PERSPECTIVE doi:10.1038/nature09659 Systemic risk in banking ecosystems Andrew G. Haldane1 & Robert M. May2 In the run-up to the recent financial crisis, an increasingly elaborate set of financial instruments emerged, intended to optimize returns to individual institutions with seemingly minimal risk. Essentially no attention was given to their possible effects on the stability of the system as a whole. Drawing analogies with the dynamics of ecological food webs and with networks within which infectious diseases spread, we explore the interplay between complexity and stability in deliberately simplified models of financial networks. We suggest some policy lessons that can be drawn from such models, with the explicit aim of minimizing systemic risk. I n the 1960s, the notion of the ‘balance of nature’ played a significant part as ecologists sought a conceptual foundation for their subject. In particular, Evelyn Hutchinson1, following Elton2, suggested that ‘‘oscillations observed in arctic and boreal fauna may be due in part to the communities not being sufficiently complex todamp out oscillations’’. He went on to state, based on a misunderstanding of MacArthur’s3 paper, that there was now a ‘‘formal proof of the increase in stability of a com- munity as the number of links in its food web increases’’. To the direct contrary, however, a closer examination of model eco- systems showed that a random assembly of N species, each of which had feedback mechanisms that would ensure the population’s stability were it alone, showed a sharp transition from overall stability to instability as the number and strength of interactions among species increased. More explicitly, for N ? 1 this transition occurs once ma2 . 1, where m is the average number of links per species, and (6) a their average strength4. In ecology this has, since the 1970s, prompted a search for special food-web structures that may help reconcile complexity with persistence or stability5–8. Along these lines there is, for example, tentative evidence for modularity9 (particularly in plant–pollinator associations, where linkages tend to be overdispersed or disassociative), and more generally for nested hierarchies in food webs10. The fact that some features of the network structure of interactions (such as predator/prey ratios) inferred from the Burgess Shale communities are similar to those in present day ones11 reinforces hopes that this is a meaningful area of research. In the wake of the global financial crisis that began in 2007, there is increasing recognition of the need to address risk at the systemic level, as distinct from focusing on individual banks12,13. This quest to understand the network dynamics of what might be called ‘financial ecosystems’ has interesting parallels with ecology in the 1970s. Implicit in much eco- nomic thinking in general, and financial mathematics in particular, is the notion of a ‘general equilibrium’. Elements of this belief underpin, for example, the pricing of complex derivatives. But, as shown below, deeper analysis of such systems reveals explicit analogies with the con- cept that too much complexity implies instability, which was found earlier in model ecosystems. There are, of course, major differences between ecosystems and financial systems. For one thing, today’s ecosystems are the winnowed survivors of long-lasting evolutionary processes, whereas the evolution of financial systems is a relatively recent phenomenon14. Nor have selective pressures been entirely dispassionate, with the hand of govern- ment a constant presence shaping financial structures, especially among institutions deemed ‘‘too big to fail’’15. In financial ecosystems, evolu- tionary forces have often been survival of the fattest rather than the fittest. In what follows, we first consider the role of the growth in intrafinancial system claims in generating bank failure and instability, focusing on the problemsinherentinprevailingmethodsofpricingcomplexderivatives,or arbitrage pricing theory (APT). Second, we sketch various ways in which such an initial bank failure, or ‘shock’, may propagate to cause cascades of subsequent failure. Third, we outline some tentative policy lessons that mightbedrawnfromthesedeliberatelyoversimplifiedmodels.Last, weask how we might reshape thefinancial system to realizethe economicbenefits individual banks can deliver, while at the same time paying deliberate and explicit attention to their system-wide stability. Potential causes of an initial shock Events external to the banking system, such as recessions, major wars, civil unrest or environmental catastrophes, clearly have the potential to depress the value of a bank’s assets so severely that the system fails. Although probably exacerbated by such events, including global imbalances (China as producer and saver, the United States as consumer and debtor), the present crisis seems more akin to self-harm caused by overexuberance within the financial sector itself. Perhaps as much as two-thirds of the spectacular growth in banks’ balance sheet over recent decades reflected increasing claims within the financial system, rather than with non- financial agents. One key driver of this explosive intrasystem activity came from the growth in derivative markets. In2002, when WarrenBuffetfirstexpressedhis viewthat‘‘derivatives are financial weapons of mass destruction’’16, markets—although booming— seemed remarkably stable. Their subsequent growth, illustrated in Fig. 1, has been extraordinary, outpacing the growth in world gross domestic product (GDP) by a factor of three. In some derivatives markets, such as credit default swaps (CDS), growth has outpaced Moore’s Law. These developments contributed significantly towards an unprecedented influx of mathematically skilled people (quantitative analysts) into the financial/ banking industry. These people produced very sophisticated tech- niques (including APT), which seemingly allowed you to put a price on future risks, and thus to trade increasingly complex derivative contracts— bundles of assets—with risks apparently decreasing as the bundles grew. However, recent empirical and theoretical studies have indicated that the trading activity associated with derivatives can have significant effects onmarkets17–19. Morespecifically, Brock and colleagues20 have shownthat proliferation of hedging instruments can destabilize markets. Building on this, Caccioli and colleagues21 note that APT makes several conventional assumptions upon which everything else depends: ‘‘perfect competition, market liquidity, no-arbitrage and market completeness’’. Crucially, this adds up to the implicit assumption that trading activity has no feedback on the dynamical behaviour of markets. And indeed, in the APT-fuelled 1 Bank of England, Threadneedle Street, London EC2R 8AH, UK. 2 Zoology Department, Oxford University, Oxford OX1 3PS, UK. 2 0 J A N U A R Y 2 0 1 1 | V O L 4 6 9 | N A T U R E | 3 5 1 Macmillan Publishers Limited. All rights reserved ©2011

boom timethatprecededthebust, APTseemed tobeverysuccessful.Inits imaginary world, market failures are caused by regulatory carelessness, resulting in a focus on creating institutional arrangements that seek to guarantee the premises upon which APT is based22. To the contrary, Caccioli and colleagues argued21 that APT is not a ‘theory’ in the sense habituallyusedinthesciences, butrather asetofidealizedassumptionson which financial engineering is based that is, APT is part of the problem itself. Caccioli and colleagues21 illustrate their point by exploring the dynamical properties of a model that gives a more realistic caricature of markets, going beyond the idealized world of APT to include the effects of individual trades on prices. Prices now depend on the balance between demand and supply. The outcome is that ‘‘the road to efficient, arbitrage-free, complete markets can be plagued by singularities which arise upon increasing financial complexity’’21. Figure 2 illustrates the main results of the analysis by Caccioli and colleagues21. Here n is essentially a measure of the proliferation of deriva- tives or similar financial instruments, and s is the overall average value of thesupplyofany onesuchderivative/financialinstrument.The parameter e encodes the risk premium that banks require for trading derivatives21. We see from Fig. 2 that if n is less than n* (here n* 5 4.14), the average supply of derivatives, s, is relatively steady and essentially independent of the banks’ risk premium (as measured by e). But as market complexity increases, so that n approaches n*, there is a sharp singularity at e 5 0. For n . n*, the average supply increases with increasing n (that is, increasing proliferation of financial instruments) if e . 0. Conversely, for e , 0 the supply decreases with increasing complexity once n . n*. It is emphasized21 that such sensitivity in market behaviour in the neigh- bourhood of the singularity can easily produce very strong fluctuations— either positive or negative—in the volume of trading in derivative markets. Note that the consequences of this singularity are not easily intuited from the competitive equilibrium setting. It seems to us that the basic process—in grossly simplified terms—is that once there are enough deri- vatives to span the space of available states of nature (the net supply of derivatives within the system necessary to meet true hedging demand from non-banks), the market is essentially complete in the sense of the Arrow–Debreu23 model. Once that happens, gross derivatives positions within the system are essentially unbounded. So long as there is an incentive to supply new instruments—a positive premium to trading— banks will continue to expand gross positions, independent of true hedging demand from non-banks. Such trades are essentially redundant, increasing the dimensionality and complexity of the network at a cost in terms of stability, with no welfare gain because market completeness has already been achieved. Caccioli andcolleagues21 alsoexaminea measure ofmarketvolatility as the risk premium parameter e varies. If they calculate this quantity under the approximation that the fluctuations in the values of the individual ‘supply variables’ (si derivatives, etc) are completely uncorrelated, they in effect recover the happy world of APT, with no singularities. This strongly indicates that the highly important singularities in their accurate and self-consistent calculations, with market dynamics included, are associated with the supplies of different derivatives being strongly corre- lated in this domain, as has found to be the case among derivatives markets in practice. In summary, Caccioli and colleagues suggest that the idealized assumptions upon which recent financial engineering has been based can give a misleading account of potential instabilities in markets. They also note that these instabilities echo those that can develop in ecosys- tems as complexity increases4,24. Propagation of shocks within financial systems In ecology’s models of food webs, aimed at qualitative understanding of their dynamical response to perturbation, the nodes are simply species, linked to other nodes/species as prey, predator, competitor or mutualist. Inepidemiologicalnetworks,thenodesaresusceptible,infected/infectious or recovered/immune individuals linked by sexual or other contacts. But in a minimally realistic caricature of financial networks—henceforth called banks—the nodes have a more complex structure. Following Nier and colleagues25 and Gai and Kapadia26, we define such a bank/node as schematically illustrated in Fig. 3. In this deliberately oversimplified scheme, a bank’s activities are partitioned among four categories. Two represent assets: interbank loans (li) and external assets (ei). The other two represent liabilities: interbank borrowing (bi) and deposits (di). The subscript i labels the specific bank (i 5 1, 2, ..., N for a total of N banks). Solvency requires that the difference between a bank’s assets and its liabilities (the capital reserve or ‘net worth’, labelled ci in Fig. 3) be positive. That is, cI (ei 1 li) 2 (di 1 bi) $ 0. These banks are now assumed to be interlinked in a random, Erdo˝s– Renyi ´ network, with any one of the N banks connected to any other as lender or borrower, or possibly both, each with probability p. A bank’s average number of incoming/borrowing or outgoing/lending links is then z 5 p(N 2 1). Various further assumptions are now made to carry these Bank of England/Federal Reserve Bank of New York models to the point where the knock-on effects of a single bank failure can be explored in numerical simulations.Muchoftheessentialfindingsofsuchstudiescanbecaptured, and made more transparent, by a ‘mean-field’ approximation in which each bank has exactly average behaviour27. This means all banks are the same size (rescaled to 1), every bank is linked to exactly z others, all loans 0.9 ε = 0.1 ε = –0.1 ε = 0.01 ε = –0.01 0.8 0.7 0.6 s 0.5 0.4 0.3 0.2 0.1 0 1 10 n Figure 2 | Discontinuous transition to instability of derivatives as complexity increases. Average supply of any one derivative, s, at competitive equilibrium as a function of the number, n, of different derivatives being traded, for various values of banks’ risk premium, e. Adapted with permission from ref. 21. For fuller discussion, see text. 700 600 500 400 US$ trillions 300 200 100 0 1998 1999 2000 2001 2002 2003 2004 Year 2005 2006 2007 2008 2009 Figure 1 | Notional principal value of outstanding derivative contracts, as recorded at year end. These include foreign exchange, interest rates, equities, commodities and credit derivatives. Data from UK Department for Business, Innovation and Skills, International Monetary Fund and Bank of England calculations. RESEARCH PERSPECTIVE 3 5 2 | N A T U R E | V O L 4 6 9 | 2 0 J A N U A R Y 2 0 1 1 Macmillan Publishers Limited. All rights reserved ©2011