Diversification disasters

Abstract

The recent financial crisis has revealed significant externalities and systemic risks that arise from the interconnectedness of financial intermediaries’ risk portfolios. We develop a model in which the negative externality arises because intermediaries’ actions to diversify that are optimal for individual intermediaries may prove to be suboptimal for society. We show that the externality depends critically on the distributional properties of the risks. The optimal social outcome involves less risk-sharing, but also a lower probability for massive collapses of intermediaries. We derive the exact conditions under which risk-sharing restrictions create a socially preferable outcome. Our analysis has implications for regulation of financial institutions and risk management.

Introduction

It is a common view that the interdependence of financial institutions, generated by new derivative products, had a large role in the recent financial meltdown. A primary mechanism is that by entering into sophisticated derivative contracts, e.g., credit default swaps (CDS) and collateralized debt obligations (CDO), financial institutions became so heavily exposed to each others’ risks that when a shock eventually hit, it immediately spread through the whole system, bringing down critical parts of the financial sector. Since it created this systemic failure, the interdependence was extremely costly from a social standpoint.¹

There is an opposite viewpoint, however, namely that such interdependence plays a much more functional role—that of diversification. To motivate this in a simple way, start with a case in which each financial firm holds a particular risk class with its unique idiosyncratic risk. Now allow the firms to form a joint mutual market portfolio, with each firm contributing its risky portfolio to the total and receiving back its proportional share of the total. Given a sufficient variety and number of risk classes, the firms may succeed in eliminating the idiosyncratic risks embedded in their individual portfolios. This is in line with the classical view in finance, that risk-sharing—i.e., diversification—is always valuable (see, e.g., Samuelson, 1967). Therefore, interdependence is valuable and, indeed, what we should expect. In practice, we may expect both effects to be present, i.e., by sharing risks, intermediaries decrease the risk of individual failure, but increase the risk of massive, systemic failure.²

Which factors determine the risks of systemic failures of financial institutions and the benefits of diversification? When do the risks outweigh the benefits? What are the policy implications of such a trade-off? In this paper, we analyze these questions by introducing a parsimonious model that combines the two aspects of diversification in an integrated analysis. Along the lines of the previous viewpoints, while individual institutions may have an incentive to diversify their risks, diversification creates a negative externality in the form of systemic risk. If all intermediaries are essentially holding the same diversified portfolio, a shock may disrupt all the institutions simultaneously, which is costly to society, since it may take time for the financial system, and thereby the economy, to recover. Specifically, the slow recovery time creates a significant and continuing social cost because the unique market-making and information analysis provided by banks and other intermediaries³ is lost until they recover; see Bernanke (1983). Indeed, Bernanke's concern with the social cost created by bank failures appears to have motivated many of the government bank bailouts.

We show that the costs and benefits of risk-sharing are functions of five properties of the economy. First, the number of asset classes is crucial: The fewer the number of distinct asset classes that are present, the weaker the case for risk-sharing. Second and third, the correlation between risks within an asset class, and the heavy-tailedness of the risks are important. The higher the correlation and the heavier the tails of the risk distribution, the less beneficial risk-sharing is. Fourth, the longer it takes for the economy to recover after a systemic failure, the more costly risk-sharing is and, fifth, lower discount rates also work against risk-sharing. We define the diversification threshold to be the threshold at which the cost to society of systemic failure begins to exceed the private benefits of diversification, and we derive a formula for the threshold as a function of these five properties.

The distributions of the risks that intermediaries take on are key to our results. When these risks are thin-tailed, risk-sharing is always optimal for both individual intermediaries and society. But, with moderately heavy-tailed risks, risk-sharing may be suboptimal for society, although individual intermediaries still benefit from it. In this case, the interests of society and intermediaries are unaligned. For extremely heavy-tailed risks, intermediaries and society once again agree, this time that risk-sharing is suboptimal.

One can argue that the focus of our study, moderately heavy-tailed distributions, is the empirically interesting case to study. It is well-known that diversification may be suboptimal in the extremely heavy-tailed case,⁴ and some risks may indeed have extremely heavy tails (e.g., catastrophic losses, in which case individual insurers may withdraw from the market precisely because the benefits of diversification are unavailable; see Ibragimov, Jaffee, and Walden, 2009). However, arguably most financial risks are moderately heavy-tailed, as shown by several empirical studies in recent years. Specifically, the rate at which a distribution decreases for large values, the so-called tail exponent, $\alpha$ (which we rigorously define in the paper), provides a useful classification of heavy-tailedness. Risks with $\alpha <1$ are extremely heavy-tailed, whereas risks with $1<\alpha <\infty$ are moderately heavy-tailed, and when $\alpha =\infty$, they are thin-tailed. Many recent studies argue that the tail exponents in heavy-tailed models typically lie in the interval $2<\alpha <5$ for financial returns on various stocks and stock indices (see, among others, Jansen and de Vries, 1991, Loretan and Phillips, 1994; Gabaix, Gopikrishnan, Plerou, and Stanley, 2006; Gabaix, 2009). Among other results, Gabaix, Gopikrishnan, Plerou, and Stanley (2006) and Gabaix (2009) provide theoretical results and empirical estimates that support heavy-tailed distributions with tail exponents $\alpha \approx 3$ for financial returns on many stocks and stock indices in different markets.⁵

Our analysis has implications for risk management and policies to mitigate systemic externalities. We show that value at risk (VaR) considerations lead individual intermediaries to diversify, as per incentives similar to those in the Basel bank capital requirements. Within our framework, however, the diversification actions may lead to suboptimal behavior from a societal viewpoint. It then becomes natural to look for devices that would allow individual firms to obtain the benefits of diversification, but without creating a systemic risk that could topple the entire financial system. In Section 4, we provide a framework to develop such solutions and provide specific proposals.

Our paper is related to the recent, rapidly expanding literature on systemic risk and market crashes. The closest paper is Acharya (2009). Our definition of systemic risk is similar to Acharya's, as are the negative externalities of joint failures of intermediaries. The first and foremost difference between the two papers is our focus on the distributional properties of risks and the number of risk classes in the economy, which is not part of the analysis in Acharya (2009). Moreover, the mechanisms that generate the systemic risks are different in the two papers. Whereas the systemic risk in Acharya (2009) arises when individual intermediaries choose correlated real investments, in our model the systemic risk is introduced when intermediaries with limited liability become interdependent when they hedge their idiosyncratic risks by taking positions in what is in effect each others’ risk portfolios. Such interdependence may have been especially important for systemic risk in the recent financial crisis. This leads to a distinctive set of policy implications, as we develop in Section 4.

Wagner (2010) independently develops a model of financial institutions in which there are negative externalities of systemic failures, and diversification therefore may be suboptimal from society's perspective. Wagner's analysis, however, focuses on the effects of conglomerate institutions created through mergers and acquisitions and the effects of contagion. Furthermore, the intermediary size and investment decisions are exogenously given in Wagner's study, and only a uniform distribution of asset returns is considered. Our model, in contrast, emphasizes the importance of alternative risk distributions and the number of risks in determining the possibly negative externality of diversification. The two studies therefore complement each other.

A related literature models market crashes based on contagion between individual institutions or markets. A concise survey is available in Brunnermeier (2009). Various mechanisms to propagate the contagion have been used. Typically it propagates through an externality, in which the failure of some institutions triggers the failure of others. Rochet and Tirole (1996) model an interbank lending market, which intrinsically propagates a shock in one bank across the banking system. Allen and Gale (2000) extend the Diamond and Dybvig (1983) bank run liquidity risk model, such that geographic or industry connections between individual banks, together with incomplete markets, allows for shocks to some banks to generate industry-wide collapse. Kyle and Xiong (2001) focus on cumulative price declines that are propagated by wealth effects from losses on trader portfolios. Kodes and Pritsker (2002) use informational shocks to trigger a sequence of synchronized portfolio rebalancing actions, which can depress market prices in a cumulative fashion. Caballero and Krishnamurthy (2008) focus on Knightian uncertainty and ambiguity aversion as the common factor that triggers a flight to safety and a market crash. Most recently, Brunnermeier and Pedersen (2009) model a cumulative collapse created by margin requirements and a string of margin calls.

The key commonalities between our paper and this literature is the possibility of an outcome that allows a systemic market crash, with many firms failing at the same time. Moreover, as in many other papers, in our model there is an externality of the default of an intermediary—in our case, the extra time it takes to recover when many defaults occur at the same time. The key distinction between our paper and this literature is, again, our focus on the importance of risk distributions and number of asset classes in an economy. Thus, a unique feature of our model is that the divergence between private and social welfare arises from the statistical features of the loss distributions for the underlying loans alone. This leads to strong, testable implications and to distinctive policy implications. In our model, these effects arise even without additional assumptions about agency problems (e.g., asymmetric information) or third-party subsidies (e.g., government bailouts). No doubt, such frictions and distortions would make the incentives of intermediaries and society even less aligned.

The paper is organized as follows. In the next section we introduce some notation. In Section 3, we introduce the model, and in Section 4 we discuss its potential implications for risk management and policy making. Finally, some concluding remarks are made in Section 5. Proofs are left to a separate appendix. To simplify the reading, we provide a list of commonly used variables at the end of the paper.