Can capital requirements induce private monitoring that is socially optimal?

Journal of Financial Stability 8 (2012) 252–262

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Journal of Financial Stability journal homepage: www.elsevier.com/locate/jfstabil

Can capital requirements induce private monitoring that is socially optimal? Kenneth J. Kopecky a,∗ , David VanHoose b a b

Richard J. Fox School of Business and Management, Temple University, Philadelphia, PA 19122, United States Hankamer School of Business, Baylor University, One Bear Place #98003, Waco, TX 76798, United States

a r t i c l e

i n f o

Article history: Received 16 November 2009 Received in revised form 16 December 2011 Accepted 24 February 2012 Available online 16 March 2012 JEL classification: G28 Keywords: Capital requirements Monitoring Social optimum

a b s t r a c t This paper develops a framework for analyzing socially and privately optimal bank loan-monitoring decisions, with and without capital regulation. In contrast to the monitoring decision of a social planner who seeks to maximize the utility of aggregate consumption, banks choose to monitor only if doing so is consistent with maximizing the market value of equity. As a consequence, socially and privately optimal monitoring choices can diverge. Under some circumstances, appropriately configured capital regulation can bring private loan-monitoring decisions into line with those of the social planner. Nevertheless, the capital ratio required to attain this outcome hinges on a number of factors that are likely to be economyspecific, including the banking system’s monitoring technology and its exposure to default. Thus, it is unlikely that a unique capital ratio will be able to induce socially optimal monitoring in all economies. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Previous research has examined the effect of capital requirements on the monitoring decision of private banks. Missing from the analysis to date, however, is a clear identification of the fundamental factors that a regulator must balance if it wishes to apply capital requirements in an attempt to generate a socially desired level of monitoring. This is the issue addressed in our paper. A review of the literature indicates that the relationship between monitoring and capital requirements has an ambiguous sign. On the negative side, Boot and Greenbaum (1993) argue that a higher capital ratio reduces monitoring incentives because it dilutes ownership. A similar argument is made by Besanko and Kanatas (1996) in their analysis of the interactions among bank insiders, bank outsiders, depositors, and a bank regulator. They conclude that meeting capital requirements induces banks to issue more equity shares to outsiders, which decreases the insiders’ incentives to engage in effective monitoring. Decamps et al. (2004), however, reach a contrary conclusion in their analysis of the interplay between capital regulation and bank monitoring. Their model focuses on continuous-time cash flows and derives the implied market and book values of a representative bank, with

∗ Corresponding author. E-mail addresses: [email protected] (K.J. Kopecky), David [email protected] (D. VanHoose). 1572-3089/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.jfs.2012.02.001

and without capital requirements. They conclude that as long as a regulator imposes Basel II-style rules requiring banks to issue subordinated debt and to be subjected to supervisory audits, a capital requirement can allow the regulator to determine when a bank’s monitoring cost is sufficiently high relative to its cash flow to induce it to monitor less effectively – and hence decide when closure of the bank is appropriate. Recent theoretical studies have broadened the scope of the analysis by investigating how the competitive environment within the loan market affects the relation between capital requirements and monitoring. Almazan (2002) examines a spatial model of bilateral bank competition in which two banks must balance their capital conditions vis-à-vis their monitoring expertise as proxied by the distance of the banks from borrowers. He obtains mixed results regarding the interplay between capital conditions and monitoring that can be summarized by four alternative equilibrium outcomes: scarce capital with no competitive interactions between the two banks, plentiful capital but with no monitoring by either bank, a case in which one bank monitors and the other does not, and a case in which both banks monitor but the bank endowed with more capital obtains a higher market share and benefits more from an expansionary monetary policy that reduces the riskless interest rate. At the other end of the competitive structure, Kopecky and VanHoose (2006) consider a perfectly competitive banking system composed of banks facing heterogeneous monitoring costs. They show that capital requirements that bind a portion of banks also raise market loan rates and thereby reduce the incentives for

K.J. Kopecky, D. VanHoose / Journal of Financial Stability 8 (2012) 252–262

some banks to monitor their loans. The consequence is that even though capital regulation reduces the overall amount of risky lending, it also reduces loan monitoring and thus loan quality. Whereas Kopecky and VanHoose examine a static banking industry, Boot and Marinˇc (2006) examine a setting in which the long-run structure of the banking industry varies as a consequence of bank entry or exit. They show that within this environment, capital regulation can eventually weed out the weakest banks. Nevertheless, at intermediate levels of bank-monitoring quality and sufficiently high degrees of competition, a banking system with open entry could experience a reduction in monitoring incentives. Thus, Boot and Marinˇc’s analysis also suggests a net ambiguous effect of capital regulation on aggregate loan quality.1 Clearly, there has been progress in developing an understanding of the various ways in which capital regulation impinges on bank monitoring decisions. Implicit in the above research, however, is the assumption that monitoring is a desirable ‘social’ activity and should be promoted by the government. The objective of this paper is to provide a critical assessment of this implicit assumption. To model the ‘social’ aspects of monitoring and capital requirements, we present and analyze two models of the lending process. The first model describes the decision-making of a social planner whose objective is to maximize the utility of aggregate consumption. The social planner provides a predetermined amount of lending to borrowers and faces both loan default and a rising marginal cost of monitoring loans. Within this framework, the planner must decide whether it is optimal to monitor the outstanding volume of loans. The second model describes the decision-making of a private banking system that is overseen by a government authority. The government has two policy instruments that it can target to influence the banking system: an interest rate on government securities and a capital requirement ratio on bank loans. We assume that the objective of the government is to emulate both the loan quantity and the optimal monitoring stance of the social planner. These two models create a framework for analyzing capital requirements that is broader than that assumed in the extant literature. Since a capital requirement is a means to an end but not an end in itself, we are able to derive explicit conditions that must be satisfied for the government to employ capital requirements as an appropriate tool for altering bank behavior. As we show below, implementing a binding capital requirement is not always an optimal policy choice. There are several important assumptions that enable us to explicitly solve the two models. First, we assume perfect certainty. Thus, the social planner, government and private banking system have complete knowledge of the effects of monitoring on borrower behavior. Although a perfect-certainty assumption constitutes a limiting case, our analysis provides meaningful insights into the qualitative comparisons between public and private decision-making and enables us to assess the important factors that determine the optimal capital requirement structure. Second, in the private bank model loan demand is specified as an ad hoc function of the loan rate. This specification is required to derive a functional representation for the targeted government security rate. The third assumption involves the funding of the private banks. Since the imposition of capital requirements

1 Consistent with these theoretical conclusions, recent empirical research finds evidence of considerable heterogeneity in bank risk responses to capital requirements (see Jokipii and Milne, 2011; Delis et al., forthcoming) and of a minimal worldwide response of measures of bank risk to generalized adherence to Basel’s supervisory principles (see Demirgüc¸-Kunt and Detragiache, 2011). For a broad review of the literature on capital regulation, see Santos (2001) and VanHoose (2007).

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necessarily involves decision-making regarding the quantity of equity, we assume that equity is endogenous and, for simplicity, that deposits are exogenous. Equity endogeneity is of course assumed in the models of Boot and Greenbaum (1993) and Besanko and Kanatas (1996). Their focus, however, is on the equity conflict that emerges when capital requirements are imposed unexpectedly. In our model capital requirements are perfectly foreseen, so a conflict between old and new owners is not an issue. Our findings are as follows: There is a broad range of values for the parameters defining the loan default rate and the marginal cost of monitoring for which capital requirements are not necessary from the perspective of a social planner. In economies characterized by these parameter values, the banks endogenously adopt the monitoring stance that is socially optimal as long as the government’s setting of the security rate correctly targets the same loan quantity as that in the social planner model. However, if the values of the loan default rate and monitoring cost parameters lie outside this range, the economy’s banks will implement a socially suboptimal monitoring position even though the government has set the security rate correctly. We show that, within this narrower parameter set, in some cases a binding capital ratio can reverse the bank’s suboptimal monitoring decision but, in other cases, binding capital requirements will fail to alter the banks’ suboptimal decision. Thus, when we use the social planner model to guide the government/private bank model, we find that in economies differentiated according to the values of the loan default rate and monitoring cost parameters, monitoring per se is not always ‘desired’ and capital requirements can be either irrelevant, needed but ineffective, or needed and highly efficacious. The outline of the paper is as follows: The social planner and government/private bank models are presented in Sections 2 and 3, respectively. Section 4 examines several cases dealing with the convergence or divergence between private bank monitoring and socially optimal monitoring. The efficacy of a binding capital requirement is analyzed in each case. Most of the mathematical details are provided in Appendix A. Section 5 concludes the paper and also offers a brief discussion of the direction for future research. 2. The social planner model How might a social planner go about deciding the appropriate level of loan monitoring? To address this question, we consider a stark economy in which a (hypothetical) social planner uses banklike lending to finance the consumption-augmenting and highly idiosyncratic projects of the economy’s borrowers. The social planner is the only source of lending, which is used to produce aggregate consumption, C, according to the following production function, C = [1 − ı(1 − M)]L( − 1) −

c f ML2 − L2 2 2

(1)

where L denotes total lending, M is an indicator variable with M = 0 when loans are not monitored and M = 1 when monitoring occurs, ı represents an assumed certain deadweight loss per dollar of loans arising from default on unmonitored loans,  − 1 represents the net productivity of non-defaulted loans, c is a parameter that determines the magnitude of loan monitoring costs, and f is a parameter that captures all of the remaining costs associated with creating and managing the social planner’s loan portfolio. Since our focus is not on the production-function aspects of lending but on the consumption productivity of monitoring loans, we set  = 2, for simplicity. We also assume that when monitoring occurs (M = 1), the loan monitoring technology available to society is quadratic in the level of loans monitored and equals (c/2)ML2 . This simplification ensures that the marginal cost of a monitored dollar of loans, given by cL, is positive. There are of course alternative monitoring specifications,

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some of which will lead to a more sophisticated outcome for the social planner.2 Our focus, however, is on the convergence of outcomes between the social planner and the government-influenced private banking system. While the quantitative aspects of this convergence process are likely to depend on the shape of the assumed monitoring cost function, our analysis provides answers to the key qualitative questions regarding whether convergence occurs and if not, whether the government can create convergence. The monitoring cost parameter, c, almost certainly varies across economies due to differing monitoring cost structures arising from country-specific factors associated with divergent endowments of capital equipment, data analysis capabilities, and managerial skill levels. In the very long run these differences may disappear. Within the time frame of our model, however, we assume that superior technological and managerial monitoring skills are not transferable across economies. To recognize that lending involves non-monitoring-related costs, we assume that these costs are also quadratic and equal (f/2)L2 . Eq. (1) indicates that if M = 0, so that no loans are monitored, each dollar of lending is subject to a per-dollar deadweight loss ı and thereby yields the aggregate level of lending-generated consumption, C = (1 − ı)L − (f/2)L2 . For the sake of expositional simplicity, we assume that when M = 1 (monitoring takes place), each dollar of monitored lending is not subject to any loss; hence (1 − ı(1 − M))L = L and the entire amount of lending is potentially available to support consumption. Monitoring costs, however, divert resources away from providing for borrowers’ consumption, so that if M = 1, total consumption is equal to aggregate bank loans, L, less the sum of monitoring and loan-servicing costs, (c/2)L2 + (f/2)L2 . Social welfare is derived from consumption that is facilitated by bank-like loans and is given by U = u(C);

u (C) > 0,

u (C) < 0

(2)

Substituting (1) into (2) yields U = u([(1 − ı(1 − M)]L −

c f ML2 − L2 ) 2 2

(3)

As discussed above, when M = 0 in (3), there is no monitoring of loans, and the quantity ıL is a deadweight-loss reduction in private consumption. When M = 1, monitoring eliminates this loss. Although more elaborate specifications of the relationship between ı and M could change the consumption pattern when monitoring occurs,3 (3) captures the essential idea that monitoring is beneficial to the economy in a very specific manner. In particular, the social value of monitoring lies in its ability to curtail the wasteful diversion of loans that reduces aggregate consumption. Since monitoring decisions are the focus of the paper, we assume that the social planner’s targeted level of L is predetermined

2 In (1), C is a linear function of M. When L is fixed (as it is below), monitoring costs are also fixed and M then indicates whether monitoring is socially optimal. A plausible extension is to assume that C is quadratic in M, which would lead to an interior solution for M where a fraction of loans is monitored. See footnote 4 for a further explanation. However, the bank model in Section 3 is essentially a fixed monitoring cost model. Thus a comparison between a social planner model with fractional loan monitoring and a fixed-cost private banking model would not be informative because an all-or-nothing private monitoring decision could never analytically replicate a fractional social monitoring decision. 3 ı can also be a negative function of M. For example, if the fraction of monitored loans were to increase, there may be an announcement effect that induces all borrowers to extend greater effort in managing their projects. Nevertheless, our focus is on a basic rationale for social monitoring rather than on a complete investigation, so we have opted for the simplest representation of the relationship between ı and M.

(L = L¯ ) and analyze only the planner’s M decision.4 Differentiating (3) with respect to M yields ∂U = u ( ) ∂M



ı−

c L¯ 2



L¯

(4)

Given the linearity of M in (1), (4) does not yield an interior solution, which implies that M may go to a limit of either 0 or 1 or may be indeterminate. Specifically, since u ( ) > 0, M = 1,

if

ı L¯ > , c 2

ı L¯ = , and M is indeterminate if c 2 L¯ ı < . M = 0, if c 2

(5)

According to (5), monitoring is socially optimal (M = 1) as long as the benefit from monitoring – the default loss per dollar of loans, ı, which full monitoring eliminates – exceeds the (now) fixed cost of monitoring, (c/2)L¯ , per dollar of loans. If the contrary inequality holds, choosing not to monitor loans is socially optimal. Although this framework is highly stylized, it captures the fundamental trade-off that a social planner faces when contemplating a desired degree of loan monitoring. On the one hand, an increased extent of loan monitoring reduces deadweight losses that are otherwise produced by borrowers who engage in activities that lead to loan default. This reduction in turn raises the amount of aggregate consumption generated by lending. On the other hand, a greater degree of loan monitoring expends resources that otherwise could be allocated to consumption. These results suggest that in economies in which the public debt market is immature and thus bank lending plays a quantitatively significant role in funding consumption-augmenting investments, (per capita) L can be sufficiently large, in relation to given values of ı and c, so that M = 0 might be the optimal solution to the social planner’s problem. Of course, these economies may also be characterized by a very high value of ı, so that monitoring may turn out to be socially optimal even when loans are quantitatively large. The role of the model is to point out that the socially optimal decision depends on the relationship among several parameters that describe the benefit-cost trade-off to monitoring loans.5 3. The private banking/government model The social planning model is a useful tool for understanding socially optimal monitoring decisions. Nevertheless, in a

4 L could be treated as an endogenous variable in a more sophisticated social planner model. Define the resulting L as LEND . However, as discussed below, the government in the private bank model would still view LEND as an exogenous variable because the government’s role is to replicate the social planner’s outcomes, so it takes these outcomes as predetermined. 5 To obtain an interior solution for M, we reinterpret M as the fraction of loans that are monitored and assume that the loan monitoring technology available to society 2 is quadratic in M, so that c/2(ML) . Then aggregate consumption becomes C = [1 − ı(1 − M)]L − (c/2)M 2 L2 , yielding the optimal solution, M ∗ = ı/cL. Here the optimal fraction of monitored loans is determined by the ratio of the marginal per-dollar default loss associated with unmonitored loans to the marginal resource expense arising from loan monitoring. Complete loan monitoring, M ∗ = 1, occurs only when ı ≥ cL, a condition that is almost identical to that in (5). However, when ı < cL, M ∗ < 1, and the socially optimal value of M* diminishes in response either to a decrease in the value of ı or increases in the values of either c or L, both of which are associated with increased aggregate resource costs of loan monitoring. As in the simpler setting in the text, in which the social planner makes a choice of either M = 0 or M = 1, the solution in the quadratic M model suggests that when bank loans play a quantitatively significant role in funding consumption-augmenting investments, (per capita) L may be sufficiently large, in relation to given values of ı and c, so that M ∗ < 1 solves the social planner’s problem. Incomplete rather than zero monitoring would then be socially optimal.

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decentralized economy, private banks monitor loans, not the social planner. Although a private banking system focuses solely on maximizing the value of equity and is not interested in socially optimal decisions, we assume that the economy’s government has the same objective as the social planner, possesses complete knowledge of the social optimum, and has a set of tools that can be used to influence the decisions of the private banks. Our goal is to analyze whether the government can devise policies that will induce the banking system to emulate both the loan quantity and monitoring decision of the social planner. When this convergence occurs, private monitoring is socially efficient; otherwise, private monitoring is wasteful from a social perspective. This section presents an elementary model of a competitive banking system and derives loan supply functions for the two cases of monitored and unmonitored loans. In both cases we examine the effect of the government’s imposition of a binding capital ratio on the banks’ loan quantity and monitoring decisions. As in the previous section, we assume a certainty environment, which rules out the case of bank insolvency because otherwise the presentday market value of equity would be zero and the banking system would not be funded in the private equity market.6 We then use the alternative specifications of the banking system’s loan supply function to determine the government’s optimal settings of the security rate. The analysis of the government’s success, or lack thereof, in emulating the social planner is conducted in Section 4. 3.1. The banking model We describe the behavior of a representative bank at two dates (t = 0, 1).7 At date 0, the bank offers equity shares to the public giving rise to the equity cash flow, EBK . Since the bank ‘represents’ the entire banking system, we need to impose conditions on the bank’s equity financing that are consistent with industry-wide constraints. From an industry perspective, equity will flow into the banking system in the form of newly created competitive banks as long as the market value of bank equity exceeds the book value of equity. The equity component of the banking system and thus of the representative bank becomes determinate only when the market and book values of equity are identical, which implies that Tobin’s Q = 1.8 This condition is, of course, a long-run competitive equilibrium condition. Since our model has only two dates, however, we explicitly assume that both the initial and long-run equity financing of the representative bank and the banking industry occur at date 0. Furthermore, at date 0, we also assume that the representative bank acquires an exogenous amount of one-period deposits, D, paying the default-free deposit rate, rD .9 The sum of exogenous D and endogenous EBK then determines the scale of the bank. The

6 Although insolvency, which is usually viewed as a ‘bad’ event because of its implicit destruction of consumption, is ruled out in our model, the present analysis deepens our understanding of the relation between social welfare and the imposition of capital requirements. Whether a stochastic framework involving bankruptcy would alter the thrust of our conclusions is a topic for ongoing research. 7 If we were to add a source of bank heterogeneity, along lines similar to Kopecky and VanHoose (2006), it would be possible to determine equilibrium shares of monitoring and non-monitoring banks. An interior solution to the social planning problem such as the one discussed in Footnote 5 could then be compared with the equilibrium shares desired by private banks. The key factors governing this comparison, however, would necessarily involve the same elements that are identified in the simpler framework of this paper. 8 Tobin’s Q = 1 implies that long-run excess bank profits are zero and thus that the rate of return on bank stock is equal to its required return. This condition ensures that the long-run net present value of an investment in bank equity is zero. 9 Bankruptcy cannot occur and thus the deposit rate is default-free. Note also that the model could be generalized to include endogenous deposits, but the resulting model becomes highly non-linear.

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bank converts both sources of funds into one-period government securities, G, paying a risk-free interest rate rG , and one-period loans L, at the loan rate, rL .10 Since deposits are not the crucial funding source in our model, we set rD = rG for simplicity. This implies that bank deposits and securities are perfect substitutes from the depositor’s point of view. At date 1, the bank receives the gross amount of loan and securities revenues, pays off its gross cost to depositors, and distributes the net proceeds to its equity owners. Under perfect certainty, the gross revenues and costs at date 1 are foreseen and discounted back to date 0, which establishes the market value of the bank, E0 . Thus, E0 =

ˆ1  , 1+ˇ

(6)

ˆ 1 represents the net cash flow at date 1 (defined below) and where  ˇ is the discount rate.11 The representative bank’s maximization of E0 in (6) at date 0 is subject to alternative sets of the following constraints, depending on whether binding capital requirements are present: Balance sheet : G + L = D + EBK :

t=0

(7)

Capital requirement : EBK ≥ L :

t=0

(8)

where  ≡ the capital requirement ratio. Eq. (7) defines the balance sheet constraint of the bank. Eq. (8) specifies a capital requirement on loans that is defined in relation to the amount of book equity. Since there are two assets that the bank can hold, (8) explicitly assumes that differential capital requirements are in force. That is, the capital ratio against government securities, G, is set equal to zero. When the capital requirement in (8) binds the bank’s decision making, we also assume that it is known at the date when the bank raises its equity capital. This assumption allows us to investigate the extent to which capital requirements alter the endogenous loan quantity and monitoring decisions of the bank. ˆ 1, The net cash flow to equity owners at date 1, denoted  depends on whether the bank endogenously chooses to monitor. Given our assumption that rD = rG under perfect foresight, the net cash flow of a monitoring (M) bank is ˆ 1M = (1 + rL )L + (1 + rG )(G − D) − 

f + c 2

L2 ;

t=1

(9.1)

If the bank chooses to be a non-monitoring (NM) institution, its net cash flow is ˆ 1NM = (1 − ı)(1 + rL )L + (1 + rG )[G − D] − 

f 2 L ; 2

t=1

(9.2)

In (9.1) and (9.2), the parameter f governs the magnitude of quadratic resource costs of lending that are unrelated to monitoring.12 Note that (9.1) and (9.2) incorporate the same

10 Bank reserves could be added to the balance sheet. However, bank reserves have never been viewed as a potential instrument in the capital requirements literature. We do, however, need a second bank asset into which the bank can invest funds whenever capital requirements become binding. One possibility is to replace G with Cash, but an interest-bearing asset such as G seems a more realistic choice as the second asset. 11 We could assume that the cost of equity depends on leverage (deposit/equity ratio). Leverage certainly plays an important role in standard banking models in which book equity is exogenous and deposits are endogenous. In these models leverage raises the equity discount rate as the bank expands its use of debt (deposits). The leverage effect, therefore, is needed to ensure a finite sized bank. In our model, deposits are exogenous and book equity is endogenous. The condition, Tobin’s Q = 1, then plays the bank-size-constraining role so that the leverage effect is redundant in our model. 12 By assumption, the loan resource cost function does not allow for economies of scope. A positive value of f ensures an upward-sloping marginal resource cost of lending. Resource costs of deposits could also be included without altering the

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specification for the bank’s loan processing and monitoring cost functions as in the social planning model.13,14 3.2. Bank behavior under a non-binding capital ratio When the capital requirement ratio does not bind the bank’s decision, it maximizes the market value of equity in (6) subject only to the balance sheet constraint in (7), as either a potential monitor or non-monitor of loans and chooses the monitoring strategy that ˆ 1 expression is  ˆ 1M gives the highest equity value. The applicable  ˆ 1NM in (9.2) in (9.1) for the representative monitoring bank and  for the representative non-monitoring bank. Substituting the balance sheet constraint for the term, G–D, in (9.1) and (9.2) gives the following loan supply functions for the representative monitoring and non-monitoring bank, respectively, rL − rG f +c

∗

LM =

(10.1)

and ∗

LNM =

rL − rG − ı(1 + rL ) f

(10.2)

Eqs. (10.1) and (10.2) indicate that the representative bank confronts the same basic trade-off that a social planner faces: the monitoring of loans to eliminate borrowers’ default requires expending resources, which reduces the amount of current lending and the future cash inflow derived from such lending, but failing to monitor loans exposes the bank to default losses that also reduce its future cash inflow. Using the balance-sheet constraint in (7) to eliminate G, substituting (10.1) or (10.2) for the optimal quantity of loans, and using the valuation Eq. (6) yields the following expressions for the equity values of the representative monitoring E0M and non-monitoring E0NM bank, conditional on a fixed amount of book equity, EBK ,15,16 E0M =

(1 + rG )EBK + ((rL − rG )2 )/(2(f + c)) 1+ˇ

E0NM =

(11.1)

2

(1 + rG )EBK + ([rL − rG − ı(1 + rL )] /2f ) . 1+ˇ

(11.2)

As discussed earlier, in long-run equilibrium, equity cash is invested in the banking industry and thus in the representative bank up to the point where the book and market values of equity are

results since under the assumption of exogenous deposits these costs are fixed during the time horizon we consider. As discussed in Elyasiani et al. (1995), the presumed absence of resource costs for securities is consistent with the assumption of portfolio separation, so that banks’ asset and liability decisions are independent. 13 The social planning model does not explicitly define the date of consumption. To be consistent with the banking model and with the notion of an ex post cost of monitoring loans, consumption is dated at t = 1 in (2), so that the right-hand side of (2) is implicitly discounted by a social discount rate. 14 In our banking model, loan monitoring costs are borne ex post at date 1 and thereby reduce the net cash flow received by equity owners when the bank is dissolved. The model can also be set up to describe screening behavior. Since screening is ex ante to the extension of loans, screening costs would be borne at date 0 and thus would reduce the quantity of funds that could be converted into assets, resulting in a relatively smaller-sized bank. Although we focus only on ex post monitoring, in a more general framework the bank would make an optimal choice between monitoring or screening at date 0. The decision would depend on which choice (if any) maximized the market value of equity at date 0. 15 The model’s solution would be significantly more complicated if resource costs of deposits were included and the bank’s supply of deposits were allowed to be endogenous. This setup would require maximizing E with respect to both loans and deposits. The resulting first-order conditions would lead to non-linear solutions for E0M . 16 Note that setting the deposit rate equal to the security rate eliminates D from (11.1) and (11.2), which significantly reduces the model’s complexity.

equalized.17 In our model, this condition determines the size of the bank.18 Imposing the conditions, E0M = EBK in (11.1) and E0NM = EBK (11.2), yields, respectively, the model’s equilibrium equity valua∗ tions for the representative monitoring (E0M ) and non-monitoring ∗ NM (E0 ) bank, ∗

E0M =

(rL − rG )2 2(f + c)(ˇ − rG )

∗

E0NM =

(12.1) 2

[rL − rG − ı(1 + rL )] . 2f (ˇ − rG )

(12.2)

Eqs. (12.1) and (12.2) indicate that the market value of equity depends positively on factors that increase the available net cash flow at date 1, such as a higher market loan rate and a reduction in the value of f that translates into lower loan resource costs. Moreover, when capital requirements are not binding, the equilibrium ratio of equity to loans is endogenous. It can be shown, for example, that this ratio depends negatively on the security rate. Since the security rate is identical to the deposit rate in our model and since deposits are exogenous to the bank, a decrease in the deposit rate, ceteris paribus, lowers the bank’s deposit costs and thus increases bank profit. The market value of equity then rises relative to its book value, which in turn leads to a desired expansion in book equity. Lending also expands, but some of the induced increase in book equity will be used to finance an increase in securities rather than loans because loan expansion is subject to increasing resource costs. Thus, when the bank is not bound by capital requirements, the ratio of equity to loans rises. A comparison of (12.1) and (12.2) also reveals the nature of the basic trade-off underlying the choice between monitoring and not monitoring loans. As a non-monitor, a private bank experiences a reduction in net revenues via the per-dollar default loss ı, yet at the same time the bank’s overall costs are lower because it does not incur any monitoring costs. In contrast, as a monitor, the bank eliminates default but incurs higher operating costs and hence issues fewer loans. Similar to the trade-offs faced by the social planner, the monitoring/non-monitoring decision for a private bank ultimately depends on the relative contribution of ı and c. In contrast to the social planner, however, the bank focuses on the impact of these factors on its market equity value, not on social welfare. 3.3. Bank behavior under a binding capital ratio Let a “tilde” denote capital-constrained values of the relevant variables. From (8), the representative bank’s supply of loans when either monitoring or not is L˜ =

E˜ BK 

(13)

Using the same procedure as in Section 3.2, we first derive the market value of equity conditional on a given amount of book equity, E˜ BK , and then equate the market and book values of equity. This gives the following expressions for the market equity values of the

17 Perfect foresight precludes, by assumption, any conflict between old and new equity owners, as in Besanko and Kanatas (1996). 18 In banking models, bank size is always limited by a specific set of assumptions. For example, Diamond (1991) assumes a fixed amount of book equity and abstracts from deposits, so the asset side of the bank is completely predetermined. In Diamond’s framework, therefore, loan-monitoring costs can be treated as fixed because the amount of loans cannot exceed the predetermined amount of book equity. In our model, the liability side of the bank’s balance sheet is endogenously determined, which requires a limiting factor on the asset side. We provide this limitation by specifying quadratic costs for both a monitoring and non-monitoring bank, where the cost functions are related to the scale of the bank’s loan portfolio.

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257

∗ ∗ representative monitoring E˜ 0M and non-monitoring E˜ 0NM bank at date 0: ∗ E˜ 0M =



and ∗ E˜ 0NM =

2 f +c



2 f



[(rL − rG ) − (ˇ − rG )]

(14.1)

 {[rL − rG − ı(1 + rL )] − (ˇ − rG )}.

(14.2)

The loan supply functions for the representative monitoring and non-monitoring banks then become ∗ L˜ M =

 2 

and ∗ L˜ NM =

f +c

2 f

[(rL − rG ) − (ˇ − rG )]

{[rL − rG − ı(1 + rL )] − (ˇ − rG )}

(14.3)

(14.4)

As in the non-binding case in Section 3.2, the essential difference in the specifications of the equity values and loan functions for the two types of representative bank lies in the inclusion of the monitoring cost parameter, c, in (14.1) and (14.3) and the default parameter, ı, in (14.2) and (14.4).19 More importantly, the presence of  in (14.1) and (14.2) is a necessary condition for inducing a change in bank behavior via the government’s implementation of a binding capital ratio.

optimal monitoring position. Although the sequencing of the government’s decisions is arbitrary because time plays no role in the model, for illustrative purposes, we assume that the government first evaluates the effect of using only its security rate instrument to attain its twin objectives. If this single-instrument policy fails to induce the socially optimal loan and monitoring outcomes, the government then employs its second instrument and selects a value of the capital ratio that, in conjunction with its security rate instrument, will lead the representative bank to the desired social outcomes. One of the following four loan market equilibrium conditions is available for the government’s use in setting the security rate: LM∗ = LD : L

NM ∗

L˜

M∗

L

We assume that the government knows the social planner’s optimal strategy and the representative bank’s alternative equity values and loan supply functions under both binding and nonbinding capital ratios. However, in contrast to the social planner, the government does not directly control the outstanding volume of loans. In a decentralized economy this quantity is determined in a private loan market. The models in Sections 3.2 and 3.3 describe the behavior of a representative competitive bank which proxies for the behavior of the banking system.20 The market supply of loans depends on whether the representative bank monitors its loans and whether it operates under a binding capital requirement. Since loan supply in the model depends on the differential between the loan rate and the security rate, setting the bank’s loan supply equal to the socially optimal loan quantity solves only for the interest rate differential between the two rates. An explicit solution for the security rate, therefore, requires an explicit solution for the loan rate, which in turn requires a loan demand function. To determine the equilibrium loan rate, we assume an ad hoc demand for loans that is a linear function of the loan rate, (15)

The government uses the security rate and a capital ratio to guide the banking system towards achieving two social objectives21 : the socially optimal quantity of loans and the socially

D

=L :

= LD :

˜ NM ∗

3.4. The government’s setting of the security rate

LD = a0 − a1 rL

Fig. 1. Alternative values of the loan default rate ı. The social planner does not monitor loans below ıSP * but does monitor loans above ıSP *. The banking system and above but not between ıNM and follows the social monitoring plan below ıNM 0 0 . Between these two ı values, binding capital requirements may play a socially ıM 0 efficacious role.

D

=L :

unbound monitoring banks unbound non-monitoring banks bound monitoring banks bound non-monitoring banks

(16.1) (16.2) (16.3) (16.4)

To illustrate, suppose monitoring is socially optimal. To induce the targeted quantity of loans demanded, the government first sets LD = L¯ and solves for the equilibrium loan rate, rL∗ .22 If the government attempts to achieve its monitoring objective without imposing a capital ratio, it substitutes rL∗ into (16.1) and solves for ∗ the targeted security rate rGM , ∗ rGM = rL∗ − (f + c)L¯

(17.1)

If the single-instrument policy in (17.1) fails, the government then imposes a capital ratio and uses (16.3) to solve for the revised security rate target r˜GM , r˜GM =

rL∗ − (L¯ /2)(f + c) − ˇ (1 − )

(17.2)

Similarly, if achieving the social optimum calls for unbound nonmonitoring banks, the government uses (16.2) to derive the targeted security rate ∗ rGNM = (1 − ı)rL∗ − (ı + f L¯ ),

(17.3)

If this policy fails, the government then imposes a capital ratio and uses (16.4) to solve for r˜GNM , the targeted security rate when the government’s goal is to bind the decision-making of nonmonitoring banks. 4. Attaining social objectives in a decentralized economy

19 In contrast to the non-binding case, a minimum value of the equity to loan ratio is predetermined under binding capital requirements. 20 Essentially, we assume that there are J banks and that the banking system’s aggregate loan supply equals J times the representative bank’s supply. For simplicity, we then set J = 1. 21 We could also assume that rG = r + , where r is the targeted short-term interest rate and reflects a term premium. However, would play no further role in our analysis. Thus setting setting = 0 (to reduce the number of parameters) together with our model’s assumption that rG = rD (due to the assumed absence of reserve requirements) implies that r = rG = rD . In our framework, therefore, control over the short-term interest rate is the same as control over rG .

Our findings depend on the specific values of ı and c that describe the default rate and marginal monitoring cost parameters governing decision-making within any banking system. The critical ı values are graphed in Fig. 1 and refer to the following propositions:

22 Note that the equilibrium loan rate is exogenous to the bank’s monitoring decision. The security rate, however, is not.

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1. If ı lies in the range, 0 → ıNM or ıM → ıMAX , the government can 0 0 achieve the socially optimal outcomes for both loans and monitoring without resorting to the imposition of binding capital requirements on banks. Note that the location of ıM depends on 0 the value of c. Within the two ı ranges above, a binding capital ratio is socially unnecessary. If ı > ıMAX , the government can achieve its twin objectives irrespective of the value of c and thus a binding capital ratio is also socially unnecessary. ∗ ∗ 2. Within the ı ranges, ıNM → ıSP and ıSP → ıM , supplementing 0 0 the targeted security rate with a binding capital ratio may induce the banks to adopt the socially optimal monitoring stance. The outcome, however, depends on the specific value of c, as discussed below. When this monitoring cost condition is satisfied, binding capital requirements are socially efficacious. When this condition is not satisfied, a binding capital ratio cannot alter the banks’ sub-optimal monitoring decision. Here capital requirements are socially ineffective. As a corollary to the above, there is no single socially optimal capital ratio for all combinations of {ı, c} and thus for all possible banking systems. Below we first discuss Proposition 1 and then analyze Proposition 2 for the under-monitoring case in which banks fail to monitor their loans even though monitoring is socially optimal. We also briefly discuss the over-monitoring case. Appendix A provides the technical details for both cases. 4.1. Achieving the social optimum with only the security rate Capital requirements are unnecessary if the setting of the security rate induces private banks to adopt the correct monitoring stance. Using the social planner’s optimal decision rule in (5), we ∗ define ıSP = c L¯ /2 as the separating value of ı. For a given value of c, non-monitoring is socially optimal when the economy is characterized by ı < ıSP *. The reverse inequality requires monitoring. Consider first the case in Fig. 1 where the banking system is characterized by ı < ıSP *. In this ı range, non-monitoring is optimal, so ∗ the government chooses the non-monitoring security rate rGNM in (17.3) to induce lending exclusively by non-monitoring banks. It can accomplish this objective as long as the banks perceive that their equity value as non-monitors exceeds their equity value as ∗ monitors, E0NM > E0M . It is easy to verify that this inequality holds as ı → 0 To determine whether it continues to hold as ı → ıSP *, we ∗ , the value of ı at which banks set E0NM = E0M and solve for ıNM 0 would become indifferent between not monitoring and monitor∗ ing their loans, given rG = rGNM . The expression for ıNM is given in 0 Appendix A, which shows that it depends importantly on c and f. Suppose that the monitoring cost parameter c is a multiple of the operational cost parameter f, i.e., c = Nf. Then, as shown in Fig. 1, ∗ ıNM < ıSP for all N > 0. Thus, as long as monitoring costs are pos0 itive, the government cannot rely solely on using the security rate to ensure that banks adopt the socially optimal non-monitoring policy for all values of ı < ıSP *. The government’s single-instrument policy fails because the banks’ perception that monitoring is more valuable than non-monitoring occurs at a smaller value of ı than the social planner’s ıSP *. When a loan defaults, the social planner loses the ı fraction of its loan portfolio. Banks also lose this fraction but, in addition, they lose the interest income that would have been earned on the defaulted loans. Thus, recouping additional loan revenue gives banks a stronger incentive to institute monitoring as ı rises relative to the evaluation of losses by the social planner. Now consider an economy characterized by the inequality, ı > ıSP *. In this economy monitoring is socially optimal. To induce lending exclusively by monitoring banks, the government sets the ∗ security rate at the monitoring targeted value rGM in (17.1). To

willingly monitor loans, however, the banks must perceive that their equity valuation is higher as monitors rather than non∗ monitors, i.e., E0M > E0NM . This occurs automatically at the value, ∗ ∗ M MAX = (rL − rG )/(1 − rL∗ ) in Fig. 1. Here E0NM = 0 and banks do ı not consider non-monitoring as a possible operational choice. The government’s targeted security rate will, however, fail to induce ∗ socially optimal monitoring if the reverse inequality, E0M < E0NM , SP MAX occurs in the range, ı * < ı . To determine whether this happens, ∗ we solve E0M = E0NM for the default rate, ıM in Fig. 1. If the economy’s 0 ∗ , banks will choose not to ı lies within the range, ıSP ≤ ı < ıM 0 monitor because their equity valuation is higher as a non-monitor. Appendix A analyzes this case and shows that when monitoring costs are sufficiently high, the government will need to consider ∗ supplementing its security rate target rGM with a binding capital ratio as it attempts to implement the socially optimal plan. 4.2. Can using a binding capital ratio induce socially optimal monitoring We first discuss the case in which banks do not monitor their loans even though monitoring is the optimal social choice. This situation arises whenever the economy’s ı lies within the range ∗ ıSP ≤ ı < ıM in Fig. 1. A detailed analysis is provided in Appendix 0 A. Here we offer an intuitive explanation for the specific example where the banking system’s loan default rate is set at ı = ıSP * and the social planner optimally prefers monitored to unmonitored loans. The following two conditions will lead to the suboptimal preference of the banks for unmonitored loans: the government initially ∗ uses only the security-rate target rGM in its attempt to induce monitoring; and monitoring costs are sufficiently high such that the inequality (A2.1) in Appendix A is satisfied. Given the government’s use of a single interest-rate instrument and sufficiently high monitoring costs, banks perceive a higher equity valuation as non-monitors of their loan portfolios. Moreover, the non-monitoring banks want to create a quantity of unmonitored loans larger than the optimal amount L¯ 23 so that at the ∗ security rate rGM the government is also not able to equilibrate the demand and supply of loans at L¯ . Thus, in this economy the government’s sole reliance on its interest-rate instrument leads to its simultaneous failure to achieve both its loan and monitoring objectives. A capital ratio that can redress these two failures in the loan market must satisfy two relationships involving several equity/loan ratios. First, the government’s capital ratio ˜ must bind the NM equity/loan ratio of the non-monitoring banks, (E/L) ; otherwise these banks would not desire to alter any of their decisions. Second, the capital ratio must lie within the range of equity/loan ratios in which capital-constrained banks prefer to monitor; otherwise these banks would choose not to monitor their loans. The capital ratio, at which the loan market is inhabited exclusively by capital-constrained monitoring banks, is derived from (17.2) and given by ˜ =

rL − r˜GM − L¯ (f + c)

(18.1)

ˇ − r˜GM

where r˜GM is the value of security rate chosen when the capital ratio is imposed. Eq. (18.1) shows that the two instruments are interdependent, so the value of one is arbitrary. Assume for simplic∗ ity that r˜GM = rGM , i.e., the government chooses to maintain the same

∗

23 To see this, substitute (17.1) for rGM into (10.2), the loan supply function of non-monitoring banks and evaluate the resulting expression at the default rate, ∗ ıSP = c L¯ /2.

K.J. Kopecky, D. VanHoose / Journal of Financial Stability 8 (2012) 252–262

security rate at the time when it imposes capital requirements. We then derive ˜ =

(f + c)L¯

(18.2)

∗

2(ˇ − rGM )

and

 E NM L

=

c L¯ (1 − rL∗ ) + 2f L¯

(18.3)

∗

4(ˇ − rGM )

NM

It is easy to verify that ˜ > (E/L) . When non-monitoring banks become bound by the capital ratio, they must reexamine their optimal strategy and decide whether to operate as constrained monitors or non-monitors. Recall that initially the non-monitoring banks had desired to supply a loan quantity in excess of L¯ . When the capital ratio is increased enough to bind the equity/loan ratio in (18.3), these banks will have insufficient equity to satisfy their capital requirements and thus will be forced to reduce the desired size of their loan portfolio while simultaneously reconsidering their monitoring decision. The need to reduce the amount of loans will eventually work in favor of the banks’ complete conversion to monitored loans because, as the quantity of loans decreases, the total cost of monitoring a loan portfolio decreases at a greater rate (since these costs are quadratic) as compared to the decrease in total loan revenues (which decline linearly). Thus, as the banks’ loan portfolio is forced to shrink in size to L¯ , the profit earned as a loan monitor rises relative to the profit of a non-monitor. The government, however, will attain its optimal loan quantity and loan monitoring targets only if the capital-constrained banks’ conversion to monitoring occurs at an equity/loan ratio smaller ˜ otherwise constrained non-monitoring banks will continue than ; ˜ to inhabit the loan market. Let (E/L) represent the equity/loan ratio at which constrained banks are indifferent regarding the monitoring of their loan portfolios,

  E˜ L

=

(f + c)L¯ (1 − rL∗ ) ∗

2(ˇ − rGM )

(18.4)

˜ in (18.4) is less than ˜ in (18.2), which implies By inspection, (E/L) that capital-constrained banks are also monitoring banks, lending the socially optimal L¯ quantity of loans. Thus, at ıSP * the government can successfully combine its capital ratio and security rate instruments to induce the private banking system to supply the quantity of monitored loans that correspond to the social planner’s optimal decision. The ability to set a binding capital ratio is therefore socially efficacious. The second potential use of capital requirements arises when the banking system’s loan default rate is less than ıSP *. As discussed in Section 4.1, private banks may monitor their loans at loan default rates below ıSP *in Fig. 1 even though non-monitoring is socially optimal. Appendix A provides a detailed analysis of this case. Essentially, there is a range of default rates where the capital ratio will not alter the banks’ sub-optimal monitoring choice, and thus imposing a capital ratio will be socially ineffective. For other default rates, it is possible that a binding capital ratio will produce the optimal social outcome. To understand how a binding capital ratio can convert a monitoring bank into a non-monitoring bank, assume the government imposes a capital ratio above the initial equity/loan ratio of the monitoring bank. One possible reaction for the bank is to reduce the size of its loan portfolio. But the government is simultaneously targeting the security rate to keep the quantity of loans constant at L¯ , so a reduced loan portfolio is not feasible for the now-constrained monitoring bank. In the absence of an escape route via a change in the quantity of loans, the bank must reexamine its monitoring strategy by comparing its equity value as a monitor or non-monitor. The bank realizes that it will have to bear

259

a net opportunity cost equal to (ˇ − rG ) because, to support its L¯ loan portfolio, the bank will be forced to invest an increased amount of equity capital (with a required return equal to ˇ) into a larger, but sub-optimal, amount of securities. For example, if the bank’s book equity is deficient by $1, an additional $1 of book equity would not result in additional lending (because loans are fixed at L¯ ) but in an additional $1 of securities, an increase that is sub-optimal from the bank’s perspective. The attendant reduction in net revenue relative to its cost of capital implies that the monitoring bank will examine other methods of organizing itself in an attempt to raise net revenue. The only path open to the bank is to reconsider the benefits and costs of monitoring. Under certain conditions, a relatively higher cost of monitoring compared to other costly types of loan operations implies that a monitoring, capital-constrained bank can reduce its overall cost of operations and thereby raise its net revenue by becoming a constrained non-monitoring bank. Binding capital requirements would be, therefore, socially efficacious.

5. Conclusion This paper has investigated the strategic choices made, on the one hand, by a social planner who has complete control of the loan market and, on the other hand, by a government that is limited to using two policy instruments, an interest rate target and a capital ratio, to influence the economy’s private banking system and thus the private loan market. Our objective has been to determine the conditions under which the government could achieve the same loan quantity and loan monitoring outcomes as the social planner. Much of the literature only focuses on whether a capital ratio alters bank behavior. While our analysis also investigates this aspect, it is nonetheless broader in scope because it first asks whether the government’s imposition of a capital ratio is the correct social policy. Capital requirements are therefore motivated within the context of an explicit government/private banking model. A basic implication of our analysis is that economies with alternative configurations of loan market interest rates, loan default rates, and loan monitoring costs can face very different socially optimal loan monitoring solutions. Within various subsets of these economies, the government may find that capital ratios are unneeded, or highly effective, or completely ineffectual in guiding the private banking system to the same optimal loan outcomes as the social planner. It is unlikely, therefore, that a single capital ratio would satisfy all economies worldwide. Thus, our analysis indicates that a government aiming to use capital requirements to align social and private monitoring (an activity promoting bank safety and soundness) must have the flexibility to determine a capital ratio, if any, that would effectively control banks’ decisions. Our analysis also suggests several avenues for further research. The present paper has examined a stark setting characterized by certainty, linearity in the social planner’s decision problem, identical banks, and no strategic interaction between bank lenders and their borrowers. Potentially useful extensions would entail a more sophisticated specification of the social welfare problem faced by the planner and a more elaborate structure for the domestic banking system that recognizes, in particular, the presence of heterogeneous banks. Modeling heterogeneity could also be extended to an international setting to determine whether and how conflicting objectives could be resolved among the social planners of different countries. This work would build on several recent papers studying the coordination of regulatory supervisory rules. For instance, Acharya (2003) analyzes cross-country spillover effects arising from different regulatory policies regarding forbearance versus bank closure. In a different vein, Holthausen and Rønde (2005) examine the range of options available to national

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regulators with potentially divergent interests who confront a single multinational bank while Dell’Ariccia and Marquez (2006) study the benefits and the costs of centralized regulation across multiple jurisdictions. In the present model, monitoring has an either/or effect on the default rate. Extending the model to the uncertainty case would permit a more insightful analysis of the influence of monitoring activity in determining loan defaults. Moreover, embedding a strategic interaction between the social planner and its borrowers, and the private banks and their borrowers would endogenize the market equilibrium loan rate. While this extension would increase the scope of the model, it would also create a far more complex control problem for the government. We expect that the framework developed in this paper will, however, be useful for analyzing these future research topics.

the banks require a stronger incentive in the form of a higher loan rate to implement monitoring. Consider the case in which monitoring and other banking costs are equalized (N = 1). Here, the loan rate would have to exceed 17.16% to violate the inequality in (A2.1). It follows therefore that in higher ı economies in which monitoring is socially optimal, the government’s success when using only a security rate target is not robust to all values of the equilibrium loan rate and the cost of monitoring. If monitoring costs are sufficiently high, the government will need to consider supplementing its security rate target with a binding capital ratio in its attempt to implement the socially optimal plan. A.3. When capital requirements are needed to induce monitoring ∗

This situation arises when ıSP < ı < ıM . Capital requirements 0 will successfully convert a non-monitoring banking system into a monitoring system when the following conditions are satisfied:

Acknowledgments We thank the referees for their supportive and constructive comments. All errors are of course our responsibility. VanHoose is grateful for partial support of this research provided by Networks Financial Institute.

˜ >

A.1. The default rate –

ıNM 0

The monitoring-indifference condition for ıNM results in a 0 quadratic equation with a positive root,

 L¯ = ( f (f + c) − f ) ∗ 1 + rL

(A1.1)

ıNM 0

The value of is crucial for Proposition 1. The government’s attempt to use the security rate as the sole policy instrument both to control the quantity of loans and to induce the banks to adopt ∗ the socially optimal non-monitoring stance will fail if ıNM < ıSP . 0 This inequality holds when 2  ( f (f + c) − f ) < 1 + rL∗ c

(A1.2)

Inequality (A1.2) depends crucially on the relative values of c and f, the monitoring and non-monitoring loan cost parameters. Suppose that c is a multiple of f, i.e., c = Nf. In this case, (A1.2) then becomes 2  ( 1 + N − 1) < 1 + rL∗ N

(A1.3) ∗

This inequality holds for all N > 0. Thus ıNM < ıSP . 0 A.2. The default rate – ıNM 0 Assuming again that monitoring and other operational banking costs are related by c = Nf, the government’s targeted security rate ∗ rGM will fail to create the social monitoring optimum if 1 + rL∗ <

˜ >

loan market equilibrium with monitoring banks

 E NM

(A3.1)

L

bound non-monitoring banks at r˜GM

(A3.2)

E˜ L

monitoring equity dominance at r˜GM , ˜

(A3.3)

 

Appendix A.

ıNM 0

∗ L˜ M = L¯

2[1 + N − (1 + N)1/2 ] N

(A2.1)

This inequality sets a boundary on the equilibrium loan rate that depends on the extent to which monitoring costs deviate from other bank-related costs. When monitoring costs are relatively small, the loan rate needed to induce monitoring will also be quite low. For example, if N = 0.01 (the monitoring cost is 1/100 the cost of other operational loan activities), (A2.1) would not be satisfied at a loan rate above 0.25%. Banks therefore would adopt monitoring and the government would never need to implement a capital requirement. However, as monitoring costs rise to a more plausible level,

where r˜GM represents the value of the government’s security rate instrument and ˜ is the capital ratio. Eq. (18.1) in the text defines the relation between the capital ratio and the security rate implied by (A3.1). The following inequality must hold to satisfy (A3.2), rL∗ − r˜GM + ı(1 + rL∗ ) > (f + c)L¯

(A3.4)

This inequality depends on the monitoring cost parameter, c. If c is too large, there is no value of r˜GM that satisfies (A3.4) since r˜GM cannot be smaller than zero. In this case, any targeted capital ratio will always be too low to bind the unconstrained non-monitoring banks. Essentially, the higher is c, the smaller is the capital ratio associated with a loan market equilibrium that has only monitoring banks. The government, therefore, would not be able to induce non-monitoring banks to simultaneously adopt the socially optimal monitoring position and the socially optimal quantity of loans. In other words, the government’s dual instrument policy would be socially ineffective. ∗ The inequality (A3.3) requires that ı(1 + rL∗ ) > c L¯ /2. Since ıSP = ¯ c L/2, this inequality is automatically satisfied within the range ∗ . ıSP < ı < ıM 0 A.4. When capital requirements are needed to induce non-monitoring ∗

Banks want to monitor in the ı range ıNM < ıSP even though 0 non-monitoring is socially optimal. As shown in Appendix A.1, this range always exists as long as the cost of monitoring is positive. For capital requirements to be efficacious, the following three conditions must be satisfied by the targeted policy values chosen by the government for the security rate/capital ratio combination, ˜ {˜rGNM , }: ˜ >

 E M L

L˜ NM∗ = L¯

bound monitoring banks at r˜GNM

(A4.1)

loan market equilibrium with non-monitoring banks (A4.2)

K.J. Kopecky, D. VanHoose / Journal of Financial Stability 8 (2012) 252–262

  E˜ L

˜ <

non-monitoring equity dominance at r˜GNM , ˜

(A4.3)

Condition (A4.1) requires that the capital ratio bind the equity/loan ratio of unbound monitoring banks; otherwise, capital requirements will never alter the banks’ decisions. Condition (A4.2) requires that the government use the constrained non-monitoring bank loan supply function to induce the socially optimal loan quantity, L¯ . Condition (A4.3) requires that the capital ratio lie in the region where the equity value of the capital-constrained nonmonitoring bank exceeds that of the constrained monitoring bank; otherwise the constrained bank will continue to monitor. When ˜ will these conditions are satisfied, the policy combination {˜rGNM , } convert unbound banks that would otherwise prefer to monitor their loans into bound banks that optimally decide against monitoring their loans. Thus, the binding capital ratio, ˜ and the targeted security rate, r˜GNM , if they exist, will allow the government to fully implement the socially optimal plan in the range, ∗ ıNM < ı < ıSP . 0 The general solution is fairly complicated. To focus on the important aspects of the solution, suppose that the economy is at the boundary loan default ratio, ıNM , in Fig. 1 and that the govern0 ment sets the targeted value of the security rate r˜GNM at zero. (Recall that the value of one of the government’s instruments is arbitrary.) Under these conditions, the capital ratio and the two equity/loan ratios in (A4.1)–(A4.3) are given by24,25 ˜ =

(1 + rL∗ ) − (f L¯ /2) rL∗ − ıNM 0 ˇ

 E M L

=

rL∗

(A4.4)

(A4.5)

2ˇ

and

  E˜ L

=

(1 + rL∗ )(f + c) crL∗ − ıNM 0 cˇ

(A4.6)

Condition (A4.3) ensures that only capital-constrained nonmonitoring banks will supply loans at the loan market equilibrium. It implies that ∗ ¯ ıNM 0 (1 + rL ) < c L/2

in economies characterized by a loan default ratio within the range, ˆ < ı < ı. ıNM 0 Condition (A4.1) requires that an unbound bank, engaged in sub-optimal monitoring at ıNM , must become capital constrained at 0 a lower capital ratio than that needed to generate the desired nonmonitoring bank equilibrium in the loan market. If the unbound monitoring bank becomes constrained at a higher capital ratio, the government cannot create an equilibrium outcome in the loan market with non-monitoring banks. Condition (A4.1) implies that ∗ ¯ rL∗ > 2ıNM 0 (1 + rL ) + Lf

(A4.8)

and assuming that c = Nf , (A4.8) is Substituting (A1.1) for ıNM 0 explicitly written as



rL∗ > c L¯

2(1 + N)1/2 − 1 N



(A4.9)

In (A4.9) when the marginal cost of monitoring equals the marginal cost of other loan operations (N = 1), the equilibrium loan rate must be 1.82 times larger than the marginal cost of monitoring c L¯ . However, as N rises (so that the relative cost of monitoring is also rising), the percentage by which the loan rate must exceed the marginal cost of monitoring declines. This percentage falls to 0.56 when N = 10, and to 0.19 when N = 100. Thus the condition on the loan rate in (A4.9) becomes less restrictive as N increases. For example, let c L¯ = 0.01. When N = 1, the loan rate must exceed 0.0182 but when N = 10, the loan rate only needs to exceed 0.0056, so the condition on the loan rate becomes weaker as N increases. The implication of (A4.9) is straightforward. In economies within the restricted range, ıNM < 0 ˆ the capital ratio ˜ in (A4.1) can quickly bind socially subı < ı, optimal monitoring banks as long as they face a non-trivial ratio of marginal monitoring to other marginal loan operational costs. Intuitively, a higher value of N increases the sensitivity of an unbound monitoring bank to a change in the capital ratio. Thus within the ıNM < ı < ıˆ range, the government can select 0 NM ˜ {˜r , } to induce the banking system to adopt the socially optimal G

monitoring position. Capital requirements are therefore socially efficacious.

(A4.7)

, inequality (A4.7) is ((1 + N)1/2 − Substituting (A1.1) for ıNM 0 , the capital 1)/N < 1/2, which is satisfied for N > 0. Thus at ıNM 0 ratio in (A4.4) is consistent with the appearance of only capitalconstrained non-monitoring banks as loan suppliers in the loan market. The feasibility of (A4.7) does not, however, apply to the ∗ entire socially sub-optimal range, ıNM < ı < ıSP . In particular, 0 the inequality does not hold at the loan default rate given by ∗ ıˆ = c L¯ /(2(1 + rL )) < ıSP . Consequently, the government is limited in its ability to use capital requirements as an instrument for converting socially sub-optimal monitoring banks into socially optimal non-monitors. A binding capital ratio is a feasible policy tool only

24

261

˜ if we assume that ˇ > r ∗ (which essentially limits In the general solution for , L

the equity growth of the bank), ˜ is negatively related to r˜GNM because an increase in the security rate induces a decrease in the non-monitoring banks’ supply of loans. Since the quantity of loans is targeted at a predetermined amount, L¯ , the capital ratio must be lowered to induce the constrained banks to restore their loan supply to L¯ . 25 In the general solution for (E/L)M , if we assume that ˇ > r˜GNM (i.e., the required return on bank equity exceeds the security rate), (E/L)M is negatively related to r˜GNM . Ceteris paribus, as the securities rate increases, the equity value of an unbound monitoring bank declines.

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