e,
(3)
While the multi-factor market model is certainly richer than the simple market model, it may
stiU exclude some relevant systematic factors. To address this concern, we also estimate a third returngenerating model. This model differs in concept from those expressed in (I) and (3). Rather than
estimating a regression based on a set of economicaUy interpretable independent variables, we apply
factor analysis to our sample of BHC stock returns and endogenously derive a set of five vectors whose
linear combination best explains the returns of our sample of stocks. Though these five vectors are not
interpretable economicaUy, they can be considered systematic factors common to the returns of the BHCs
in our sample.• The factor analysis approach has the advantage that systematic risk is not constrained to
be a function of observable economic variables. Consequently, this third approach provides a check on
the robustness of results derived from return-generating models that may have omitted variables.
The R2 derived from factor analysis (the fraction of return variability explained by the five
endogenously derived factors) has a somewhat different interpretation from those derived using the
market model or multi-factor model. This R2 measures the extent to which a particular BBC's stock
tracks those systematic elements common to the BHC stocks in our sample, as opposed to economy-wide
systematic factors. One advantage of this approach is that risks systematic to banks, including regulatory
risks such as changes in capital requirements or deposit insurance premia, will now contribute to the
systematic component of risk, whereas in the earlier models such risks would likely be absorbed by the
firm-specific component. On the other hand, if most BHCs in our sample have a common risk, such as
large exposure to problems in the commercial real estate sector, a BHC with that exposure may appear
to be more diversified than one lacking that exposure.
• We have also estimated a six-factor decomposition of BHC stock returns and found nearly identical
results to those generated using the five-factor model.
5
We estimate each of the three return-generating models using data from the Center
for Research
in Securities Prices (CRSP). We identified over 150 publicly traded BHCs by
referring to the 1985
Bank Compustat database. Our analysis is based on those BHCs that traded for
at least 30 weeks in a
given calendar year between 1980-1993 and for which we could retrieve boch
CRSP return data and data
from Y-9C reports describing BHC characteristics. 7 As Column (I) of Table
1 indicates, our sample
size ranges from 80 in 1993 to 133 in 1986. This variability arises because several
of the BHCs in our
sample did not have traded equity in every year between 1980 and 1993. In the
case of mergers, we
dropped acquired companies from the sample after the date of acquisition. Acquire
rs remain in the
sample.
We constructed weekly (Friday-to-Friday) BHC stock returns using daily CRSP
return data.'
Columns (2) and (3) of Table I provide return summary statistics by year. The
lowest mean weeklyreturn of-0.85 percent occurred in 1990, amidst growing concern regarding bank
asset quality. The
mean weekly-return for 1987, the year of the stock market "crash," is also low,
at -0.36 percent. The
highest mean weekly-return of 0.89 percent occurred in 1991, as the economy
pulled out of its recent
recession. Return standard deviations tend to rise over most of the sample period,
illustrating an
increasing trend in overall bank risk.•
Table I also presents the median R2 for each yearly sample, based on the market
model, multi-
7
Compustat and CRSP both use "cusip" numbers as company identifiers. Howeve
r, the only
identifier common to Compustat and the Y-9C is the company's name. Our sample
includes only those
BHCs for which the match between the name provided by Compustat and that
appearing on the Y-9C
was unambiguous.
8
Daily returns are adjusted by CRSP to account for dividend payouts and stock
splits. In cases
where Friday was a holiday and no stocks were traded, we used the Thursday-to-Fri
day or Friday-toThursday returns instead. In cases where the return was not available for a given
stock on a given
Friday, that stock's weekly return was coded as missing.
• Neuberger (1993) reports a similar pattern for his sample of 84 large BHCs.
6
factor model, and factor analysis approaches (Columns (4), (5), and (6), respectively). 10 We estimate
these models by year to account for the possibility that the estimated parameters change over time." In
the multi-factor model, yield, term, and credit quality spread are each measured on a Friday-to-Friday
basis, to maintain consistency with our weekly return variables. In both the market model and multifactor model, R,., is measured using the weekly return on an equally-weighted portfolio of stocks trading
on the NYSE or AMEX. Since we are particularly interested in the relationship between the market
2
model R and asset size we use an equally-weighted index rather than a value-weighted index. 12
Column (4) reports the median market model R2s, which range from 2.5 percent in 1993 to 36.2
percent in 1987. A relatively high median R2 is observed in 1987 because the "crash" of October 1987
resulted in a large degree of explained variance (systematic risk) for most stocks. Adding the interest
rate and credit risk variables to the return-generating model increases the median R2s for each year by
about 5 to 10 percentage points (column (5)). The median R2 for 1987 continues to exceed those for
other years. 13
We attribute the low R2s from both the market and multi-factor models in 1992 and 1993 to
recent changes in the regulatory environment, including full implementation of risk-based capital
standards and increases in deposit insurance premia. These events represent risks particular to banks;
i.e., risks that will not be captured by the systematic factor(s) underlying the market model and multifactor model. Note that the R2 derived using factor analysis (column (6)) remains high in 1992-1993,
since it reflects systematic factors common to the BHCs in our sample. In general, the factor analysis
10
We do not report estimates for the multi-factor model in 1980-1982 because the data used to construct
the credit quality spread variable were unavailable for those years.
11
Using a switching regression model, Kane and Una! (1988) find that the parameters of returngenerating models for depository institutions changed significantly in 1979 and 1982.
12
A market model estimated using a value-weighted index will induce some degree of correlation
between R2 and asset size.
13
Coefficient estimates from the multi-factor return-generating model are available from the authors.
7
approach results in higher R's and greater stability of R's over
time.
m.
Diversification, Size, and Risk
Use of the market model R2 as a diversification index facilitates
a straightforward investigation
of the relationship between BHC size and diversification. Table
2 reports the correlation between asset
size and the R's estimated using each of the return-generating
models described above. Asset size is
measured as of the beginning of each year in order to avoid a
potential simultaneity problem. 14 Since
we look only at publicly traded BHCs, our sample asset size
distribution is not representative of all
BHCs. It does provide ample size variation, however. In 1993,
the asset sizes of the BHCs in our
sample ranged from $340 million to $214 billion, with a media
n of $10 billion. Taken as a group, these
BHCs held close to half of all banking assets in the U.S.
In Table 2, colum ns(]), (3), and (5) report the Pearson correla
tion coefficient between the log
of total assets and each of the three R' statistics for each year.
Columns (2), (4), and (6) report the
Spearman correlation coefficient. Both correlation coefficients
are positive and highly significant for
almost every year in o_ur sample period regardless of which return
-generating model was used to derive
R2 • We conclude that the positive relationship between BHC
size and diversification is both strong and
robust. 15
Some interesting differences emerge when comparing the correla
tion coefficients based on the
market model R2 with those based on the factor analysis R'.
When either of the two market models are
used, correlation coefficients for 1988 and 1992 are somewhat
lower than those calculated for other
14
For instance, abnormally strong performance by a set of BHC
stocks may lead to a spurious
negative correlation between asset size and the market model 2
R if that strong performance promotes
asset growth.
15
Roll (1988) compares firm size to the R's from a market model
and a five-factor analysis, using
the stocks of all companies traded on the New York and Ameri
can Stock Exchanges from September,
1982 through August, 1987. He finds a positive relationship
between firm size and the market model R'
but finds "a much less perceptible positive relation" when compa
ring firm size with the R' from the fivefactor analysis.
8
years and coefficients for 1993 are not significantly different from zero. When we use factor
analysis to
estimate a return-generating model, however, 1988, 1992, and 1993 correlation coefficie
nts rise to levels
comparable to those of the other years in the sample period. We attribute differences in
results derived
from the alternative return-generating models to events that had systematic effects on BHC
stock returns
but are not reflected in our multi-factor model. Possibilities for 1988 include the S&L crisis,
the full
impact of which may not have been felt until that year; BBC-specific events from 1992 and
1993 include
changes in the regulatory environment for banks, such as the implementation of risk-base
d capital
requirements and risk-based deposit insurance, and rising deposit insurance premia.
These findings leave unanswered an important question: Does the enhanced diversification
of
large BHCs result in lower overall risk? Columns (7) and (8) of Table 2 present the Pearson
and
Spearman correlation coefficients between asset size and total stock return variability (i.e.
return
variance) over the 1980-1993 period. In each year between 1980 and 1991, these two correlatio
n
coefficients were either positive or not statistically significant, suggesting that large BHCs
did not exhibit
lower total risk than small BHCs. Between 1980 and 1984 we estimate positive and significa
nt
correlation coefficients between size and total risk. 16 Between I 985 and 1991 there is no
relationship
between size and risk. Only in 1992 and 1993 do we estimate a negative relationship. Overall,
there is
little evidence that asset size is associated with lower BHC risk. 17
Table 3 presents the correlation coefficients between asset size and the two components of
total
risk. As shown in Panel A, there is a negative relationship between firm-specific risk and
BHC size,
16
Along similar lines, Samolyk (1994) finds that very large banks (asset greater than $10 billion)
had
return on assets exhibiting greater variability than other banks between 1984 and 1992.
These
institutions also had substantially higher levels of non-performing assets over this period.
17
Demsetz and Strahan (1995) consider the impact of recent regulatory reforms on changes
in the
size/risk relationship since 199 I.
9
particularly after 1984. 18 Panel B shows, however, that large BHCs
consistently hold more systematic
risk than small BHCs in every year throughout the sample, with the
exception of 1992 and 1993. This
relationship holds for the Spearman correlation across all three returngenerating models. The Pearson
correlation displays a similar pattern, although these results are less robust.
Between 1984 and 1991,
then, large BHCs exhibited smaller firm-specific risk but greater system
atic risk than small BHCs.
These offsetting relationships are consistent with the positive relation
ship between size and diversification
and the generally insignificant difference between the total risk of smaller
and larger institutions.
The results of Table 3 have two interesting implications. First, they
suggest that from the
standpoint of the safety and soundness of the banking system, large size
may not be as desirable a
characteristic as we may have believed. While larger BHCs appear
more diversified than smaller BHCs,
they also appear to have higher levels of systematic risk. This sensitiv
ity to systematic shocks induces
correlation among the returns of large BHCs. Consequently, adverse
events could lead simultaneously to
poor performance on the part of many large BHCs.
Our results also suggest that a large BHC sho.uld not be characterized
simply as a collection of
smaller BHCs. If large banking institutions were nothing more than
scaled-up versions of smaller
institutions, then portfolio theory would imply that large institutions
hold less firm-specific risk than
smaller institutions, but similar levels of systematic risk. Our finding
s confirm the predicted negative
relationship between BHC size and firm-specific risk but run counter
to the prediction that there is no
relationship between BHC size and systematic risk, suggesting that there
are important differences
between the characteristics and activities of large and small BHCs. 19
It may be that the heightened
18
This result is consistent over time, though most of the yearly correla
tion coefficients are not
statistically significant.
19
Boyd and Gertler (1993) provide some examples, including differences
in the C&J lending, core
deposits, and capital-to-assets ratios of banks of different sizes. They
discuss the role of these
characteristics in explaining differences in two accounting measures
of bank performance: net loan
charge-offs and net income to assets.
systematic risk associated with larger BHCs is a natural and necessary byproduct of certain large-BHC
activities valuable to the financial system: We explore this hypothesis in the next section.
IV. Bank Activities, Diversification, and Risk
This section explores differences in the characteristics and activities of large and small BHCs in
order to explain the unconditional correlations between BHC size and the components of risk. Our
framework of analysis is a multiple regression in which systematic or firm-specific risk is a function of
BHC size and a series of variables describing BHC characteristics and activities. In particular, we
estimate the following regressions:
(4)
(5)
where E/ A equals the BHC's equity capital-asset ratio; 'l'' and 'l'' are kxl vectors of parameters; and Z,.,,;
is a vector of BHC characteristics and activities, dated as of the beginning of the period over which the
diversification index is measured. 20 This timing convention ensures that the direction of causality flows
from the set of variables in Z to the dependent variables. We estimate each equation using pooled time
series/cross-section data from 1987 to 1993 and include a set of time fixed-effects to control for changes
in risk common to all BHC stocks in our sample. 21
The purpose of these regressions is to determine how BHC size is related to the components of
risk, controlling for the characteristics and activities included in Z. We are particularly interested in
whether the coefficient on size remains positive and significant in the systematic risk regression, equation
20
The derivation of the specification relating the log of risk to the log of total assets and the log of
(E/ A)2 is available from the authors.
21
We begin this analysis in 1987 because the data reported to regulators, particularly for off-balance
sheet activities, increased significantly after 1986. Also, we include only the pooled regressions since
preliminary analysis of the individual cross-sectional regressions corresponding to these equations suggest
little evidence that the parameters changed over the sample period.
II
(4). That is, we are interested in the extent to which the elements of Z help explain the
unconditional
positive correlation between size and systematic risk illustrated above. If we consider a
BHC to be a
portfolio of assets, liabilities, and off-balance sheet instruments, then increases in scale should
lead to
reductions in firm-specific risk but not systematic risk. Thus, to the extent that Z captures
all relevant
attributes, portfolio theory predicts that {J'
= 0.
Also, by controlling for BHC characteristics and
activities, we can estimate the partial effect of size on firm-specific risk, fl'. To the extent
that the
smallest BHCs in our sample have not exhausted the bulk of the risk reducing potential
of
diversification, portfolio theory predicts that fJ' < 0.
BHC Characteristics and Activities
The Z vector in equations (4) and (5) includes variables describing each BHC's assets, liabilities
,
subsidiary diversification and off-balance sheet positions, to the extent that meaningful measures
of these
characteristics are available from the Y-9C report. This sub-section describes each variable.
The last
sub-sections describe and interpret our empirical findings.
Asset Composition. We measure asset composition by including the ratio of commercial and
industrial (C&I) loans to assets, real estate loans to assets, consumer loans to assets, agricultu
ral loans to
assets and trading-account assets to total assets. We also include an index of loan concentr
ation to
control for the dispersion in lending across the following categories: real estate loans, C&I
loans,
consumer loans, agricultural loans and other loans. Our index is modelled after the Herfinda
hlHirschman index (HHI), commonly used in analyses of market concentration for antitrust
purposes. As
applied to the case at hand, the HHI equals the sum of the squared share of each loan category
relative
to total loans. Increases in this index represent increases in specialization in a given loan
category.
Liability Composition. We include the ratio of total deposits to total assets as a measure of the
contribution of the banking subsidiaries to the total business of the BHC. Also, as a crude
description of
the attributes of the funding of the bank portion of the holding company, we include the
ratio of noninterest bearing deposits to total deposits. The non-interest bearing deposits variable includes
demand
12
deposits, non-interest bearing passbook savings accounts, and non-interest bearing time deposit accounts.
We also include the ratio of deposits from foreign sources to total deposits as a measure of geographical
diversification outside the U.S. 22
Subsidiary Diversification. Beginning in 1987 the Federal Reserve Board began allowing BHCs
to hold subsidiaries engaged in limited underwriting under Section 20 of the Glass-Steagall Act of 1933.
These "Section 20 subsidiaries" are granted permission to underwrite corporate and government
securities such as commercial paper, corporate debt and equity, and municipal bonds. Proponents of
universal banking have argued that diversification associated with combining commercial and investment
banking functions in a single financial services holding company could contribute to stability of the
financial system. To test the hypothesis that investment banking activities significantly enhance BHC
diversification and reduce risk, we include an indicator variable equal to one for BHCs with Section 20
subsidiaries. Since Section 20 subsidiary powers were significantly increased between 1987 and 1990,
we include this variable only in a separate estimation of equations (4) and (5) based on data from 1991 to
1993.
We also differentiate between BHCs operating broadly throughout the nation from those
operating only within one region by including an indicator variable equal to one for BHCs with
commercial bank subsidiaries located in more than one census region. While the secondary loan market
permits BHCs to hold loans originated in many regions, BHCs operating more widely throughout the
country may have increased protection against regional downturns. 23
Off-Balance Sheet Activities. Two variables are included to measure each BHC's use of
derivative instruments: (I) the ratio of the notional principal on interest rate swaps to total assets; and (2)
22
Recent research shows that geographical diversification can be more effective than industrial
diversification in reducing risk. See Heston and Rouwenhorst (1994).
23
Levonian ( I 994) shows that bank accounting profits across states exhibit low correlations,
suggesting that banks operating in many states may be able to reduce risk through diversification.
13
the ratio of the notional principal on foreign exchange futures to total assets. We use the
notional
principal amounts of these derivatives to reflect the scale of derivatives activities, while
acknowledging
that they do not represent either the marked-to-market value or the risks associated with
the contracts.
No better measures of derivatives activities were available throughout the sample period. 24
We also include the ratio of non-interest income to net interest income as a measure of each
BBC's reliance on off-balance sheet activities. Boyd and Gertler (1994) show that the ratio
of noninterest income to net interest income may be interpreted as the ratio of the present value
of the BBC's
off-balance sheet contracts to the present value of its on-balance sheet assets. This interpreta
tion holds
under two assumptions: (I) the average returns to off- and on-balance sheet assets are equal;
and, (2)
non-interest expenses are allocated to off- and on-balance sheet assets in proportion to their
value.
Capital. The Z vector controls for differences in the characteristics and activities of
BHCs
which influence risk through their effects on variabilities of asset, liability and off-balan
ce sheet returns.
We also include the log of the square of the equity capital-asset ratio (based on book values)
to control
for the effects of leverage on the levels of both systematic and firm-specific risk." Since
a given
fluctuation in asset value translates into larger variation in the value of equity at more highly
leveraged
firms, we expect to find a negative relationship between the equity capital-asset ratio and
both firmspecific and systematic risk.
Trading Frequency. Since the variables in Z may not fully account for the underlying variances
of the BBC's assets, liabilities, and off-balance sheet positions, we also include a measure
of the
frequency with which each BBC's stock is traded. Trading frequency proxies for the rate
of arrival of
new information about the economic. value of a BHC's stock, which is presumably correlate
d with the
24
Since collection of data on foreign exchange swaps began only in the third quarter of 1990,
we do
not include a measure of foreign exchange swap activity in the regression analysis.
" A simple model of bank diversification that motivates the use of this particular functiona
l form is
available from the authors.
14
underlying variances of the BHC's assets, liabilities and off-balance sheet positions. We measure trading
frequency using the variable turnover, defined as total trading volume divided by the average shares
outstanding over the year in which stock return variability is measured. Data on both volume and shares
outstanding are from CRSP.
Empirical Results
Table 4 contains simple summary statistics for all of the independent variables included in
equations (4) and (5). Since we are interested in explaining the correlation of size and risk based on
differences in portfolio attributes of large and small BHCs, Table 4 also presents the Pearson correlation
of the log of asset size with each of the other variables included in equations (4) and (5). As shown,
large BHCs engage more heavily in C&I lending, hold more assets in the trading account, hold more
foreign deposits, and hold more derivative instruments than small BHCs. More large BHCs operate
across multiple census regions than small BHCs. Small BHCs are more likely to be dominated by their
banking subsidiaries than large BHCs, as measured by the ratio of deposits to total liabilities. Table 4
also shows that large BHCs tend to have smaller capital ratios than small BHCs. 26
Table 5 presents the results from estimating equations (4) and (5). Columns (I) and (2) present
the estimated coefficients and t-statistics when the dependent variable is systematic risk; columns (3) and
(4) present the estimated coefficients and t-statistics when the dependent variable is firm-specific risk.
Columns (I) and (3) include the regression results with only the size variable (along with time fixed
effects), while columns (2) and (4) contain the results including size along with the control variables.
We present the results using the factor analysis model to divide risk into systematic and firm-specific
components; similar results were obtained using the multi-factor return-generating model.
The results in Table 5 show quite clearly that size affects risk according to the predictions of
portfolio theory, once the attributes of the BHCs' portfolios have been held constant. The coefficient on
26
Boyd and Runkle (1993) also find that large banks have smaller capital ratios than small banks.
15
size in the firm-specific risk regressions is negative and statistically significant at the one
percent level in
both specifications including the control variables. By contrast, the positive, unconditional
relationship
between size and systematic risk becomes insignificant when the BHC characteristics and
activities are
included in the regression. 27
Comparing the results of columns (I) and (3) with those of columns (2) and (4), it is evident
that
controlling for a BHC's activities leads to an increase in the magnitude of the negative size
coefficient in
the firm-specific risk regression and a decrease in the magnitude of the positive size coefficie
nt in the
systematic risk regression. The size coefficient falls from 0.17 to 0.07 in the systematic
risk regression
and falls from -0.14 to -0.25 iri the firm-specific risk regression. The coefficient on size
is biased
upward when the control variables are omitted because, on average, the activities and character
istics of
large BHCs increase both firm-specific and systematic risk, offsetting the potential for risk
reduction
associated with increased diversification.
These results explain the observed unconditional correlations between size and the compone
nts of
risk. As noted above, portfolio theory suggests that increases in scale will reduce non-syste
matic risk,
holding constant portfolio attributes. Consistent with this theory, we find that large BHCs
hold less
firm-specific risk than small BHCs, controlling for assets, liabilities and off-balance sheet
activities
available from the Y-9C Report. In fact, the results indicate that large BHCs achieve economic
ally
important reductions in risk through diversification. For instance, doubling the size of a
BHC while
holding its portfolio attributes fixed would lead to a reduction in firm-specific risk of about
25 percent.
Portfolio theory also implies that increases in scale should have no direct impact on systemat
ic risk,
again holding constant portfolio attributes. Consistent with this prediction, we find no correlatio
n
between size and systematic risk when we control for the attributes of the BHC's portfolio
.
Rosenberg and Perry (1981) reach similar conclusions using data from 1969 to 1977. They
find
that asset size is a good predictor of the market beta but that the relationship between size
and systematic
risk can be explained by other variables in their model.
27
16
Comparing; Table 4 (the correlation of the regressors with log size) with Table 5 (the regression
results), it becomes clear why large BHCs exhibit more systematic risk than small BHCs. First, the
stock returns of large BHCs exhibit greater trading frequency, as measured by the variable turnover,
which is positively correlated with measured stock price variance, both syst~matic and firm-specific.
Perhaps more interesting, large BHCs engage in more C&I lending, are more active in derivatives and
hold less capital than small BHCs. Table 5 shows that C&I lending, derivatives holdings (as measured
by FX futures) and low capital ratios all are positively associated with systematic risk. The
unconditional positive correlation between BHC size and systematic risk results from these large-BHC
activities. Table 5 also shows that C&I lending and low capital ratios are positively related to firmspecific risk. 28 As a consequence, the effect of size on firm-specific risk is understated when these
variables are omitted from the regression.
It is worth noting, however, that not all of the attributes of large BHCs are positively related to
risk. For example, large BHCs are more likely to operate in more than one census region. In 1987, for
example, 72 percent of BHCs in the top quartile of the size distribution operated across census regions
while only 12 percent of BHCs in the s!llallest quartile operated across census regions.
Table 6 presents the regression results using data from 1991-1993 and including the Section 20
subsidiary indicator. These results provide no support for the idea that universal banking will lead to
reductions in risk through diversification. In fact, we estimate that both systematic risk and firm-specific
risk are about 25 percent higher at BHCs holding Section 20 subsidiaries. The coefficients are
statistically significant at the IO percent level in both models. It is important to emphasize, however,
that these results may not apply to universal bank holding companies which could emerge following
major Glass-Steagall reform. Currently, a BHC must receive explicit approval from the Federal Reserve
28
Samolyk (1994) finds that banks with a higher proportion of C&I loans relative to total assets
have higher levels of nonperforming assets and net charge-offs, providing further evidence from
accounting information that this category of loans is high risk relative to other kinds of bank lending.
17
Board to set up a Section 20 subsidiary and can generate no more than 10 percent of its gross revenues
from that subsidiary. Whether similar results would hold in a different regulatory environment remains
an open question. However, these results do raise questions about the validity of the argument that
universal banks could reap significant risk reduction from diversification.
The other results in Table 6, which are based on the 1991-1993 period, are quite consistent with
the results estimated over the whole 1987-1993 sample period. In particular, we continue to find a
positive and significant relationship between size and systematic risk when portfolio characteristics are
omitted but no relationship between size and systematic risk when portfolio attributes are included in the
regression. The negative relationship between size and firm-specific risk also becomes stronger (more
negative) when these portfolio controls are included in the model.
BHC Growth and Risk
Because of data limitations, the regressions reported in Table 5 are based only on the last seven
years of our sample period. In this section, we estimate an alternative specification capable of
accommodating data from 1983 to 1993 (when the multi-factor model is used to decompose risk) or from
1980 to 1993 (when factor analysis is used to decompose risk). In particular, we pool data across time
and regress the log of systematic or firm-specific risk on log of total assets, a set of time fixed effects
and a set of firm fixed effects. Because the Z vector, the equity capital-assets ratio, and the turnover
variable are not included in this specification, the size coefficients quantify the total derivative of each
risk component with respect to BHC size. Since the model includes firm fixed effects, however, the
estimate of this total derivative is driven solely by within-firm variation over time; the asset size
coefficient reflects both the direct effect of asset growth (either by expansion or acquisition) and the
indirect effect of portfolio changes associated with asset growth.
We expect the size coefficient to be negative when the dependent variable measures firm-specific
risk, since both the unconditional and conditional relationships between asset size and firm-specific risk
were negative and significant. Our prior on the size coefficient in the systematic risk regression is less
18
clear. To the extent that non-size characteristics and activities are stable over time, we would expect to
observe an insignificant size coefficient. However, if changes in asset size over the sample period are
accompanied by changes in non-size characteristics and activities, we may observe either a positive or
negative size coefficient.
Results based on a risk decomposition derived using the multi-factor model are presented in the
top half of Table 7. Those based on a risk decomposition derived using factor analysis are presented in
the bottom half of Table 7. In both cases, the size coefficient is negative and significant when the
dependent variable measures firm-specific risk, as expected. When the dependent variable measures
systematic risk, we observe mixed results. Asset growth is associated with significant declines in
systematic risk when the risk decomposition is based on the multi-factor model and insignificant changes
in systematic risk when the risk decomposition is based on factor analysis. Taken as a whole, the results
in Table 7 confirm that asset growth, either through acquisition or simple expansion, led to risk
reduction for the BHCs in our sample. Coupled with our earlier results, this suggests that asset growth
of BHCs in our sample was not accompanied by significant risk-enhancing changes in portfolio
composition.
V. Conclusions
This paper provides strong evidence of a link between size and diversification at large, publiclytraded bank holding companies. The correlation between the R2 from three return-generating models and
the log of total assets is consistently positive and significant from 1980 to 1993. This result is important
but not surprising, since it is generally accepted that large banks have better opportunities to diversify.
We show, however, that the positive relationship between BHC size and diversification does not result in
a negative relationship between BHC size and total risk during most of the sample period. This second
result has important implications for the policymaker, who must fully understand the risks that particular
institutions introduce into the banking system. Our results suggest that larger BHCs exhibit less firmspecific risk than smaller BHCs but greater systematic risk. These relationships are approximately
19
offsetting, so that larger BHCs and smaller BHCs exhibit similar total
risk.
The positive relationship between BHC size and systematic risk may
be troublesome from the
standpoint of the safety and soundness of the banking system as a whole.
Systematic risk is particularly
undesirable since it increases the likelihood that many BHCs will simulta
neously face difficulties in a
given economic environment. Our results suggest that the positive correla
tion between BHC size and
systematic risk is in large part due to the differences in the characteristics
and activities of large and
small BHCs. Specifically, large BHCs hold more C&I loans, hold more
derivative instruments, and
operate with lower capital ratios than small BHCs. Each of these charact
eristics and activities tends to
be associated with increases in systematic risk. Despite their riskier
portfolios, however, large BHCs do
not exhibit greater overall risk than small BHCs because of the effects
of diversification. In fact, our
results show that increases in size may lead to large reductions in firm-sp
ecific risk, provided these
increases in size are not accompanied by risk-enhancing changes in portfol
io composition.
20
References
Amihud, Yakov and Baruch Lev. "Risk Reduction as a Managerial Motive for Conglomerate
Mergers." Bell Journal of Economics 12 (1981), 605-617.
Amihud, Yakov, Jacob Y. Kamin and Joshua Ronen. "Managerialism, Ownerism and Risk." Journal
of Banking and Finance 7 (1983), 189-196.
Barnea, Amir and Dennis E. Logue. "Stock-Market Based Measure of Corporate Diversification."
Journal of Industrial Economics (1973), 51-60.
Boyd, John H. and Mark Gertler. "U.S. Commercial Banking: Trends, Cycles, and Policy." In
NBER Macroeconomics Annual, edited by Olivier Jean Blanchard and Stanley Fischer, pp. 319368, Cambridge: The MIT Press, 1993.
Boyd, John H. and Mark Gertler. "Are Banks Dead? Or Are the Reports Greatly Exaggerated?"
Quarterly Review, Federal Reserve Bank of Minneapolis, Summer 1994.
Boyd, John H. am:I-David E. Runkle. "Size and Performance of Banking Firms." Journal of Monetary
Economics 31 (1993), 47-67.
Demsetz, Rebecca S. and Philip E. Strahan. "Historical Patterns and Recent Changes in the
Relationship between Bank Size and Risk." Federal Reserve Bank of New York mimeo, 1995.
Heston, Steven L. and K. Geert Rouwenhorst. "Does Industrial Structure Explain the Benefits of
International Diversification?" Journal of Financial Economics 36 (1994), 3-27.
Kane, Edward J. and Haluk Una!. "Change in Market Assessments of Deposit-Institutions
Riskiness." Journal of Financial Services Research I (1988), 207-229.
Levonian, Mark E. "Interstate Banking and Risk." Weekly Letter, Federal Reserve Bank of San
Francisco 94:26 (1994).
Neuberger, Jonathan. "Risk and Return in Banking: Evidence from Bank Stock Returns." Federal
Reserve Bank of San Francisco mimeo,. I 993.
21
Roll, Richard. "R,2 ," Journal of Finance XLIII (1988), 541-566.
Rosenberg, Barr and Philip Perry. "The Fundamental Determinants of Risk in Banking."
In Risk and
Capital Adequacy in Commercial Banks, edited by Sherman Maisel, pp. 367-407. Chicago:
University of Chicago Press, 1981.
Samolyk, Katherine A. "U.S. Banking Sector Trends: Assessing Disparities in Industry
Performance." Economic Review, Federal Reserve Bank of Cleveland 30:2 (1994), 2-17.
22
Table 1
Summar y Statistics for BHC Returns and Return-G enerating Model R2 Statistics
Weekly Returns
Median R2
N
Mean•
Standard
Deviationb
Market
Model
Multi-Factor
Model
Factor
Analysis
(1)
(2)
(3)
(4)
(5)
(6)
1980
118
0.35%
3.49%
21.1%
48.4%
1981
118
0.41%
3.14%
12.3%
36.1%
1982
119
0.29%
3.75%
21.2%
41.0%
1983
129
0.62%
3.31%
8.3%
14.5%
33.3%
1984
131
0.25%
2.96%
11.9%
17.9%
37.0%
1985
132
0.70%
3.28%
10.3%
16.0%
35.3%
1986
133
0.11%
4.10%
21.0%
26.2%
44.9%
1987
129
-0.36%
4.90%
36.2%
44.5%
55.6%
1988
119
0.17%
3.92%
10.4%
18.2%
35.7%
1989
111
0.28%
3.71%
11.2%
16.5%
33.7%
1990
105
-0.85%
5.54%
21.2%
25.7%
48.2%
1991
98
0.89%
5.51%
22.1%
27.9%
44.2%
1992
89
0.74%
4.12%
6.8%
12.7%
41.0%
1993
80
0.06%
3.60%
2.5%
10.6%
46.6%
• Annual means, averaged over the BHCs in our sample.
• Annual standard deviations, averaged over the BHCs in our sample.
Table2
Pearson and Spearman Correlation of BHC Size with R2 and with Overall Stock Return Variance'
Market Model R 2
Multi-Factor Model R'
Factor Analysis R'
Overall Stock Return Variance
Pearson
Spearman
Pearson
Spearman
Pearson
Spearman
Pearson
Spearman
(I)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
1980
0.13
0.24•
0.44-
0.48-
0.20·
0.33-
1981
0.39-
0.46-
0.61-
0.59-
0.18
0.29-
1982
0.33-
0.38-
0.42-
0.35-
0.45-
0_4g-
1983
0.64-
o.5r
0.60-
0.54-
0.55-
0.48-
0.24-
0.24·
1984
0.56-
o.sr
0.57-
o.51-
0.59-
0.53-
0.45-
0.43-
1985
0.59-
0.56-
o.5r
0.53-
0.46-
0.40-
-0.01
0.09
1986
0.65-
0.64-
0.62-
0.61-
o.6r
0.56-
0.03
-0.02
1987
0.31-
0.35-
0.30-
0.32-
0.44-
o.4r
0.05
0.10
1988
0.25•
0.28-
0.18
0.19
0.42-
0.40-
-0.07
-0.04
1989
o.6r
0.68-
0.62-
0.60-
0.54-
0.56-
-0.13
0.01
1990
0.42-
0.48-
0.32-
0.37-
0.49-
0.53-
-0.03
0.20·
1991
0.55-
0.50-
0.46-
0.39-
o.s1·
0.48-
-0.03
0.12
1992
0.26-
o.2r
0.28-
0.31-
0.6S-
0.63-
-0.30-
-0.21"
0.69-
o.6r
-0.35-
-0.14
1993
-0.13
-0.15
-0.01
-0.03
Toe Pearson correlation is computed using the log of total assets as of the end of the prior year and R' or total stock return
variance over the current year.
Values marked "**" are statistically significant at the 1 % level; those marked "*" are significant at the 5% level.
0
Table 3, Panel A
Pearson and Spearman Correlation of BHC Size with Finn-Specific Risk•
Market Model
Multi-Factor Model
Return-Generating Model
Based on Factor
Analysis
N
Pearson
Spearman
Pearson
Spearman
Pearson
Spearman
(1)
(2)
(3)
(4)
(5)
(6)
(7)
1980
118
0.18
0.33-
-0.01
0.15
1981
118
0.12
0.23"
-0.02
0.08
1982
119
0.34-
0.39-
0.20·
0.30-
1983
129
0.11
0.14
0.09
0.12
-0.06
-0.01
1984
131
0.30-
0.29-
0.25-
0.26-
0.11
0.17
1985
132
-0.12
-0.05
-0.12
-0.07
-0.18
-0.16
1986
133
-0.10
-0.29-
-0.10
-0.28-
-0.18
-0.36-
1987
129
-0.01
-0.06
-0.01
-0.06
-0.08
-0.22·
1988
119
-0.07
-0.08
-0.07
-0.07
-0.08
-0.19
1989
111
-0.16
-0.13
-0.16
-0.13
-0.16
-0.14
1990
105
-0.07
0.10
-0.06
0.11
-0.08
-0.07
1991
98
-0.07
0.02
-0.07
0.04
-0.12
-0.03
1992
89
-0.31-
-0.23"
-0.31-
-0.24*
-0.35-
-0.4r
80
-0.34-
-0.12
-0.35-
-0.11
-0.37-
-0.4r
1993
•The Pearson correlation is computed using the log of total assets as of the end of the prior year and firm-specific risk (total residual variance from the three
return-generating models) over the current year. Values marked '**" are statistically significant at the 1% level; those marked '*' are significant at the 5%
level.
Table 3, Panel B
Pearson and Spearman Correlation of BHC Si:re with Systematic Risk'
Market Model
Multi-Factor Model
N
Pearson
Spearman
(1)
(2)
(3)
1980
118
0.15
1981
118
1982
•
Return-Generating Model
Based on Factor
Analysis
Pearson
Spearman
Pearson
Spearman
(4)
(5}
(6)
(7)
0.34-
0.43-
o.51-
o.3r
0.46-
o.4r
0.49-
119
0.55-
0,59-
0.64-
0.61-
1983
129
o.6r
0.54-
0.53-
0.48-
0.54-
0.44-
.1984
131
0.72-
0.66-
0.74-
0.64-
0.68-
0.64-
1985
132
0.54-
o.5r
0.49-
0.46-
0.36-
0.33-
1986
133
0.65-
0.65-
0.59-
0.59-
0.39-
0.44-
1987
129
0.21·
0.29-
0.18
0.26-
0.24·
0.38-
1988
119
0.06
0.23*
-0.06
0.05
0.02
0.26-
1989
Ill
0.40-
o.so-
0.02
0.35-
0.11
o.3r
1990
105
o.4r
0.54-
0.13
0.46-
0.10
1991
98
0.26-
0.46-
0.15
0.31-
0.21·
1992
89
0.03
0.05
-0.03
1993
80
-0.23-
-0.26-
-0.33-
,
0.42o.2r
0.02
-0.06
0.12
-0.17
-0.06
0.17
'The Pearson correlation is computed using tlie log of total assets as of the end of the prior year and systematic
risk (total explained variance from the three
return-generating models) over the current year. Values marked "**" are statistically significant at the
1 % level; those marked "*" are significant at the 5%
level.
Table 4
Summary Statistics for BHC Portfolio Attributes
Mean
(Standard Deviation)
with Log Size'
(!)
(2)
Correlation
Log Size
16.2946
(1. 1568)
C&I Loans/Assets
0.1863
(0.0641)
Real Estate Loans/Assets
0.2299
(0.0947)
-0.2i-
Agricultural Loans/Assets
0.0046
(0.0062)
-0.02
Consumer Loans/Assets
0.1209
(0.0537)
0.01
Loan Concentration
0.3206
(0.0536)
Trading Assets/Assets
0.0124
(0.0346)
Total Deposits/Assets
0.7595
(0.0944)
Non-interest Deposits/Deposits
0.2165
(0.0675)
Foreign Deposits/Deposits
0.0853
(0. 1571)
Multi-Census Indicator
0.5009
(0.5005)
Notional Principal on Interest Rate
Swaps/Assets
0.1842
(0.4768)
Notional Principal on FX
Futures/Assets
0.2974
(0.7842)
Non-interest Income/Net Interest
Income
0.6651
(3.7104)
Book Value of Capital/Assets
0.0619
(0.0138)
Stock Market Turnover
0.7049
(0.4856)
-0.50•
o.5r
o.5i-
0.01
'Correlations are based on pooled data over the 1987-1993 period. These correlations are adjusted for time fixed
effects. That is, the table presents the correlations of the difference between the variable and the annual means of
that variable using the pooled data. Values marked ''"'"" are statistically significant at the I% level; those marked
"'"" are significant at the 5 % level.
Table 5
Regressions of Factor-Analysis Firm-Specific Risk and Systematic Risk on BHC Size, Characteristics and
Activities. Pooled Data from 1987 to 1993, with Time Fixed Effects. T-statistics in parentheses; "**" indicates
statistical significance at the 1% level, "*" indicates statistical significance at the 5% level.
Dependent Variable:
Ln(Systematic Risk)
Ln(Firm-Specific Risk)
Log of Assets
Without
Controls
With
Controls
Controls
With
Controls
(I)
(2)
(3)
(4)
0.171
(6.06)-
0.067
(1.72)
Without
-0.142
(-4.33)-
-0.248
(-5.74)""
C&I Loans/Assets
2.537
(5.31)""
2.333
(4.43)""
Real Estate Loans/Assets
1.044
(2.20)"
0.758
(1.45)
Agricultural Loans/Asset
1.102
(0.23)
12.388
(2.38)"
Consumer Loans/Assets
0.523
(0.74)
1.384
(1.77)
-0.194
(-0.25)
1.807
(2.10)"
Trading Assets/Assets
-0.013
(-0.01)
-1.282
(-0.82)
Total DeposiJs/Assets
-0.072
(-0.16)
0.489
(1.00)
-0.510
(-1.34)
-0.374
(-0.74)
-0.431
(-0.80)
-0.368
(-0.88)
-0.212
(-3.33)-
-0.260
(-3.71)-
Interest Rate Swaps/Assets
-0.029
(-0.19)
FX Futures/Asset
0.170
(2.03}"
0.043
(0.26)
0.063
(0.68)
Non-interest Income/Net Interest
Income
0.009
(1.21)
0.016
(2.00}"
Log (CapiJal/Assets Squared)
-0.644
(-8.88)-
-1.056
(-13.20)""
Turnover
0.423
(6.31)-
0.332
(4.50)-
Loan Concentration
Non-interest D.!Tota/ D.
Foreign D.!Total D.
Multi-Census Indicator
N
551
551
551
551
R'
35.70%
59.37%
14.16%
51.31%
Table 6
Regressions of Factor-Analysis Firm-Specific Risk and Systematic Risk on BHC Size, Characteristics and
Ac.tivities. Pooled Data from 1991 to 1993, with Time Fixed Effects. T-<itatistics
in parentheses; "**" indicates
statistical significance at the 1% level, "*" indicates statistical significance at the 5% level.
Dependent Variable:
Ln(Systematic Risk)
Ln(Firm-Specijic Risk)
Without
(I
With
Controls
(2)
0.135
(3.04)..
0.076
(1.32)
Controls
Log of Assets
C&I Loans/Assets
Without
Controls
With
Controls
(3)
(4)
-0.215
(-5.01)""
-0.262
(-4.29)""
3.930
(5.18)..
1.308
(1.68)"
4.281
(0.57)
0.760
(0.59)
-0.333
(-0.23)
2.764
(3.44)..
1.445
(1.75)"
12.913
(1.62)
0.668
(0.49)
0.728
(0.47)
Trading Assets/Assets
-0.405
(-0.22)
-1.168
(-0.61)
Total Deposits/Assets
0.959
(1.10)
0.276
(0.36)
-0.247
(-0.41)
0.063
(0.07)
0.198
(0.25)
0.337
(0.53)
0.266
(1.87)"
-0.389
(-3.72)..
0.256
(1.70)"
-0.323
(-2.92)..
0.169
(0.82)
0.026
(0.25)
0.042
(1.10)
0.104
(0.48)
-0.028
(-0.25)
0.033
(0.83)
-0.969
(-7 .22)""
0.408
(3.82) ..
-1.022
(-7 .20)..
0.233
(2.06) ..
Real Estate Loans/Assets
Agricultural Loans/Asset
Consumer Loans/Assets
Loan Concentratwn
Non-interest D./Tota/ D.
Foreign D./Total D.
Section 20 Subsidiary Indicator
Mu/Ji-Census Indicator
Interest Rate Swaps/Assets
FX Futures/Asset
Non-interest Income/Net Interest
Income
Log (CapiJal/Assets Squared)
Turnover
N
R'
212
17.78%
212
62.15%
212
26.39%
212
59.68%
Table 7
Regressions of Firm-Specific Risk and Systematic Risk on BHC Size. Pooled Data from 1983 to 1993 for
MultiFactor return-generating model and 1980 to 1993 for Factor Analysis return-generating model, with Time
and
Firm Fixed Effects. T-statistics in parentheses;
statistical significance at the 5% level.
11
*"'" indicates statistical significance at the 1 % level, "*"
indicates
Log(Systematic Risk)
Log(Firm-Specific Risk)
(1)
(2)
-0.200
(-2.08)"
--0.277
(-3.80)"'
N
1256
1256
R'
61.07%
51.21 %
-0.057
(-0.88)
-0.133
(-2.17)'
N
1611
1611
R'
56.31 %
42.92%
Multi-Factor Return-Generating
Model
Coefficient on Log(Size)
Factor Analysis Return-Generating
Model
Coefficient on Log(Size)
Addendum: A Model of Bank Asset Risk and Diversification
This appendix describes a model of economies of scale in diversification based on asset risk.
We use this model to motivate our choice of functional form in Equations (4) and (5). For this purpose,
we assume that liabilities are non-stochastic and that the bank has no off-balance sheet positions.
Suppose bank i holds n assets.
The return on assetj held by bank i is generated as follows:
j=l, ... ,n
where R'; equals the systematic component of the return (to simplify notation, this is assumed to be the
same for all assets held by bank i); E;; is the diversifiable component of the return, assumed to be
independent of R', and independent of the other E;;• So, the return on bank i's assets equals the
following:
R,"
=
"
R:+ E
j=l
where w,;
(w11 e1)
= the portfolio weight for assetj at bank i.
The return on equity, then, may be written as follows:
n
R1
=
(A/E) 1[R,' +
E (w J e,)]-(L/E)J{/
1
j=l
where A = total assets; L = total liabilities; E = total equity capital; and (R~
bank's liabilities (assumed to be non-stochastic).
Systematic risk may be written as follows:
o 21(s) = (A/E)22s)
1 o (R1
= the return on
the
Assuming that the ;bank holds equal proportions of each of its assets (w;J
be expressed as follows:
a:(fJ =
= 1/n), firm-specific risk may
•
(AIE>7E ((l/n)2a7;)
}"l
2 -2
= (,4/E)1 a 1(fJ I n
-2
where a 1(fJ "
•
't""
.t...,
/•I
2
a 1J I n
a:; " the variance of e1J
We allow n (the number of assets with independent risks held by bank i) to vary with size, as follows:
n = K(Assets)"
so that
2
2 -2
a,(fJ = (A/E), a,(fJ
I
(K(Assets)")
The parameter a describes the relationship between asset size and the number of assets with
independent risks. If a = 0, then large and small banks hold the same number of assets with
diversifiable risks (i.e. large banks achieve no reduction in risk from diversification); if O < a < 1,
large banks achieve reductions in risk from diversification but there are diminishing returns; if c, > = 1,
large banks achieve reductions in risk from diversification and there are constant or increasing returns
(this case seems implausible). This model motivates the log specification in equations (4) and (5), as
follows:
Since a 2(R;') and a';(t) may vary with the configuration of each bank's portfolio, we include a set of
portfolio characteristics (Z) in both the systematic and firm-specific risk regressions. These variables are
included to permit unbiased estimation of a. Both regressions also include the log of the squared equity
capital-asset ratio and the log of total assets, allowing us to test the hypothesis that c, = 0 when the
dependent variable is systematic risk. This functional form will not fully reflect the relationship between
risk and size if equity return variance is generated by risks associated with the bank's liabilities as well
as its assets. However, this simplification is unlikely to introduce a significant bias whenever the bulk of
a bank's risks stem from its assets.
2
@FedFRASER