A theory of systemic risk and design of prudential bank regulation

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Abstract

Systemic risk is modeled as the endogenously chosen correlation of returns on assets held by banks. The limited liability of banks and the presence of a negative externality of one bank’s failure on the health of other banks give rise to a systemic risk-shifting incentive where all banks undertake correlated investments, thereby increasing economy-wide aggregate risk. Regulatory mechanisms such as bank closure policy and capital adequacy requirements that are commonly based only on a bank’s own risk fail to mitigate aggregate risk-shifting incentives, and can, in fact, accentuate systemic risk. Prudential regulation is shown to operate at a collective level, regulating each bank as a function of both its joint (correlated) risk with other banks as well as its individual (bank-specific) risk.

Introduction

A financial crisis is “systemic” in nature if many banks fail together, or if one bank’s failure propagates as a contagion causing the failure of many banks. At the heart of bank regulation is a deep-seated concern that social and economic costs of such systemic crises are large. It is thus broadly understood that the goal of prudential regulation should be to ensure the financial stability of the system as a whole, i.e., of an institution not only individually but also as a part of the overall financial system.1 Different reform proposals such as the ones by the Bank of International Settlements (1999) have been made with the objective of improving bank regulation, and in the aftermath of the global financial crisis of 2007–2009, many more proposals will come to the fore. A central issue is to examine these proposals under a common theoretical framework that formalizes the (often implicit) objective of ensuring efficient levels of systemic failure risk. This paper seeks to fill this important gap in the literature.

The standard theoretical approach to the design of bank regulation considers a “representative” bank and its response to particular regulatory mechanisms such as taxes, closure policy, capital requirements, etc. Such partial equilibrium approach has a serious shortcoming from the standpoint of understanding sources of, and addressing, inefficient systemic risk. In particular, it ignores that in general equilibrium, each bank’s investment choice has an externality on the payoffs of other banks and thus on their investment choices. Consequently, banks can be viewed as playing a strategic Nash game in responding to financial externalities and regulatory mechanisms. Recognizing this shortcoming of representative bank models, this paper develops a unified framework with multiple banks to study the essential properties of prudential bank regulation that takes into account both individual and systemic bank failure risk.

Our analysis has two features: one positive and one normative. The positive feature of the analysis provides a precise definition and an equilibrium characterization of systemic risk. Unlike most of the extant literature on systemic risk (see Section 2) that has focused on bank liability structures, we define systemic risk as the joint failure risk arising from the correlation of returns on asset side of bank balance sheets. Moreover, we give a characterization of conditions under which in equilibrium, banks prefer an inefficiently high correlation of asset returns (“herd”), giving rise to systemic or aggregate risk.

The normative feature of the analysis involves the design of optimal regulation to mitigate inefficient systemic risk. To this end, we first demonstrate that the design of regulatory mechanisms, such as bank closure policy and capital adequacy requirements, based only on individual bank risk could be suboptimal in a multiple bank context, and may well have the unintended effect of accentuating systemic risk. Next, we show that optimal regulation should be “collective” in nature and should involve the joint failure risk of banks as well as their individual failure risk. In particular, (i) bank closure policy should exhibit little forbearance upon joint bank failures and conduct bank sales upon individual bank failures, and (ii) capital adequacy requirements should be increasing in the correlation of risks across banks as well as in individual risks.

In our model, banks have access to deposits that take the form of a simple debt contract. Upon borrowing, banks invest in risky and safe assets. In addition, they choose the “industry” in which they undertake risky investments. The choice of industry by different banks determines the correlation of their portfolio returns. Systemic risk arises as an endogenous consequence when in equilibrium, banks prefer to lend to similar industries.2

Since deposit contract is not explicitly contingent on bank characteristics, the depositor losses resulting from bank failures are not internalized by the bankowners. This externality generates a role for regulation. The regulator in our model is a central bank whose objective is to maximize the sum of the welfare of the bankowners and the depositors net of any social costs of financial distress.

In this setting with multiple banks, when one bank fails, there are two conflicting effects on other banks. First, there is a reduction in the aggregate supply of funds (deposits) in the economy, and hence, in aggregate investment. This results in a recessionary spillover (a negative externality) to the surviving banks through an increase in the market-clearing rate for deposits, that reduces the profitability of banks.3 Second, surviving banks have a strategic benefit (a positive externality) from the failure of other banks due to an increase in scale or an expansion, resulting from the migration of depositors from the failed banks to the surviving banks, or, due to strategic gains from acquisition of failed banks’ assets and business.

Over a robust set of parameters, the negative externality effect exceeds the positive externality effect, in which case banks find it optimal to increase the probability of surviving together, and thus failing together, by choosing asset portfolios with greater correlation of returns. This would arise, e.g., if (i) the reduction in aggregate investment is substantial upon a bank’s failure, i.e., banks are ‘large’; or (ii) the depositors of the failed bank do not migrate to the surviving banks, i.e., banks are ‘essential’; or (iii) other banks cannot benefit from acquiring the business facilities of the failed bank, i.e., banks are ‘unique’ or anti-trust regulations prevent such acquisitions. The preference for high correlation arises as a joint consequence of the limited liability of the banks’ equityholders and the nature of the externalities described above. This equilibrium characterization of systemic risk is the first contribution of the paper. We call such behavior as systemic risk-shifting since it can be viewed as a multi-agent counterpart of the risk-shifting phenomenon studied in corporate finance by Jensen and Meckling (1976), and in credit rationing by Stiglitz and Weiss (1981).

In the first-best allocation, however, different banks undertake investments in assets with lower correlation of returns. This is because losses to depositors, and to the economy, from a joint failure exceed those from individual failures. In individual bank failures, depositors of the failed bank migrate to surviving banks and intermediation role played by the failed bank is not fully impaired. However, such a possibility does not exist in a joint failure and there is a greater reduction in aggregate investment compared to states of individual bank failures.

The central bank attempts to mitigate systemic and individual risk-shifting incentives of bankowners through its design of bank closure policy and capital requirements. Our second contribution is to illustrate the design of bank closure policies that takes into account the collective investment policies of banks. We model the closure policy as a bail out of the failed bank with a dilution of bankowners’ equity claim, greater dilution implying a less forbearing closure policy. A bank bail out eliminates the financial externalities discussed above but also induces moral hazard depending upon the extent of forbearance exercised. The optimal ex ante closure policy is shown to be “collective” in nature: it exhibits lower forbearance towards bankowners upon joint failure than upon individual failure. The costs of nationalizing a large number of banks however may render such a policy suboptimal from an ex post standpoint, i.e., time-inconsistent and hence, lacking in commitment. The resulting (implicit) “too-many-to-fail” guarantee, where bankowners anticipate greater forbearance upon joint failure than in individual failure, accentuates systemic risk by inducing banks to make correlated investments so as to extract greater regulatory subsidies.

Further, a “myopic” closure policy that does not take into account the collective response of banks and hence, does not distinguish between forbearance in individual and joint failures, also fails to mitigate systemic risk-shifting behavior. It is strictly dominated by collective regulation that counteracts any residual systemic moral hazard induced through “too-many-to-fail” guarantee, by conducting bank sales (possibly subsidized) upon failure of individual banks. This increases the charter value of banks, in a relative sense, in the states where they survive but other banks fail, in turn, inducing a preference for lower correlation.4

Our third important contribution concerns the design of capital adequacy regulation. The current BIS capital requirement is a function only of a bank’s individual risk and does not penalize banks for holding asset portfolios with high correlation of returns. We show that under such a structure, each bank may optimally reduce its individual failure risk, but systemic risk arising from high correlation remains unaffected. To remedy this, we propose a “correlation-based” capital adequacy requirement that is increasing, not only in the individual risk of a bank, but is also increasing in the correlation of a bank’s asset portfolio returns with that of other banks in the economy. We propose an intuitively appealing implementation by considering a portfolio theory interpretation. The risks undertaken by banks can be decomposed into exposures to “general” risk factors and “idiosyncratic” components. For any given level of individual bank risk, correlation-based regulation would encourage banks to take idiosyncratic risks by charging a higher capital requirement against exposure to general risk factors.

Many financial institutions already employ a collective approach to capital budgeting (see Section 2) and regulators have also acknowledged the role of intra-bank correlations by proposing a long-term shift towards portfolio models for credit risk measurement (BIS, 1999). The proposed reforms of the BIS regulation appears however to have focused too much on the portfolio risk of each bank and ignored the inter-bank correlation effects for diversification of the economy-wide banking portfolio. Given the attention being devoted to possible reforms of the capital adequacy regulation and lender-of-last-resort policies, we believe our advocacy of collective regulation of systemic risk is particularly germane.

The remainder of the paper is structured as follows. Section 2 discusses the related literature. Section 3 describes the model setup for the multiple-bank economy. Section 4 characterizes the systemic risk-shifting phenomenon in the intermediated economy. Section 5 considers the design of bank closure policies and Section 6 looks at the design of capital adequacy requirements. Finally, Section 7 concludes with a brief discussion of possible avenues for related research and the relevance of our results for other economic phenomena. Appendix A contains certain regularity assumptions and Appendix B contains proofs. The Addendum contains appendices from the unabridged version, Acharya (2001), referred to in the text.

Section snippets

Related literature

A discussion of the seminal papers in banking regulation can be found in Dewatripont and Tirole (1993), and Freixas and Rochet (1997).

There is a burgeoning literature on models of contagion among banks: Rochet and Tirole (1996), Kiyotaki and Moore (1997), Freixas and Parigi (1998), Freixas et al. (1999), and Allen and Gale (2000c), to cite a few. The primary focus of these studies is on characterizing the sources of contagion and financial fragility. These studies examine the liability

Model

We build a multi-period general equilibrium model with many agents, viz. banks and depositors, and many markets, viz. markets for safe and risky assets, and market for deposits. In order to study systemic risk and its prudential regulation, the model incorporates (i) the likelihood of default by banks on deposits; (ii) financial externalities from failure of one bank on other banks; (iii) regulatory incentives; and (iv) the interaction of these features. The model builds upon the Allen and Gale

Systemic risk-shifting in the intermediated economy

We demonstrate a systemic risk-shifting phenomenon where both banks undertake correlated investments by investing in similar industries at t=0. In the presence of standard debt contract, there is risk-shifting at collective level in addition to the risk-shifting behavior at individual bank level. This collective behavior aggravates joint failure risk in the economy.

Design of bank closure policies

In the previous section, we showed that risk-shifting arises at individual and collective levels because deposit contracts that are conditional upon observable bank characteristics cannot be written. This creates a “missing market.” (as discussed in Gorton and Mullineaux, 1987). The central bank can design mechanisms to overcome the inefficiencies arising due to this “missing market”. However, we show below that lack of judicious regulation could also induce such risk-shifting behavior. We will

Design of capital adequacy regulation

As discussed in the previous section, optimal closure policy designs may be difficult to implement as they are time-inconsistent. Hence, we examine the ex ante mechanism, viz. the capital requirements. We first discuss the suboptimality of myopic capital adequacy that is based only a bank’s own risk, and next show that the optimal capital adequacy also takes into account the joint risk of banks, in particular, their correlation.

Consider the economy as described in Section 3. We continue to

Conclusion

We have developed a positive theory of systemic risk and a normative theory of its prudential regulation in a multi-period general equilibrium model with many banks and depositors that incorporates (i) the likelihood of default by banks on deposits; (ii) financial externalities from failure of one bank on other banks; (iii) regulatory incentives; and (iv) the interaction of these features. Several applications of our analysis and results are immediate.40

Acknowledgements

This paper is an essay from my PhD dissertation. I have benefited from encouragement and guidance of Yakov Amihud, Douglas Gale, Marty Gruber, Kose John, Anthony Saunders, Marti Subrahmanyam, and Rangarajan Sundaram. I am especially indebted to Rangarajan Sundaram for introducing me to the topic, to Douglas Gale for many insightful discussions, and to Ken Garbade and Anthony Saunders for detailed comments on an earlier draft. In addition, I am grateful to Franklin Allen, Edward Altman, Allen

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