Bank spread with uncertain deposit level and risk aversion

Journal of Banking and Finance 13 ( 1989) 797-810. North-Holland

BANK SPREAD WITH UNCERTAIN DEPOSIT LEVEL AND RISK AVERSION Emilio R. ZARRUK* Florida Atlantic University,Boca Raton,FL 33431, USA

Received September1986;final version receivedNovember 1988 This paper derives a model of the banking firm under uncertainty and risk aversion. The selection of the bank’s optimal spread between loan and deposit rates is emphasized. The model’s results provide some implications for bank asset quality, capital regulationand deposit insurance.For example, it is shown that increases in the level of equity capital tend to increase the bank’s spread mmderDARA. This implies an improvement in bank asset quality. On the other hand, as the deposit supply function becomes more volatile, the bank’s spread narrows, which implies a decline in the quality of the bank’sassets.

1. Introduction

The recent deregulation of deposit markets has narrowed the size of bank interest spreads. As a result, the ability of banks to sustain loan losses has been diminished. As the trend toward deregulation continues, banks will have to devote more attention to the management of their spreads. A clear understanding of how banks adjust their spreads in response to changes in their environment is an important issue for bankers and regulators. The literature on models of the banking firm is extensive.’ Several authors have provided deposit rate-setting models of bank behavior [e.g., see Kareken (1967), Klein (1971), Monti (1972), Sealey (1980), Flannery (1982)]. The bank’s optimal determination of’the spread between loan and deposit rates have drawn relatively less attention in the literature. Ho and Saunders (1981) applied a model of bid-ask prices for security dealers [see Stoll (1978)] to the analysis of bank interest margins. McShane and Sharpe (1985) used a modified version of Ho and Saunders’ model to conduct empirical tests of the spreads of Australian trading banks. o and Saunders was Since the particular framework employed by origin;llly intended for the analysis of the trading activities of security dealers, it fails to consider some relevant aspects of a bank’s operation. For *The author acknowledges the helpful comments of Robert E. artin and the anonymous referees. altensperger(1980) and Santomero( 198 ‘For reviews of models of the bankin 0378-4266/89/$3.500 1989, ElsevierScience PublishersB.V.(North-Holland)

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example, administrative costs associated with the maintenance of loan and deposit c:@ntra%:ts as well as costs incurred in the provision of banking services 8 -;I:: I ;,i.x>red. This ye 5~ examines The choice of the optimtil spread betwsn loan and deposi_ -: ,:$,-;.5 51~ ,I. bank under uncertainty and risk aversion. A model of bank behavi.x iii;developed using ?. framework similar to the one employed by Se&y (I?;+>). Comparative statics of the model yield a number of significant pi,.:positions concerning the behavior of the bank’s spread. For example, given decreasing absolute risk aversion DARA), the results indicate the bank’s spread increases with the amount of quity capital and decreases with deposit variability (in the sense of a mean preserving spread). The results also show the bank’s spread is an incre sing function of the deposit insurance premium for the case of nondecreasing absolute risk aversion. Under DARA, however, the effect on the bank’s spread from a change in the cost of deposit insurance is ambiguous. The aforementioned results have some imp ications for the regulation of bank capital, deposit insurance and the quality of bank assets. Specifically, if the narrowing of bank spreads lowers the quality of bank assets as suggested by Chan, Greenbaum and Thakor (1986), then strengthening bank capital requirements will have a positive effect on bank asset quality; since higher -levels of equity capital lead to increases in bank spreads. In contrast, changes in the bank deposit markets which increase the variability of the bank’s supply of deposits will reduce the bank’s spread and thereby lower the quality of bank assets. The model also predicts that increases in deposit insurance premiums will not necessarily lead to an improvement in bank asset quality. The paper is organized as follows. A spread-setting model of bank “behavioris introduced in section 2. The solution of the model is considered in section 3. Section 4 develops the comparative statics properties of the model. Finally, section 5 contains the implications of the model and conclusions. 2. The model Consider a bank which makes decisions in a single period horizon. The bank can acquire two kinds of assets: loans (L) and loans in the Federal funds market. The bank holds three types of claims: deposits (D), borrowings in the Federal funds market, and equity capital (E). The balance sheet constraint can be written as: L+B=D’+E,

(1)

where B is a composite variable representing the bank’s net position in the

E.R. Zamk, Bank spread

199

Federal funds market (B>O implies a net lender), and D* is the amount of ex-ante deposits. The model abstracts from legal reserve requirements and equity capital is assumed fixed throughout the decision period.2 The bank is assumed to be a rate setter in the loan market. Loan demand is a known downward-sloping function of the rate on loans, &. The loan demand is given by L = L(R,); L’( RL) < 0,

(2)

where primes denote derivatives of functions of a single-argument. The assumption of a downward-sloping loan demand has been used in a number of models of the banking firm [e.g., see Klein (1971), Pringk (M/3), Sealey and Lindley (1977), Mason (1980), Santomero and Blackwell (198611. Empirical evidence which rsupports loan rate-setting behavior by banks haa been provided by Slovin and Sushka (1983) and Hancock (1986). Following Sealey (1980), \hse bank faces an uncertain supply of deposits which is assuined to. be an increasing function of the rate of interest on deposits, RD. Uncertainty in TVe bank’s deposit supply function is modeled through the use of an additive random term (p) having a known probability density function, g(&.3 The deposit supply is given by D’=D(RD)+p,

D’(R,)>O.

(3)

The expected value of p over its domain ( -m, 00) is assumed to be equal to zero. The timing of decisions is as follows. At the start of its decision period, the bank chooses the rates on loans and deposits. Following the selection of RL and RD (and thus the spread), the uncertainty surrounding the supply of deposits is revealed. At this point, the bank makes the necessary balancesheet adjustments by using the Federal funds market. This determines the value of the bank’s profits for its decision period. The bank’s objective is to maximize the expected utility of profits (7~) subject to the balance sheet constraint given by eq. (1). Let U(a) be a von Neumann-Morgenstern utility function defined in terms of the end-of-period profits, with properties such that U’(z)>O and U”(n) SO, depending on whether the bank is risk neutral or risk averse. The bank’s optimization problem can be stated as

2The inclusion of legal reserve requirements complicates the structure of the model without significantly altering the results of the model. jThe assumption of an additive uncertain term in the deposit supply function is made in order to make the model tractable for analybfs. Sealey (1980) made use of this assumption throughout his paper.

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Maximize 0 = E[ U(Z)] = 7

U(lt)g(p)

(4)

dp,

-m

RLRD

subject to L+B=D*+E. The bank’s profits are defined as follows’

a=R,L-C,L-R,[D+/i]-C,[D+j+P[D+Cc)+RB,

(5)

where P is the deposit insurance premium. Solving for B out of (l), and substituting into (5) yields

Using the definition of profits given Sy (6),the objective function can be written as

bank’s unconstrained

Maximize i U{R,L-C,L-R,[B+Cr)-CDID+~J.-PID+Czl RLRD

-m

ution to the model The first-order conditions for the optimal values of RL and RD ares

~=&=E{u+I)[L+L~(R,--c,--R)]}=o, L

(8)

4The arguments of the functions L and D are omitted in the equations of the rest of the paper. ‘Letter subscripts indicate partial derivatives.

E.R. Zamk, Bank spread

801

-order conditions for a maximum are6

DLL=E{U’(R)[~L’+L”(R~-C~--R)]+U”(~~)(X~)~}<~,

(10)

U,=E{U’(lr)[-20’+Dtt(R-R,-C,-P)]+Utt(n)(n,)2}<0,

(11)

H= O,J,,-

o;,>o,

(12)

where O,, = E[zJE[U”(lt)~_,J =0 since E[nJ =0 by the first-order condition for RL. Eq. (8) can be rewritten as 8,=E[U’(lr)]L+E[U’(rc)]L’(R,-CL-R)=O.

(13)

It is clear from (8) that a necessary condition for an interior value of RL is that (R,- CL-R) >O. In similar fashion 0, can be rewritten as ir,=E[U’(a)(-D-c()]+E[U’(Z)]D’(R-RD-CD-P)=O.

(141

The first term in (14) is negative since U’(lt)>O and [ -D-p) ~0. Therefore, a necessary condition for an interior value: of RD is that (R - RD- CD- P) >O. Combination of the necessary conditions for RL and RI, reveals that an interior solution requires the spread between RL and R1, to be strictly greater than the deposit insurance premium plus the sum of the marginal administrative costs of loans and deposits. 3.1. Risk averse versw risk-neutral spread It is possible to compare the risk-neutral spread to the optimal spread under risk aversion. The first-order condition for RD cm be expressed as ( i 5)

&,=E[U’(~)]E[R~]+COV(U’(~~),R,,)=~,

where q,=[-D-p+D’(R-RD-CD-P)]. Given risk aversion, i.e., U”(w)< 0, Cov( U’(x),a,) > 0, since dn,l8~ < 0 and XJ’(@/& SO as U”(z) SO. From eq. (15), it follows that E[Q,]=[-D+D’(R-RD-

CD-P)] SO

as

U”(7r)
(16)

Eq. (16) reveals that the risk-neutral bank (U”(a) = 0) sets its deposit rate, RL, by equating the sum of the marginal resource costs of deposits, including the deposit insurance remium, to the rate on Federal funds. his is the familiar %ufficientconditions for a maximumexpected utility are that L”~0 and D”
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Bank

spread

result for the case of risk neutrality derived by Klein (1971).’ In essence, the Federal funds rate becomes the exogenous ‘peg’ of the system, i.e., the exogenous rate to which R”, is pegged [see Pringle (1973)]. Under risk aversion, the bank sets its deposit rate, R%, by eq sting the sum of the marginal resource costs of deposits plus the deposit insurance premium to a quantity which exceeds the Federal funds rate. Consequently, the risk-averse deposit rate! is set at a higher level than the risk-neutral rate, R$> R”,. Consider now the optimal loan rate under risk aversion (Ri) and the risk-neutral loan rate ( eL). Dividing eq. (13) by L’E[ U’(x)] yields L

c+k

-CL=R.

(17)

Eq. (17) implies the risk-averse loan rate is set at a level such that the expected marginal revenue on loans (L/L’ + RL) less the marginal administrative costs of loans equals the Federal Funds rate. When the bank is risk neutral, U(z) is linear and U’(lt) is a constant. Eliminating E[U’(lt)] from (13) and dividing through by L’, the first-order condition for the optimal value of R;). is given by (17). Hence, it is clear that the optimal risk-neutral and risk-averse rate are set at the same level. What implications do these results have for bank spreads? Risk aversion implies the bank sets a smaller spread than under risk neutrality. Intuitively, this result can be interpreted in terms of the variability of the bank’s profits. When deposits are random, the variance of profits is given by (18) where Var(& is the variance of the random term p. From (18) it is easily verified that dVar(a)/dR,
omparative statics of the model

aving examined the solution to the bank’s optimal spread under risk

‘In Klein’s model, the Federal funds rate is replaced by the rate on government securities.

E.R. Zamh, Brrrnkspread

803

aversion, this section considers the effects on the spread from changes in the parameters of the model.* 4.1. The effect of changes in administrativecosts o~ilthe bank’s sptead First, look at the impact on the spread from a change in the administrative cost of deposits. Implicit differentiation of &c first-order conditions with respect to CDyields 8RD D’E[U’(n)]-E(U”(z)(-D-p)[-D-p+D’(R-R,-CD-P)]} -= 0,

a

-= G

-E{U”(n)(-D-p)}

E[L+L’(R,-C,-R)] &

9

(19

.

By the first-order condition for RL, it is clear from (20) that the optimal loan rate is invariant with respect to changes in Co. Therefore, the overall effect on the bank’s spread depends upon the sign of (19). The first term in the numerator of (19) is positive, and after some algebra, the second term in the numerator of (19) can be rewritten as E{U”(lc)(-D-p)[-D-p+D’(R-RR,-Cc,--P)]}

xE{U”(lr)[(-D--p+D’(R--R,--Q-P]}.

(21)

In general, eq. (21) is indeterminate. However, if the index of absolute risk aversion as developed by Arrow (19M) and Pratt (1964) is decreasing (DARA), constant (CARA) or increasing (IARA), the sign of (21) can in some cases be established. Proposition 1. The effect on the bank’s optimal risk-averse spread from an increase in the administrative cost of deposits is (i) ambiguous given DAR A, (ii) positive under nondecreasing absolute risk aversion (CARA or SARA). ProoJ:

Since the first term of (21) is clearly negative for all risk averse utility functions, part (i) of this proposition can be verified by showing the last term of (21) to be positive. Under DARA, the term 8Formal proafs and derivations of the propositions presented in this section are available

upon requestfrom the author.

E.R. Zarruk, Bank spread

804

(22) is negative.g This implies that eq. (21) is indeterminate and part (i) of Proposition 1 follows. The second part of this proposition can be verified by showing that, under CARA or IARA, eq. (19) is negative. Thus given the sign of (22) under CARA and IARA, part (ii) @fProposition 1 follows. iven nondecreasing absolute risk aversion (CARA or IARA), the bank es on to depositors any changes in the cost of administering deposits ugh changes in the rate of interest paid on deposits. An increase in CD lowers the deposit rate and increases the bank’s spread. Under DARA, however, the impact of a change in CD on the banC”s spread is ambiguous. This ambiguity is significant in the sense that DARA is the most reasonable hypothesis concerning risk-taking behavior. It implies that as wealth increases, the bank is more apt to take on added risk. An intuitive interpretation of Proposition 1 can be given in terms of the risk premium (2) or the maximum amount which makes the bank indifferent between the random profit it, and the certain sum (E[z] -2). Let Z be a function of at least the first two moments of the bank’s profits, i.e., 2 =Z(E[z], Var(lc)). The totai effect on 2 from a change in Co can be expressed as dZ 82 8ECz-J+ dZ 8 Var(a) -z!!!!5 dCD-dE[lr] ijc, dVar(lt) XD

(23) l

The terms of eq. (23) capture the mean profit effect and the risk effect on 2 from a change in CD.It is easily verified that +!WO, aVar(7r) =-2(&R,G

(24)

CD-P) Var@) CO.

(24) and (25) it is apparent that an increase in the administrative cost of deposits reduces both the expected value and the risk of the bank’s profits. Since &Z/aVar(z)>O, the second term of (23) is negative for any risk-averse utility function. In the case of CARA or IARA, it can be shown that the mean profit effect on 2 from a change in C, is zero (CARA) or negative (IARA) and, thus Z/X,cO. Since the bank’s deposit rate increases with the risk premium (see section 3.1), 8R,IdC,cOand part (ii) of Proposition 1 follows. The 9A detailed proof of the sign of (22) under DARA, CARA, and IARA is contained in an t.ppendix available upon request from the author.

E.R. Zmuk, Bank spread

805

ambiguity under DARA is due to the conflicting signs of the mean profit and variance effects on 2 [the terms of (23) have opposite signs]. Given DARA, 2: decreases as E[z] increases, and the mean profit erect on Z is positive. The overall impact on RD and thus on the spread will depend on the relative magnitude of the two effects operating on 2. Consider now the effects of a change in the administrative cost of loans on the bank’s optimal spread. Differentiation of the first-order conditions with respect to CL yields 8R,, LE(U”(~~)C-D-C~~D’(R-R~-C~-P~~} -= aCL ha

9

BRL L’E[U’(x)]+LE[U”(z)]E[L+L’(R,-C,-R)]>O -= ac, . * &,

.

(29

Eq. (17) implies the optimal loan rate is an increasing function of CL for all degrees of risk aversion. Proposition 2. The effect of an increase in the administrativecost of loans on the bank’s optimalspread is (5) ambiguousunder DARA, (ii) positiveunder nondecreasing absolute risk aversion(CARA or SARA). Proof. Recall that under DARA the sign of (22) is negative. Inspection of (26) reveals that the optimal deposit rate is an increasing function of CL.

Thus given DARA, the bank increases both its loan and deposit rates in response to an increase in CL. As a result, the overall effect on the spread from a change in CL is ambiguous under DARA. Alternatively, the bank’s deposit rate is invariant (for CARA) or decreasing (for IARA) with Ct. This follows directly from the sign of (22) under CARA and IARA. Hence, the bank’s spread is an increasing function of CL for IARA or CARA. The results of Propositions 1 and 2 can be summarized as follows. Nondecreasing absolute risk aversion implies the spread is an increasing function of CL and CD. If absolute risk aversion is decreasing (DARA), the impact from a change in either CL or Co is indeterminate. Next, examine the impact on the spread from a change in the deposit insurance premium. 4.2. The effect of a change in the deposit insurancepremium on the bank’s spread

Implicit differentiation of the first-order condiGons with respect to

p

gives

E.R. Zamk, Bank spread

806

a& -E(U”(x)(-= aP

D-p)}E[L+L'(R,-C,9 bt

(28)

8RD D'E[U'(~)]--E{U"(~)(-D-~)[--D-_I~+D'(R-R,-C~-P)]} -= . (29) 8P

0,

Note the expressions for dRJdP and aR&?P are identical to eqs. (20) and (19). Hence, changes in thz cost of deposit‘ insurance have the same effect on the bank’s spread as changes in the marginal administrative cost of deposits. Proposition3. The effect on the bank’s spreadfrom an increase in the deposit insurance premium is (i) ambiguousunder DARA, (ii) positiveunder CARA or SARA. Pro05

The proof of this proposition follows directly from section 4.1. Under nondecreasing absolute risk aversion the spread is an increasing function of the deposit insurance premium. In the case of DARA, the impact on the spread from a change-in the deposit insurance premium is ambiguous. The results of Proposition 3 have an analogous interpretation to that provided for Proposition 1. .#

4.3. The effects of a change in equity capital on the bank’s spread

Differentiation of the first-order conditions with respect to E yields

a& -=

aE

- RE[U”(a)]E[L+ L’(R&-CL- R)]

G,

(30)

9

aRD -RE{U”(w)[-D-p+D’(R-R,-CD-P)]} -= dE Qm

.

(31)

The numerator of (30) equals zero by the first-order condition for RL. Therefore, the bank’s optimal loan rate is invariant with respect to changes in the amount of equity capital. Proposition4.

An increase in the amount of equity capital

(i) increases the bank’s spread under DAR A, (ii) has no effect on the bank’s spread under CA (iii) decreases the bank’:?:;3rwd under IA

EA. Zarmk, Bank spread

807

ProoJ: The results of Proposition 4 follow directly from the sign of (22) and the second-order conditions. Proposition 4 suggests that under DARA an increase in bank’s equity capital will lead to an increase in the bank’s spread. Consider the risk premium 2 which makes the bank indifferent between R and (E[z]-2). If 2 is assumed a function of E[z] and Var[z], then dZ -= dE

82 aE[n] + 82 Mar(z) aVar(lt) 3E aEc7r-J 8E

l

Note that the second term in (32) vanishes since a change in equity has only a mean profit effect on the risk premium. Under DARA, it is easily verified that an increase in the bank’s equity capital has a negative mean on 2. From section 3.1, RD increases with increases in 2, and thus part (i) of Proposition 4. Finally, it is of interest to consider the enects on the bank’s spread from a mean preserving spread in the distribution of the random term.

4.4. The effects on thz bank’s spreadfrom a mean preserving spread of p A mean preserving spread in the distribution of a random variable redistributes the we,& of the distribution from the center to the tails while keeping the mxn of the distribution unchanged [see Rothschild znd Stiglitz (29’7Q)].In this model, a mean preserving spread of p increases the dispersion of the bs&‘s &posit supply function. Define a mean preserving spread of p as

Writing the first-order conditions in terms of p*, and differentiating with respect to il and evaluating at i?= i Y;._ aRL (R-RD- CD-P)E[L+L'(R-R,-C,)]E[U"(z)p] -= 9 aA &,

E[U’(n)p]+(R- R,,- CD-P)E{U”(n)(-p)[-D-p+ Gm

EA. Zmuk, Bank spread

808

Inspection af (34) reveals dRLldil=0 by the first-order condition for RL. Thus, the sign of (35) determines the efiect on the spread from a mean preserving spread of CL.The first term in the numerator of (35) is negative. The second term of the numerator of (35) can be rewritten as

+(R-R,-C,-2)(D-D”(R-R,-C,-P)) x E(U”(x)[. Proposition 5.

D-p+

. WJ

D’(R-R,-C,-p)]}.

A mean preserving spread in the distribution

ofp

(i) decreases the bank’s spread under DA RA and CARA, (ii) is indeterminate in sign mder SARA.

Urooj The first term in (36) is negative by the first-order condition for RD. The second term of (36) is also negative under DARA [see eq. (22)]. Thus, (36) is negative and aR,lan >O. Under CARA, the second term of (36) is zero and part (i) of this proposition follows. Given IARA, the sign of (36) is ambiguous, and the effect on the spread cannot be determined. Proposition 5 suggests that, under nonincreasing absolute risk aversion, an increase in the variability of the distribution of deposits (a mean preserving spread), reduces the bank’s optimal spread. Intuitively, a mean preserving spread of ~1increases the variability or ‘risk’ of profits while having no effect on expected profits. Since d Var(lr;)/dRD~0, the bank neutralizes the increased variability by increasing RD. As a result, the bank’s spread is reduced.

5.Implicationsof the model and conclusions This paper presents a model of the banking firm under uncertainty and aversion. By viewing the bank as a spread setter (simultaneous loan and deposit rate setting), a number of results concerning the behavior of the bank’s spread are obtained. It is shown that, while the risk-averse bank operates with a smaller spread than the risk-neutral bank, the expected size or scale of operation is larger in the case of risk aversion.‘O Comparative statics results of the model have implications for bank l”Sealey ( 1980) derives a similar result (see pp. 1~8-1~9, and 2).

Proposition

6 and Corollaries

1

E.R. Zmk,

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869

quality of bank assets. As recently demonstrated by Thakor (1986), the narrowing of interest rate spreads expenditures on bank loans which leads to asset quality. Part (i) of Proposition 4 sho s in the bank’s equity capital increase the 1 of capital reduces the interest cost of te), which leads to a larger spread. This result implies that strengthening capitai requirements improves bank asset quality, pmvi or is characterized by DAM, The model also s about the impact of changes in the cost of bank asset quality. For example, it’ absofute risk aversion is nondecreasing, part (ii) of Proposition 3 shows that the bank’s spread is a positive function of the deposit insurance premium. This implies that increasing the deposit insurance premium has a positive effect on bank asset quality. However, under DARA, the most reasonable hypothesis concerning risk-taking behavior, an increase in the deposit insurance premium has an amt+guous effect on the bank’s spread. Therefore, raising the cost of osit insurance to banks does not necessarily improve the quality of th oreover, Propositions 1 and 3 show that changes in the deposit insurance premium have the same effect on the spread as changes in the administrative costs of deposits. The. implications of the foregoing results are quite interesting in light of the lengthy debate on capital adequacy and the reform of the FDIC system. Currently, the discussion on reform issues centers around ways of reducing the uncompensated liability imposed by banks on the FDIC [e.g., see Pyle (1986)]. Two of the proposed alternatives (not mutually-exclusive) are risksensitive deposit insurance premiums and capital regulation. The implications of the model in this paper suggest bank capital requirements are effective in adjusting the FDIC’s liability. Finally, the results have implications concerning the effects of deposit variability on bank asset quality and bank size. The result in part (i) of Proposition 5, concerning a mean preserving spread of the deposit supply function, indicates the bank’s spread is negatively related to deposit variability. As the bank’s supply of deposits becomes more variable, the size of the bank’s spread is reduced. As noted previouslly, a narrowing of the bank’s spread implies a decline in asset quality. Note also that part (i) of Proposition 5 suggests a positive relationship between dep sit variability and bank size? ’ AS the bank%supply of deposits becomes more volatile, the bauk raises the deposit rate under DARA or CARA. This implies a larger expected bank size. “For an extensive discussion of the literature dealing with deposit variability and bank size, see Kaufman (1972).

810

E.R. Zarmk, Bank spread

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