Bank capital regulation, asset risk, and subordinated uninsured debt

Journal of Economics and Business 56 (2004) 443–468

Bank capital regulation, asset risk, and subordinated uninsured debt Susanne Homölle∗ Department of Banking, University of Münster, Universitätsstr. 14-16, 48143 Münster, Germany Received 20 June 2002; received in revised form 16 March 2004; accepted 2 June 2004

Abstract Whether more stringent capital requirements lead to reduced or to increased bank risk-taking has been discussed intensively in the academic literature. In this paper we drop the common but unrealistic assumptions that banks only issue deposits whose returns are guaranteed by a subsidised deposit insurance and that deposit insurance is free. We prove that with uninsured debt and a flat-rate deposit insurance premium the reaction of a bank to a higher capital requirement may change substantially. In some scenarios banks increase asset risk due to the enforcement of a more stringent capital requirement. © 2004 Elsevier Inc. All rights reserved. JEL classification: G21; G28 Keywords: Capital regulation; Bank risk-taking; Subordinated uninsured debt

1. Introduction The risk-taking of banks is limited by regulatory capital requirements to prevent bank insolvency. Recently, the Basel Committee on Banking Supervision (2003) issued a revised proposal for a capital adequacy framework (“Basel II”) replacing the 1988 Basel Capital Accord. Due to this development a question which has been widely discussed since the eighties of the last century (e.g., Besanko & Kanatas, 1996; Blum, 1999; Flannery, 1989; Gjerde & Semmen, 1995; Lam & Chen, 1985; Kendall, 1991; Rochet, 1992; Santos, 1999;

∗

Tel.: +49-251-8322879; fax: +49-251-8322882. E-mail address: [email protected] (S. Homölle).

0148-6195/$ – see front matter © 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jeconbus.2004.06.002

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Zarruk, 1989, and the survey article of Santos, 2001) is again high on the agenda: what are the effects of changing and especially rising capital requirements? The answers vary with the different analytical frameworks. For example, Kahane (1977), Koehn and Santomero (1980), and Kim and Santomero (1988) apply portfolio theory to show that higher capital requirements can give a utility-maximising bank the incentive to increase the risk of its portfolio. The increase in asset risk may offset the desired effect of reduced leverage. Thus the regulatory authority cannot be sure to achieve a reduction in insolvency risk. Furlong and Keeley (1989, henceforth FK89) point out the relevance of the deposit insurance guarantee for the so-called asset-substitution effect. Deposit insurance schemes with risk-insensitive premiums encourage banks to shift risk from bank owners to deposit insurers as neither the insured depositors nor the deposit insurance react to this risk-shifting behaviour. In models based on state-preference theory and option pricing theory, respectively, FK89 prove that a higher capital requirement does not give banks additional incentives to substitute riskier assets for less risky assets. Thus the asset-substitution effect cannot be observed. The rather optimistic conclusion of FK89 has got other authors to analyse whether it can still be drawn in a modified framework. For example, Gennotte and Pyle (1991) assume an imperfect instead of a perfect market for assets and demonstrate that the probability of insolvency may rise due to a higher capital requirement. Our paper focuses on the implications of the liabilities side of the balance sheet for the asset-substitution effect of a higher capital requirement, thus closing a gap in the discussion mentioned above. Firstly and most importantly, the capital structure of the bank is often assumed to be quite simple, consisting only of equity and insured deposits. Knowing that in reality banks do not only hold insured deposits, we allow for subordinated uninsured debt as well. The issue of subordinated uninsured debt is an important aspect in the discussion about “Basel II”. The Basel Committee on Banking Supervision (2003) promotes enhanced transparency, e.g., of the level of seniority, to support market discipline as the third pillar of its framework. The Shadow Financial Regulatory Committees (1999) even recommend “a minimum subordinated debt requirement as a means to bring market discipline to bear on bank risk and capital management” (for an overview of various subordinated debt proposals see Board of Governors of the Federal Reserve System, 2000). In our model, however, where banks are allowed but do not have to hold subordinated uninsured debt, the issue of subordinated debt may also induce some negative effects.1 Our model confirms the result of FK89 for a bank that only issues insured deposits and has a fixed amount of equity. It will decrease asset risk as a reaction to more stringent capital requirement. However, a bank that in addition holds subordinated uninsured debt may raise asset risk if it has to pay a positive, fixed-rate deposit insurance premium. Moreover, the response of a bank with variable equity to a tightened capital requirement is not as clear as stated by FK89. In this case an increased risk-taking cannot be rejected either. The driving force behind these results is that the bank has got more possibilities to fulfill the prescribed reduction in leverage. Moreover, there is no longer a fixed relationship between changes in insured deposits and in assets. Whereas a bank with constant equity

1

For potential benefits, see for example, Evanoff and Wall (2000), Lang and Robertson (2002).

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and insured deposits must sell assets and pay back insured deposits to reduce leverage, a bank that also holds uninsured debt or may raise additional equity can change these instead of changing insured deposits. A decline in asset size is no longer a necessary condition to fulfill the new capital requirement. It will become clear later in this paper that if the amount invested in assets increases, it is optimal for the bank to raise asset risk as well. The second new aspect discussed in our paper is the role of imperfections on the markets for liabilities. We are going to show in which market scenario one must drop the assumption of banks with only insured deposits. Thirdly, FK89 say that their conclusions are the same whether deposit insurance is free or the bank has to pay a positive, fixed-rate premium. Their statement, however, does not hold any more if we allow for subordinated uninsured debt. Although capital regulation and deposit insurance schemes are rather specific for banks, the key issue of this paper can easily be set in a wider context. Risk-taking incentives are also known in corporate finance theory. Jensen and Meckling (1976), e.g., describe the incentive conflicts between owner–managers of a firm and its bondholders. Because of their limited liability the owner–managers have the incentive to change their announced investment policy investing only in the riskiest projects. This strategy is optimal as long as the bondholders do not react to the increase in risk, i.e. do not require a higher return on their capital invested in the firm. A deposit insurance may serve as a reason why the creditors (insured depositors) do not take care about the investment policy of the firm (bank). The question is how the owner–managers’ risk-shifting incentives alter due to a change in the exogenous restrictions of their investment policy, i.e. a more stringent capital requirement. The role of additional capital provided by creditors who do react to the owner–managers’ decisions is the special focus of this paper. In Section 2, the structure of our model is introduced. We use a state-preference model to distinguish between situations with different repayments to the creditors due to bank solvency and insolvency. The optimal levels of asset risk, insured deposits, and uninsured debt of a regulated, profit-maximising bank are determined in Section 3. Section 4 presents the effects of a more stringent leverage restriction on these optimal levels, especially on asset risk. Section 5 concludes.

2. The model The analytical framework is a model based on state-preference theory (e.g., Arrow, 1964; Myers, 1968). As a starting point of our analysis, we use the approach of FK89, which is altered in several aspects. We consider two dates, time 0 and time 1, and three possible states of nature at time 1. The time 0 price of one dollar payout in state i, pi , i = 1, 2, 3, is taken as given where Σi pi < 1 indicating a positive riskless rate of return. At time 0 a bank decides on the structure of its capital and assets. The bank raises equity, C0 , which has to exceed a certain level.2 Bank owners have limited liability. The bank can 2

A minimum amount of regulatory capital (equity) is often one of the requirements to obtain a bank charter. For example, in the EU banks have to raise at least 5 million as initial core capital.

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issue insured deposits, E0 , and uninsured debt, U0 . Insured deposits are senior to uninsured debt. For the deposit insurance guarantee the bank pays a fixed-rate premium, I0 , depending on the amount of insured deposits:3 I0 = vE0 ,

1 > v ≥ 0.

(1)

FK89 assume a free deposit insurance. This case is included in our model by setting the deposit insurance premium rate, v, equal to zero. The capital invested in assets, A0 , consists of equity, insured deposits, diminished by the insurance premium, and uninsured debt: A0 = C0 + E0 (l −v) + U0 . The bank can allocate its funds among the securities X and Y. Let α, 0 ≤ α ≤ 1, be the share of X in the asset portfolio and 1 − α the share of Y.4 At time 1 the bank gets a return of xi (yi ) per invested dollar in X (Y) if state i occurs. The capital market is assumed to be complete and perfect. The net present values of X and Y equal zero: p1 x1 + p2 x2 + p3 x3 − 1 = p1 y1 + p2 y2 + p3 y3 − 1 = 0.

(2)

By assumption security X is riskier than security Y. The spread of returns on X is higher than that of Y:5 x1 < y1 < x2 < y2 < y3 < x3 .

(3)

From (3) we know that, independent of α, asset returns in state 1 are lower than those in state 2 which in turn fall below asset returns in state 3: A1 < A2 < A3 , where Ai = A0 (αxi + (1 − α)yi ), i = 1, 2, 3. Whether asset returns in state i are higher or lower than the liabilities of the bank depends on the concrete values of the parameters and variables (C0 , E0 , U0 , α) of our model so that we could analyse quite a lot of scenarios. For example, one might expect that the bank is not able to meet its obligations thus going bankrupt in any state or in states 1 and 2 or only in state 1. However, anticipating the optimal behaviour of the bank and adding one additional assumption on the level of the return x2 leads to a reduction in complexity. Thus we can restrict our analysis to the following subsample of all possible scenarios one might think of: in state 1 the bank is insolvent and insured depositors make a claim on the deposit insurance, in state 2 the bank may be solvent or insolvent but the deposit insurer does not have to make any payment, and finally in state 3 the bank is solvent. In state 1 asset returns are assumed to be so low that the bank cannot even fully repay the promised returns on insured deposits, E0 e. For E0 > 0 and U0 ≥ 0 it holds 3 For example, the deposit insurance of German private banks (Einlagensicherungsfonds des Bundesverbandes Deutscher Banken) required a uniform, yearly premium of 0.03% of liabilities to non-banks until 1998. In that year some kind of risk-adjusted premium was adopted by introducing three risk classes for banks. So the premium rates differ according to the risk class, but they still refer to the amount of (insured) liabilities to non-banks. 4 The bank cannot sell short X and Y. There exists a conflict between this implicit assumption of FK89 and the state-preference theory because a capital market with these two securities and short-selling restrictions is not complete. Thus one should at least implicitly suppose that there exists a third security which may be sold short. For a complete capital market with short-selling restrictions, cf. Raab and Schwager (1993). 5 Compare the comment of Sharpe (1978) on the definition of “risk” in a state-preference model: “risk is generally considered to have increased when a set of returns becomes more ‘spread’ out”.

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A0 (αx1 + (1 − α)y1 ) ≤ A0 y1 < E0 e ≤ E0 e + U0 u

447

(4)

where U0 u denotes the promised returns on uninsured debt.6 The deposit insurance guarantees that insured depositors receive full repayment at time 1 whatever state occurs. Hence, in state 1 the deposit insurer has to pay to insured depositors the difference between promised returns on insured deposits and asset returns. Uninsured creditors and bank owners get no return. The net present value (NPV) of the deposit insurance guarantee, I0PV − I0 , is assumed to be positive:7 I0PV − I0 = p1 [E0 e − A0 (αx1 + (1 − α)y1 )] − vE0 > 0.

(5)

From (3) we know that the return on the asset portfolio in state 2 is higher than the return in state 1. Moreover, we assume that (1 − v)x2 ≥ l/(p1 + p2 + p3 ) so that the bank can fully meet its obligations to insured depositors:8 E0 e ≤ A0 x2 ≤ A0 (αx2 + (1 − a)y2 ).

(6)

Whether the bank is also able to pay the contracted returns on uninsured debt, U0 u, cannot generally be determined. If the level of uninsured debt, U0 , is very low, the bank is solvent in state 2 and uninsured creditors receive U0 u2 = U0 u. If U0 is rather high, the bank might become insolvent (E0 e + U0 u > A0 (αx2 + (1 − α)y2 )) so that uninsured creditors only get U0 u2 = A0 (αx2 + (1 − α)y2 ) − E0 e. From (6) it follows that the bank is solvent in state 3, i.e.9 E0 e + U0 u ≤ A0 y3 ≤ A0 (αx3 + (1 − a)y3 ). Following Kareken and Wallace (1978) we assume that the bank has some monopoly power on the market for insured deposits, e.g., because it supplies payment services together with insured deposits (see also Dothan & Williams, 1980). The depositors’ demand for insured 6 If uninsured debt was sufficiently high, inequality (4) would be violated (E e < A (αx + (1 − α)y )). The 0 0 1 1 net present value of the deposit insurance guarantee would be negative (see also (5)) because insured depositors would never claim on the deposit insurance. As the profit of the bank would be lower than in the scenario in this paper (see also (9)), the profit-maximising bank chooses its variables so that the conditions in (4) are fulfilled. 7 This assumption is made, often implicitly, by all authors who model a free deposit insurance, e.g., FK89, Gennotte and Pyle (1991), Gjerde and Semmen (1995). A reason for assuming a positive NPV in a world without asymmetric information is given by Buser, Chen, and Kane (1981). For the calculation of a theoretically correct premium see, e.g., Merton (1977), Chan, Greenbaum, and Thakor (1992), Dermine and Lajeri (2001). The value of the deposit insurance for the bank at time 1 is identical to the value of a put option on assets with an exercise price of E0 e, max[0, E0 e − A0 (αxi + (1 − α)yi )], because the deposit insurer has to pay E0 e − A0 (αx1 + (1 − α)y1 ) in state 1 and nothing in states 2 and 3. 8 From (7) it follows e < 1/(p + p + p ) so that E e ≤ E (1 − v)x + U x + C x = A x with U ≥ 0 1 2 3 0 0 2 0 2 0 2 0 2 0 and C0 ≥ 0. Alternatively, one might assume that (1 − v)x2 < 1/(p1 + p2 + p3 ) so that E0 e > A0 (αx2 + (1 − α)y2 ) at least for some values of E0 , U0 , C0 , and α. Then the deposit insurer has to make a payment in states 1 and 2. Although the profit function of the bank must be extended by p2 [E0 e − A0 (αx2 + (1 − α)y2 )], this would not change our results substantially. 9 A scenario with E e + U u > A (αx + (1 − α)y ), U > 0, does not make sense because uninsured creditors 0 0 3 3 0 0 do not invest in a bank that goes bankrupt in any state and thus can never pay the promised return. For U0 = 0 it follows from (6) that the bank is solvent in state 3.

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deposits is a strictly increasing, convex function of (the present value of) the return per invested dollar, e: E0 = f(pe)

(7)

where p = (p1 , p2 , p3 ,), e = (e, e, e) , pe ∈ [0, 1[, f(0) = 0, limpe→1 f(pe) = ∞, f  (pe) > 0, f  (pe) > 0. Thus the bank earns a positive NPV on the imperfect market for insured deposits: E0 − E0PV = E0 − (p1 + p2 + p3 )E0 e = E0 (1 − f −1 (E0 )) > 0.

(8)

The market for subordinated uninsured debt is assumed to be perfectly competitive (Kareken & Wallace, 1978). The NPV of uninsured debt, U0 − U0PV , equals zero: U0 (l − p2 u2 − p3 u) = 0. In the EU the market for inter-bank deposits, which are not insured, is an example for an (almost) perfectly competitive market for uninsured debt.

3. A regulated bank At time 0 the bank chooses the optimal share of security X, α, and the optimal levels of equity, C0 , insured deposits, E0 , and of uninsured debt, U0 . As it acts on behalf of the shareholders, it maximises the NPV of equity, C0PV − C0 , put differently (the present value of) its profit, G0 : − A0 + I PV − I0 + E0 − EPV + U0 − U PV G0 = C0PV − C0 = APV
0  
0  
 0 
 0   = p1

=0

>0

>0

=0



1 E0 f −1 (E0 ) − A0 (αx1 + (1 − α)y1 ) p1 + p 2 + p 3

−vE0 + E0 (1 − f −1 (E0 ))

(9)

where A0 = C0 + E0 (l − v) + U0 . APV 0 − A0 denotes the (zero) NPV of assets. According to (9) the profit is equal to the positive NPVs of the deposit insurance guarantee, I0PV − I0 , and of insured deposits, E0 − E0PV . In this paper the effects of a more stringent leverage constraint are analysed. The bank has to comply with a lower bound, s, on the ratio of assets to liabilities which guarantees that equity (minus the deposit insurance premium) is positive:10 A0 ≥ s > 1. E0 + U 0

(10)

Of course, (10) defines a quite simple form of capital constraint. However, the inclusion of asset risk-weights would not bring about fundamental changes in our results as long as the regulatory authority prescribes the same risk-weight for two or more assets with (slightly) different risk. Even in the latest proposal of the Basel Committee on Banking Supervision 10

By assumption subordinated debt, U0 , does not show the characteristics of tier 2 capital so that regulatory capital only consists of equity, C0 .

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(2003) an approach with such standardised risk-weights is included (besides the new internal ratings based approach). The misrepresentation of asset risk in pure leverage constraints as well as in standardised risk-weighted capital regulation has been a reason for the discussion about banks’ risk-shifting behaviour to which we want to add another aspect in this paper. Assuming that both securities X and Y belong to the same risk class would be the easiest way to bring both kinds of capital regulation into line.11 A profit-maximising, unregulated bank would raise leverage risk ad infinitum holding only the prescribed minimum amount of equity and expanding insured deposits.12 Therefore the capital rule (10) is binding: 1 s−1 A0 = s ⇔ E0 = C0 − U0 . E0 + U 0 s−1+v s−1+v

(11)

Referring to FK89, we assume that—in addition to the leverage constraint—there exist some (regulatory) cost which limit the incentive of the bank to increase asset risk. Thus we get an interior solution for the optimal share of security X. Let Kα be a strictly increasing, strictly convex function of α: dKα > 0, dα

d2 K α > 0. dα2

Actual regulatory restrictions are influenced by the idea of limiting the share of a single asset to guarantee that bank portfolios are sufficiently diversified. In the EU, a loan to a single company is restricted by law to 25% of the bank’s regulatory capital. Banks are allowed to go beyond this limit if they accept further restrictions. In Germany, e.g., capital covering 100% of the exceeding amount cannot be used any more to fulfill other regulatory rules. Therefore loan concentration does indeed lead to some cost. The bank raises the share of the riskier security X until the marginal gain equals the marginal cost:  ∂G0  dKα = , where 0 < α = α¯ < 1. (12)  ∂α s dα For A0 = s(E0 + U0 ), E0 > 0, and U0 = 0 the marginal gain from increasing asset risk is positive:  ∂G0  = −p1 s(E0 + U0 )(x1 − y1 ) > 0. (13) ∂α s

11 A risk-weighted capital requirement may be written as: (αr + (1 − α)r )A /(E + U ) = s, where r (r ) X Y 0 0 0 X Y is the risk-weight of security X (Y). For rX = rY = r the risk-weighted capital requirement may be reduced to rA0 /(E0 + U0 ) = s, i.e. the left-hand side of the leverage constraint (10) must only be multiplied by a constant factor r. 12 Notice that ∂G /∂C < 0. Raising insured deposits is optimal because the positive NPV of the deposit 0 0 insurance subsidy is a strictly increasing function of E0 (see Appendix A(a)) and the NPV of insured deposits is always positive.

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Without Kα the bank would only invest in the riskier asset X (even if it voluntarily holds subordinated uninsured debt).13 This result, which also holds for an unregulated bank, is widely known as the moral hazard problem of a subsidised deposit insurance (e.g., Dothan & Williams, 1980; Gjerde & Semmen, 1995; Kareken & Wallace, 1978; Sharpe, 1978). The returns to the bank owners in case of solvency rise whereas the bank owners do not lose more than bank equity in state 1 due to limited liability and do not have to pay a higher return on insured deposits in states 2 and 3. As insured depositors know that the promised return per dollar, e, is guaranteed by the deposit insurance, they do not react to changes in asset risk. In case of insolvency in state 2 the bank compensates uninsured creditors for its decreasing return, u2 , by a higher promised return, u, so that the present value of uninsured debt, U0PV , and thus the profit are kept constant. For the calculation of the optimal levels of equity and liabilities we are going to differentiate between banks that can raise their equity and banks whose equity can be viewed as constant at least in the short run (see also FK89). For example, publicly owned savings banks are unable to issue new equity in the capital market. So it may be difficult for them to respond to a new capital regulation by increasing equity. Fixed equity If equity is supposed to be fixed, the leverage constraint (11) defines a negative relation between changes in insured deposits and subordinated uninsured debt and, since U0 cannot be negative, an upper bound for E0 . The question is whether insured deposits should be expanded up to this upper limit, E0 = C0 /(s − 1 + v). The marginal gain from increasing insured deposits (while reducing uninsured debt) can be expressed by the total derivative of the profit function (9) subject to (11) (where ds = 0 and dC0 = 0):    dG0  1 vs −1 (αx = −p f (E ) + + (1 − α)y ) 1 0 1 1 dE0 s,C0 p1 + p 2 + p 3 s−1 p 2 + p3  −v + 1 − f −1 (E0 ) − E0 f −1 (E0 ) (14) p1 + p 2 + p 3 The first part of the profit function, the NPV of the deposit insurance guarantee, strictly increases in E0 (see also Appendix A(b)) because the contracted returns on insured deposits, E0 e, increase. Besides the positive sign of the ratio vs/(s − 1)—if v > 0—shows that assets shrink as long as the bank issues insured deposits while reducing uninsured debt.14 Increasing insured deposits has a higher positive impact on liabilities than on assets because a part of the money additionally raised, v dE0 , has to be paid for the deposit insurance and cannot be invested in assets. To compensate the decline in the ratio of assets to liabilities the regulated bank has to reduce uninsured debt more than it increases insured deposits. Therefore asset size, A0 , and hence asset returns in state 1 decline. Since this effect cannot 13 A suitable mandatory subordinated debt requirement, however, could sufficiently discipline the bank (Evanoff & Wall, 2000). The lower bound of uninsured debt must be so high that asset returns in state 1 exceed the contracted returns on insured deposits. See also footnote 6. 14 For v > 0 and dC = 0, it holds that |dE | < |dU | because dE = −(s − l)/(s − 1 + v)dU (see (11)). Hence 0 0 0 0 0 for dE0 > 0 it follows that dA0 = (1 − v)dE0 + dU0 < 0.

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Fig. 1. Profit function G0 of a bank with fixed equity (C0 = 0.2, p1 = 0.05, p2 = 0.1, p3 = 0.7, (αx1 + (1 − ␣)y1 ) = 0.4, s = 1.1, v = 0, f(pe) = ln(1 − pe)).

be observed if the premium rate, v, equals zero, the assumption of a free deposit insurance is not innocuous. The second part of the profit function, the NPV of insured deposits, may be strictly increasing in E0 or it may show an interior maximum (see the examples in Appendix B). In the second case the possible decrease in the NPV of insured deposits may just be equal to the increase in the NPV of the deposit insurance subsidy so that the marginal gain from increasing insured deposits, dG0 /dE0 |s,C0 , equals zero at a certain level of insured deposits, E0∗ < C0 /(s − 1 + v). Fig. 1 shows an example where this local maximum is also the global maximum, i.e. G0 (E0∗ ) > G0 (C0 /(s − 1 + v)). The question whether this interior solution exists so that the bank holds both insured and uninsured liabilities or whether the bank issues (only) insured deposits up to E0 = C0 /(s − 1 + v) can only be answered if the values of the parameters of (9) are known. In Section 4 we will treat the two cases separately. Variable equity A regulated bank that can raise additional equity does not issue subordinated debt and tends to expand insured deposits ad infinitum, thus increasing the payments of the deposit insurer in state 1. This can be seen from the total derivatives of (9) subject to (11):  dG0  = −p1 s(αx1 + (1 − α)y1 ) < 0, (15) dU  0 s,E0

   dG0  1 −1 = p1 f (E0 ) − s(αx1 + (1 − α)y1 ) dE0 s,U0 p1 + p 2 + p 3 p 2 + p3  −v + 1 − f −1 (E0 ) − E0 f −1 (E0 ). p1 + p 2 + p 3

(16)

Issuing uninsured debt (accompanied by an increase in equity) would only lead to an increase in assets. It would not improve the profit, for the NPVs of uninsured debt and of assets equal

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Fig. 2. Profit function G0 of a bank with fixed equity (U0 = 1, p1 = 0.1, p2 = 0.1, p3 = 0.7, (αx1 + (1 − ␣)y1 ) = 0.4, s = 1.1, v = 0.001, f(pe) = ln(1 − pe)).

zero. Instead the profit would shrink because more assets would be available in state 1 to pay off the claims of creditors. The bank owners would have to take over more of the losses in state 1 so that the (net) present value of the deposit insurance guarantee would decline. The regulated bank increases its insured deposits (and its equity) as far as possible because the NPV of the deposit insurance subsidy is again positively related to E0 (see Appendix A(c)) and the NPV of insured deposits is always positive (see (8)). Although there may exist a local maximum, it holds that limE0 →∞ G0 = ∞ (e.g., see Fig. 2). FK89 also describe a bank’s incentive to increase insured deposits and equity, thus expanding asset size. They argue that the bank invests in assets until the marginal gain is balanced by the marginal cost and regard regulatory restrictions of asset size as a main source of this cost. In our more general analytical framework this crucial assumption is no longer convincing because the increase in insured deposits and not the rise in assets is responsible for the rising deposit insurance subsidy and because a rise in insured deposits does not necessarily lead to an increase in assets (due to the possible reduction in uninsured debt). That is why we assume that the incentive of the bank to expand insured deposits is limited by some (regulatory) cost, KE . So we get a certain optimal level of insured deposits, ¯ 0 , as a starting point of our following analysis of a change in the capital constraint. Let E KE be a function of E0 , where dKE > 0, dE0

d2 K E > 0. dE0 2

¯ 0 where the Taking the cost into account the regulated bank issues insured deposits up to E marginal cost equals the marginal gain:  dKE dG0  ¯ 0. = , where E0 = E (17) dE0 dE0 s,U0 The assumption about restrictions of insured deposits is not only important for our analysis. In reality such limitations exist as well. For example, the EU directive on deposit guarantee

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Table 1 Optimal behaviour of a regulated bank Bank with fixed equity Insured deposits

Increases insured deposits up to the upper limit (given by leverage constraint): E0 = C0 /(s − 1 + v)

Uninsured debt

Does not hold uninsured debt: U0 = 0

Asset risk

Bank with variable equity Increases insured deposits up to ¯0 E0 = E

Does not increase insured deposits up to the upper limit (given by leverage constraint): E0 = E0∗ < C0 /(s − 1 + v) Holds uninsured debt: Does not hold U0 = −(s − 1 + v)/(s − uninsured debt 1)E0∗ + 1/(s − 1)C0 > 0 U0 = 0 Increases asset risk up to α = α¯ < 1

schemes requires an upper bound of 20,000 per person. Such an upper bound is not directly connected with regulatory cost but we can imagine that the circumvention of this limit, e.g., opening (insured) deposit accounts under the name of relatives if a person’s insured deposits reach the limit, may induce some transaction cost. The results referring to the structure of assets and liabilities of a regulated bank are summarised in Table 1. The three different profit maxima are used as starting points for the analysis of the impact of a higher capital requirement in the next section.

4. The enforcement of a more stringent capital requirement 4.1. A bank with fixed equity Suppose the regulatory authority enforces a more stringent capital requirement (ds > 0), i.e. a decrease in leverage. We analyse whether the (marginal) rise in s can induce the bank to take higher asset risk. For this analysis we once again distinguish between banks with fixed and variable equity, starting with fixed equity. 4.1.1. No uninsured debt In the previous section we have shown that under certain circumstances it may be optimal for a bank with fixed equity not to hold uninsured debt but solely insured deposits. Applying the envelope theorem we know that if the regulatory authority increases the capital ratio (ds > 0), the bank changes its decision variables E0 and α so that the (first-order) conditions for a maximum (11) and (12) still hold (remember U0 = 0 and dC0 = 0): dE0 = −

C0 d2 K α ds and dα = −p1 s(x1 − y1 )dE0 − p1 E0 (x1 − y1 )ds. (s − 1 + v)2 dα2

After some transformations we get: C0 dE0 =− < 0, ds (s − 1 + v)2

(18)

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p1 C0 (1 − v)(x1 − y1 ) dα < 0. =  2 α d K ds (s − 1 + v)2

(19)

dα2

Since the bank cannot raise additional equity and liabilities only consist of insured deposits, the bank has to sell assets and pay back insured deposits to comply with the new capital regulation (see (18)). A change in asset size, however, is an important factor in the effect of the higher capital requirement on asset risk. A marginal increase in the share of the riskier asset X implies a reduction in the return on assets in state 1 of (y1 − x1 ) per dollar invested at time 0. The lower the amount invested in assets at time 0, A0 , the lower is the decline in total asset returns in state 1, A0 (y1 − x1 ), which is identical to the additional payments of the deposit insurer in state 1. In other words, a change in asset risk of a large bank has a greater impact on the deposit insurer than the same change of a small bank. Therefore the reduction in assets which is necessary to comply with the new capital rule leads to a decline in the additional payments of the deposit insurer in state 1 that are induced by a rise in asset risk. For the bank it means that the marginal gain from increasing asset risk, ∂G0 /∂α|s , is reduced. As the marginal cost, dKα /dα, is unaffected by the change in the capital requirement, the bank reduces the share of the riskier security X until the thus decreasing marginal cost equals the reduced marginal gain, i.e. until the optimum condition (12) holds again. Result 1. A more stringent capital requirement gives a bank with fixed equity and insured deposits the incentive to reduce asset risk. This conclusion is quite similar to that described by FK89.15 We are now going to show that it does not necessarily hold for a regulated bank which also issues subordinated uninsured debt. 4.1.2. Uninsured debt The bank issues a certain amount of insured deposits, E0∗ < C0 /(s − 1 + v), for which the following holds:  dG0  = 0. (20) dE  0 s,C0

The optimal level of asset risk is again defined by (12). Let the profit function (including regulatory cost) be strictly concave in E0∗ and α¯ (see Appendix C). The optimal variation in the share of security X when s is raised is calculated by inserting (13) and (14) into (12) and (20) and totally differentiating:   2 1 v s ∗ + U )a + p −p − y )(−(E ¯ 1 )) (x (αx ¯ 1 + (1 − α)y 1 1 1 0 1 0 s−1 dα (s−1)2 = (21)  2 α  2 ds d K vs 2 2 a − p1 s−1 (x1 − y1 ) dα2 15

FK89 say that “a bank would not be expected to respond to higher capital requirements by increasing the riskiness of its asset portfolio”. The difference is caused by our assumption about the (regulatory) cost Kα . FK89 do not argue using a continuous, twice differentiable cost function but only consider the upper part of the cost function. They state that the marginal cost of exceeding α¯ should be at least equal to the marginal gain.

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     1 2 +p3 where p1p+p [2f −1 (E0∗ ) + E0∗ f −1 (E0∗ )] > 0, − s−1+v E0∗ + s−1 C0 > 0. s−1 2 +p3 The denominator of (21) must be positive to fulfill the second-order conditions for an initial maximum at E0∗ and α¯ (see Appendix C). Thus the sign of dα/ds depends on the numerator: p1

v2 s > dα > (αx ¯ 1 + (1 − α)y ¯ 1 ) (E0∗ + U0 )a ⇔ 0. 2 < ds < (s − 1)

(22)

Whether the bank increases asset risk or not depends on how the marginal gain from increasing asset risk, ∂G0 /∂α|s , is affected by a rise in s. This effect can be split up into two effects: one direct effect due to the required reduction in leverage and one indirect effect via adjustments in the structure of liabilities (for an overview over direct and indirect effects see Table 2). The direct effect of a change in s induces the bank to reduce asset risk. This is because the bank sells assets and pays back liabilities to achieve the required decrease in leverage. Again the marginal gain from increasing asset risk decreases due to the reduction in asset size so that decreasing the share of the riskier asset X is optimal. If the bank does not have to pay any deposit insurance premium, there are no other effects of the more stringent capital requirement so that dα/ds < 0. For v = 0 the marginal gain from increasing insured deposits (see (14)) and thus the first-order condition (20) are independent of s. Hence for the bank it does not matter whether it reduces insured deposits or uninsured debt to decrease leverage as required. An adjustment of the structure of liabilities under compliance with the capital rule does neither change the level of liabilities nor does it have an impact on asset size because increasing (decreasing) insured deposits by one dollar must be accompanied by a reduction (rise) in uninsured debt of exactly one dollar (dE0 = − dU0 (see (11)) and dA0 = dE0 + dU0 = 0 if v = 0, ds = 0, dC0 = 0). Therefore the marginal gain from increasing asset risk is not affected by restructuring liabilities but only by the reduction in leverage itself. Result 2. A bank with insured deposits, subordinated uninsured debt, and fixed equity decreases asset risk as a response to a higher capital requirement if deposit insurance is free. Even if the premium rate is positive, v > 0, we can identify conditions under which the bank decreases asset risk. As distinct from the first scenario with fixed equity and no uninsured debt, an increase in the capital ratio, s, does not automatically lead to a lower level of insured deposits. It might even be optimal to increase insured deposits while reducing uninsured debt thus choosing another structure of liabilities. We already know that if additional insured deposits are issued, the necessary decrease in uninsured debt is higher than the rise in insured deposits (see the discussion of (14)). The new structure of liabilities (under compliance with the capital requirement) causes a lower level of assets (and of liabilities). Thus the change in s shows another, indirect effect (via the increase in insured deposits, dE0 > 0) on the marginal gain from increasing asset risk,∂G0 /∂α|s . As both the direct and indirect effect lead to a reduction in this marginal gain, it is optimal for the bank to decrease asset risk.

456

Effects on

Bank with fixed equity No uninsured debt

Uninsured debt

Bank with variable equity (no uninsured debt)

Marginal gain from increasing asset risk ∂G0 /∂α|s

Direct Indirect (via dE0 )

−p1 E0 (x1 − y1 )ds > 0 −p1 s(x1 − y1 ) dE0  


p1 (1/(s − 1))(E0∗ + U0 )(x1 − y1 )ds < 0 p1 (vs/(s − 1))(x1 − y1 ) dE0  


¯ 0 (x1 − y1 )ds > 0 −p1 E −p1 s(x1 − y1 ) dE0  


Marginal gain from increasing insured depositsa

Direct Indirect (via dα)

– –

−p1 (v/(s − 1)2 )(αx ¯ 1 + (1 − α)y ¯ 1 )ds ≤ 0 p1 (vs/(s − 1))(x1 − y1 ) dα
 

−p1 (αx ¯ 1 + (1 − α)y ¯ 1 )ds < 0 −p1 s(x1 − y1 ) dα
 

Asset risk decreases

Asset risk decreases if v = 0, asset risk may decrease or increase if v > 0

Asset risk may decrease or increase

>0

≤0

≤0

Total effect a

Bank with fixed equity: (dG0 /dE0 )|s,C0 , bank with variable equity: (dG0 /dE0 )|s,U0 .

>0

>0

S. Homölle / Journal of Economics and Business 56 (2004) 443–468

Table 2 Effects of a higher capital requirement (ds > 0)

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Result 3. A bank with insured deposits, subordinated uninsured debt, and fixed equity decreases asset risk as a response to a higher capital requirement if insured deposits do not shrink (see Appendix D). Whether an increase in insured deposits is actually optimal depends on the marginal gain from increasing insured deposits, dG0 /dE0 |s,C0 . Raising s directly induces a reduction in this marginal gain (see Table 2), which can be explained as follows. From the discussion of (14) we know that increasing insured deposits while reducing uninsured debt under compliance with the leverage constraint leads to a reduction in assets and thus in asset returns in state 1 so that the payments of the deposit insurer in state 1 increase. If the regulatory authority tightens up this leverage constraint, uninsured debt has to be decreased less than before to compensate for the increase in insured deposits: ∂(dU0 ) v dE0 > 0, = ∂s (s − 1)2

where dU0 = −

s−1+v dE0 , s−1

dE0 > 0. (23)

Asset size declines less if uninsured debt is substituted by insured deposits so that the payment of the deposit insurer in state 1 does not increase as much as before. The marginal gain from increasing insured deposits is reduced. Besides this direct effect of a change in s, the marginal gain from increasing insured deposits is influenced by the variation in asset risk. Since y1 > x1 , a decrease in the share of the riskier asset, α, has a positive impact on asset returns in state 1 (per dollar invested at time 0). Then the decline in assets induced by substituting insured deposits for uninsured debt leads to a higher reduction in total asset returns in state 1. The additional payment of the deposit insurer and finally the marginal gain from increasing insured deposits increase if the share of the riskier asset declines. In total, the marginal gain from increasing insured deposits rises if this indirect effect more than offsets the direct effect of raising s. Only under these circumstances the bank issues additional insured deposits until the increasing marginal cost, dKE /dE0 , again equals the marginal gain. However, if the more stringent capital requirement leads to a decline in insured deposits (and an increase in uninsured debt), it may be optimal for the bank to expand asset risk. Since the substitution of uninsured debt for insured deposits leads to an increase in assets, the marginal gain from increasing asset risk increases if this indirect effect (via the adjustments of insured deposits) dominates the opposing direct effect of changing s, which implies decreasing asset size. Whether increasing asset risk is actually optimal depends on the values of the parameters of our model (see also (22)), e.g. • the curvature of the profit function at the initial level of insured deposits; • the premium rate, v, and • the initial capital ratio, s. According to (22) the lower a, the more likely is an increase in asset risk because the indirect effect on the marginal gain from increasing asset risk, ∂G0 /∂α|s , gets stronger. a denotes the negative value of the second derivative of the profit function with respect to E0 at the initial level of insured deposits, E0∗ (see (21) and Appendix C). It reflects the curvature of the profit function. With a small a the strictly concave curve is almost flat. Provided that the new leverage constraint leads to a decrease in the marginal gain from increasing

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insured deposits, the bank reduces insured deposits (and increases uninsured debt). Since the profit function is almost flat, the reduction in insured deposits must be relatively large until the marginal gain from increasing insured deposits is again equal to zero thus fulfilling the optimum condition (20). The higher the decrease in insured deposits, the higher is the increase in uninsured debt and in assets so that it is more likely that, in total, asset size increases as well. Finally the marginal gain from increasing asset risk increases and the bank raises asset risk. A higher premium rate, v, also induces a stronger indirect effect. As ∂(dU0 ) 1 =− dE0 > 0, ∂v s−1

where dU0 = −

s−1+v dE0 , s−1

dE0 < 0.

a rise in v causes a higher increase in uninsured debt if insured deposits shrink. This leads to a more expanded asset size. Besides it is more likely that the bank raises asset risk if the prescribed lower bound of the capital ratio, s, is low. From (23) we know how the relationship between uninsured debt and insured deposits depends on s. If insured deposits are reduced and s is rather low, the increase in uninsured debt must be rather high thus leading to a significant increase in asset size. To sum up, for a bank holding both insured deposits and uninsured debt and paying a deposit insurance premium depending on its insured deposits it is not generally clear whether it decreases or increases asset size as a reaction to a more stringent capital requirement. Decreasing insured deposits while issuing uninsured debt by the same amount leaves the level of liabilities unchanged whereas there is more capital available to be invested in assets due to the decreasing deposit insurance premium. Therefore changing the structure of liabilities is another possibility to increase the ratio of assets to liabilities thus fulfilling the more stringent capital requirement. Because of this possibility, which is the more important, the higher the deposit insurance premium is (or other comparable cost of insured deposits), a bank need not sell assets to comply with this capital requirement even if it cannot raise additional equity. Instead it may vote for a substitution of uninsured debt for insured deposits thus increasing asset size. As asset size positively affects the marginal gain from increasing asset risk, it is then optimal to increase the share of the riskier security in the asset portfolio. 4.2. A bank with variable equity To analyse the reaction of a bank that can raise additional equity and issues insured deposits but no uninsured debt we use the first-order conditions (12) and (17). Additionally, ¯ 0 and α¯ are also fulfilled we assume that the second-order conditions for a maximum at E (see Appendix E). Inserting (13) (with U0 = 0) and (16) into (12) and (17) and totally differentiating lead us to

  ¯ 0 d2 KE2 + a¯ − p1 s(αx −p (x − y )( E ¯ 1 + (1 − α)y ¯ 1 )) 1 1 1 dα dE  2 α  2 0 E  = (24) d K ds + d K + a¯ − p2 s2 (x − y )2 dα2

dE0 2

1

1

1

 ¯ 0) + E ¯ 0 f −1 (E ¯ 0 )]. where a¯ = (p2 + p3 /p1 + p2 + p3 )[2f −1 (E

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The denominator of dα/ds is positive by assumption (see Appendix E) whereas the numerator is not obviously of a given sign (notice that d2 KE /dE0 2 + a¯ > 0 (see Appendix E)). It is remarkable that (24) is independent of the premium rate, v. Result 4. The reaction of a bank with insured deposits and variable equity to a higher capital requirement is independent of the deposit insurance premium. This result differs from the reaction of a bank with constant equity (see Result 2) because with variable equity there is no longer a fixed relationship between changes in insured deposits and uninsured debt, which automatically causes adjustments of asset size.16 Again we can differentiate between two effects of a more stringent leverage constraint on the marginal gain from increasing asset risk, ∂G0 /∂α|s (see also Table 2): one direct effect via the reduction in leverage and one indirect effect via the adjustment of insured deposits. Leaving insured deposits unchanged for the moment the bank has to raise equity and invest in new assets to comply with the new capital constraint. Using the same argument as above we can say that the marginal gain from raising asset risk increases due to the positive change in asset size. The bank tends to increase asset risk. This reaction is optimal if insured deposits do not decrease, which is (again) a feasible reaction—as distinct from the first case with fixed equity and no uninsured debt. Result 5. A bank with insured deposits and variable equity increases asset risk as reaction to a higher capital requirement if insured deposits do not shrink (see Appendix D). The optimal adjustment of insured deposits depends on the change in the marginal gain from increasing insured deposits (see (16)). This marginal gain decreases in s so that it seems to be optimal to decrease insured deposits as a reaction to a higher capital requirement. The intuition is straightforward. The first, relevant part of the marginal gain from increasing insured deposits is equal to the NPV of the deposit insurance guarantee per insured deposit (see (16)). An increase in s (= A0 /E0 ) implies that more assets per insured deposit must be hold thus increasing asset returns in state 1 per insured deposit. For this reason the payments of the deposit insurer per insured deposit and finally the NPV of the deposit insurance guarantee per insured deposit decrease. However, the marginal gain from increasing insured deposits is also influenced by the variation in asset risk. A rise in asset risk leads to a reduction in asset returns in state 1 (per insured deposit). Thus the first, direct effect of a rise in s on the marginal gain from increasing insured deposits and this indirect effect (via dα > 0) point in different directions (see also Table 2). Due to the interdependence of the optimal adjustments of insured deposits and asset risk and due to the possible increase in insured deposits, the reaction of a bank with variable equity and insured deposits to a tightened capital requirement is not as clear as for a bank with fixed equity and no uninsured debt. As opposed to such a bank, a bank with variable equity can raise insured deposits even though the capital constraint is tightened. To achieve the required reduction in leverage issuing new equity must overcompensate for the increase The independence of dα/ds from v would still hold if we assumed that the bank with variable equity also held subordinated uninsured debt. 16

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in liabilities. The statement of FK89 that a bank which can issue new equity “would not hold more assets when required to reduce leverage” cannot be confirmed in our extended analytical framework. Without knowing the actual data of the bank, it is by no means clear whether the bank sells assets and reduces insured deposits or invests in new assets (mainly) financed by equity. Therefore the direction of the change in assets is ambiguous so that the question whether the marginal gain from increasing asset risk, ∂G0 /∂α|s , increases or decreases (or remains constant) cannot be answered in general. Looking at (24) we can identify some parameters which influence the sign of dα/ds, e.g. • the curvature of the profit function at the initial level of insured deposits and • the initial capital ratio, s. The sign of dα/ds tends to be positive if the curvature of the profit function at the initial ¯ 0 , is very strong, i.e. the second derivative of the profit function volume of insured deposits, E (including regulatory cost KE ) with respect to E0 , [−d2 KE /dE0 2 + α], ¯ is very low (see (24) and Appendix E). Then the bank does not have to alter the insured deposits very much to maintain the optimum condition (17). Therefore the indirect effect on the marginal gain from increasing asset risk is quite small. Because of the dominating direct effect of increasing s, the marginal gain from increasing asset risk rises and it is optimal to increase the share of the riskier security in the asset portfolio. If the initial leverage constraint is not too strict, which means that the ratio of assets to insured deposits, s, is rather low, a change in insured deposits does not have a great impact on asset size. Once again the direct effect of a change in s on the marginal gain from increasing asset risk is the dominating effect. Taking into account all the possible effects of a tightened capital requirement we cannot confirm the conclusion of FK89 that a bank which can issue new equity does not react with increased risk-taking to the new capital requirement. At least in some situations the possibility of rising asset risk may not be denied.

5. Conclusion In this paper we have analysed the impact of a higher capital requirement on bank risk-taking. Using a state-preference model we have identified at least three points which are worth emphasizing because they are crucial for the answer to this problem. First and most importantly, in our model the profit-maximising bank can issue not only insured deposits but also subordinated uninsured debt. The issuance of uninsured debt heavily influences the reaction of the bank to a more stringent capital requirement. A bank which only holds a fixed amount of equity and insured deposits decreases asset risk. However, this clear result cannot generally be obtained if a bank holds both, insured and subordinated uninsured liabilities, or if a bank can issue new equity. In these cases the bank increases asset risk under certain conditions. An important reason for this deviation from the result of FK89 is that a bank with different kinds of liabilities has more possibilities to react to a tightened capital requirement. Including other than insured liabilities allows us to cut the interdependence between changes in insured deposits and in asset size, which plays a crucial role in the model of FK89.

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Second, the model includes assumptions about perfect as well as imperfect markets for liabilities. We showed under which conditions it is reasonable to deviate from the usual assumption of a bank with only equity and insured deposits. With imperfections on the market for insured deposits, a regulated bank with fixed equity may not only hold insured deposits but also issue subordinated uninsured debt. In this paper we refrain from modelling imperfections on the market for uninsured debt because it would hardly change our main results. If we assumed such imperfections, it might even be optimal for a regulated bank with variable equity to issue subordinated uninsured debt. In this case the reaction to a higher capital requirement would be ambiguous as well. Finally and perhaps most surprisingly, the question whether a subsidised deposit insurance charges a fixed-rate premium (depending on the amount of insured deposits) or does not charge a premium at all is sometimes essential for our results. Therefore the usual assumption of free deposit insurance is not as innocuous as its wide use may seem to suggest. For example, a bank with fixed equity that issues insured deposits and subordinated uninsured debt decreases the risk of its assets if deposit insurance is free, but with a premium that rises proportionally to the insured deposits the reaction is not clear. The fact that not all capital raised from insured depositors can be invested in assets due to the deposit insurance premium whereas uninsured debt is fully available for investment plays a crucial role for these results. In this paper we have extended the framework of FK89. Nevertheless, the model is still quite simplifying. Some other variations may be useful. For example, the assumption that insured deposits are senior to uninsured debt is dropped in a companion paper. If insured and uninsured liabilities are both of equal seniority, the response of a bank to an enforcement of a more stringent capital requirement is ambiguous as well. The asset-substitution effect is only one effect of a more stringent bank capital regulation. If a bank increases asset risk, one can ask whether this effect exceeds the effect of decreased leverage so that insolvency risk is higher than before. To solve this problem within our analytical framework, it must be examined whether the number of insolvency states tends to vary because of the changing structure of bank capital and assets. But that is still another question left for further research. Acknowledgments I thank Andreas Pfingsten, Franz Hubert, and the participants of the 61st Scientific Conference of the Association of University Professors of Management and the 6th Annual Meeting of the German Finance Association for helpful comments on an earlier version of this paper. Appendix A. Net present value of the deposit insurance subsidy I0PV − I0  = p1

 1 E0 f −1 (E0 ) − A0 (αx1 + (1 − α)y1 ) p1 + p 2 + p 3   1 A0 −vE0 > 0 ⇔ p1 f −1 (E0 ) − (αx1 + (1 − α)y1 ) − v > 0. p1 + p 2 + p 3 E0

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It follows



(a) p1



1  (f −1 (E0 ) + E0 f −1 (E0 )) − (1 − v)(αx1 + (1 − α)y1 ) p1 + p 2 + p 3 −v > 0 ⇔

 (b)

p1

∂(I0PV − I0 ) > 0, ∂E0 

1 vs  (αx1 + (1 − α)y1 ) (f −1 (E0 ) + E0 f −1 (E0 )) + p1 + p 2 + p 3 s+1  d(I0PV − I0 )  −v > 0 ⇔ > 0,   dE0 s,C0

 (c) p1



1  (f −1 (E0 ) + E0 f −1 (E0 )) + s(αx1 + (1 − α)y1 ) p1 + p 2 + p 3  d(I0PV − I0 )  −v > 0 ⇔ > 0,   dE0 s,U0

where 

f −1 (E0 ) > 0, 1 > v ≥ 0, s = A0 /(E0 + U0 ), E0 > 0, U0 ≥ 0.

Appendix B. Demand equations First example f(pe) =

1 − 1, (1 − pe)n

n ≥ 1,

where n n(n + 1) > 0, f  (pe) = > 0, (1 − pe)n+1 (1 − pe)n+2 f(0) = 0, lim f(pe) = ∞. pe→1

f  (pe) =

The inverse function f−1 (E0 ) and its first and second derivatives can be written as:

((1+n)/n) 1 1 1 n −1 −1 f (E0 ) = 1 − , , f (E0 ) = E0 + 1 n E0 + 1 ((1+2n)/n)

1 1+n −1 f (E0 ) = − 2 . E0 + 1 n If this demand equation holds, the NPV of insured deposits, E0 (l − f−1 (E0 )), is a strictly increasing and concave function of E0 ≥ 0:

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463



1−f

−1

(E0 ) − E0 f



−1

((1+n)/n) 1 1 1 − E0 E0 + 1 n E0 + 1 (1/n) 

1 E0 1 1− > 0, = E0 + 1 n E0 + 1

(E0 ) =

n



−2f −1 (E0 ) − E0 f −1 (E0 )

((1+n)/n) ((1+2n)/n)

1 1+n 1 2 + E0 2 =− n E0 + 1 E0 + 1 n

((1+n)/n)  1 + n E0 1 1 2− =− < 0 q.e.d. n E0 + 1 n E0 + 1 Second example f(pe) = |ln(1 − pe)|n ,

n ≥ 1,

where n |ln(1 − pe)|n−1 > 0, (1 − pe) n f  (pe) = |ln(1 − pe)|n−2 ((n − 1) + |ln(1 − pe)|) > 0, (1 − pe)2 lim f(pe) = ∞. pe→1

f  (pe) =

f(0) = 0,

The inverse function f−1 (E0 ) and its first and second derivatives: f −1 (E0 ) = 1 − e− 

f −1 (E0 ) =

√ nE 0

,



f −1 (E0 ) =

1 −√ n (1−n)/n e E0 E0 , n

1 −√ n (1−2n)/n 1/n e E0 E0 (1 − n − E0 ). 2 n

If this demand equation holds, the NPV of insured deposits reaches its absolute maximum at E0 = nn . First-order condition: 

1 − f −1 (E0 ) − E0 f −1 (E0 ) = 0,

1 √ n 1/n − e− E0 (n − E0 ) = 0 ⇔ E0 = nn . n

Second-order condition: 



−2f −1 (E0 ) − E0 f −1 (E0 ) < 0, where E0 = nn , √ 1 1 n (1−n)/n 1/n − 2 e− E0 E0 (n + 1 − E0 ) = − n+1 n < 0 q.e.d. n n e

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Appendix C. Optimisation problem of regulated banks with fixed equity (and subordinated uninsured debt) Maximise GK 0 = p1





1 E0 f −1 (E0 ) − A0 (αx1 + (1 − α)y1 ) p1 + p 2 + p 3

−vE0 + E0 (1 − f −1 (E0 )) − Kα subject to s =

A0 . E0 + U 0

¯ First-order conditions for a maximum at E0∗ and α:   ∂GK 1 vs 0 = p1 f −1 (E0∗ ) + ¯ 1) (αx ¯ 1 + (1 − α)y ∂E0 p1 + p 2 + p 3 s−1  p 2 + p3 dG0   −v + 1 − f −1 (E0∗ ) − E0∗ f −1 (E0∗ ) = 0 ⇔ = 0, p1 + p 2 + p 3 dE0 s,C0  ∂GK dKα dKα ∂G0  0 ∗ = = −p1 s(E0 + U0 )(x1 − y1 ) − =0⇔ . ∂α dα ∂α s dα Second-order conditions: p2 + p3   0>− [2f −1 (E0∗ ) + E0 f −1 (E0∗ )], p1 + p 2 + p 3 d 2 K α p2 + p 3   0< [2f −1 (E0∗ ) + E0 f −1 (E0∗ )] − p1 2 dα2 p1 + p2 + p3

vs s−1

2 (x1 − y1 )2 .

Appendix D. Relation between variations of insured deposits and asset risk Optimal variations in the share of the riskier asset, α, and of insured deposits, E0 , when s is raised:    

∂2 GK ∂2 GK ∂2 GK ∂2 GK 0 0 0 0 − ∂E0 ∂s ∂E0 ∂α ∂α∂s ∂E0 2 dα = ,   2

ds 2 GK 2 GK ∂2 GK ∂ ∂ 0 0 0 − ∂E0 ∂α ∂α2 ∂E 2 0

dE0 = ds

∂2 GK 0 ∂α∂s



∂2 GK 0 ∂α2

∂2 GK 0 ∂E0 ∂α





−

∂2 GK 0 ∂E0 2



where ∂ 2 GK 0 < 0, ∂α2

∂ 2 GK 0 < 0, ∂E0 2

∂2 GK 0 ∂E0 ∂s

−



∂2 GK 0 ∂E0 ∂α

∂2 GK 0 ∂α2

2

 ,

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2 K ∂ 2 GK 0 ∂ G0 − ∂α2 ∂E0 2



∂ 2 GK 0 ∂E0 ∂α

2 >0⇔

∂2 GK 0 ∂E0 ∂α

465

2

∂ 2 GK 0 < K . ∂α2 ∂2 G0 ∂E0 2

A bank with fixed equity: dα dE0 ≥0⇒ < 0. ds ds Proof: dE0 ≥0 ds ⇔

2 K 2 K ∂ 2 GK ∂ 2 GK 0 ∂ G0 0 ∂ G0 ≥0 − ∂α∂s ∂E0 ∂α ∂E0 ∂s ∂α2 2

∂2 GK 0

2 K ∂E0 ∂α ∂ 2 GK ∂ 2 GK 0 ∂ G0 0  ≥0 − ⇒ ∂α∂s ∂E0 ∂α ∂E0 ∂s ∂2 GK0 ∂E0 2

⇔

∂2 GK 0 ∂E0 ∂α ∂2 GK 0 ∂E0 2

 



2 K 2 K ∂ 2 GK ∂ 2 GK 0 ∂ G0 0 ∂ G0 − 2 ∂α∂s ∂E0 ∂E0 ∂s ∂E0 ∂α

⇒

2 K 2 K ∂ 2 GK ∂ 2 GK 0 ∂ G0 0 ∂ G0 − > 0, ∂α∂s ∂E0 2 ∂E0 ∂s ∂E0 2 ∂α

⇔

2 K 2 K ∂ 2 GK ∂ 2 GK 0 ∂ G0 0 ∂ G0 <0 − ∂E0 ∂s ∂E0 ∂α ∂α∂s ∂E0 2

⇔

dα < 0 q.e.d. ds

 ≥0

where

A bank with variable equity (and no uninsured debt): dα dE0 ≥0⇒ > 0. ds ds Proof: dE0 ≥0 ds ⇔

2 K 2 K ∂ 2 GK ∂ 2 GK 0 ∂ G0 0 ∂ G0 ≥0 − ∂α∂s ∂E0 ∂α ∂E0 ∂s ∂α2

∂ 2 GK 0 < 0, ∂α∂s

∂ 2 GK 0 ≤0 ∂E0 ∂α

䊐

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⇒

∂2 GK 0 ∂E0 ∂α

2

2 K ∂ 2 GK ∂ 2 GK 0 ∂ G0 0  > 0, − ∂α∂s ∂E0 ∂α ∂E0 ∂s ∂2 GK0

where

∂ 2 GK 0 <0 ∂E0 ∂s

∂E0 2

⇔

∂2 GK 0 ∂E0 ∂α ∂2 GK 0 ∂E0 2







2 K 2 K ∂ 2 GK ∂ 2 GK 0 ∂ G0 0 ∂ G0 − ∂α∂s ∂E0 2 ∂E0 ∂s ∂E0 ∂α

⇒

2 K 2 K ∂ 2 GK ∂ 2 GK 0 ∂ G0 0 ∂ G0 > 0, − ∂E0 ∂s ∂E0 ∂α ∂α∂s ∂E0 2

⇔

dα > 0 q.e.d. ds

 >0

where

∂ 2 GK 0 >0 ∂E0 ∂α 䊐

Appendix E. Optimisation problem of regulated banks with variable equity Maximise GK 0

 = p1

−vE0 + E0 (1 − f −1 (E0 )) − Kα − KE subject to s =



1 E0 f −1 (E0 ) − A0 (αx1 + (1 − α)y1 ) p1 + p 2 + p 3

A0 . E0 + U 0

Notice: U0∗ = 0 because ∂GK 0 = −p1 s(αx1 + (1 − α)y1 ) < 0. ∂U0 ¯ 0 and α: First-order conditions for a maximum at E ¯   K ∂G0 1 ¯ 0 ) − s(αx = p1 f −1 (E ¯ 1 + (1 − α)y ¯ 1) ∂E0 p1 + p 2 + p 3 p 2 + p3 dKE ¯ 0) − ¯ 0) − ¯ 0 f −1 (E −v + 1 − f −1 (E E p1 + p 2 + p 3 dE0  E dG0  dK =0 ⇔ = . dE0 s,U0 dE0  ∂GK ∂G0  dKα dKα 0 ¯ 0 (x1 − y1 ) − =0⇔ . = = −p1 sE  dα ∂α s dα ∂α Second-order conditions  2 E  d K p 2 + p3 −1 ¯ −1 ¯ ¯ 0>− + [2f ( E ) + E f ( E )] , 0 0 0 p1 + p 2 + p 3 dE0 2

S. Homölle / Journal of Economics and Business 56 (2004) 443–468

0<

d2 K α dα2



467



d2 K E p 2 + p3  ¯ 0) + E ¯ 0 f −1 (E ¯ 0 )] + [2f −1 (E p1 + p 2 + p 3 dE0 2

−p1 2 s2 (x1 − y1 )2 .

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