Deposit rates, credit rates and bank capital

Journal of Banking and Finance 10 (1986) 99-114. North-Holland

DEPOSIT RATES, CREDIT RATES AND BANK CAPITAL The Klein-Monti Model Revisited J. D E R M I N E * INSEAD, F-77305 Fontainebleau Cedex, France Received May 1984, final version received December 1984 The paper expands the Klein-Monti model with bankruptcy risk and deposit insurance. The well-known result of independence between deposit and credit rates is shown to be lost; the causal relationship becomes recursive and the direction of the recursivity depends on the existence or the absence of a deposit insurance mechanism.

1. Introduction

The object of this article is to analyse the relationship between deposit and credit rates and the optimal level of bank capital in an expanded KleinMonti (K-M) model [Klein (1971), Monti (1972)]. Bankruptcy and deposit insurance are introduced in the K - M model and a detailed discussion of their role for deposit and credit rates and for bank capital is provided. A principal motivation for expanding the K - M model is that the elegance and simplicity of their approach has led many authors to develop similar micromodels of the banking firm. In their survey on bank modeling, both Baltensperger (1980) and Santomero (1984) group a large segment of the literature in the neoclassical category originated by Klein and Monti. The main characteristic of these models is that the bank maximises an expected profit which includes revenues on loans and government securities net of the expenses incurred on deposits and real resources. Despite the limits of the expected profit assumption which can only be justified by risk neutrality or by the ability of banks' shareholders to diversify away the risk, 1 it seems useful to generalise the K - M model and to revisit some of their results, *The current draft of the manuscript was written while the author was visiting assistant professor at the Wharton School of the University of Pennsylvania in the fall of 1984. The author is thankful to Professors J. Dreze, J. Haubrich and H. Langohr for helpful comments. 1A recent paper in the risk neutrality tradition is by Desmukh-Greenbaum-Kanatas (1983). As is discussed by Baltensperger (1980) and Santomero (1984), other types of models exist which rely on risk aversion of managerial origin, on a CAPM valuation equation or on a contingent claim approach. 0378--4266/86/$3.50 © Elsevier Science Publishers B.V. (North-Holland)

1O0

J. Dermine, Deposit rates, credit rates, bank capital

because of the insights which are provided, because our results are shown to hold under several sets of assumptions (state preference and CAPM framework) and, finally, because its elegance has made this model popular in the literature. A main result of the paper is that the independence property of the K - M model is lost; the causal relationship between deposit and credit rates is recursive and the direction of the recursivity depends on the existence or the absence of a deposit insurance mechanism. The paper is organised as follows. The K - M model is reviewed briefly in section 2. The eventual failure of the borrowers or of the bank to honor their debt commitment is introduced in section 3 and, in the fourth section, the optimal level of equity and the causal relationship between deposit and credit rates are discussed under alternative regimes of deposit insurance.

2. The Klein-Monti model The asset side of the balance sheet of the bank consists of government securities (B) and loans (L) and the liability side of deposits (D) and equity (E). The supply of securities (yielding an interest g) is perfectly elastic. The loan demand by borrowers [L(-)] is a decreasing function of the interest rate p and the deposit supply [D(.)] is an increasing function of the interest rate d. All parameters are known with certainty. 2 The opportunity cost of equity in a certain world is the exogenous security rate g. The bank chooses deposit and credit rates and equity to maximise its end-of-period net value (NV),

max NV=(1 +p)L+ (1 +g)B--(1 + d)D-(1 +g)E, d,p,E

(1)

subject to

L+B=D+E. Substituting the balance sheet constraint into the objective function, one has

max NV=(p-g)L +(g-d)D. d,p,E

(2)

The net value is the sum of two terms: income on loans net of an opportunity cost (g) and net income on deposits invested in securities. The 2Klein (1971) introduces also a liquidity cost function to reflect the fact that deposits are withdrawable. As this does not affect the causal relationship between rates [Langhor (1982)], we have chosen the simpler model of Monti (1972).

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first-order conditions for deposit and credit rates are

(g-d)D'-D=O,

(p-g)L'+L=O.

(3a, b)

Denoting by r/o and r/L the interest rate elasticity of deposits and loans, one obtains d(1 + r/o x) =g,

p(1 + r/L x) =g.

(4a, b)

Relations (4) express the neoclassical equality between the marginal cost of deposits, the marginal revenue on loans and the exogenous opportunity cost g. The security rate g is the peg of the system which brings independence between the deposit and credit rate decisions. 3'4 One observes from relation (2) that the equity level is indeterminate because the opportunity cost of equity equals the exogenous market rate g. Additional equity invested in market securities does not yield any net value. This last result shows the limits of the K - M model. Given the current debate on the capital adequacy for banks, it seems that this model is missing the important feature of bankruptcy risk. Additional equity invested in some assets reduces the probability of the intermediary's bankruptcy. This will increase the expected return on deposits and their volume when there is no insurance mechanism and, if there is one, the insurance premium will be eventually lowered. If one ignores bankruptcy risk, it is not surprising that the equity level becomes indeterminate. One would like to model the fact that the borrower and the bank might not honor their debt commitment. This is the object of section 3.

3. The K-M model with risk of bankruptcy We assume that the bank faces an aggregate borrower which finances an asset partly with a loan. The end-of-period value of the asset of the borrower is stochastic so that the eventual failure of the borrower to honor his debt commitment depends on the value of the asset at the end of the period. Is it larger or smaller than the borrower's promised debt repayment? If it is smaller, will it force the intermediary into bankruptcy or not? Following the aAs has been discussed and eriticised by Baltensperger (1980), the presence of monopoly power is essential to ensure the finite size of the intermediary. A real cost function for loans and deposits can be added to the model and the inelastic market assumption becomes unnecessary as the net marginal revenue (the exogenous market interest rate net of a marginally increasing cost) is decreasing. We have kept the inelastic market hypothesis for simplicity of exposition and we deal with the price taking case in appendix. References to real costs will be made in the discussion as they impinge on the results. 4The K-M model is a partial equilibrium model of the banking firm which takes as given the demand and supply elasticities. In a general equilibrium model, the elasticities would depend on the market structure and, in particular, on the costs of entry.

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lender-borrower model of Jaffee-Modigliani (1969), we assume that the size of the asset financed by the loan is fixed 5 and we denote by f(a) the borrower's asset value marginal density function defined over the interval (k,K) and by F(a) the cumulative density function. Three situations can occur at the end of the period: (i) the borrower repays the loan: a > (1 + p)L, (ii) the borrower defaults but the bank meets its obligations: a* < a < ( 1 +p)L where the break-even value a* is equal to

(5)

a* =( l + d)D-( l + g)B, (iii) the borrower defaults and the bank fails: a < a*.

In case of bank failure, it will be the depositors or the deposit insurance agency which will bear the loss. The deposit insurance is financed by a premium P paid by the bank at the beginning of the period. Following the K - M neoclassical class of models, we assume risk neutrality 6 but it must be pointed out that our results hold in a state preference version of the model or with a CAPM valuation approach (see appendix B). The bank maximises its expected end-of-period net value [relation (6)] where the first integral represents the end-of-period value when the borrower can repay his debt and the second integral the value in case of borrower's bankruptcy. In this last case, the intermediary does not recover the principal (plus interest) but only the borrower's asset a: K

E(NV)=

J"

((1 +p)L+(1 + g ) B - ( 1 +d)D)f(a)da

(1 + p)L (1 + p)L

+

~

a*

(a+(l+g)B-(l+d)D)f(a)da-(1

+g)E,

(6)

subject to

B+L+P=D+E. The second integral is evaluated from a lower bound a* since the shareholders' liabilities are limited. 5A m o r e realistic assumption would be that the potential outcome is a function of the size of the loan. O u r results hold if, following Jaffee (1971, p. 158), we assume that the marginal return on investment is a non-increasing function of the size of the asset. 6 O n e m a y wonder why a deposit insurance system would exist in a risk neutral world. The existence is usually justified on the grounds that it prevents bank runs. Bank runs are likely to occur also in a risk neutral world as in the case of bad rumours it seems logical for depositors to run and convert (temporarily) their deposits into currency.

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We assume a general form for the insurance premium (7). It is the sum of two terms: an independent term 6 and a fraction a of the present value of the insurer's expected liabilities (i.e., the bank's expected end-of-period losses), a*

P =6-af-~g

! (a+(1 + g ) B - ( 1

+d)D)f(a)da.

(7)

The integral is bounded upward since above a* the intermediary is solvent. This general formulation takes as a special case the fair insurance premium (0~= 1,6=0) and the case observed in many countries where 0~ equals zero and where the independent term is proportional to deposits (6=cD). 7 We show in appendix A how relation (6) can be simplified into relation (8), (1 + p)L

E(NV)=(p-g)L+(g-d)D-

I

a*

F(a)da

a*

- a ~ F(a)da--(1 +g)&

(8)

k

The expected net value (8) is similar to the maximand of the Klein-Monti model (2) except that three terms have been added: one to measure the loss of income due to the borrower's default and two terms to measure the cost of the variable insurance premium (a fraction a of the bank's expected losses) and the cost of the independent premium 6. 8 The next section deals with the optimal choice of equity, deposit and credit rates and with their causal relationship.

4. Deposit rates, credit rates and bank capital A preliminary question concerns the determinants of the supply of deposits. In the certain case, deposits were an increasing function of the deposit rate. When the failure of the bank is a possible event, it becomes necessary to consider two institutional settings. In the first one, public intervention or insurance mechanisms protect deposit holders and the relevant rate is the posted deposit rate. In the second one, there is no insurance mechanism and we assume that the deposit supply is a function of the expected return d. The two institutional settings are successively examined. VWhen parameter ~ is less than one, it is implicitly assumed that the shareholders of the bank will not (or perceived that they will not) bear the liabilities of the underfunded insurance. sit can be shown that the intermediary's losses j'~,*(a +(1 +g)B--(1 + d)D)f(a)da are equal to - $7,*F(a) da. The last term is well defined in dollars; in the case of rectangular density function defined over the inverval (0, K), it is equal to --a*2/2K.

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J. Dermine, Deposit rates, credit rates, bank capital

4.1. The 'deposit insurance' case In this case, the deposit supply is an increasing function of the posted deposit rate. The first-order conditions to maximise the expected net value (8) are (assuming ctF(a*)< 1) 9

OE(NV) (.1-F(a*) ) ~d =((g-cOD'-D)\l_aF(a,) =0,

(9a)

= ( ( g - d ) D ' - D)(1- F(a*)) (g-d)D'-D , + , F ( a * ) ( '-~F-(-a--~ )(~-F(a*)),

(9a')

c~E(NV) =((p-g)L' + L ) - F((1 + p)L)(1 +p)L' +L) Op +(1-~)(1 +g)L'

F(a*) 1 - ~F(a*)

-0,

(9b)

=((p-g)L' + L)(1 - F((1 + p)L)) -(1 + g)L'(F((1 + p)L)- F(a*)) aF(a*)(1 + g)L' 1 -aF(a*)

E(NV) OE

-

(1--~)(l+g)

(1-F(a*)) =0,

F(a*) -_~0 1 - aF(a*)

as

(9b')

a~-l,

(9c)

= - ( 1 +g) +(1 +g)(1-F(a*))

+

aF(a*)(1 + g) (1 - F(a*)). 1-aF(a*)

(9c')

To ease the economic interpretation of the first-order conditions, we have decomposed them in a series of meaningful terms [(9a')-(9c')]. The expected profit due to a change in the deposit rate is the sum of two terms: the expected marginal revenue on deposits when the intermediary does 9This plausible assumption is necessary to ensure that an exogenous shock to the break-even value a* carries a t'mite effect, as one can "~bserve an infinite sequence of effects caused by the insurance premium (7), the balance sheet constraint and the break-even value a* (5).

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not default and the expected income due to the investment of the premium reduction, OP _ c3d

~(1 +g)-lF(a*) ( ( g - d ) D ' --D) 1 - F(a*)

.

The financial intermediary considers its revenue on deposits only when it does not default because its liabilities are limited. In case of bankruptcy, it is the deposit insurer that will benefit. However, to force the bank to take into account its expected liabilities, the insurer can impose a risk-related premium. For instance in the case of a fairly priced premium (~= 1), the bank will consider also the net revenue which accrues to the insurer in case of bankruptcy [see the last term in (9a') with ~ = 1]. One will observe that the optimal deposit rate is identical to the one obtained in the certain case of section 2. The optimal deposit rate is independent of the probability of bankruptcy F(a*) and of the loan rate p. Real costs incurred in servicing deposits have been ignored in the analysis; they are considered in appendix C. The independence of the deposit rate requires in this case two additional assumptions: the real cost function is separable in deposits and loans and the return on bonds remains the exogenous peg of the system. The second first-order condition (9b') for the optimal loan rate can be interpreted in a similar way. The expected net profit due to a change in the loan rate is the sum of three terms: the expected marginal revenue on the loan when the borrower meets his obligation, an expected marginal loss on the loan when the borrower defaults (but not the bank) and the income on the investment of the premium reduction,

OP

o~F(a*)L'

c3p

1 - ,F(a*)"

One will observe that, with a parameter ~ different from unity, the optimal loan rate is a function of the probability of bankruptcy F(a*) and, given the definition of the break-even value a* in (5), the loan rate will depend on the optimal deposit rate d. The origin of this relationship is the limited liability of the bank and the 'unfair' premium that permits the bank to underestimate (~<1) or to overestimate (~>1) the losses in case of bankruptcy, x° Two cases must be distinguished: 1°In fact, there is an additional reason for the fact that we observe a dependence with the probability of bankruptcy F(a*) in the case of the loan rate but not in the case of the deposit rate. In this last case, the net revenue on deposits ((g-d)D'-D) is equal in the bankruptcy and non-bankruptcy states, so that this term can be factored out of the first-order condition. But this symmetry is not observed in the case of the loan rate. The net income on the loan in case of non-bankruptcy depends on the contractual rate p but the net income in case of bankruptcy does not depend on this rate since only the assets of the borrower are recovered.

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106

Case 1. ~ < 1.

Differentiating implicitly relation (9b), one observes a positive relationship between the optimal credit rate p and the level of equity E or the net profit on deposits ( g - d ) D . A reduction in the probability of failure of the bank (caused by additional equity or net profit on deposits) decreases the benefits derived from the limited liability (i.e., the loss on the loan in bankruptcy state is underestimated). Loans become less attractive and the loan rate is raised. Case 2. ~> 1.

The last term in (9b) becomes positive and converse con-

clusions apply. The causal relationship between deposit and credit rates is recursive when the insurance parameter ~ differs from unity. The deposit rate is independent of the loan rate but the loan rate is a function of the deposit rate. However, if the sign of the effect of the net deposit profit [ ( g - t 0 D ] on the loan rate can be determined, the effect of the deposit rate remains ambiguous. One can easily construct cases where shifts in the deposit supply imply a higher deposit rate with lower or higher net profit on deposits. The third first-order condition (9c') relates to the optimal level of equity. 11 The expected net profit due to an increase in equity is the sum of three terms: the opportunity cost, the expected income on equity in case of nonbankruptcy and the revenue obtained by the reduction of the insurance premium, Op

- eF(a*)

~?E

1 -- eF(a*)"

When the insurance premium is underpriced (~< 1), the optimal level of equity is close to zero. When the premium is fairly priced, the equity level is indeterminate and for ~ > 1, the equity level will be such that the probability of failure is zero. The costs associated with bankruptcy (such as lawyers' fees) have been ignored in the analysis. They would raise the value of the fair insurance premium and, in this case (~= 1), the level of equity would be raised to reduce to zero the probability of bank failure, la The case of imperfect competition on the deposits and loan markets has been analysed in this section. As is discussed in appendix C, one observes a similar recursive relationship between the volumes of loans and deposits in the competitive price taking case. lXThis results can be compared to Baltensperger's (1980, section 2.2) study of the liability structure of the banking firm. Modeling aspects set aside, there is a major difference: we assume a limited liability for the bank when Baltensperger assumes implicitly full liability in his relations (10) and (17). 12This result is also mentioned by Kareken-Wallace (1978).

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107

If the deposit rate is chosen before the loan rate in the 'insurance' case, it is shown below that the recursivity is reversed in the 'no insurance' case: the loan rate is chosen first.

4.2. The 'no insurance' case When the deposit holders assume themselves the risk of bankruptcy, the deposit supply is assumed to be a function of the expected return on deposits d which can be expressed as K

a*

(1 +d)D= .[ ((1 +d)D)f(a)da+ ~ (a+(1 +g)B)f(a)da. a*

(10)

k

The first integral represents the depositors' income when the intermediary can cope with its obligation and the second integral, income in case of bankruptcy. Relation (10) can be expressed as a*

(1 + ~ D = ( 1 +d)D- ~ F(a)da.

(11)

k

The expected return on deposits is equal to the posted deposit rate minus the losses in case of bank failure (see footnote 8).

Proof. Substitute the balance sheet constraint and add and subtract from (10) ~*(1 +d)Df(a)da, a*

(1 +aT)D=(1 +d)D+ .~ (a+(g-d)D+(1 + g ) E - ( 1 +g)L)f(a)da. k

The second term can be integrated by parts to yield (11).

(12)

Q.E.D.

As in section 4.1, we assume that the bank maximises its expected end-ofperiod net value, (1

max E ( N V ) = ( p - g ) L + ( g - d ) D d,p,E

+p)L

I

a*

F(a)da.

(13)

This relation can be transformed by adding and subtracting ~7,*F(a)da and by taking into account the definition of the expected return d (11), max E ( N V ) = ( p - g ) L +(g-eTa)D-

(1 + p ) L

I k

F(a)da.

(14)

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J. Dermine, Deposit rates, credit rates, bank capital

The problem can be interpreted as the optimal choice of equity, of the loan rate p and of the expected return on deposits d. 13 The first-order conditions for value maximisation are ~D dE(NV) = (g-- d)-7-~ -- O =0, Od

,E(NV) c~p

=(p-g)L'+ L - F((1 + p)L)((1 +p)L'+L)=O,

8E(NV) =0.

( 15a) 05b) (15c)

dE Once the optimal loan rate p and expected return d have been calculated, one can use relation (11) to obtain the posted deposit rate d. One will observe the indeterminacy of the equity level, the independence between the loan rate and the expected return on deposits and, therefore, the independence between the optimal volumes of loans and deposits. This result is similar to the one obtained in the K - M model discussed in section 2, except that now independence concerns the expected return on deposits. If the loan rate is independent of the posted deposit rate (15b), the reverse is not true. The posted deposit rate (11) is an increasing function of the breakeven value a* and, via its definition in (5) and the balance sheet constraint, a decreasing function of the loan rate. A smaller volume of loans reduces the probability of bankruptcy and the posted deposit rate can be lowered. In the 'no insurance' case, the relationship between rates runs through the expected return on deposits and the effect of the loan rate on the posted deposit rate is clearly negative. The indeterminacy of the equity level can be understood by looking at relations (11) and (13). The loss due to an increase in equity (which decreases the break-even value a*) is identical to the gain accruing to depositors so that the bank can reduce the posted deposit rate and recover exactly the initial loss?* Here too, the presence of bankruptcy costs would encourage the bank to increase equity so as to reduce to zero the probability of bankruptcy. Imperfect competition on the deposit and credit markets have been assumed in this section. We show in appendix C that similar results hold in the competitive price taking case. 13Relation (14) is formally equivalent to the deposit insurance case with 0t= 1, 6 = 0 (8), except that d replaces d. 14One will note that the indeterminacy of the equity level causes the indeterminacy of the deposit rate [relation (11)]. This may explain why different banks post different rates d while offering the same expected return a.

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109

5. Conclusions The determinants of deposit rates, credit rates and equity have been discussed in a neoclassical model of the Klein-Monti variety. Independence between the rate decisions is lost by the introduction of risk in the K - M model but recursivity permits to study the determinants of one rate independently of the other one. Table 1 summarizes the results. Table 1 The relationship between deposit and credit rates.

Klein-Monti Deposit insurance case (0t= 1) Deposit insurance case (a# 1) No insurance case

d independent of p d independent of p d causes p p causes d

The recursivity between the loan and deposit rates is caused by the limited liability of the bank and its derived benefit in the 'insurance' case. In the 'no insurance' case, it is the expected return on deposits which causes the dependence of the posted deposit rate on the loan rate. As is shown in appendix B, these results hold in a state preference version of the model and with a CAPM valuation equation. The elegance of the Klein-Monti model has led many authors to develop neoclassical models of the banking firm. Expanded to include the risk of bankruptcy and a deposit insurance mechanism, the K - M model still provides a simple exposition of the determinants of deposit and credit rates and equity. The cost of its simplicity lies, in the author's view, not so much in that it is a one-period risk neutral model but rather in the fact that the origin of deposits and loans is exogenous to the model. One would like to understand how the explicit introduction of transactions costs and imperfect information - which are the raison d'etre of financial intermediation - would change the nature of the results. 15

Appendix A: The expected net value The derivation of relation (8) is explained in this appendix. We start from relation (6) and we substitute the financing constraint for B, K

E(NV)=

(1 + p)L

((p-g)L+(g-d)O+(l+g)E-(l+g)P)f(a)da

(1 +p)L

+ --(1

~

a*

(a-(l+g)L+(g-d)D+(l+g)E

+g)P)f(a)da-(1 +g)E.

(A.1)

15Recently, partial models of the banking firm have been developed to study the nature of bank assets [Diamond (1984), Ramakrishnan-Thakor (1984)].

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110

It,~,+P)~((p-g)L+(g-d)D+(l+g)E-

We add and subtract from (A.1) (1 +g)P)f(a)da: K

E(NV) = I

a*

((p-g)L+(g-tOD+(1 (1 +p)L

+

I

a*

((1

+g)E-(1

+g)P)f(a)da

af(a)da+((g--p)L-(1 +g)L)

+p)L

x t ,,,I f(a)da

) - ( 1 +g)e.

(A2)

We integrate by parts the second integral in (A.2) and we take into account the definition of a* in (5) to obtain (A.3),

E(NV)=(p--g)L+(g-d)D--l(+g)P-

(1 +p)L

~ F(a)da.

(A.3)

a*

A similar procedure can be used to solve the value of the insurance premium so that we obtain finally (A.4) [i.e., (8)],

E(NV)=(p-g)L+(g-d)D-

(1 + p)L

I

a*

F(a) da

a*

-a~ F(a)da-(1 +g)6. k

(A.4)

Appendix B: The state preference and the CAPM models We present the model in a state preference and in a CAPM framework, respectively:

B.1. 'State preference" modeP6 Let us define a state of the world (0 ~ O) with probability f(0), A(O): the value of the borrower's asset in state 0, s'(O): the price of a primitive security that yields $1 in state 0 and $0 otherwise, g: the risk free rate: ~ s'(0)=(1 + g ) - l , s(0)= s'(0)(1 +g). 0:

16A state preference model of the banking firm is presented in Kareken-WaUace (1978).

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111

Three situations can occur at the end of the period, (1) the borrower does not default: A(0)>(1 +p)L VOE01, (2) the borrower defaults but not the bank: (I+d)D-(I+g)B
The insurance premium is defined as

P= -o~ ~ s'(O)(A(O)+(1+g)B-(1 +d)D)f(O).

(B.1)

O~@ 3

We will present the 'insurance' case; a similar methodology can be applied to the 'no insurance' case. We assume that the bank maximises its net value,

max ~ s'(O)((1+p)L+(1 +g)B-(1 +d)D)f(O) d,p,E O~O1

+ ~ s'(O)(A(O)+(l+g)B--(1 +d)D)f(O)-E,

(B.2)

Oe@ 2

subject to ~

B+L+P=D+E. We multiply (B.2) by (1 +g), substitute the financing constraint and the insurance premium (B.1), and add ~0~e2ue3 s(0)((1 +p)L+(1 + g ) B - ( 1 + d)D)f(O):

max(p-g)L+(g--d)D+ ~ s(O)(A(O)-(1+p)L)f(O) d,p,E

OEO 2

-- ~ s(O)((p-g)L+(g-d)D+(l +g)E)f(O) 0~0 3

+ ( o~~03 s(O)(A(O) -- ( 1 + g)L + (g - d)D + ( 1 + g)E) f (0))( 1 - ~03 s(O))

1-~ ~o3 s(O) The net value maximisation gives similar results to those obtained in the neoclassical model as far as the level of equity and the relationship between deposit and credit rates are concerned.

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J. Dermine, Deposit rates, credit rates, bank capital

B.2. 'CAPM' model In a CAPM valuation model, the expected value of the bank and the insurance premium are adjusted by the market price for risk, 2cov(-,Rm), where 2 denotes the market price per unit of risk and where R m is the market return. The adjustments to the net value (6) and to the insurance premium (7) follow in order, (1 + p )L

J'c°v(a, RnOtaI*+p)L=

((a- E(a))(Rm- E(R,O)f (a, Rm) da dRm,

I

a*

Rm

and 1 2 cov (a, Rm)~,*. l+g

The risk-adjusted net value becomes in this case, (1 + p ) L

E(NV) =(p--g)L +(g-cOD-

I

a*

F(a)da

a*

--o~ F(a)da-(1 +g)6--2cov(a, RnOt,,1,+P)r-ct2 coy (a, Rm)~,*. k

Similar results for the optimal level of equity and for the relationship between the deposit and credit rates follow with a CAPM valuation equation. Appendix C: The price taking case with real costs The price taking case is developed in this appendix. Real cost functions for loans [K(L)] and deposits [G(D)] are introduced in the model. The interest rates on loans and deposits are exogenous and the perfect market security rate g is assumed net of real cost. xv The break-even value a* and the insurance premium px8 are defined in (C.1) and (C.2), 17The assumed cost functions are restrictive as they imply independence between loans and deposits on the cost side [see the discussion in Adar-Agrnon-Orgler (1975)-]. They have been chosen because we want to focus exclusively on the dependence created by the limited liability of the intermediary and by the deposit insurance. lSWe implicitly assume that the intervention of the deposit insurer occurs after that the payment of real costs have been made. An alternative procedure would be to model explicitly the real costs as a function of the probability of bankruptcy. For instance, the wages should increase with risk in the same way as the posted rate did in the risk bearing case.

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113

(c.1)

a*=(1 +d)O+ K(L)+G(D)-(1 +g)B, a*

! (a+(1 +g)B--(1 +d)D-K(L)-G(D))f(a)da.

P= - ~-~

(c.2)

Following the same procedure used in appendix A, one obtains the expected net value, a*

E(N V) = (p -- g)L + ( g - d)D - K ( L) - G( D) - o~~ F( a) da -

(1 +p)L

k

I F(a)da.

a*

(c.3) The 'deposit insurance' case is considered first. The optimal volumes of deposit and loans are given by the following first-order conditions:

OE(NV) , (1-F(a*) ) OO = ( g - d - G ) i ~ a - ~ ) =0,

(C.4a)

0E(NV) OL =(p-g-K')-F((I +p)L)(1 +p)

l+g+K')=O.

+(1 -~)F(a*) \ i ----~(~))

(CAb)

One observes the independence of the volume of deposits but the dependence of the volume of loans on the break-even value a* (in the case of ~ 1) and, via its definition, on the volume of deposits. The relationship is recursive and the direction of the recursivity runs from the deposits to the volume of loans. The 'no insurance' case follows. The expected return on deposits is defined a s 19 K

(1 + ~ D = ~ (1 +d)Df(a)da Q*

a*

+ 5 (a+(1 +g)B-K(L)-G(D))f(a)da.

(C.5)

k

Following the procedure used in the text, one obtains E(NV) =(p--g)L + ( g - D O - - K(L)- G(D)-

(1 + p)L

I F(a)da. k

x9it is assumed implicitly that the assets will cover the real costs.

(C.6)

J. Dermine, Deposit rates, credit rates, bank capital

114

T h e o p t i m a l volumes of l o a n s a n d d e p o s i t s are given by the first-order conditions, ~3E(NV) ~D

= g - d - G' = 0,

c~E(NV) ~L -p-g-K'-F((l+p)L)(l+p)=O.

(C.7a)

(C.7b)

T h e v o l u m e of loans is i n d e p e n d e n t of d e p o s i t s b u t t h e p o s t e d deposit rate is positively related to the v o l u m e of l o a n s via t h e d e f i n i t i o n o f the expected r e t u r n o n deposits.

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