Portfolio diversification and systemic risk in interbank networks

https://doi.org/10.1016/j.jedc.2017.01.013Get rights and content

Abstract

This paper contributes to a growing literature on the ambiguous effects of risk diversification. In our model, banks hold claims on each other’s liabilities that are marked-to-market on the individual financial leverage of the obligor. The probability of systemic default is determined using a passage-problem approach in a network context and banks are able to internalize the network externalities of contagion through their holdings. Banks do not internalize the social costs to the real economy of a systemic default of the banking system. We investigate the optimal diversification strategy of banks in the face of opposite and persistent economic trends that are ex-ante unknown to banks. We find that the optimal level of risk diversification may be interior or extremal depending on banks exposure the external assets and that a tension arises whereby individual incentives favor a banking system that is over-diversified with respect to the level of diversification that is desirable in the social optimum.

Introduction

The folk wisdom of “not putting all of your eggs in one basket” has been a dominant paradigm in the financial community in recent decades. Pioneered by the works of Markowitz (1952), Tobin (1958) and Samuelson (1967), analytic tools have been developed to quantify the benefits derived from increased risk diversification. However, recent theoretical studies have begun to challenge this view by investigating the conditions under which diversification may have undesired effects (see, e.g., Battiston, Gatti, Gallegati, Greenwald, Stiglitz, 2012b, Ibragimov, Jaffee, Walden, 2011, Wagner, 2011, Stiglitz, 2010, Brock, Hommes, Wagener, 2009, Wagner, 2010, Goldstein, Pauzner, 2004). These works have found various types of mechanisms leading to the result that full diversification may not be optimal. For instance, Battiston et al. (2012b) assume an amplification mechanism in the dynamics of the financial robustness of banks; Stiglitz (2010) assumes that the default of one actor implies the default of all counterparties.

Our paper is closely related to Wagner (2010). They study how the trade-off between diversification and the diversity of portfolio held by different banks may lead to full diversification being suboptimal from society’s perspective. The underlying assumption of their model is that in the case of a single default, the insolvent bank can sell its assets to the solvent bank and avoid physical (and more costly) liquidation. Since liquidation costs are not internalized by the banks, payoff maximizing and social optimal levels of diversification not necessarily coincide. This implies the existence of negative externalities among the banks, whereby by increasing a bank’s diversification level it also increases the possibility of costly liquidation of assets by the other, therefore augmenting the joint failures of systemic crises.

Differently from Wagner and from the aforementioned literature, this paper sheds light on a new mechanism under which a tension may arise between individual risk diversification and systemic risk. We demystify the effects of diversification in a complex context, where banks not only hold overlapping portfolios but also hold claims on each other liabilities. In this setting, a bank’s payoff not only depends on the bank own financial condition, but also on the financial conditions of the other banks to which it is interconnected. We model the default of a bank as a problem of first passage time in a network-based stochastic diffusion process where the valuation of a bank liabilities relies also on the liabilities of the other banks in its network. We can show that the ambiguous role of diversification on systemic risk is much more complex and pronounced than in the setting proposed by Wagner and it involves an amplification of losses along the chain of lending relations, which - depending on the interbank network configuration - may lead the optimal degree of diversification not to be internal, as in Wagner, but to be at the corner: either minimum or maximum.

In particular, this paper assumes an arbitrage-free market where price returns are normally distributed and uncorrelated. Moreover, there are negative externalities arising from the fact that banks interbank assets are negatively affected when a counterparty’s leverage increases. Banks internalize these externalities since their utility function is computed using a default probability model that accounts for the contagion from counterparties. However, the market may follow positive or negative trends that are ex-ante unpredictable and persist over a certain period of time. This incomplete information framework leads to a problem of portfolio diversification under uncertainty. In fact, portfolio returns display a bimodal distribution resulting from the combination of two opposite trends weighted by the probability of being either in a bad or in a good state of the world. We find that optimal diversification can be interior. This result holds both at the individual and at the social welfare level. Moreover, we find that individual incentives favor a financial system that is over-diversified with respect to what is socially efficient.

More in detail, we consider a banking system composed of leveraged and risk-averse financial institutions (hereafter, “banks”) that invest in two asset classes. The first class consists of debts issued by other banks in the network (hereafter, “interbank claims”). The value of these securities depends, in turn, on the leverage of the issuers. The second class represents risky assets that are external to the financial network and may include, e.g., mortgages on real estate, loans to firms and households and other real economy-related activities (hereafter, “external assets”). The underlying economic cycle is the primary source of external asset price fluctuations, but it is unknown ex ante to the banks and, with a certain probability p, it may be positive (hereafter, “uptrend”) or negative (hereafter, “downtrend”). In this paper, we focus on the effect of varying levels of diversification across external assets, while the interbank network is considered as given.

As a general property, diversification of idiosyncratic risks lowers the volatility of a bank’s portfolio of external assets and increases the likelihood of the portfolio to follow the economic trend underlying the price movements. Therefore, if the future economic trend is unknown and banks cannot divest for a certain period, risk diversification is beneficial when the economic trend happens to be positive, because it reduces the downside risk. In contrast, risk diversification is detrimental when the economic trend happens to be negative because it reduces the upside potential. As a result, the intuition would suggest that the optimal level of risk diversification is always interior and depends on the probability of the market trend to be positive or negative. However, this intuition arises from the logical fallacy that maximizing a convex combination of functions is equivalent to take the convex combination of the maxima, which is not correct in general.

As a first result, we show that optimal risk diversification is interior under certain conditions, but is not interior in general. In particular, it can be optimal for individual banks to pursue a full diversification strategy even when the downtrend is almost as probable as the uptrend. Because full diversification implies that all banks are exposed to the same shocks, the probability of a systemic default conditional to an individual default tends to one. This fact leads to our second result: in a wide range of parameters there exists a tension between the individual bank’s incentive to fully diversify and social optimum due to social costs associated to simultaneous defaults. Interestingly, the tension exists both when the optimal is interior and when it is an extreme point of the feasible range of m.

The network of interbank claims exposes banks to shocks on external assets held by their counterparties. Therefore, the network amplifies the effect of the negative trend, when it occurs, and the impact of shocks when banks have largely overlapping portfolios due to extensive diversification on external assets.1

One of the novelties of our work is the fact that the result about interior optimal diversification holds even in the absence of asymmetric information, behavioral biases or transaction costs and taxes. Moreover, we do not need to impose ad hoc asset price distributions as in the literature on diversification pitfalls in portfolios with fat-tailed distributions (to name a few, Zhou, 2010, Ibragimov, Jaffee, Walden, 2011, Mainik, Embrechts, 2012).

In our model, because external assets carry idiosyncratic risks, banks have an incentive to diversify across them. In this respect, similar to Bird and Tippett (1986), Elton and Gruber (1977), Evans and Archer (1968), Johnson and Shannon (1974), Statman (1987), we measure how the benefit of diversification vary as the external assets in an equally weighted portfolio is increased. This benchmark is the so-called 1/n or naive rule. However, because banks are debt financed, we depart from the methods of those previous studies by modeling risk not in terms of a portfolio’s standard deviation but in terms of the default probability. Indeed, in that literature the relationship between default probability and portfolio size has not been investigated in depth.

In order to investigate the notion of default probability in a system context we develop a framework in which banks are connected in a network of liabilities, similarly to the stream of works pioneered by Eisenberg and Noe (2001). However, that literature considers only the liquidation value of corporate debts at the time of default. In particular, in the works based on the notion of “clearing payment vector” (e.g., Cifuentes, Ferrucci, Shin, 2005, Elsinger, Lehar, Summer, 2006), the value of interbank claims depends on the solvency of the counterparties at the maturity of the contracts and it is determined as the fixed point of a so-called “fictitious sequential default” algorithm. Starting from a given exogenous shock on one or more banks, one can measure ex-post the impact of the shock in the system and investigate, for instance, which structure are more resilient to systemic risk (Battiston, Delli Gatti, Gallegati, Greenwald, Stiglitz, 2012a, Roukny, Bersini, Pirotte, Caldarelli, Battiston, 2013).

Our objective instead is to derive the default probability of individual banks, in a system context, that can be computed by market players ex-ante, i.e. before the shocks are realized and before the maturity of the claims. A related question was addressed in (Shin, 2008) where one assumes that asset values are random variables that move altogether according to a same scaling factor. The expected value of the assets is plugged into the Eisenberg–Noe fixed point algorithm yielding an estimate of the values of the liabilities before the observation of the shocks. However, the latter approach does not apply if assets are independent random variables and, more importantly, it does not address the issue of how the default probability of the various banks are related.

Strictly speaking, default means that the bank is not able to meet its obligations at the time of their maturity. Therefore, in principle, it does not matter whether, any time before the maturity of the liabilities, the total asset value of a firm falls beneath the book value of its debts as long as it can recover by the time of the maturity. In practice, however, it does matter a lot. This is the case, for instance, if the bank has also some short-term liabilities and short-term creditors decide to run on the bank. Indeed there is a whole literature that building on Black and Cox (1976) investigates the notion of time to default in various settings. Such notion extends the framework of Merton (1974) by allowing defaults to occur at any random time before the maturity of the debt, as soon as the firm assets value falls to some prescribed lower threshold.

Drawing inspiration from these approaches, in this paper we model the evolution over time of banks assets as stochastic processes where, at the same time, the value of interbank claims is a function of the financial leverage of the counterparties as reflected by the credit-liability network. Although from a mathematical point of view, the framework requires to deploy the machinery of continuous stochastic processes, this work offers a valuable way to compute the default probability in system context under mild assumptions. The default probability can be written in analytical form in simple cases and it can be computed numerically in more complicated cases. An underlying assumption in the model is to consider the credit spread of counterparties as an increasing function of their leverage, i.e. the higher the leverage the higher the credit spread. As a benchmark, in this paper we assume that such a dependence is linear.

In general, the framework developed here allows to investigate how the probability of defaults depends on certain characteristics of the network such as the number of interbank contracts and the number of external assets. In this paper, we focus on the diversification level across external assets and we look at the limit in which analytical results can be obtained. The assumption we make is that the interbank market is relatively tightly knit and banks are sufficiently homogeneous in balance sheet composition and investment strategies. Indeed, it has been argued that the financial sector has undergone increasing levels of homogeneity, Haldane (2009). Moreover, empirical evidence shows that bank networks feature a core-periphery structure with a core of big and densely connected banks and a periphery of smaller banks. Thus, our hypothesis of homogeneity applies to the banks in such a core (see, e.g., Elsinger et al., 2006; Iori, Jafarey, Padilla, 2006, Battiston, Puliga, Kaushik, Tasca, Caldarelli, 2012c; Fricke and Lux, 2015).

The paper is organized as follows. In Section 2, we introduce the model. Section 3 adopts a marginal benefit analysis by formalizing the single bank utility maximization problem with respect to the number of external assets in the portfolio. In Section 4 we formulate the utility maximization problem in the “social optimum” and compare to the utility maximization problem of individual banks. Section 5 concludes the paper and considers some policy implications.

Section snippets

Model

Let time be indexed by t ∈ [0, ∞] in a system of N risk-averse leveraged banks with mean-variance utility function. To ensure simplicity in notation, we omit the time subscripts whenever there is no confusion. For the bank i{1,,N}, the balance sheet identity conceives the equilibrium between the asset and liability sides as follows: ai=li+ei,t0where a:=(a1,,aN)T is the column vector of bank assets at market value. l:=(l1,,lN)T is the column vector of bank debts at book face value. There

Benefits of diversification in external assets

Similar to Bird and Tippett (1986), Elton and Gruber (1977), Evans and Archer (1968), Johnson and Shannon (1974), Statman (1987), in this section we measure the advantage of diversification by determining the rate at which risk reduction benefits are realized as the number m (≤ M) of external assets in an equally weighted portfolio is increased. In contrast with those studies, rather than minimizing the variance of the banks’ assets, we maximize their expected utility with respect to m. The

Private incentives vs. social optimum

In the following, we assume that there are social costs associated to the default of one or more banks due to negative externalities to the real economy that limited-liability banks commonly do not account for. We also assume that it is more costly to restore the functioning of the market in case of simultaneous defaults than in the case of isolated defaults. We intentionally leave the definition of social costs general as it may depend on the characteristics of the financial system under

Concluding remarks

This paper develops a new modeling framework in order to investigate the probability of default in a system of banks holding claims on each others as well as overlapping portfolios due to bank diversification in a market with a finite number of external assets.

While our framework, uniquely combining together network and portfolio theories, allows for the investigation of several cross-disciplinary questions, we focus on the core of tightly knit banks, i.e., w=1. In this setting, we study how

Acknowledgments

We are grateful to Andrea Collevecchio, Co-Pierre Georg, Martino Grasselli, Christian Julliard, Helmut Helsinger, Rahul Kaushik, Iman van Lelyveld, Moritz Müller, Paolo Pellizzari, Loriana Pelizzon, Didier Sornette, Claudio J. Tessone, Frank Schweitzer, Joseph Stiglitz, Jean-Pierre Zigrand and participants at various seminars and conferences where preliminary versions of this paper have been presented.

The first version of this work was written while S.B. and P.T. were employed at the Chair of

References (40)

  • H. Windcliff et al.

    The 1/n pension investment puzzle

    North Am. Actuar. J.

    (2004)
  • A.G. Atkeson et al.

    Measuring the Financial Soundness of US Firms 1926–2012

    (2013)
  • S. Battiston et al.

    Credit default cascades: when does risk diversification increase stability?

    J. Financial Stab.

    (2012)
  • P. Billingsley

    Convergence of Probability Measures

    (1968)
  • R. Bird et al.

    Note—Naive Diversification and Portfolio Risk—A Note.

    Manag. Sci.

    (1986)
  • F. Black et al.

    Valuing corporate securities: some effects of bond indenture provisions

    J. Finance

    (1976)
  • W. Brock et al.

    More hedging instruments may destabilize markets

    J. Econ. Dyn. Control

    (2009)
  • P. Collin-Dufresne et al.

    The determinants of credit spread changes

    J. Finance

    (2001)
  • J. Danielsson et al.

    Equilibrium asset pricing with systemic risk

    Econ. Theory

    (2008)
  • L. Eisenberg et al.

    Systemic risk in financial systems

    Manag. Sci.

    (2001)
  • Cited by (29)

    • Research on systemic risk in a triple network

      2023, Communications in Nonlinear Science and Numerical Simulation
    • Does diversification promote systemic risk?

      2022, North American Journal of Economics and Finance
      Citation Excerpt :

      However, data are limited to listed banks so that results depend on the effectiveness of the securities market. The models based on contagion risk simulate the contagion process among banks when faced with external shocks (Acemoglu et al., 2015; Roukny et al., 2018; Tasca et al., 2017). The results depend on the effectiveness of common risk exposures captured by the correlation of investment assets.

    • Network structure, portfolio diversification and systemic risk

      2021, Journal of Management Science and Engineering
      Citation Excerpt :

      Therefore, the first contribution is that we analyze the effect of portfolio diversification in different interbank networks. One kind of interbank network structure is studied in the existing literature on this topic (Maeno et al., 2012; Kobayashi, 2012, 2013; Tasca et al., 2017), and we expand this kind of research. As a second contribution, this paper considers two types of risk sources to investigate the effect of portfolio diversification.

    View all citing articles on Scopus
    View full text