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«Abstract This paper argues that in environments that exhibit non-convexities, risk diversification by financial institutions may be socially ...»

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Inefficient Diversification∗

Kostas Bimpikis† Alireza Tahbaz-Salehi‡

This version: April 2014

First version: November 2012


This paper argues that in environments that exhibit non-convexities, risk diversification by

financial institutions may be socially inefficient. To this end, we study a stylized, micro-founded

model in which individual banks have an incentive to hold diversified portfolios in order to mini-

mize the likelihood of bank runs. Yet, at the same time, diversification may increase the aggregate risk faced by the banks’ depositors, creating a negative externality. The increase in systemic risk in such an environment is due to the fact that even though diversification decreases the proba- bility of each bank’s failure, it may increase the probability of joint failures, which may be socially inefficient when the depositors are risk-averse. The presence of such externalities suggests that financial innovations that enable banks to engineer more diversified portfolios may have non- trivial welfare implications.

Keywords: Diversification, financial intermediation, systemic risk, bank runs.

JEL Classification: G01, G11, G21.

∗ We are particularly thankful to Daron Acemoglu and Asuman Ozdaglar for their invaluable help and feedback. We also thank Paul Glasserman, Martin Oehmke, Ali Shourideh, Gerry Tsoukalas, Wolf Wagner, and Pierre Yared for helpful comments. All remaining errors are ours.

† Graduate School of Business, Stanford University.

‡ Columbia Business School, Columbia University.

1 Introduction The proliferation of new complex financial instruments can be considered as one of the most signif- icant events in finance over the past two decades. According to the conventional wisdom, financial innovations in the form of credit default swaps and similar products have enabled financial institu- tions to diversify risk more effectively and as a result, have increased the efficiency of the system as a whole. In fact, the potential benefits of such instruments served as one of most important rationales for deregulation in the decade prior to The Financial Crisis of 2007–2009.1 On the other hand, and more recently, it has also been argued that these instruments may have contributed to the fragility of the financial system by reducing the diversity among different institutions. For example, as argued by Andrew Haldane, the Executive Director of Financial Stability in the Bank of England, “diversifica- tion strategies by individual firms generated a lack of diversity across the system as a whole,” leading to a financial system that exhibited “both greater complexity and less diversity,” at the detriment of the system’s stability (Haldane (2009, p. 8)).

In this paper, we argue that the presence of non-convexities in the economy (e.g., due to the possibility of costly bank runs), creates a wedge between the investment incentives of financial institutions and their depositors’ welfare, thus, implying that risk diversification may indeed be inefficient from a social welfare point of view.

We study a stylized economy consisting of two competitive banking sectors, owned by riskneutral bankers, and a mass of risk-averse consumers, who are subject to idiosyncratic preference shocks as in the canonical model of Diamond and Dybvig (1983). In order to ensure themselves against these shocks, the consumers deposit their funds in the banks, who would then invest in a common pool of assets on their behalf in exchange for standard demand deposit contracts. We further assume that even though the final returns on the banks’ investments are observable, their investment decisions are not contractible.

Given that the banks are subject to runs when the returns on their investments are below a certain level, bankers have an incentive to choose diversified portfolios, as diversification decreases the

probability of a run on each of the banks. Yet, such diversified portfolios may be socially inefficient:

more diversification implies that the returns on the banks’ portfolios would become more correlated, and hence, the probability of simultaneous bank runs would increase. We show that if the depositors are risk-averse, the welfare loss due to joint failures may outweigh the gains in reducing the probability of individual bank runs, implying an inefficient equilibrium.

Excessive equilibrium diversification in our model relies on a number of key ingredients. First and foremost, the possibility of costly bank runs creates non-convexities in the mapping from the banks’ investment returns to the payments to the depositors upon withdrawal. In the absence of such non-convexities, the incentives of the banks and their depositors are always fully aligned, and thus, the equilibrium and efficient levels of diversification would coincide. The second ingredient is the assumption that the banks invest in a common pool of assets, which implies that they cannot 1 See, for example, Greenspan (1997) and Financial Crisis Inquiry Commission (2011).

2 construct diversified portfolios without increasing the correlation between their returns. Clearly, if each bank has access to a different set of assets with independent returns, diversification does not lead to correlated portfolios. The third ingredient is the risk-aversion of the depositors. If the depositors are risk-neutral, their incentives (as far as the extent of the diversification is concerned) would be aligned with those of the bankers. Yet, risk-aversion guarantees that, all else equal, the depositors would be worse off as the probability of simultaneous runs increases. Finally, the fact that the banks’ investment decisions are not contractible implies that bankers do not internalize the adverse effects of their diversification on depositors.

We remark that in our model, there are no externalities among the banks. Rather, it is the negative externality of the banks’ decisions on the depositors that is the source of inefficiency. In particular, with diversified portfolios, a negative return on only one of the assets may lead to simultaneous runs on both banks; an outcome that would have been avoided with no diversification. Thus, effectively, by choosing more diversified (and hence, more similar) portfolios, the banks reduce the set of contingencies in which the depositors are (at least, partially) paid above the liquidation value.

Our analysis highlights that regulatory mechanisms that focus on each bank’s risk in isolation may not be sufficient for mitigating risks at a systemic level. Rather, an effective regulatory policy may need to take the endogenous correlations between different banks’ portfolios into account.2 Our results also suggest that financial innovations that enable banks to engineer more diversified portfolios, may indeed lead to lower social welfare, as the probability that several financial institutions default together during periods of financial distress may increase. This observation thus suggests that the well-known benefits of financial innovations (such as more efficient levels of risk-sharing) may be inseparable from a “curse of financial engineering” manifested in the form of higher systemic risk.

Related Literature Several papers, such as Acharya (2009), Ibragimov, Jaffee, and Walden (2011) and Wagner (2010, 2011) study the possible adverse effects of diversification on increasing systemic risk. Like the current paper, this literature emphasizes the fact that joint failures of financial institutions may create a higher social cost compared to that of individual failures. The key common assumption in these papers is that a systemic market crash creates direct externalities between different intermediaries, leading to welfare losses. For example, Ibragimov et al. (2011) study a model in which the rationale for diversification is weakened in the presence of heavy-tailed risks and high correlations between risks within an asset class. The key assumption in their model is that joint failures of intermediaries would lead to slower recovery of the financial system, hence creating larger social costs. In contrast to this literature, no such inter-bank externalities exist in our model. Rather, the welfare loss is due to the fact that the banks do not internalize the impact of their joint investment decisions on the depositors.

Recently, Wagner (2010) shows that diversification may not be desirable if liquidation is more 2 We emphasize that similar observations have been made by others. See, for example, Acharya, Pedersen, Philippon, and Richardson (2010), Adrian and Brunnermeier (2011) and Brunnermeier, Gorton, and Krishnamurthy (2011).

3 costly in a systemic crisis in which multiple banks have to liquidate assets at the same time. In particular, he shows that if the banks’ liquidation costs depend on the extent of liquidation by other banks — for example, due to pecuniary externalities — then the negative externality that banks impose on one another guarantees that banks’ portfolio choices would be inefficient. In a related paper, Wagner (2011) explores the diversification-diversity tradeoff in the presence of pecuniary externalities, and shows that the risk of joint liquidation creates an incentive for investors to forgo the benefits of diversification and hence, optimally choose heterogenous, not fully diversified portfolios. In contrast to these works, our model exhibits no such pecuniary externalities as liquidation costs do not depend on other banks’ portfolios, returns, or liquidation decisions. Rather, it is the non-convexity of the mapping from the assets’ returns to the payments to the depositors that is driving our results.

Furthermore, in contrast to Wagner’s, banks in our model find it optimal to hold fully diversified portfolios to minimize the probability of a run.

The paper is also related to recent works by Stiglitz (2010a,b), who, by the means of a reducedform model, argues that full integration of global financial markets can exacerbate contagion, whereas capital controls can be welfare enhancing. We build a micro-founded model which properly accounts for the benefits and costs of diversification in a banking context and show that non-convexities create a wedge in the incentives of banks and depositors. As a result, under fairly general conditions, the equilibrium level of diversification does not coincide with the efficient level.

Our paper is also related to Acharya and Yorulmazer (2007, 2008), who study a model in which banks have an ex ante incentive to herd and increase the likelihood of joint failures in order to induce a bailout by the government. Somewhat relatedly, Farhi and Tirole (2012) argue that untargeted policy instruments used by a central bank during the times of financial distress, such as lowering the Fed Funds rate, would incentivize the banks to take on too much correlated risk. A different strand of literature, such as Kubler and Schmedders (2012) and Simsek (2013), studies the potential negative implications of financial innovations on portfolio risk and asset price volatility when agents have heterogenous beliefs. Also relevant is Buffa (2013), who develops a structural model of credit risk in which asset value dynamics are endogenously determined by optimal portfolio allocation. He shows that in the presence of systemic externalities, strategic considerations lead financial institutions to adopt polarized and stochastic risk exposures, but without sacrificing full diversification. Finally, our paper is related to the broader literature on systemic risk and financial contagion, such as Shaffer (1994), Allen and Gale (2000), Goldstein and Pauzner (2004), Dasgupta (2004), and more recently, Acharya et al. (2010) and Allen, Babus, and Carletti (2012), among others.

Outline of the Paper The rest of the paper is organized as follows. We first present a simple reducedform variant of the model in order to illustrate the key economic forces at play. Section 3 contains the full-fledged micro-founded model. Our main results are presented in Section 4. Section 5 concludes. All proofs and some other mathematical details can be found in the Appendices.

42 Reduced-Form Model

This section contains a simple reduced-form version of the model, which we use to illustrate the key economic forces at play. The complete micro-founded model is presented and analyzed in the subsequent sections.

Consider an economy consisting of two risk-neutral financial institutions (henceforth, banks for short), indexed by a and b, and a risk-averse representative consumer who has deposited a unit of endowment in each of the banks. Each bank can invest the deposits in a distinct, “non-overlapping” set of finitely many assets with independent returns. The banks are subject to default (for example, due to a run) if the returns on their investments are below some exogenously given threshold d, in which case they make zero profits. On the other hand, if the realized return of the bank’s investment is larger than d, it obtains a constant payoff of R 0. Thus, bank i ∈ {a, b} chooses a portfolio that minimizes its default probability, pi.3 The payoff to the representative depositor also depends on the returns of the banks’ portfolios.

In particular, if the realized returns of a bank is larger than d, the depositor receives a return of c, whereas she only gets α c in case of default.4 Thus, the expected utility of the representative depositor can be written as

Vn = u(2c) + (pa + pb ) u(c + α) − u(2c) + pa pb u(2c) + u(2α) − 2u(c + α),

which is strictly decreasing in both pa and pb. This immediately implies that when the sets of assets in which the banks invest in do not overlap, the incentives of the banks and the depositor are fully aligned: all parties prefer investments that reduce the probability of individual defaults. In particular, if diversified portfolios decrease the probability of an individual failure, diversification is desirable both from the banks’ and the social welfare points of view.

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