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Portfolio analysis of asset and liability management in small-, medium- and large-sized banks
Journal of Monetary Economics 4 (1978) 459--480. © North-Holland Publishing Company
PORTFOLIO
ANALYSIS OF ASSET AND LIABILITY MANAGEMENT'
IN SMALL-, MEDIUM- AND LARGE-SIZED BANKS Jack Clark F R A N C I S * Bernard M. Baruch College, CUNY, Nel~ York, N Y 10010, U.S.A.
The paper presents an analysis of the commercial banking firm based on Markowitz portfolio analysis. A bank is treated as a portfolio of five banking assets and three banking liabilities. The average rate of return and risk of each asset and liability is estimated empirically for groups of banks categorized by size - small, medium and large. Banks' rates of return on equity are defined as the weighted average of the assets' rates of return less the liabilities' rates of return. Quadratic programming is used to delineate the set of banking portfolios which have the maximum rate of return on equity at each level of risk.
1. Introduction In Crosse and Hempel (1973) the authors, one o f w h o m was a Federal Reserve b a n k examination officer, state: 'Small banks tend to be director-managed because they do not earn enough to be able to affJrd top-grade operating m a n a g e m e n t . . . ' (p. 43). The author goes on to say that in ' m o s t of these banks the examiner rated the m a n a g e m e n t as ranging from fair to p o o r . ' The purpose of this study is to confront statements such as the preceding with empirical data a n d analyze the asset a n d liability m a n a g e m e n t of average small-, medium- a n d large-sized banks in the risk return context suggested by M a r k o w i t z (1959). 1 Some recent banking studies tend to direct their attention to banking competition [see Gies a n d Apilado (1971, section 1)] a n d various economies of scale in banks [see Benston (1965), Daniel et al. (1973) a n d Silverberg (1973)]. But, banking studies should also consider the ability of banks of various sizes to manage their portfolios of assets a n d liabilities as they endeavor to evaluate *This paper benefited from helpful comments offered by George Oldfield, Amos Tuck School of Business Administration, Dartmouth College, and an anonymous referee. The author is also indebted to the Federal Reserve, where he was an Economist, for the data and computer programming assistance. 1Parkin's work (1970) is a somewhat similar contribution which precedes this study. However, the approaches differ. Unlike Parkin's model, this model abstracts from dollar quantities to facilitate comparing the asset and liability holdings of different sized banks and graphing the analysis in risk-return space. This analysis deals with more assets and liabilities than Parkin's, recognizes constraints, and differs in other respects. This model extends and tests Pyle's (1971) model; it is essentially the model Black (1975) described informaify.
460
J. C. Francis, PortJ~4io analysis
the desirability of branching, merging, and the optimal expansion path for the bank/ng system. Therefore, this study holds economies of scale constant and compares and contrasts the risk and return of the principal balance sheet items across average banking statistics gathered from a cross-section of hundreds of banks categorized by' size- small, medium and large. The principle hypotheses being examined is that, ceteris paribus, large banks manage money better than small banks. Because aggregate data is used in this study, some individual banks can be found which operate differently than the aggregate (or average) data suggests. However, individual banks are not relevant to this macro-analysis of banks" asset and liability management. Section 2 redefines the financial statements of a bank in symbolic terms suitable for analysis and discusses the proportions that various asset and liability items occupy on actual banks' balance sheets. The empirical sample is explained in the third section. In section 4 the types of risk peculiar to individual asset and liability items are specified in terms of variance and average return or interest rate statistics. Section 5 explains the total risk of each asset and liability item and discusses the interaction of systematic market forces and unsystematic default risk. In ~ection 6 the unconstrained efficient frontier of banking assets and liabilities are discussed. Positive economic analysis is undertaken in the seventh section by constraining the banking efficient frontiers of the average small-, medium- and large-banks to obtain efficient frontiers similar to empirically observed average bank portfolios. The findings are summarized and conclusions are suggested in section 8. 2. Aeeoa~ag deCmltions principal assets and liabilities of a commercial ba.nking firm are listed on its balance sheet. Table 1 shows what financial analysts call a common-sized ~ n c e sheet, with all items stated as a percentage of equity, averaged over hundreds of smalb ($), medium- (M), and large-sized (L) banks. The balance sheet ide~tity is stated in eq. (1), where the letters denote dollar quantities for the items in table 1. (c+g+h+b+l)+(-d-s-f
) = e,
O)
(Total assets)-(Total liabilities) = Equity.
(la)
Diving eq. (I) through by equity yields the common-sized balance sheets shown in eqs. (2) and (2a). The symbol w~ denotes the weight of the ith balance sheet item sta'ed as a percentage of equity (that is, wc = c/e, for example), =
=
1.0.
(2)
$
w~ = 1.0 = w,. |ffil
(2a)
J.C. Francis, Portfolio analysis
461
Constraining equity to equal unity is the standard convention in Markowitz (1959) portfolio analysis. This constraint is no way limits equity. Equity may assume any positive dollar amount and vary in every period. Equity is defined as unity, or equivalently 100°/.,o, for all sizes of banks to facilitate calculation of the rate of return on equity and owners" risk for cross-sectional comparisons.
Table 1 Common-sized balance sheet for the average small-, medium- and large-sized banks, 1971, Weights Assets (symbol) Cash and uncoil. funds (c) Gov't. bonds (g) H o m e and other mortgages (h) Business loans (b) Installment loans (1)
Total assets
S
M
Weights
L
1.3 3.7
1.5 3.5
1.8 2.9
2.0 1.5 2.4
1.8 1.3 2.6
1.5 3.3 1.0
10.9
10.7
10.5
Liabilities (symbol) D e m a n d deposits (d) Savings deposits (s) Cert. of deposit (f) Equity (e)
Total liabilities and equity Sample size
S
M
L
-4.2 - 2.4
-4.5 - 2.4
-4.7 - 2.3
- 3.3 - 1.0
- 2.8
- 2.5
-
-
- 10.9
- 10.7
- 10.5
684
231
79
1.0
1.0
2,1. The average b~lance sheet proportions
Perusal of table ! reveals some interesting asset and liability patterns for the different sized banks.-" First, the average banks' ratios reveal that small banks have a preference for installment lending while the larger banks make relatively more business loans. The analysis which follows suggests that all banks may make too many business loans and insufficient installment l o a n s - in this respect the small banks asset portfolio proportions are more optimal. The existing pattern probably results from the lending opportunities readily available to banks of different sizes- large business headquarters do tend to be located in cities near the larger banks. A second interesting pattern is the small banks heavier reliance upon deposits purchased by issuing Certificates of Deposit (CD). The Markowitz portfolio analysis suggests that the smaller banks should use CDs more than larger banks. 2The ratios shown in Exhibit 1 differ slightly from the ratios in the Functional Cost Analysis because minor Balance Sheet items are omitted to expedite this presentation. And, all the statistics vary over the business cycle somewhat.
462
J.C. Francis, Portfolio analysis
This may result from the lower cost of CDs available to the smaller banks, on average (see top of tables 2, 3 and 4 for average return data). Calculation of the return on equity necessary for Markowitz analysis is exptlained next. 2.2. Rate o f return o f bank equity
The rate of return on equity for a bank may be stated symbolically as the weighted average rates of return on the five assets and three liabilities, as shown in eq. (3), where rj denotes the net rate of return or one-period yield on the /th balance sheet asset or liability. wcr¢+wgrg+wsrh+wvr~+w:r~+wdrd.eWsrs+Wyrf+k = r e.
(3)
The symbol k represents fixed costs as a percent of equity. Assets' weights are non-negative (we, % , w~, w~, w~ _>_0) and the weights of liabilities are non~ t i v e ( ~ . %; wj. _<_0). Eq. (3) could be restated as an income statement in dollar quantities ~,~y~ult.',piying b,')th sides of the equation by the dollar value of equity. However, banks' financial ~,tatemevts are formulated in common-sized per&~tag~ to facilitate analysis an,:l compvrison of different banks.
3. Six years of empirical data from 1966 through 1971 inclusive are analyzed [Functional cost analysis (FCA) (1971)]. Over 68 large banks (with over $200 million in deposits), over 198 medium banks (with deposits between $50 and $2~.g) rqillioa) and over 603 small banks (with less than 550 million in deposits) coml~se the sample underlying the average banking statistics used here. Rates of return (annual interest expense) on each of the six years were averaged over all banks in the three bank-size categories to obtain average annual returns (interest costs). These six annual rates of return (rates of interest ¢ x ~ : ) were all net of the direct expenses incurred in managing each category ~ 0labilities). The averages and variances shown in tables 2, 3 and 4 were computed from the six years of average returns net of their direct expenses for each asse*, a.~d liability category. 3 Economies of scale associated with bank ~A~,erage rates of return per year over six years were used to measure the typical returns &~ ~ over ~'~ecomplete businesscycle. This facilitates drawing general conclusions about f~¢ ~ and liability management abilities of the average small-, medium- and large-sized ~ . Only t ~ direct (that is, marginal) costs of labor, related executive salaries, departm e n ~ fum/ua~, computer usage, postage, telephoning, printing, insurance, etc., were deducted /n ~ / ~ g tim net rate of return (e~qx'nse) associated with each asset (liability) category. | ~ readers may obtain the annual Functional Cost Analysis booklets for free from the Federal ~ e if ff~y wish to see a de, a/led breakdown of these direct expenses. Fixed costs
J.C. Francis, Portfolio analysis
463
management, with the exception of check processing, 4 have been held constant in calculating the net rates of return (rate of expense) associated with each asset (liability). And, regional considerations peculiar to one area of the nation have also been eliminated by using large samples of banks from across the United States. Table 2 Risk and return statistics for the average small bank.
Mean annual rate of return (or cost) Credit risk
e
g
h
1
b
d
s
f
0 0
5.86 0
5.57 0.03
6.80 0.35
5.62" 0.24
1.89 0
4.51 0
4.87% 0
Var~ance-covariance matrix with credit risk and market risk included:
Cash & reserves (c) Government bonds (g) Mortgages on homes, etc. (h) Installment loans (1) Business loans (b) Demand deposits (d) Savings deposits (s) Certificates of deposit (f)
c
g
h
1
b
d
s
f
0 0
0 0.92
0 0.45
0 0.23
0 0.66
0 0
0 0.43
0 0.47
0 0 0 0 0 0
0.42 0.23 0.66 0 0.43 0.47
0.26 0.11 0.30 0 0.24 0.24
0.I 1 0.48 0.16 0 0.11 0.13
0.30 0.16 0.73 0 0.31 0.33
0 0 0 0 0 0
0.24 0.11 0.31 0 0.27 0.24
0.24 0.13 0.33 0 0.24 0.26
"This statistic was adjusted upward for compensating balances as explained in section 3.
(for example, the bank President's salary, the cost of the bank's building, etc.) were not allocated or deducted in determining the net rates of return and expense in order to minimize the vagaries of cost accounting for joint costs. Sampling and aggregation bias problems appear to be minimal. The items under the various asset and liability headings for each size bank are highly homogeneous because of (a) bank regulation, and, (b) the commensurate market power of banks which are similar in size ensures additional uniformity. Essentially, the sample of transactions underlying each balance sheet item for each bank size category differ only with respect to the date on which they were execute,~. Since 'banks which are in trouble" usually decline to participate in the Federal Reserve°s Functional Cost Analysis unusually low rates of return (viz., losses associated with a bankruptcy) are not likely to downbias the arithmetic average returns. Furthermore, the sample is such a large portion of the population of all 'sound banks' that the sample is highly representative. 4Economies of scale in check processing are attainable and do enter this analysis through the cost of demand deposits. The cost of check processing as a percent of demand deposits at small, medium and large banks is 1.89%, 1.73 % and 1.71 % in 1971, respectively, as shown in tables 2, 3 and 4. These differences are essentially the result of economies of scale. These economies could have been eliminated simply by setting the different banks' costs of demand deposits all equal. This was not done, however, because (a) the differences are small, and, (b) check processing costs are a type of money management cost.
464
1.C. Francis, Portfolio analysis
One of the statistics in tables 2, 3 and 4 has been adjusted. The recipients of business loans frequently are not allowed to use the full amount of the loan. Many borrowers are required to keep a percent of thei~ loan unused at the bank as a compensating bal~nce that increases the bank's effective yield. Using approximately the number published by a Federal Reserve bank examination officer, it is assumed that 18 % of all business loans must be kept in the bank as a ~ompensating balance [Cross and Hempel (1973, p. 20)]. As a result, the effective net yield the average large-, medium- and small-sized bank receives on business loans is estimated to be 7.4 % as shown below: s Gross rate of return u~nadj.
7.05 % = net yield + dir. exp. x 1.22 = 1/(1.0-0.18)
times adjustment factor:
8.62 %
Effective gross rate ei ' return less direct expenses:
after comp. bal.
- 1.22 %
from FCA
7.40%
Net effective yield
Table 3 Risk and return :~tistics for the average medium bank.
Mean annual rate of return (or--_cost) Credit .:isk
0 0
g
h
1
b
d
s
f
5.97 0
5.68 0.02
6.38 0.36
5.85" 0.25
1.73 0
4.64 0
4.99% 0
Variance-covariance matrix with credit risk included:
Cash and rese~,es (e) bonds (8) M ~ on homes, eu:. (h) Inualinm~t loans (1) loans (hi dqx~hs (d) Say/rigsdeposits (s) C_.m'lifimt~ of deposit ( f )
c
g
h
1
b
d
s
f
0 0
0 0.70
0 0.38
0 0.24
0 0.70
0 0
0 0.35
0 0.39
0 0 0 0 0 0
0.38 0.24 0.70
0.28 0.14 0.30 o 0.23 0.23
0.14 0.45 0.20 o 0.13 0.13
0.30 0.20 0.93 o 0.30 0.33
0 0 0 o 0 0
0.23 0.13 0.30 o 0.22 0.20
0.23 0.13 0.33 ,o 0.20 0.22
o
0.35 0.39
q'his statistic was adjusted upward for compensating balances as ,explained in section 3. make-up of the etficient frontier was somewhat sensitive to the rate of return (interest) on the ,,arious ~ (liabilities). To evaluate the sensitivity of, say, the risk of the efficient fr~ wilh respect m a change in the effective yield on business loathe;the following derivatives may h e ~ :
[~ var(r,)]~w~][~wd~E(r~)] = ~.b[awdaE(r~)]. Io ~ event, the conclusions with regard to the asset and iiabiltity management of small-, ~i~and large-sized banks are not sensitive to the effective yield used for business loans. See F t ' a ~ and Archer (1979, ch. 13) ¢ ~ . e m i n g the mathematical portfolio analysis.
J.C. Francis, Portfolio analysis
465
This adjustment was necessary because the FCA (1971) reported average rate of return on business loans ignored compensating balance effects. Table 4 Risk and return statistics for the average large bank.
Mean annual rate of return (or cost) Credit risk
c
g
h
1
0 0
6.21 0
5.89 0.03
6.16 0.3;7
b
d
6.15a 1.17 0.30 0
s
f
4.78 0
0
5.35%
Variance--covariance matrix with credit risk included:
Cash and reserves (c) Government bonds (g) Mortgages on homes, etc. (h) Installment loans (1) Business loans (b) Demand deposits (d) Savings deposits (s) Certificates of deposit (f)
c
g
h
1
b
d
s
f
0 0
0 0.77
0 0.38
0 0.32
0 0.69
0 0
0 0.28
0 0.46
0 0 0 0 0 0
0.38 0.32 0.69 0 0.28 0.46
0.34 0.19 0.22 0 0.18 0.19
0.19 0.51 0.25 0 0.12 0.18
0.22 0.25 1.05 0 0.20 0.45
0 0 0 0 0 0
0.1~ 0.1.2 0.20 0 0.16 0.14
0.19 0.18 0.45 0 0.14 0.30
"This statistic was adjusted upward for compensating balances as explained in section 3.
4. Risk-return analysis of items Before delineating the efficient banking frontiers for the three size categories of banks it is instructive to consider the risk and return of the individual asset and liability items.
4.]. Bushwss and mortgage loans Figure 1 shows graphically that the standard deviation of six annual rates of return or expense and the 1971 average rate of return or expense were positively related across the three size categories of banks for business loans (denoted b) and mortgage loans (denoted h). This is the positive relationship between risk and return suggested by economic theory. It appears likely that the disproportionately large a m o u n t of business loans made by large banks, on average, (see table 1), is obtained by making risky loans at high rates of interest.
4.2. CDs and installment loans Certificates of deposit (denoted f ) and installment loans (denoted l) demonstrate neither the expected positive nor a negative relationship between risk and
466
J.C. Francis, Portfolio analysis
return in figure 1. The higher returns earned on installment loans by the smaller banks helps explain why they hold relatively more installment loans in their asset portfolios than large banks, on average. And the smaller banks' relatively Installment loans
7.0
rI
#
I
I M X
'
~ \
•
Government \
bond\
~
j
,~/
.'*'L'""
6,0 t
,'Mn, ' ~--~-
! I
!
,~-~
1
t
>< °
~. ~
t.1"
j
Business loans
s+/
('+
S."~__
")" - ~
1
"~.~
S'W
-...'""
:#s
Certificates / ~ L "1-~ of deposit -'~ M "
L.L. 5 . 0 0
,"
j
~
Savings deposits
~- 4.0.0
3.0~ i
O)
¢J
~
2.0
1.0 ! i i
ti
0
/
Cash O'.1
I
.__1
0.3
0.5
t . . . . .
0,7
i
0,q
m
l
1.1
Standard deviation of returns =,~/ar(r)=Fig. 1. Individual balance sheet items plotted in risk-return space.
heavy use of CDs as a source of funds (see table 1) probably results from the lower average interest expense small banks pay for these deposits (see figure I). One reason that small banks pay lower interest rates than larger banks fairly consistently may be that they sell their CDs to wealthy savers in small towns who
d.C. Francis, Portfolio analysis
467
are unaware that better returns are paid for CDs at larger b a n k s - which are presumably big city banks. 6
4.3. Savings deposits and government bonds Figure 1 shows a perverse negative relationship between risk and return for savings deposits (denoted s) and government bonds (denoted g) across the banksize categories. The small (so-called country) banks are often able to pay uncompetitively low interest rates on savings because their deposits are not highly mobile unless wide savings rate differentials become obvious to the small depositors; this is advantageous for the small banks. But, their dominated position in government bond investing tends to offset this advantage. The higher risk and lower net return small banks receive from government bonds may be the result of inept trading as the market interest rates and prices of these bonds change (since default losses were essentially nonexistent). Small banks lack of quantity discounts at brokerages may also hinder them. To obtain additional insights into banks' asset and liability management practices, the interaction of these items must be considered simultaneously. 5. Risk statistics
Two main types of risk will be included in this analysis- market (or interest rate) risk and the risk of loss from loans which are never repaid (that is credit or default risk).
5.1. Market risk Market risk may be viewed as variability of the banks' return caused by systematic exogenous forces which cause changes in market interest rates and direct operating expense. The first step in appraising market risk is to calculate the variances of net rates of return and interest expense for the five assets and three liabilities across the six year sample for the average small-, medium- and large-sized banks. These statistics are shown in tables 2, 3 and 4.
5.2. Credit risks Credit risk is measured by the variability of return resulting when creditors who have obtained mortgage, business or installment loans from a bank 6The loan losses appear to be statistically independent of the rates of return on loan assets ~according to page AI4 of the 1970 FCA issue). These loan loss rates are summed with other inite rates of return (cost) and the central limit theorem suggests that they will tend to be aormally distributed. Thus, the two-parameter analysis is appropriate for analyzing rates of •eturn on equity. IMonE-- B
468
J.C. Francis, Portfolio analysis
dishonor their debts. These three categories of loans are three Bernoulli populations. The variance of return from credit risk is calculated by multiplying the probability of loss for the ith loan category, p;, times its complement, var(r~) = ( p ) ( l - P i ) , as shown in table 5. Table 5 Credit risk statistics by bank size. a Business loans (b)
Percent of loans repaid, p~, Percent lost (1-pD Credit risk, var (r)
Small
Medium
Large
99.760 ~o 0.240 ~ 0.2394
99.746% 0.254 % 0.2533
99.796 0.304 ~o 0.3033
Small
Medium
Large
99646 °" /o ,'k254 ~ 0 3527
99.373o/ /o 0.363 ,%o 0.3620
99.621°" /o 0.369 ~/. 0.3676
Installment loans (1)
Percent of loans repaid, pi Percent lost, (1-p,) Credit risk, var (r)
Home anu edwr mortgages (h) '
Percent of loans repaid, ph Percent lost, (I-Ph) Credit risk, var (r)
S,mall
Medium
Large
0,/ 99.971/o 0.029 ,"~ 0.0289
99.976°/ 0.t~24 % 0.0239
/t o 99.969°; 0.031 ~'o 0.309
aSce footnote 6 and Yamane (1967).
5.3. Return generating function
Rates of return are random variables presumed to be the sum of a linear additive process, shown in eq. (4), for the j t h asset or liability in the tth time period: r jr -- i + p j + m jr + Cjt.
(4)
The proposed bank model is in no way dependent upon this breakdown of each item's one-period rate of return or expense; it was assumed merely to explain how the Federal Reserve data was adapted for use in the model. The four quantities on the right-hand side of eq. (4) are explained below.
J.C. Francis, Portfolio analysis
(1)
(2) (3) (4)
469
The real or pure rate of interest, not including inflation allowances or risk premiums, is presumed constant and equal to the marginal physical productivity of capital. It is denoted i and is the same for all assets and liabilities. The risk premium on the jth asset or liability [Sharpe (1967)] is presumed constant, it is represented by the symbol pj. For loans, the average ralc of credit losses is includea in pj. And, pj includes an inflation allowance. Market risk for the jth item in the tth period is a random variable with an expected value of zero, E(mj) = 0. The variance in m is measured around the value of (i +pj). The credit risk associated with thejth balance sheet asset in period t hv~san expected value of zero, E(cj) = 0, because the actuarially determined ~rat-.; of average credit losses has already been included in the risk premium. Thus, cj fluctuates from positive to negative from period to period as credit conditions vary.
Market and credit risk are presumed independent, cov (m, c) = 0, so that the variance of the jth item's return generating function, eq. (4), is shown in eq. (5), without covariances. 7 var(rj) = 0 + 0 + var(mj)+var(cj).
(5)
The variances in the variance--covariance matrices in tables 2, 3 and 4 contain both market and credit risk combined as shown in eq. (5).
6. Unconstrained banking efficient frontiers The asset and liability management opportunities of the three size categories of banks will be analyzed using Markowitz portfolio analysis. The combinations of banking assets and liabilities (that is, balance sheets) that maximize the bank's return on equity at each level of risk will be delineated and referred to as the efficient banking frontier. Of course, the overall efficient frontier which could be generated using a wider assortment of securities would dominate the efficient banking frontier. But, this analysis is limited to only bank assets and liabilities. The portfolio owners' risk, as measured by quadratic eq. (6), was minimized at each rate of return on equity, eq. (3), subject to the constraint that the assets 7Page AI4 of the 1970 edition of FCA says installment loan losses 'tend to occur regularly.' This implies homoscedasticity and independence. The same paragraph explained that loss rates on other loan categories varied, but no statistical dependence was suggested. Therefore, the covariance was presumed zero. It would be simple to use a nonzero covariance and include it in the analysis if additional information so indicated.
J.C. Francis, Portfolio analysis
470
less the liabilities equal equity of 100~o, that is, 1.0: 8
8
The portfolio analysis problem is represented mathematically by the Lagrangian objective function, z = var(re)+21[E(re)-E*]+22 1
.
i=t
w , - 1.
el .
(7)
In eq. (7) vat (re) is the variable to be minimized; z has no financial interpretation. The variables 2t and 22 are Lagrangian multipliers. Standard calculus techniques can be used to solve eq. (7) for the efficient frontier [Francis ~nd Archer (1979, ch. 13) shows the mathematics]. However, it was more expeditious to use a quadratic programming algorithm to perform the computations. Thus, a computer program [Karash (undated)] was used to minimize the bank owners' risk, eq. (6), iteratively at each rate of return on equity, eq. (3), to delineate the efficient banking frontiers shown in table 6 and graphed in figure 2. To nonbank financial analysts, one of the striking features of this analysis is the extent to which banks use leverage and its effects on their equity returns and risk. The risk and return of the individual banking assets and liabilities are shown graphically in figure I. Comparing figures 1 and 2 reveals that figure 1 and thus all the individual balance sheet items occupy only a tiny lower-left portion of figure 2 (see figure 2). The risk and return on equit) capital for the three sizes of average banks represented optimally by the efficient frontiers is expanded several times beyond the risk and return of the underlying individual assets and liabilities. The efficient frontiers shown in tat'le 6 and figure 2 have significant differences between their risk at each rate of return, a The average small bank's unconstrained efficient frontier (denoted US) in figure 1 dominates the average medium (UM) and the large (UL) bank's unconstrained optimum investment opportunities. This dominance exists in spite of the fact that the large bank's assets are spread more evenly across the three assets held, and, in spite of the fact that the Sin comparing the various efficient banking frontiers, the question arises as to whether they are significantly different. If the standard error implied by the central limit theorem applied to all banks in the sample is used, then ~the three efficient frontiers would be judged to be significantly different over most of their ranges at the 0.05 level of significance. That is, for the n = 944 (= 684+231 +79) banks which comprised the total sample using a standard deviation of the efficient medium-sized bank of E(r)= 25.0~ suggests ¢r,,,e= 2.29 (from column 3 of table 6). Using these values, the central limit theorem suggests a standard error of 0.0726 (= 2.29/~/(994)= o'm,t~/n) is appropriate for comparing efficient frontiers. This discussion proceeds on the assumption that the efficient frontiers are sufficiently different to ,.varrant investigating their differences.
J.C. Francis, Portfolio analysis
471
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J.C. Francis, Porifolio analysis
473
average large bank utilizes relatively more low-cost demand deposits than the other banks. Examination of the bank's average risk statistics (shown in tables 2, 3 and 4) reveals that the large banks tend to take the largest credit risk and market risk. The average large bank also has some sizable positive covariance between its various assets and negligible covariances between its assets and liabilities which reduce the opportunity to diversify effectively [Pyle (1971)].9 The smaller bank's stronger negative correlations between assets and certificate of depc.sit rates highlights a subtle risk-reducing benefit which helps rationalize the smaller banks' relatively higher use of CDs (shown in table I). The average large bank does earn higher net rates of return on four of its five assets than the other banks (see tables 2, 3 and 4). These superior returns might compensate for the high risks the average large bank assumes, except for the fact that the large bank also pays the highest interest rates to obtain deposits. After deducting interest costs, the average large bank's investment opportunity set in risk-return space would probably be dominated even if the risks undertaken by small and large average banks were equal. Essentially, many large banks' problems seem to stem from their locations in competitive urban areas where the net yield spreads are smaller and more variable. 7. Positive economics
As a practical matter, banks cannot hope to attain the unconstrained efficient banking frontier because of certain limitations. First, there is the legal constraint that banks hold some of their assets in nonearning required reserves. The average small bank, for example, holds cash and other non-earning assets equal to 1.3 times its equity, t° To obtain a realistic efficient banking frontier such constraints must be considered, as shown in inequality (8)"
1.3.
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91t might have been informative to examine all the banks in a given size category crosssectionally for the same time interval rather than analyze only time-series data for average annual returns and interest expenses. Unfortunately, however, the necessary disaggregated data for individual banks was unattainable. 1°As a practical matter, reserve requirements are stated as a percentage, denoted R, of deposits. This implies that a bank's reserves are R d dollars, where d is the dollar amount of deposit~;. In the formulation employed here, deposits are stated as some proportion, P, of equity, E. Thus, deposits are P E = d and the banks reserves are R P E = R d where R P > 1.3 in eq. (8). This shows that stating reserves relative to equity rather than relative to deposits, as is customary, are equivalent procedures mathematically. The structure of reserve requirements which req,lires reserves to vary with the dollar volume of a bank's deposits and the breakdown of deposits between various deposit categories is simplified to expedite comparing the average small-, medium- and large-sized banks. If one or more individual banks were being analyzed, it would be necessary to increase the number of constraints to reflect the full structure of reserve requirements. However, when the full set of capital adequacy constraints was added to this analysis, they were not binding after the liquid reserve constraint, inequality (8), was added.
474
J.C. Francis, Portfolio ar alysis
The medium and large banks were likewise constrained to provide insurance against insolvency by maintaining, on average [FCA (1971)], reserves of 1.5 and 1.8 times their respective equities. Competition is a second major constraint which limits banks. Banks cannot obtain all the interest-free demand deposits they desire because of competing banks which the depositor may find more convenient. The average small bank, for example, has checking account deposits equivalent to 4.2 times its equity. This average condition is treated as a constraint, as shown in eq. (9), in order to be able to compare the three sizes of average banks realistically with the actual average banks' risk and return statistics: wa ->- - 4.2.
(9)
The average medium and large banks are also constrained to hold less than or equal to the average amount of demand deposits for a bank their size, that is, 4.5 and 4.7 times equity, respectively. The Federal Reserve bank examiners encourage banks to maintain certain capital adequacy ratios [Chambers and Charnes (1961), Cross and Hempel (1973, ch. 5), McKinney and Browl~. (1974, ch. 14)]. These ratios were imposed on the quadratic program as constraints too. However, none of the capital adequacy constraints encumbered any efficient portfolio in any way. So, they aren't discussed further. The required liquid reserves, inequality (8), was the only portion of the capital adequacy ratios which constrains efficient portfolios of banking assets and liabilities. 7.1. Constrahled efficient frontier To determine if the constraints which face banks' asset and liability managers alter any of the previous findings, efficient frontiers were delineated subject to inequalities (8) and (9). The resulting statistics are shown in table 7. The constrained banking efficient frontiers- denoted CS, C M and C L - are graphed in figure 2 for the average small, medium and large banks, respectively. The reserve requirement constraint brought nonearning assets into solution but no more than the minimum requirement. And, the constraint on demand deposits caused banks to obtain more deposits from savings deposits and by selling more certificates of deposit. These changes reduced the efficiency of all three sizes of banks' investment opportunities and forced them to use m6re leverage to attain any level of return in an efficient manner. The average large bank used leverage, as measured by the debt to equity ratio, the least. However, the higher risk inherent in the assets and liabilities of the average large bank combined with its higher liability costs caused the constraints to reduce its efficient frontier (which was dominated before the constraints were added) even more, as shown graphically in figure 2.
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7.2. The average small, medium and large banks The empirically observed average proportions for small, medium and large banks are shown in tables c,~and 7 and represented graphically in figure 2. The actual portfolio holdings of the three average banks are roughly similar to their constrained efficient portfolios with the same level of return, with one major exception. None of the efficient portfolios contain government bonds. 1~ Casual empiricism suggests that the federal and municipal bonds [which have had their yields adjusted upward to compensate for their tax-exemption before this analysis, see FCA (1971, p. 10)] banks actually hold may result from five factors not included in the bank model: (1) indirect pressure from bank exzeminers to hold primary reserves in government bonds, (2) banks which are privileged to have a Tax and Loan Account from the U.S. Treasury can use these deposits to pay for purchases of Treasury bonds but not other issuer's bonds; (3) some bank's willingness to buy municipal bonds to help the local community, (4) the bank may obtain the municipality's demand deposits if it buys the municipality's bonds, and (5) municipal bonds may be counted as reserves at some state chartered banks. These subjective considerations are beyond the scope of this analysis of money management capabilities of different sized banks. Their effect is not always as powerful as was the case when analyzing the 1971 returns, however. Government bonds did enter efficient banking portfolios (which are not shown to save space) generated with other years' statistics.12 Only the average large bank is near its efficient frontier. The small and medium banks are far below their constrained frontiers ~presumably because of inept management and limited opportunities. Nevertheless, the average large bank is still dominated by the smaller banks because the large bank's opportunity set is inferior, on average. 111n preparing the Functional Cost Analysis books each year, the Federal Reserve endeavors to present data which is not distorted by bookkeeping gimmickry and tax tricks. Thus, for example, the Federal Reserve increases each bank's returns on tax-exempt bonds so that they are comparable with the returns on taxable bonds and publishes these adjustments. The upward adjusted returns on the tax-exempt bonds were used in this analysis so that tax considerations would not distort the analysis. See FCA (1971, p. 10) for an explanation of the adjustment for tax-exempt bonds' income. " In actuali'y, the adeptness with which a bank avoids paying income and other taxes can affect its aftec-tax earnings. However, analysis of tax-dodging procedures was ignored in this study of av~:rage money management capabilities. If tax-avoidance is a significant determinant of banks' holdings then the cenclusions of this study are biased accordingly. However, discussions with commercial bankers and central bankers suggests that tax-avoidance considerations are at befit a minor consideration m the management of most banking assets. And, tax considerations effect liability management decisions even less. 12Nine years of annual data from 1966 to 1974 were analyzed, although only the 1966-71 results are shown. The credit crunches in 1969-70 and 1973-4 kept those years' returns from being representative and thus precluded their use as a final year, that is, for use in eq. (3). The year 1971 was selected for discussion here because it was not distorted by Regulation Q ceilings, Phase I or Phase II price controls, or double-digit inflation.
J.C. Francis, Portfolio analysis
477
8. Summary and conclusions The preceding evidence indicates that, on average, large banks managed their assets and liabilities better than small- and medium-sized banks since the average large bank is practically on its constrained efficient frontier while the other banks are significantly below theirs. Nevertheless, the average large bank, and even its constrained efficient frontier, is dominated by the average small and medium banks. There are several reasons for this situation to exist. The large banks earn a better return on assets than the smaller banks, on average. However, the average large banks also pay higher interest rates on their liabilities. As a result, large banks average returns on equity are not above the smaller banks. And, the increased risks the average large bank must assume to achieve higher returns on assets make it easy for the average small bank to dominate in risk-return space even though the smaller banks do not appear to be as aggressively managed. 13 An examination of the correlation matrices (not shown here for brevity) associated with the variance--covariance matrices in tables 2, 3 and 4 revc:ds that the large banks' average returns from different assets are less correlated than smaller bank's returns. This suggests large banks may be able to diversi[,; more effectively across different assets. However, Pyle (1971) has shown that strong negative correlations between assets and liabilities are necessary to effectively utilize the risk reducing hedging effects available through intermediation. Unfortunately, the average large bank's assets and liability returns were not significantly correlated. Thus, the overall ability of large banks to reduce their owners' risk through hedging is not superior partly because their liabilities and assets' returns have the least significant correlations, on average. 1"~ Another reason that the average small bank's efficient frontier dominated the better managed large bank's efficient frontier is because of the higher installment loan returns available to the smaller banks. Figure 1 shows this, and also shows that the small banks were able to obtain savings deposits and CDs on more desirable terms than the larger banks, on average. The Functional Cost Accounting (FCA) booklets which provided the raw data for this analysis also contain a few nonfinancial descriptive statistics which help clarify the typical markets in which the different sized banks operate. The averages for each bank size category shown in table 8 help define the nature 13The more steeply progressive reserve requirements for large city banks which were implemented in 1966 and 1968 are reflected in the three different liquid asset constraints, inequality (8), and obviously place the larger banks' rates of return on equity at a disadvantage relative to banks of lesser size. 1,,The results of this study which suggest that large banks manage money better than small banks, on average, is evidence which weights in favor of liberalized bank expansion regulations. More specifically, if large banks were allowed to open branches state-wide and/or branch across state lines this would accelerate the acquisition of small country banks by large city banks and thus bring more banking assets and liabilities under better management.
1.17. Francis, Portfolio analysis
478
of the average bank in each category, and are useful in interpreting the analytical results and hypothesizing causes for different banking practices. It seems reasonable to suppose that the smaller banks are located in the rural small towns, This supposition is suggested by the well-known phrase 'small country bank' [Cross and Hempel (1973, pp. 14-16)] and the income percentages from 'farm management' revenue (whatever that may be) shown in table 8. Furthermore, differences in the average number of branches implies that the large banks are located in large cities where, on average, they face more banking Table 8 Selected descriptive statistics, 1971.
Number of branches Total personnel Percent of total revenues from farm management Deposits (D) Sample size (no. of banks)
Small
Medium
Large
1.25 45.96
5.54 204.07
32.05 1225.49
6.53~ g D < $50M 684
10.05% $50M < D < $200M 231
0.26% $200M < D 79
Source: Functional Cost Accounting (1971).
competition and more interest-elastic supply a , d demand functions for funds. In contrast, the banks in smaller towns apparently enjoy depositors that are less concerned with their yields, and, give the local banks their deposits at uncompetitive low interest rates. This low-risk, low-cost deposit environment in which the ~majority of small banks apparently operate insulates their owners' risk and return from the poor portfolio management which seems to exist, on average, in these banks. This implies that many of the differences between the average small-, medium- and large-sized banks result from the different environments in which they operate instead of explicitly different management policies. ~5 I SThe author of this paper was a Federal Reserve Economist who had considerable contact with bank executives in large cities (namely, Philadelphia) and small towns (for example, southern New Jersey and central Pennsylvania) alike. Through numerous business lunch and dinner contacts, speech-making at local bankers associations, supervision of surveys of the intentions of bank managers in big city banks, weekly telephone calls to key banking executives, and other contacts over a two-year period (1972-1973) the author developed some understanding of the differences between small-, medium-, and large-sized banks which cannot and should not be reflected in this analytical (that is, objective rather than subjective) study. In particular, the author found that bank executives at the rank of vice-president and above spend large amounts of time on formal and informal personal communications with their corporate customers, other commercial bankers, central bankers, and their colleagues. Very little time is spent on economic research (beyond newspaper reading) c: creative planning. In view of the large amount of time bankers spend cultivating 'the ,,grapevine' and their "customer relations' instead of doing research or planning, the author finds the explanatory power of the bank portfolio model surprisingly high.
J.C. Francis, Portfolio analysis
479
References Black, Fischer, 1975, Bank funds management in an efficient market, Journal of Financial Economics, December. Benston, George L, 1965, Economies of scale and marginal costs in banking operations, National Bankh~g Review, June, 507-549. Chambers, D. and A. Charnes, 1961, Inter-temporal analysis and optimization of bank portfolios, Management Science 7, July, 393-410. Chen, A.H.Y., F.C. Jen and S. Zionts, 1971, The optimal portfolio revision policy, Journal of Business 44, no. 1, 51-61. Cohen, K.J. and F.S. Hammer, 1966, Analytical methods in banking (R.D. Irwin, Homewood, IL). Cootner, P.H., 1961, Common elements in futures markets for commodities and bonds, American Economic Review, May 173-183. Crosse, howard and G.H. Hempel, 1973, Management policies for commercial banks (Prentice-Hall) second edition. Daellenback, H.G. and S.H. Archer, 1969. The optimal bank liquidity: A multi-period stochastic model, Journal of Finance and Quantitative Analysis, Sept., 329-393. Daniel, Donnie L., William A. Longbrake and Niel B. Murphy, 1973, The effects of technology in bank economies of scale for demand deposits, Journal of Finance, March, 131-146. Fama, E.F., 1970, Multi=period consumption-investment decisions, American Economic Review, March, 163-174. Francis, J.C., 1974, Helping Americans get mortgages, Business Review of the Philadelphia Federal Reserve Bank, January, 14-21. Francis, Jack Clark and Stephen H. Archer, 1979, Portfolio analysis (Prentice-Hall, Englewood Cliffs, N J) second edition. Fried, Joel, 1970, Bank portfolio selection, Journal of Financial and Quantitative Analysis, June, 203-227. Functional Cost Analysis: 1971 Average Banks (Board of Governors, Federal Reserve System, Washington, DC) other years also. Gies, T.G. and V.P. Apilado, 1971, Banking markets and federal institutions (R.D. Irwin, Homewood, ILL Jacob, Nancy L., 1971, The measurement of systematic risk for securities and portfolios: Some empirical results, Journal of Financial and Quantitative Analysis, March, 815-833. Karash, Richard, undated, QPS, a quadratic programming system (MSC Associates, Boston). Klein, Michael A., 1971, Theory of the banking firm, Journal of Money, Credit and Banking, May. Levine, Sumner N., 1975, Financial analysts handbook II: Analysis by industry (Dew-JonesIrwin, Homewood, iLL McKinney, G.W. and W.J. Brown, 1974, Management of commercial bank funds (American Instit¢!e of Banking, New York). Markowitz, Harry M., 1959, Portfolio selection (Wiley, New York). Michaelsen, J.B. and R.C. Goshay, 1967, Portfolio selection in financial intermediaries: A new approach, Journal of Financial and Quantitative Analysis, June, 166-199. Parkin, M., 1970, Discount house portfolio and debt selection, Reviexx of Economic Studies, October, 469-497. Pierce, J.L., 1969, An empirical model of commercial bank portfolio management, in: D.D. Hester and James Tobin, eds., Studies of portfolio behavior (Wiley, New York). Pyle, David H., 1972, Descriptive theories of financial institutions under uncertainty, Journal of Financial and Quantitative Analysis, December, 2009-2029. Pyle, David H., 1971, On the theory of financial intermediation, Journal of Finance, June, 737-7,~7. Roll, Richard, 1970, The behavior of interest rates (Basic Books, New York). Schweitzer, Stuart A., 1974, Bank liability management: For better or for worse?, Business Review, December (Philadelphia Federal Reserve Book). Sharpe, William F., 1967, Capital asset prices" A theory of market equilibrium under conditions of risk, The Journal of Finance, September.
480
J.C. Francis, Portfolio analysis
Shull, Bernard, undated, Nonlocal competition for time deposits in isolated one and two bank towns, Staff Economic Studies No. 40, Board of Governors of The Federal Reserve System. Silverberg, Stanley C., 1973, Deposit costs and bank portfolio policy, Journal of Finance, September, 881-895. Wolfe, Phillip, 1959, The simplex method for quadratic programming, Econometrica 27, no. 3, 382-398. Yamane, Taro, Statistics, 1967 (Harper and Row, New York), 2rid ed.
PORTFOLIO
ANALYSIS OF ASSET AND LIABILITY MANAGEMENT'
IN SMALL-, MEDIUM- AND LARGE-SIZED BANKS Jack Clark F R A N C I S * Bernard M. Baruch College, CUNY, Nel~ York, N Y 10010, U.S.A.
The paper presents an analysis of the commercial banking firm based on Markowitz portfolio analysis. A bank is treated as a portfolio of five banking assets and three banking liabilities. The average rate of return and risk of each asset and liability is estimated empirically for groups of banks categorized by size - small, medium and large. Banks' rates of return on equity are defined as the weighted average of the assets' rates of return less the liabilities' rates of return. Quadratic programming is used to delineate the set of banking portfolios which have the maximum rate of return on equity at each level of risk.
1. Introduction In Crosse and Hempel (1973) the authors, one o f w h o m was a Federal Reserve b a n k examination officer, state: 'Small banks tend to be director-managed because they do not earn enough to be able to affJrd top-grade operating m a n a g e m e n t . . . ' (p. 43). The author goes on to say that in ' m o s t of these banks the examiner rated the m a n a g e m e n t as ranging from fair to p o o r . ' The purpose of this study is to confront statements such as the preceding with empirical data a n d analyze the asset a n d liability m a n a g e m e n t of average small-, medium- a n d large-sized banks in the risk return context suggested by M a r k o w i t z (1959). 1 Some recent banking studies tend to direct their attention to banking competition [see Gies a n d Apilado (1971, section 1)] a n d various economies of scale in banks [see Benston (1965), Daniel et al. (1973) a n d Silverberg (1973)]. But, banking studies should also consider the ability of banks of various sizes to manage their portfolios of assets a n d liabilities as they endeavor to evaluate *This paper benefited from helpful comments offered by George Oldfield, Amos Tuck School of Business Administration, Dartmouth College, and an anonymous referee. The author is also indebted to the Federal Reserve, where he was an Economist, for the data and computer programming assistance. 1Parkin's work (1970) is a somewhat similar contribution which precedes this study. However, the approaches differ. Unlike Parkin's model, this model abstracts from dollar quantities to facilitate comparing the asset and liability holdings of different sized banks and graphing the analysis in risk-return space. This analysis deals with more assets and liabilities than Parkin's, recognizes constraints, and differs in other respects. This model extends and tests Pyle's (1971) model; it is essentially the model Black (1975) described informaify.
460
J. C. Francis, PortJ~4io analysis
the desirability of branching, merging, and the optimal expansion path for the bank/ng system. Therefore, this study holds economies of scale constant and compares and contrasts the risk and return of the principal balance sheet items across average banking statistics gathered from a cross-section of hundreds of banks categorized by' size- small, medium and large. The principle hypotheses being examined is that, ceteris paribus, large banks manage money better than small banks. Because aggregate data is used in this study, some individual banks can be found which operate differently than the aggregate (or average) data suggests. However, individual banks are not relevant to this macro-analysis of banks" asset and liability management. Section 2 redefines the financial statements of a bank in symbolic terms suitable for analysis and discusses the proportions that various asset and liability items occupy on actual banks' balance sheets. The empirical sample is explained in the third section. In section 4 the types of risk peculiar to individual asset and liability items are specified in terms of variance and average return or interest rate statistics. Section 5 explains the total risk of each asset and liability item and discusses the interaction of systematic market forces and unsystematic default risk. In ~ection 6 the unconstrained efficient frontier of banking assets and liabilities are discussed. Positive economic analysis is undertaken in the seventh section by constraining the banking efficient frontiers of the average small-, medium- and large-banks to obtain efficient frontiers similar to empirically observed average bank portfolios. The findings are summarized and conclusions are suggested in section 8. 2. Aeeoa~ag deCmltions principal assets and liabilities of a commercial ba.nking firm are listed on its balance sheet. Table 1 shows what financial analysts call a common-sized ~ n c e sheet, with all items stated as a percentage of equity, averaged over hundreds of smalb ($), medium- (M), and large-sized (L) banks. The balance sheet ide~tity is stated in eq. (1), where the letters denote dollar quantities for the items in table 1. (c+g+h+b+l)+(-d-s-f
) = e,
O)
(Total assets)-(Total liabilities) = Equity.
(la)
Diving eq. (I) through by equity yields the common-sized balance sheets shown in eqs. (2) and (2a). The symbol w~ denotes the weight of the ith balance sheet item sta'ed as a percentage of equity (that is, wc = c/e, for example), =
=
1.0.
(2)
$
w~ = 1.0 = w,. |ffil
(2a)
J.C. Francis, Portfolio analysis
461
Constraining equity to equal unity is the standard convention in Markowitz (1959) portfolio analysis. This constraint is no way limits equity. Equity may assume any positive dollar amount and vary in every period. Equity is defined as unity, or equivalently 100°/.,o, for all sizes of banks to facilitate calculation of the rate of return on equity and owners" risk for cross-sectional comparisons.
Table 1 Common-sized balance sheet for the average small-, medium- and large-sized banks, 1971, Weights Assets (symbol) Cash and uncoil. funds (c) Gov't. bonds (g) H o m e and other mortgages (h) Business loans (b) Installment loans (1)
Total assets
S
M
Weights
L
1.3 3.7
1.5 3.5
1.8 2.9
2.0 1.5 2.4
1.8 1.3 2.6
1.5 3.3 1.0
10.9
10.7
10.5
Liabilities (symbol) D e m a n d deposits (d) Savings deposits (s) Cert. of deposit (f) Equity (e)
Total liabilities and equity Sample size
S
M
L
-4.2 - 2.4
-4.5 - 2.4
-4.7 - 2.3
- 3.3 - 1.0
- 2.8
- 2.5
-
-
- 10.9
- 10.7
- 10.5
684
231
79
1.0
1.0
2,1. The average b~lance sheet proportions
Perusal of table ! reveals some interesting asset and liability patterns for the different sized banks.-" First, the average banks' ratios reveal that small banks have a preference for installment lending while the larger banks make relatively more business loans. The analysis which follows suggests that all banks may make too many business loans and insufficient installment l o a n s - in this respect the small banks asset portfolio proportions are more optimal. The existing pattern probably results from the lending opportunities readily available to banks of different sizes- large business headquarters do tend to be located in cities near the larger banks. A second interesting pattern is the small banks heavier reliance upon deposits purchased by issuing Certificates of Deposit (CD). The Markowitz portfolio analysis suggests that the smaller banks should use CDs more than larger banks. 2The ratios shown in Exhibit 1 differ slightly from the ratios in the Functional Cost Analysis because minor Balance Sheet items are omitted to expedite this presentation. And, all the statistics vary over the business cycle somewhat.
462
J.C. Francis, Portfolio analysis
This may result from the lower cost of CDs available to the smaller banks, on average (see top of tables 2, 3 and 4 for average return data). Calculation of the return on equity necessary for Markowitz analysis is exptlained next. 2.2. Rate o f return o f bank equity
The rate of return on equity for a bank may be stated symbolically as the weighted average rates of return on the five assets and three liabilities, as shown in eq. (3), where rj denotes the net rate of return or one-period yield on the /th balance sheet asset or liability. wcr¢+wgrg+wsrh+wvr~+w:r~+wdrd.eWsrs+Wyrf+k = r e.
(3)
The symbol k represents fixed costs as a percent of equity. Assets' weights are non-negative (we, % , w~, w~, w~ _>_0) and the weights of liabilities are non~ t i v e ( ~ . %; wj. _<_0). Eq. (3) could be restated as an income statement in dollar quantities ~,~y~ult.',piying b,')th sides of the equation by the dollar value of equity. However, banks' financial ~,tatemevts are formulated in common-sized per&~tag~ to facilitate analysis an,:l compvrison of different banks.
3. Six years of empirical data from 1966 through 1971 inclusive are analyzed [Functional cost analysis (FCA) (1971)]. Over 68 large banks (with over $200 million in deposits), over 198 medium banks (with deposits between $50 and $2~.g) rqillioa) and over 603 small banks (with less than 550 million in deposits) coml~se the sample underlying the average banking statistics used here. Rates of return (annual interest expense) on each of the six years were averaged over all banks in the three bank-size categories to obtain average annual returns (interest costs). These six annual rates of return (rates of interest ¢ x ~ : ) were all net of the direct expenses incurred in managing each category ~ 0labilities). The averages and variances shown in tables 2, 3 and 4 were computed from the six years of average returns net of their direct expenses for each asse*, a.~d liability category. 3 Economies of scale associated with bank ~A~,erage rates of return per year over six years were used to measure the typical returns &~ ~ over ~'~ecomplete businesscycle. This facilitates drawing general conclusions about f~¢ ~ and liability management abilities of the average small-, medium- and large-sized ~ . Only t ~ direct (that is, marginal) costs of labor, related executive salaries, departm e n ~ fum/ua~, computer usage, postage, telephoning, printing, insurance, etc., were deducted /n ~ / ~ g tim net rate of return (e~qx'nse) associated with each asset (liability) category. | ~ readers may obtain the annual Functional Cost Analysis booklets for free from the Federal ~ e if ff~y wish to see a de, a/led breakdown of these direct expenses. Fixed costs
J.C. Francis, Portfolio analysis
463
management, with the exception of check processing, 4 have been held constant in calculating the net rates of return (rate of expense) associated with each asset (liability). And, regional considerations peculiar to one area of the nation have also been eliminated by using large samples of banks from across the United States. Table 2 Risk and return statistics for the average small bank.
Mean annual rate of return (or cost) Credit risk
e
g
h
1
b
d
s
f
0 0
5.86 0
5.57 0.03
6.80 0.35
5.62" 0.24
1.89 0
4.51 0
4.87% 0
Var~ance-covariance matrix with credit risk and market risk included:
Cash & reserves (c) Government bonds (g) Mortgages on homes, etc. (h) Installment loans (1) Business loans (b) Demand deposits (d) Savings deposits (s) Certificates of deposit (f)
c
g
h
1
b
d
s
f
0 0
0 0.92
0 0.45
0 0.23
0 0.66
0 0
0 0.43
0 0.47
0 0 0 0 0 0
0.42 0.23 0.66 0 0.43 0.47
0.26 0.11 0.30 0 0.24 0.24
0.I 1 0.48 0.16 0 0.11 0.13
0.30 0.16 0.73 0 0.31 0.33
0 0 0 0 0 0
0.24 0.11 0.31 0 0.27 0.24
0.24 0.13 0.33 0 0.24 0.26
"This statistic was adjusted upward for compensating balances as explained in section 3.
(for example, the bank President's salary, the cost of the bank's building, etc.) were not allocated or deducted in determining the net rates of return and expense in order to minimize the vagaries of cost accounting for joint costs. Sampling and aggregation bias problems appear to be minimal. The items under the various asset and liability headings for each size bank are highly homogeneous because of (a) bank regulation, and, (b) the commensurate market power of banks which are similar in size ensures additional uniformity. Essentially, the sample of transactions underlying each balance sheet item for each bank size category differ only with respect to the date on which they were execute,~. Since 'banks which are in trouble" usually decline to participate in the Federal Reserve°s Functional Cost Analysis unusually low rates of return (viz., losses associated with a bankruptcy) are not likely to downbias the arithmetic average returns. Furthermore, the sample is such a large portion of the population of all 'sound banks' that the sample is highly representative. 4Economies of scale in check processing are attainable and do enter this analysis through the cost of demand deposits. The cost of check processing as a percent of demand deposits at small, medium and large banks is 1.89%, 1.73 % and 1.71 % in 1971, respectively, as shown in tables 2, 3 and 4. These differences are essentially the result of economies of scale. These economies could have been eliminated simply by setting the different banks' costs of demand deposits all equal. This was not done, however, because (a) the differences are small, and, (b) check processing costs are a type of money management cost.
464
1.C. Francis, Portfolio analysis
One of the statistics in tables 2, 3 and 4 has been adjusted. The recipients of business loans frequently are not allowed to use the full amount of the loan. Many borrowers are required to keep a percent of thei~ loan unused at the bank as a compensating bal~nce that increases the bank's effective yield. Using approximately the number published by a Federal Reserve bank examination officer, it is assumed that 18 % of all business loans must be kept in the bank as a ~ompensating balance [Cross and Hempel (1973, p. 20)]. As a result, the effective net yield the average large-, medium- and small-sized bank receives on business loans is estimated to be 7.4 % as shown below: s Gross rate of return u~nadj.
7.05 % = net yield + dir. exp. x 1.22 = 1/(1.0-0.18)
times adjustment factor:
8.62 %
Effective gross rate ei ' return less direct expenses:
after comp. bal.
- 1.22 %
from FCA
7.40%
Net effective yield
Table 3 Risk and return :~tistics for the average medium bank.
Mean annual rate of return (or--_cost) Credit .:isk
0 0
g
h
1
b
d
s
f
5.97 0
5.68 0.02
6.38 0.36
5.85" 0.25
1.73 0
4.64 0
4.99% 0
Variance-covariance matrix with credit risk included:
Cash and rese~,es (e) bonds (8) M ~ on homes, eu:. (h) Inualinm~t loans (1) loans (hi dqx~hs (d) Say/rigsdeposits (s) C_.m'lifimt~ of deposit ( f )
c
g
h
1
b
d
s
f
0 0
0 0.70
0 0.38
0 0.24
0 0.70
0 0
0 0.35
0 0.39
0 0 0 0 0 0
0.38 0.24 0.70
0.28 0.14 0.30 o 0.23 0.23
0.14 0.45 0.20 o 0.13 0.13
0.30 0.20 0.93 o 0.30 0.33
0 0 0 o 0 0
0.23 0.13 0.30 o 0.22 0.20
0.23 0.13 0.33 ,o 0.20 0.22
o
0.35 0.39
q'his statistic was adjusted upward for compensating balances as ,explained in section 3. make-up of the etficient frontier was somewhat sensitive to the rate of return (interest) on the ,,arious ~ (liabilities). To evaluate the sensitivity of, say, the risk of the efficient fr~ wilh respect m a change in the effective yield on business loathe;the following derivatives may h e ~ :
[~ var(r,)]~w~][~wd~E(r~)] = ~.b[awdaE(r~)]. Io ~ event, the conclusions with regard to the asset and iiabiltity management of small-, ~i~and large-sized banks are not sensitive to the effective yield used for business loans. See F t ' a ~ and Archer (1979, ch. 13) ¢ ~ . e m i n g the mathematical portfolio analysis.
J.C. Francis, Portfolio analysis
465
This adjustment was necessary because the FCA (1971) reported average rate of return on business loans ignored compensating balance effects. Table 4 Risk and return statistics for the average large bank.
Mean annual rate of return (or cost) Credit risk
c
g
h
1
0 0
6.21 0
5.89 0.03
6.16 0.3;7
b
d
6.15a 1.17 0.30 0
s
f
4.78 0
0
5.35%
Variance--covariance matrix with credit risk included:
Cash and reserves (c) Government bonds (g) Mortgages on homes, etc. (h) Installment loans (1) Business loans (b) Demand deposits (d) Savings deposits (s) Certificates of deposit (f)
c
g
h
1
b
d
s
f
0 0
0 0.77
0 0.38
0 0.32
0 0.69
0 0
0 0.28
0 0.46
0 0 0 0 0 0
0.38 0.32 0.69 0 0.28 0.46
0.34 0.19 0.22 0 0.18 0.19
0.19 0.51 0.25 0 0.12 0.18
0.22 0.25 1.05 0 0.20 0.45
0 0 0 0 0 0
0.1~ 0.1.2 0.20 0 0.16 0.14
0.19 0.18 0.45 0 0.14 0.30
"This statistic was adjusted upward for compensating balances as explained in section 3.
4. Risk-return analysis of items Before delineating the efficient banking frontiers for the three size categories of banks it is instructive to consider the risk and return of the individual asset and liability items.
4.]. Bushwss and mortgage loans Figure 1 shows graphically that the standard deviation of six annual rates of return or expense and the 1971 average rate of return or expense were positively related across the three size categories of banks for business loans (denoted b) and mortgage loans (denoted h). This is the positive relationship between risk and return suggested by economic theory. It appears likely that the disproportionately large a m o u n t of business loans made by large banks, on average, (see table 1), is obtained by making risky loans at high rates of interest.
4.2. CDs and installment loans Certificates of deposit (denoted f ) and installment loans (denoted l) demonstrate neither the expected positive nor a negative relationship between risk and
466
J.C. Francis, Portfolio analysis
return in figure 1. The higher returns earned on installment loans by the smaller banks helps explain why they hold relatively more installment loans in their asset portfolios than large banks, on average. And the smaller banks' relatively Installment loans
7.0
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Standard deviation of returns =,~/ar(r)=Fig. 1. Individual balance sheet items plotted in risk-return space.
heavy use of CDs as a source of funds (see table 1) probably results from the lower average interest expense small banks pay for these deposits (see figure I). One reason that small banks pay lower interest rates than larger banks fairly consistently may be that they sell their CDs to wealthy savers in small towns who
d.C. Francis, Portfolio analysis
467
are unaware that better returns are paid for CDs at larger b a n k s - which are presumably big city banks. 6
4.3. Savings deposits and government bonds Figure 1 shows a perverse negative relationship between risk and return for savings deposits (denoted s) and government bonds (denoted g) across the banksize categories. The small (so-called country) banks are often able to pay uncompetitively low interest rates on savings because their deposits are not highly mobile unless wide savings rate differentials become obvious to the small depositors; this is advantageous for the small banks. But, their dominated position in government bond investing tends to offset this advantage. The higher risk and lower net return small banks receive from government bonds may be the result of inept trading as the market interest rates and prices of these bonds change (since default losses were essentially nonexistent). Small banks lack of quantity discounts at brokerages may also hinder them. To obtain additional insights into banks' asset and liability management practices, the interaction of these items must be considered simultaneously. 5. Risk statistics
Two main types of risk will be included in this analysis- market (or interest rate) risk and the risk of loss from loans which are never repaid (that is credit or default risk).
5.1. Market risk Market risk may be viewed as variability of the banks' return caused by systematic exogenous forces which cause changes in market interest rates and direct operating expense. The first step in appraising market risk is to calculate the variances of net rates of return and interest expense for the five assets and three liabilities across the six year sample for the average small-, medium- and large-sized banks. These statistics are shown in tables 2, 3 and 4.
5.2. Credit risks Credit risk is measured by the variability of return resulting when creditors who have obtained mortgage, business or installment loans from a bank 6The loan losses appear to be statistically independent of the rates of return on loan assets ~according to page AI4 of the 1970 FCA issue). These loan loss rates are summed with other inite rates of return (cost) and the central limit theorem suggests that they will tend to be aormally distributed. Thus, the two-parameter analysis is appropriate for analyzing rates of •eturn on equity. IMonE-- B
468
J.C. Francis, Portfolio analysis
dishonor their debts. These three categories of loans are three Bernoulli populations. The variance of return from credit risk is calculated by multiplying the probability of loss for the ith loan category, p;, times its complement, var(r~) = ( p ) ( l - P i ) , as shown in table 5. Table 5 Credit risk statistics by bank size. a Business loans (b)
Percent of loans repaid, p~, Percent lost (1-pD Credit risk, var (r)
Small
Medium
Large
99.760 ~o 0.240 ~ 0.2394
99.746% 0.254 % 0.2533
99.796 0.304 ~o 0.3033
Small
Medium
Large
99646 °" /o ,'k254 ~ 0 3527
99.373o/ /o 0.363 ,%o 0.3620
99.621°" /o 0.369 ~/. 0.3676
Installment loans (1)
Percent of loans repaid, pi Percent lost, (1-p,) Credit risk, var (r)
Home anu edwr mortgages (h) '
Percent of loans repaid, ph Percent lost, (I-Ph) Credit risk, var (r)
S,mall
Medium
Large
0,/ 99.971/o 0.029 ,"~ 0.0289
99.976°/ 0.t~24 % 0.0239
/t o 99.969°; 0.031 ~'o 0.309
aSce footnote 6 and Yamane (1967).
5.3. Return generating function
Rates of return are random variables presumed to be the sum of a linear additive process, shown in eq. (4), for the j t h asset or liability in the tth time period: r jr -- i + p j + m jr + Cjt.
(4)
The proposed bank model is in no way dependent upon this breakdown of each item's one-period rate of return or expense; it was assumed merely to explain how the Federal Reserve data was adapted for use in the model. The four quantities on the right-hand side of eq. (4) are explained below.
J.C. Francis, Portfolio analysis
(1)
(2) (3) (4)
469
The real or pure rate of interest, not including inflation allowances or risk premiums, is presumed constant and equal to the marginal physical productivity of capital. It is denoted i and is the same for all assets and liabilities. The risk premium on the jth asset or liability [Sharpe (1967)] is presumed constant, it is represented by the symbol pj. For loans, the average ralc of credit losses is includea in pj. And, pj includes an inflation allowance. Market risk for the jth item in the tth period is a random variable with an expected value of zero, E(mj) = 0. The variance in m is measured around the value of (i +pj). The credit risk associated with thejth balance sheet asset in period t hv~san expected value of zero, E(cj) = 0, because the actuarially determined ~rat-.; of average credit losses has already been included in the risk premium. Thus, cj fluctuates from positive to negative from period to period as credit conditions vary.
Market and credit risk are presumed independent, cov (m, c) = 0, so that the variance of the jth item's return generating function, eq. (4), is shown in eq. (5), without covariances. 7 var(rj) = 0 + 0 + var(mj)+var(cj).
(5)
The variances in the variance--covariance matrices in tables 2, 3 and 4 contain both market and credit risk combined as shown in eq. (5).
6. Unconstrained banking efficient frontiers The asset and liability management opportunities of the three size categories of banks will be analyzed using Markowitz portfolio analysis. The combinations of banking assets and liabilities (that is, balance sheets) that maximize the bank's return on equity at each level of risk will be delineated and referred to as the efficient banking frontier. Of course, the overall efficient frontier which could be generated using a wider assortment of securities would dominate the efficient banking frontier. But, this analysis is limited to only bank assets and liabilities. The portfolio owners' risk, as measured by quadratic eq. (6), was minimized at each rate of return on equity, eq. (3), subject to the constraint that the assets 7Page AI4 of the 1970 edition of FCA says installment loan losses 'tend to occur regularly.' This implies homoscedasticity and independence. The same paragraph explained that loss rates on other loan categories varied, but no statistical dependence was suggested. Therefore, the covariance was presumed zero. It would be simple to use a nonzero covariance and include it in the analysis if additional information so indicated.
J.C. Francis, Portfolio analysis
470
less the liabilities equal equity of 100~o, that is, 1.0: 8
8
The portfolio analysis problem is represented mathematically by the Lagrangian objective function, z = var(re)+21[E(re)-E*]+22 1
.
i=t
w , - 1.
el .
(7)
In eq. (7) vat (re) is the variable to be minimized; z has no financial interpretation. The variables 2t and 22 are Lagrangian multipliers. Standard calculus techniques can be used to solve eq. (7) for the efficient frontier [Francis ~nd Archer (1979, ch. 13) shows the mathematics]. However, it was more expeditious to use a quadratic programming algorithm to perform the computations. Thus, a computer program [Karash (undated)] was used to minimize the bank owners' risk, eq. (6), iteratively at each rate of return on equity, eq. (3), to delineate the efficient banking frontiers shown in table 6 and graphed in figure 2. To nonbank financial analysts, one of the striking features of this analysis is the extent to which banks use leverage and its effects on their equity returns and risk. The risk and return of the individual banking assets and liabilities are shown graphically in figure I. Comparing figures 1 and 2 reveals that figure 1 and thus all the individual balance sheet items occupy only a tiny lower-left portion of figure 2 (see figure 2). The risk and return on equit) capital for the three sizes of average banks represented optimally by the efficient frontiers is expanded several times beyond the risk and return of the underlying individual assets and liabilities. The efficient frontiers shown in tat'le 6 and figure 2 have significant differences between their risk at each rate of return, a The average small bank's unconstrained efficient frontier (denoted US) in figure 1 dominates the average medium (UM) and the large (UL) bank's unconstrained optimum investment opportunities. This dominance exists in spite of the fact that the large bank's assets are spread more evenly across the three assets held, and, in spite of the fact that the Sin comparing the various efficient banking frontiers, the question arises as to whether they are significantly different. If the standard error implied by the central limit theorem applied to all banks in the sample is used, then ~the three efficient frontiers would be judged to be significantly different over most of their ranges at the 0.05 level of significance. That is, for the n = 944 (= 684+231 +79) banks which comprised the total sample using a standard deviation of the efficient medium-sized bank of E(r)= 25.0~ suggests ¢r,,,e= 2.29 (from column 3 of table 6). Using these values, the central limit theorem suggests a standard error of 0.0726 (= 2.29/~/(994)= o'm,t~/n) is appropriate for comparing efficient frontiers. This discussion proceeds on the assumption that the efficient frontiers are sufficiently different to ,.varrant investigating their differences.
J.C. Francis, Portfolio analysis
471
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472
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J.C. Francis, Porifolio analysis
473
average large bank utilizes relatively more low-cost demand deposits than the other banks. Examination of the bank's average risk statistics (shown in tables 2, 3 and 4) reveals that the large banks tend to take the largest credit risk and market risk. The average large bank also has some sizable positive covariance between its various assets and negligible covariances between its assets and liabilities which reduce the opportunity to diversify effectively [Pyle (1971)].9 The smaller bank's stronger negative correlations between assets and certificate of depc.sit rates highlights a subtle risk-reducing benefit which helps rationalize the smaller banks' relatively higher use of CDs (shown in table I). The average large bank does earn higher net rates of return on four of its five assets than the other banks (see tables 2, 3 and 4). These superior returns might compensate for the high risks the average large bank assumes, except for the fact that the large bank also pays the highest interest rates to obtain deposits. After deducting interest costs, the average large bank's investment opportunity set in risk-return space would probably be dominated even if the risks undertaken by small and large average banks were equal. Essentially, many large banks' problems seem to stem from their locations in competitive urban areas where the net yield spreads are smaller and more variable. 7. Positive economics
As a practical matter, banks cannot hope to attain the unconstrained efficient banking frontier because of certain limitations. First, there is the legal constraint that banks hold some of their assets in nonearning required reserves. The average small bank, for example, holds cash and other non-earning assets equal to 1.3 times its equity, t° To obtain a realistic efficient banking frontier such constraints must be considered, as shown in inequality (8)"
1.3.
(8)
91t might have been informative to examine all the banks in a given size category crosssectionally for the same time interval rather than analyze only time-series data for average annual returns and interest expenses. Unfortunately, however, the necessary disaggregated data for individual banks was unattainable. 1°As a practical matter, reserve requirements are stated as a percentage, denoted R, of deposits. This implies that a bank's reserves are R d dollars, where d is the dollar amount of deposit~;. In the formulation employed here, deposits are stated as some proportion, P, of equity, E. Thus, deposits are P E = d and the banks reserves are R P E = R d where R P > 1.3 in eq. (8). This shows that stating reserves relative to equity rather than relative to deposits, as is customary, are equivalent procedures mathematically. The structure of reserve requirements which req,lires reserves to vary with the dollar volume of a bank's deposits and the breakdown of deposits between various deposit categories is simplified to expedite comparing the average small-, medium- and large-sized banks. If one or more individual banks were being analyzed, it would be necessary to increase the number of constraints to reflect the full structure of reserve requirements. However, when the full set of capital adequacy constraints was added to this analysis, they were not binding after the liquid reserve constraint, inequality (8), was added.
474
J.C. Francis, Portfolio ar alysis
The medium and large banks were likewise constrained to provide insurance against insolvency by maintaining, on average [FCA (1971)], reserves of 1.5 and 1.8 times their respective equities. Competition is a second major constraint which limits banks. Banks cannot obtain all the interest-free demand deposits they desire because of competing banks which the depositor may find more convenient. The average small bank, for example, has checking account deposits equivalent to 4.2 times its equity. This average condition is treated as a constraint, as shown in eq. (9), in order to be able to compare the three sizes of average banks realistically with the actual average banks' risk and return statistics: wa ->- - 4.2.
(9)
The average medium and large banks are also constrained to hold less than or equal to the average amount of demand deposits for a bank their size, that is, 4.5 and 4.7 times equity, respectively. The Federal Reserve bank examiners encourage banks to maintain certain capital adequacy ratios [Chambers and Charnes (1961), Cross and Hempel (1973, ch. 5), McKinney and Browl~. (1974, ch. 14)]. These ratios were imposed on the quadratic program as constraints too. However, none of the capital adequacy constraints encumbered any efficient portfolio in any way. So, they aren't discussed further. The required liquid reserves, inequality (8), was the only portion of the capital adequacy ratios which constrains efficient portfolios of banking assets and liabilities. 7.1. Constrahled efficient frontier To determine if the constraints which face banks' asset and liability managers alter any of the previous findings, efficient frontiers were delineated subject to inequalities (8) and (9). The resulting statistics are shown in table 7. The constrained banking efficient frontiers- denoted CS, C M and C L - are graphed in figure 2 for the average small, medium and large banks, respectively. The reserve requirement constraint brought nonearning assets into solution but no more than the minimum requirement. And, the constraint on demand deposits caused banks to obtain more deposits from savings deposits and by selling more certificates of deposit. These changes reduced the efficiency of all three sizes of banks' investment opportunities and forced them to use m6re leverage to attain any level of return in an efficient manner. The average large bank used leverage, as measured by the debt to equity ratio, the least. However, the higher risk inherent in the assets and liabilities of the average large bank combined with its higher liability costs caused the constraints to reduce its efficient frontier (which was dominated before the constraints were added) even more, as shown graphically in figure 2.
4"]5
J.C. Francis, Portfolio analysis
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J.C. Francis, Portfolio ar~lysis
7.2. The average small, medium and large banks The empirically observed average proportions for small, medium and large banks are shown in tables c,~and 7 and represented graphically in figure 2. The actual portfolio holdings of the three average banks are roughly similar to their constrained efficient portfolios with the same level of return, with one major exception. None of the efficient portfolios contain government bonds. 1~ Casual empiricism suggests that the federal and municipal bonds [which have had their yields adjusted upward to compensate for their tax-exemption before this analysis, see FCA (1971, p. 10)] banks actually hold may result from five factors not included in the bank model: (1) indirect pressure from bank exzeminers to hold primary reserves in government bonds, (2) banks which are privileged to have a Tax and Loan Account from the U.S. Treasury can use these deposits to pay for purchases of Treasury bonds but not other issuer's bonds; (3) some bank's willingness to buy municipal bonds to help the local community, (4) the bank may obtain the municipality's demand deposits if it buys the municipality's bonds, and (5) municipal bonds may be counted as reserves at some state chartered banks. These subjective considerations are beyond the scope of this analysis of money management capabilities of different sized banks. Their effect is not always as powerful as was the case when analyzing the 1971 returns, however. Government bonds did enter efficient banking portfolios (which are not shown to save space) generated with other years' statistics.12 Only the average large bank is near its efficient frontier. The small and medium banks are far below their constrained frontiers ~presumably because of inept management and limited opportunities. Nevertheless, the average large bank is still dominated by the smaller banks because the large bank's opportunity set is inferior, on average. 111n preparing the Functional Cost Analysis books each year, the Federal Reserve endeavors to present data which is not distorted by bookkeeping gimmickry and tax tricks. Thus, for example, the Federal Reserve increases each bank's returns on tax-exempt bonds so that they are comparable with the returns on taxable bonds and publishes these adjustments. The upward adjusted returns on the tax-exempt bonds were used in this analysis so that tax considerations would not distort the analysis. See FCA (1971, p. 10) for an explanation of the adjustment for tax-exempt bonds' income. " In actuali'y, the adeptness with which a bank avoids paying income and other taxes can affect its aftec-tax earnings. However, analysis of tax-dodging procedures was ignored in this study of av~:rage money management capabilities. If tax-avoidance is a significant determinant of banks' holdings then the cenclusions of this study are biased accordingly. However, discussions with commercial bankers and central bankers suggests that tax-avoidance considerations are at befit a minor consideration m the management of most banking assets. And, tax considerations effect liability management decisions even less. 12Nine years of annual data from 1966 to 1974 were analyzed, although only the 1966-71 results are shown. The credit crunches in 1969-70 and 1973-4 kept those years' returns from being representative and thus precluded their use as a final year, that is, for use in eq. (3). The year 1971 was selected for discussion here because it was not distorted by Regulation Q ceilings, Phase I or Phase II price controls, or double-digit inflation.
J.C. Francis, Portfolio analysis
477
8. Summary and conclusions The preceding evidence indicates that, on average, large banks managed their assets and liabilities better than small- and medium-sized banks since the average large bank is practically on its constrained efficient frontier while the other banks are significantly below theirs. Nevertheless, the average large bank, and even its constrained efficient frontier, is dominated by the average small and medium banks. There are several reasons for this situation to exist. The large banks earn a better return on assets than the smaller banks, on average. However, the average large banks also pay higher interest rates on their liabilities. As a result, large banks average returns on equity are not above the smaller banks. And, the increased risks the average large bank must assume to achieve higher returns on assets make it easy for the average small bank to dominate in risk-return space even though the smaller banks do not appear to be as aggressively managed. 13 An examination of the correlation matrices (not shown here for brevity) associated with the variance--covariance matrices in tables 2, 3 and 4 revc:ds that the large banks' average returns from different assets are less correlated than smaller bank's returns. This suggests large banks may be able to diversi[,; more effectively across different assets. However, Pyle (1971) has shown that strong negative correlations between assets and liabilities are necessary to effectively utilize the risk reducing hedging effects available through intermediation. Unfortunately, the average large bank's assets and liability returns were not significantly correlated. Thus, the overall ability of large banks to reduce their owners' risk through hedging is not superior partly because their liabilities and assets' returns have the least significant correlations, on average. 1"~ Another reason that the average small bank's efficient frontier dominated the better managed large bank's efficient frontier is because of the higher installment loan returns available to the smaller banks. Figure 1 shows this, and also shows that the small banks were able to obtain savings deposits and CDs on more desirable terms than the larger banks, on average. The Functional Cost Accounting (FCA) booklets which provided the raw data for this analysis also contain a few nonfinancial descriptive statistics which help clarify the typical markets in which the different sized banks operate. The averages for each bank size category shown in table 8 help define the nature 13The more steeply progressive reserve requirements for large city banks which were implemented in 1966 and 1968 are reflected in the three different liquid asset constraints, inequality (8), and obviously place the larger banks' rates of return on equity at a disadvantage relative to banks of lesser size. 1,,The results of this study which suggest that large banks manage money better than small banks, on average, is evidence which weights in favor of liberalized bank expansion regulations. More specifically, if large banks were allowed to open branches state-wide and/or branch across state lines this would accelerate the acquisition of small country banks by large city banks and thus bring more banking assets and liabilities under better management.
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of the average bank in each category, and are useful in interpreting the analytical results and hypothesizing causes for different banking practices. It seems reasonable to suppose that the smaller banks are located in the rural small towns, This supposition is suggested by the well-known phrase 'small country bank' [Cross and Hempel (1973, pp. 14-16)] and the income percentages from 'farm management' revenue (whatever that may be) shown in table 8. Furthermore, differences in the average number of branches implies that the large banks are located in large cities where, on average, they face more banking Table 8 Selected descriptive statistics, 1971.
Number of branches Total personnel Percent of total revenues from farm management Deposits (D) Sample size (no. of banks)
Small
Medium
Large
1.25 45.96
5.54 204.07
32.05 1225.49
6.53~ g D < $50M 684
10.05% $50M < D < $200M 231
0.26% $200M < D 79
Source: Functional Cost Accounting (1971).
competition and more interest-elastic supply a , d demand functions for funds. In contrast, the banks in smaller towns apparently enjoy depositors that are less concerned with their yields, and, give the local banks their deposits at uncompetitive low interest rates. This low-risk, low-cost deposit environment in which the ~majority of small banks apparently operate insulates their owners' risk and return from the poor portfolio management which seems to exist, on average, in these banks. This implies that many of the differences between the average small-, medium- and large-sized banks result from the different environments in which they operate instead of explicitly different management policies. ~5 I SThe author of this paper was a Federal Reserve Economist who had considerable contact with bank executives in large cities (namely, Philadelphia) and small towns (for example, southern New Jersey and central Pennsylvania) alike. Through numerous business lunch and dinner contacts, speech-making at local bankers associations, supervision of surveys of the intentions of bank managers in big city banks, weekly telephone calls to key banking executives, and other contacts over a two-year period (1972-1973) the author developed some understanding of the differences between small-, medium-, and large-sized banks which cannot and should not be reflected in this analytical (that is, objective rather than subjective) study. In particular, the author found that bank executives at the rank of vice-president and above spend large amounts of time on formal and informal personal communications with their corporate customers, other commercial bankers, central bankers, and their colleagues. Very little time is spent on economic research (beyond newspaper reading) c: creative planning. In view of the large amount of time bankers spend cultivating 'the ,,grapevine' and their "customer relations' instead of doing research or planning, the author finds the explanatory power of the bank portfolio model surprisingly high.
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479
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Shull, Bernard, undated, Nonlocal competition for time deposits in isolated one and two bank towns, Staff Economic Studies No. 40, Board of Governors of The Federal Reserve System. Silverberg, Stanley C., 1973, Deposit costs and bank portfolio policy, Journal of Finance, September, 881-895. Wolfe, Phillip, 1959, The simplex method for quadratic programming, Econometrica 27, no. 3, 382-398. Yamane, Taro, Statistics, 1967 (Harper and Row, New York), 2rid ed.