Systemic financial risk: agent based models to understand the leverage cycle on national scales and its consequences

Transcription

1 IFP/WKP/FGS(2011)1 MULTI-DISCIPLINARY ISSUES INTERNATIONAL FUTURES PROGRAMME OECD/IFP Project on Future Global Socks Systemic financial risk: agent based models to understand te leverage cycle on national scales and its consequences Stefan Turner Section for Science of Complex Systems, Medical University of Vienna, Austria International Institute for Applied Systems Analysis (IIASA) Laxenburg, Austria Tis report was written by Stefan Turner as a contribution to te OECD project Future Global Socks. Te opinions expressed and arguments employed erein are tose of te autor, and do not necessarily reflect te official views of te OECD or of te governments of its member countries. Contact persons: Pierre-Alain Scieb: +33 (0) , pierre-alain.scieb@oecd.org Anita Gibson: +33 (0) , anita.gibson@oecd.org 14t January

2 TABLE OF CONTENTS EXECUTIVE SUMMARY... 5 ABSTRACT... 5 Wat is at stake in a financial crisis?... 5 To do someting about crises you need to understand tem... 5 Te need for Agent Based Models (ABM)... 6 Agent Based Models... 6 Mecanics of ABM for te Financial Market... 7 How Does a Cras Occur?... 8 Main findings... 8 Main conclusions Inerent limits of ABM Recommended researc directions INTRODUCTION Wat is leverage? Te danger of leverage Scales of leverage in te financial industry Scales of leverage on national levels Failure of economics and te necessity of agent based models of financial markets Secondary Impacts and te Global Context A SIMPLE AGENT BASED MODEL OF FINANCIAL MARKETS Overview Te specific model economy Variables in te agent based model Price formation Uninformed investors (noise traders) Informed investors (edge funds) Investors to investment fund Setting maximum leverage Banks extending leverage to funds Defaults Return to edge fund investors Simulation procedure Summary of parameters and teir default values An ``ecology'' of financial agents Generic results

3 3 EVOLUTIONARY PRESSURE FOR INCREASING LEVERAGE HOW LEVERAGE INCREASES VOLATILITY Te danger of pro-cyclicality troug prudence Te process of de-leveraging LEVERAGE AND SYSTEMIC RISK: WHAT HAVE WE LEARNED? Triggers for systemic failure Counter intuitive effects IMPLICATIONS - FUTURE RESEARCH QUESTIONS Te need for global leverage monitoring Te need for understanding network effects in financial markets Te need for linking ABMs of financial markets to real economy and social unrest Systemic risk is not priced into margin requirements Extensions to leverage on national levels Data for leverage-based systemic risk on national scales OUTLOOK: TRANSPARENCY AND CONSEQUENCES FOR A NEW GENERATION OF RISK CONTROL Breaking te spell: need for radical transparency Self-regulation troug transparency: an alternative regulation sceme Toward a 'National Institute of Finance' SUMMARY REFERENCES APPENDIX A: SOCIAL UNREST AND AGENT BASED MODELS A.0.1 Patways toward social unrest: Linking financial crisis and social unrest troug agent based frameworks A.0.2 Patways to social unrest not directly related to financial crisis Figures Figure 1: (a) Demand function of a fund equation. (b) Market clearing Figure 2: Wealt time series Figure 3: Time series of a fund for its demand, wealt, leverage, capital flow, loan size and bonds Figure 4: (a) Mean of a fund conditioned on its wealt. (b) average mispricing Figure 5: Distribution of log returns Figure 6: Log-return time series Figure 7: Anatomy of a cras Figure 8: Results of a numerical experiment Figure 9: Comparison between constant maximum leverage (red circles) vs. adjustable leverage based on recent istorical volatility, Eq. 24 (blue squares) Figure 10: Comparison of regulated leverage providers to an unregulated one size of investment firms. investment forms Figure 11: Comparison of te effect of regulated (red) versus unregulated (blue) banks on aggressive funds

4 Figure 12: Davies J-curve Table 1: Summary of parameters used in te model

5 EXECUTIVE SUMMARY ABSTRACT Dr. Stefan Turner s report analyses te development, buildup and unfolding of financial market crases using a dynamic agent based model. Te model simulates relevant conditions under wic agents beave in financial markets and allows one to see ow excessive leverage (te use of credit for speculative investments) is a prime source of vulnerability and can act as te catalyst for a cras. Te model demonstrates ow crases result as a consequence of syncronization effects of te different actors in financial markets. Te report concludes wit recommendations, most notably te need for radical improvements in te transparency of creditor- debtor relationsips between financial market participants, wic would reduce te cance of syncronization. Wat is at stake in a financial crisis? Tis report analyses te results of simulations using an agent based model of financial markets to sow ow excessive levels of leverage in financial markets can lead to a systemic cras. In tis scenario, plummeting asset prices render banks unable or unwilling to provide credit as tey fear tey migt be unable to cover teir own liabilities due to potential loan defaults. Weter an overleveraged borrower is a sovereign nation or major financial institution, recent istory illustrates ow defaults carry te risk of contagion in a globally interconnected economy. Te resulting slowdown of investment in te real economy impacts actors at all levels, from small businesses to omebuyers. Bankruptcies lead to job losses and a drop in aggregate demand, leading to more businesses and individuals being unable to repay teir loans, reinforcing a downward spiral tat can trigger a recession, depression or bring about stagflation in te real economy. Tis can ave a devastating impact not only on economic propserity across te board, but also consumer sentiment and trust in te ability of te system to generate long-term wealt and growt. To do someting about crises you need to understand tem Tere is no global consensus on ow best to manage or prevent financial crises from appening in te future. Noneteless, policymakers need to develop strategies tat are designed to at least reduce te severity of teir impacts. In order to stregten te financial system, making it more resilient to potential abuses in te future, it is critical tat we understand te mecanism by wic pervasive practice like 5

6 excessive leverage can present system-wide danger. Unlike forest fires or oter sudden-onset natural disasters, financial markets are complex social systems, built on te repeated interaction of millions of people and institutions, and are subject to systemic failure,currency crises, bank runs, stock market collapses, being just a few of te terms indicative of te recent fallout. Often, tis risk arises not troug te failure of individual components in te system, suc as te closing or collapse of a single bank or major financial institutions, but rater due to te erd beavior and network effects contained in te actions of large fractions of market participants. Suc syncronization in beavior leads to many people underaking te same actions simultaneously, wic can exacerbate price movements and te overall level of volatility in te system. Tis can significantly weaken a well-functioning financial market, potentially causing te booms, busts, and bankruptcies tat are now familiar. Te need for Agent Based Models (ABM) Recent experience as called into question te previously widely-accepted belief tat te economy traveled on a steady-state growt pat, wit minor flucuations above and below a stable growt rate. Te global financial crisis as led many to suggest tat te booms and busts tat ave caracterized our global economy over te past two decades may be te norm rater tan te exception. As a result, te traditional general equilibrium models, wic ave guided our economic tinking are increasingly suspect. Tese models prove vulnerable to te erd beavior, information assymetries, and externalities tat may quickly catapult minor fluctuations into widespread market failure. Systemic risk factors like tose embedded in cases of ig leverage, suc as syncronization effects, strong price correlations, and network effects are of great concern to investors, and sould concern policymakers wen attempting to re-design policies capable of avoiding future disasters. Te approac of many financial institutions, politicians, and investors to risk management in te industry as been built upon suc traditional equilibrium concepts, wic ave been proven to fail spectacularly wen most needed. Among te most promising approaces to understanding systemic risk in complex systems are agent based models (ABM), a class of models used to explain certain penomena via a bottom-up approac wic, contrary to general equilibrium teory, does not require a steady state, but rater structures interactions between non-representative agents via a set of beavior rules. Tis report demonstrates tat ABM offer a way to systematically understand te complicated dynamics of financial markets, including te potential for downward spirals to be generated by te syncronised actions of economic actors. In particular, te report sows tat ABM could explicity describe te peculiar danger of using leverage to fund speculative investments. It also demonstrates ow levels of leverage in te system are directly linked to te probability of large scale crases and collapse, suc tat price volatility rises wit leverage and even minor external events can trigger major systemic collapses in tese environments. It suggest te importance of understanding te endogenous social dynamics of traditionally financial arena in order to more fully grasp te systemic risk for wic gradual improvements in traditional economics cannot yet account. Agent Based Models To better approximate real-world outcomes wit stylized models, ABM run repeated simulations, generating data weter tere currently is too little for meaningful analysis, allowing agents to cange teir 6

7 beavior based on interactions wit oter agents, leading to a ost of important findings. By running and analysing millions of simulations, eac approximating millions of potential interactions between actors, insigt is gained into te collective outcome of te system. Tis can be furter analyzed by canging te decision rules affecting tose interactions and even applying different regulatory measures to gauge teir potential effects. Applied to te financial system, tis process reveals ow te likeliood of default among investment funds depends eavily on tese regulations, specifically a maximum permissible leverage level. Similarly, te analysis also sows ow banking regulations suc as te Basel accords can influence default rates. Te models demonstrate tat suc regulatory measures can, under certain circumstances, lead to adverse effects, wic would be ard to foresee oterwise. Tese simulations also allow te possibility to follow te progress of a new regime installed after a sockoccurs, and analyze its effect on recovery rates, typical wealt re-distributions, etc. In sort, ABM are scaled-down, stylized versions of igly intricate and interdependent systems werein one can develop a better understanding of te dynamics of complex systems, suc as financial markets. Mecanics of ABM for te Financial Market Te model includes several different types of agents or economic actors wose interactions result in te economic activity being modelled. Tese actors include investors, banks, investment companies, and regulators, among oters. One of te allmarks of ABM in general is te use of non-representative agents. In oter words, tese agents will differ in several key respects, including teir tolerance for risk, size of endowment, and teir general investment strategy. Tese differences weig on te interactions between te agents producing te types of outcomes tat ave macro-effects. In tis model, tere are tree basic types of investors: informed investors, uninformed investors, and investors in investment management companies. Informed investors (e.g. edge funds) do researc to guide teir investment decisions and typically try to discover assets tat are under- or overpriced by te market to take advantage of arbitrage opportunities. Intuitively, uninformed investors do not carry out any analysis to discover te true price of an asset, but rater, for te sake of argument, place essentially random orders. Te tird type of investor places is or er funds in investment companies tat ten invest te money in financial markets. Tese investors monitor te performance of te company and adjust teir strategy based on tis performance relative to teir alternatives. Tese investors all compete in te same market; teir relative sares depend on teir relative performance. Banks and regulators play a different role in tis model. Banks provide te liquidity, providing credit tat allows investors to build leverage. Regulators present te constraints wic restrict te extent of leverage allowable. In tis model market, te different types of investors are buying and selling a single asset, wic pays no dividends for te purpose of simplicity. Bot uninformed investors and informed investors compute teir respective demands for te asset (i.e. ow many sares tey would like to purcase). Te difference lies in te metod. Te uninformed traders are relatively unsopisticated, wile informed traders attempt to determine a mispricing signal, namely by ow muc te asset is under-valued or overvalued by te marketplace (i.e. ow big is te opportunity for arbitrage). Investors survey te performance of eac fund and invest or witdraw money from tese funds accordingly. Meanwile, banks and te regulatory 7

8 constraints set te maximum levels of leverage available to traders. Tis process continues and firms wose wealt falls below a certain level fold and are quickly replaced. Wit an ABM, tis process is repeated many times over. How Does a Cras Occur? In scenarios of ig leverage, investors can overload on risky assets, betting more tan wat tey actually ave to gamble. Altoug tis is an obviously dangerous practice, it also creates a tremendous level of vulnerability in te system as a wole. Two events in particular can lead to a devastating collapse of a system under te weigt of significant levels of leverage: 1. Small, random fluctuations in te demand of an asset by uninformed investors can cause te asset price to fall below its true value, leading to te development of a mispricing signal. 2. In order to exploit tis opportunity for arbitrage, te investment funds capitalize on te ig allowable leverage levels to take massive positions. Tis combination is essentially a potential recipe for disaster. If te uninformed or noise traders appen to sell off a bit of tis asset and te price drops furter, te funds stand to lose large amounts of money. Tis prompts te firms to take even greater amounts of leverage, wile oter firms are even forced to begin selling off more of te asset to satisfy teir margin requirements. Naturally, tis will ave te effect of furter depressing te price, resulting in a vicious cycle of price drops, greater leverage, and more enforced selling of te asset (margin calls). Over a sort period of time, wat seemed like a stable system can cascade into a scenario in wic te asset price crases, causing major losses and bankruptcies by igly leveraged firms. Importantly, only in systems caracterized by ig levels of leverage can suc small canges trigger suc catastropic collapses. Main findings Repeated observations resulting from millions of controlled simulations, varying key parameters along te way, lead to several key findings. 1. In an unregulated world tere are market pressures leading to iger leverage levels, i.e. tere are incentives for bot investors and banks to increase leverage. 2. Tere exist counterintuitive effects of regulation: implementing different regulation scenarios, suc as te Basel accords, are demonstrated to work well in times of moderate leverage, but deepen crisis wen leverage levels are ig. Tis is due to enanced syncronization effects induced by te regulations. 3. Te actions and relative performance of different market participants influences te actions of oters, creating an ecology of market participants, wic as effects on asset prices. Tese 8

9 mutual influences cause fluctuations wic are observed in reality, but are ignored in traditional economics settings. 4. Volatility amplification: even value-investing strategies, wic are supposed to be stabilizing in general, can massively increase te probability of systemic crases given tat leverage in te system is ig. 5. Wile ig levels of leverage are directly correlated wit te risk of major systemic draw-downs, moderate levels of leverage can ave stabilizing effects. 6. Dynamics of crases: by studying te unfolding of crases over time, triggers can be identified. Small random events, wic are completely armless in situations of moderate leverage can become te triggers for downward spirals of asset prices during times of massive leverage. 9

10 Main conclusions Te findings ave several potential ramifications for public policy and te re-organization of te financial system into a more resilient and more stable generator of long-term wealt for society: 1. Global monitoring of leverage levels at te institutional level is needed and could be performed by central banks. Tis data sould be made more accessible for researc, peraps wit a time delay. Witout tis knowledge, te practice of imposing and executing maximum leverage levels is less effective. 2. Continuous monitoring and analysis of lending/borrowing networks is needed, bot of major financial players and of governments. Exactly wo olds wose debt sould be made known to te public, and not just te financial institutions involved. Witout tis surveillance, imposing meaningful and credible maximum leverage levels is difficult to implement. 3. Imposition of maximum leverage levels sould depend on debt structure, trading strategies and relative positions in lending/borrowing networks. It is clear tat government regulation can play an important role in controlling portfolio risk levels by limiting leverage. Yet, tis is no trivial undertaking as fully analyzing te trend-following strategies of investors would be very difficult due to te seer variety of market participants and instruments currently being traded, particularly under today's disclosure standards. Tere may be more scope in providing more regulatory guidance to banks on disclosing teir leverage allowances to certain types of investors. Inerent limits of ABM Agent based models are undoubtedly useful tools in engineering a unique approac to understanding complex, dynamic systems. Yet, tere are sortcomings to tis approac, ranging from te ig degree of tecnical expertise necessary to issues of data, quality standards, validation, qualitative understanding. 1. Data requirements: ABMs elp to identify relevant data. Usually tis data is not readily available in te required form. Data is needed often in te form of networks, suc as asset/liability networks, and tose of ownersip and financial flows, etc. 2. Quality standards: te seer number of parameters usually necessary for robust results makes ABM opaque. It can be ard to judge weter: a. Unrealistic ad-oc assumptions were made tat could lead to unrealistic effects b. Parameters are introduced tat are inaccessible in reality c. Initial conditions for computer simulations are actually realistic 1. Linking to reality: ABMs presently elp to understand systemic properties qualitatively. To make tem quantitatively useful ABMs must be scaled wit real data. Tis requires tremendous joint 10

11 efforts in scientific researc, data generation, and institutional cooperation. Unless tese efforts are undertaken, ABMs will remain largely descriptive. 2. Validation: ABMs sould not be expected to make actual predictions of financial crases or collapse. Using massive computer simulations, ABMs may elp to clarify levels of risk under given circumstances. Tey may illustrate relevant mecanisms tat represent legitimate precursors to a cras tat could oterwise go undetected. Recommended researc directions ABMs ave been elpful in acieving a qualitative understanding of financial system dynamics and unexpected systemic effects, as well as spotting lurking dangers witin te regulatory framework or lack tereof witin a system. To take advantage of te full potential of ABMs (i.e. to make tem effective tools for assisting actual decision-making processes) it is necessary to scale tem up and combine tem wit real data. Tis implies a significant coordinated scientific effort, bot in econometric modelling and te generation and collection of quality data. Some of te tangible steps tat must be undertaken include: 1. Data collection, storage, quality and availability: massive efforts sould be undertaken in economic data collection, under strict quality requirements wic ten remain available for analysis by a large scientific community. 2. Network studies in economics: muc of te systemic effects in economics ave teir origin in networks linking agents. Tese networks are largely left unexplored, but tey are an essential input for modelling dynamic systems. Networks need to be analyzed according to te standards of network teory, wic ave been developed over te past decade. 3. Simulation platforms of financial regulation: consequences of exogenous events suc as 'canges of rules' or disclosure requirements are only te beginning in te development of a systematic understanding. Tis field of researc sould be significantly boosted. 4. Linking to te real economy: ABMs of te financial markets sould be paired wit ABMs of oter aspects of te real economy, to understand teir mutual influences and feedback loops. Critical points of interface between financial markets and te real economy can be identified as potential avenues for beneficial regulation. 11

12 1 INTRODUCTION 1.1 Wat is leverage? Leverage on te personal scale, in te financial markets and on a national scale - as contributed muc - if not most - to te financial crisis , for scientific literature, see [1, 2, 3, 4]. Leverage is generally referred to as using credit to supplement speculative investments. It is usually used in te context of financial markets, owever te concept of leverage covers a range of scales from personal life, suc as buying a ome on credit, to governments issuing public debt. Leverage - in te sense of making speculative investments on credit - is used in a wide variety of situations. It can be used for realizing ideas wit credit, start up a company, finance a freeway or job-less programs, etc. Te outcome of tese investments is in general not predictable, tey are speculative. Te price of te ome canges according to trends in te ousing market, a new company can fail, and a freeway migt not deliver te economic stimulus wic was initially anticipated. No matter of ow tese investments pay off, te associated debt as to be re-paid, in form of principle and interest. In principle te risk associated wit leverage is te risk of unrecoverable credits. Tis by itself is a trivial observation. Te nontrivial aspects of leverage arise troug its potential to cause, amplify and trigger systemic effects, wic can turn into extreme events wit severe implications on e.g. te world-wide economy. Leverage introduces 'interactions' between market participants wic - under certain circumstances - can become strong, so tat te system starts to syncronize. Syncronized dynamics can lead to large-scale effects. Te risk associated wit a potential downturn dynamics is te systemic risk of collapse. Tis makes te issue of leveraged investments a typical complex systems problem, wic can not be treated wit te traditional tools of mainstream economics. Te term financial leverage is owever mostly used in te context of corporations, financial firms and financial markets. Tere it is mostly associated wit te assets of a firm wic are financed wit debt instead of equity. One of te important motivations for tis is tat returns can be potentially amplified. Imagine a edge fund wit an endowment of 100 $ makes investments wic yield an annual return of 10 %, i.e. 10$. Te wealt of te fund is now 110$. Imagine now a scenario wit leverage: te fund approaces a bank, and asks for a credit of 900$ at a rate of 5% p.a. It ten makes te same investments wit 1000$. At te end of te year te fund earned 100$, i.e. it now manages 1200$. It pays back te 900$ to te bank plus 45$ of interest. Te wealt of te fund is now 255$ - it grew by a factor of 255 %. Te situation looks less brilliant if in case of te leveraged scenario te investments would turn out to be -10 %. At te end of te year te fund would manage 900$, it would pay back te 900$ but could not service te interest, te fund is bankrupt, te bank as to write off te loss of 45$. For notation purposes, let us discuss some widely used measures and notions of financial leverage. Measures of leverage: Debt to equity ratio Tis is te ratio of te liabilities (of e.g. a company) wit respect to its (e.g. te 900 sareolders') equity. In our example above, te debt to equity ratio would be = 9. Anoter 100 frequently used measure is te Debt to value ratio, relating te total debt to te total assets. Total assets are composed of equity and debt. Assets to equity ratio Tis is referred to as te leverage level wic is denoted by. leverage = = Assets Equity. 12

13 Tis definition IS USED in all of te following. Talking about values of portfolios and cas positions te same definition reads = portfoliovalue portfoliovalue cas. In tis notation it is always assumed tat te cas reserves of an investor will always be used up for te investment first. Only wen tese reserves are used up, te investor searces for credit. In te above notation credit is seen as a negative cas position (loan). In te above example te leverage of te fund is = = = 10. As anoter example consider a omeowner taking out a loan using say a ouse as collateral. If te ouse costs $100 and e borrows $80 and pays $20 in cas, te margin (aircut) is 20%, and te loan to value is $80/$100 = $80%. Te leverage is te reciprocal of te margin, namely te ratio of te asset value to te cas needed to purcase it, or =100/20= 5. Or in te oter notation 100 = = Debt to GDP ratio In macroeconomics on national scales, a key measure of leverage is te (sovereign) debt to GDP ratio. Construction leverage Leverage tat is obtained troug specific combinations of underlyings and derivatives [9]. De-levering is te act or sum of acts to reduce borrowings. It is noteworty tat taking leverage can be associated to a moral azard problem: Managers can leverage to increase stock returns of teir companies wic amplifies gains (and losses) wic are related e.g. to teir bonuses. Gains in stock are often rewarded regardless of metod [10]. 1.2 Te danger of leverage If you speculate wit more tan you own, you can lose more tan you ave Witout leverage, by speculative investments one can lose all one as, te maximum loss is 100%. Te danger of leverage is tat one can lose more tan tis. Tis leads to a loss wic can drive te debtor out of business. Under normal conditions tis loss is ten taken (realized or paid) by someone, te creditor. In case te creditor becomes illiquid as a consequence of tis loss, it is passed on. Often to some oter institution or te public. Tis risk of generating losses exceeding te wealt of te debtor originates traditionally from te following fact: credit for speculative investments (leverage) requires collateral. Tis collateral is often given in te form of te acquired investments. For example if a edge fund uses leverage from a bank to buy stocks, tese stocks are eld by te bank as te collateral for te credit. Or if a omeowner buys a ome wit a mortgage, is bank will usually (co-)own te ome as collateral. Imagine te omeowner buys a ome wit a value today at 100$. He finances it wit a down payment of 10%. He owes te bank 90 $, is wealt in form of te ouse is 10$. Suppose te ousing market goes bad and witin a year te value of te ouse falls to 80$. His wealt is now 10 $, and declares bankruptcy. Te bank takes te loss by selling te ouse for 80$ and by writing off te 10$ of credit as not recoverable. As a note on te side, a classical story during te last decade was worse. It could ave been te following. A omeowner buys a ome wit a value at 100$ under te same conditions as above. Te value constantly goes up, say to 150$ 13

14 witin several years. He tinks real estate is a good investment. His wealt is now 60$ and e decides to buy a bigger ome wit tese 60$ as a 10% down payment. He sells te old ouse and takes 540$ credit for te new one. Te bank is perfectly willing to do te deal. Te same price drop appens as before, te value of te ouse drops from 600 to 480$. Te bank becomes nervous and asks te credit back, te omeowner sells te ouse for 480$ and owes te bank 120$, more tan te cost at ig time of is first ouse! He files for bankruptcy. Te bank takes te loss. Te example sows an important aspect to bear in mind for later consideration: during boom times, e.g. a ousing bubble, credit is easy to obtain. Prolonged growt and increase in value of collateral is tacitly assumed. Credit is provided - credit providers participate in te boom. In times of crisis as nervousness increases, leverage tigtens and as a consequence of de-levering losses are realized. During boom times te sustainable value of collateral can be strongly distorted. Its combination wit leverage becomes especially dangerous Te role of leverage in normal times Let us now focus on leverage take by financial institutions, suc as edge funds, insurances, investment banks etc. Usually leverage is used for professional, expert investments by informed investors. By informed investors is meant actors wose investments follow a period of researc on financial assets. For example a value investing edge fund finds out troug researc tat a given company as better prospects tan te financial assumes. Te fund determines te (subjective) 'true' value of te company. If tis value is iger tan te observed market price, te fund will try to go long in te assets of te firm until te financial market as a wole prices te assets correctly. Te fund will profit from te increase of asset price, and - if it was a rational fund - would sell te assets as te market price its te 'true' value. In tis case te researc invested in estimating te true value better tan te oters will result in profit for te fund. Note ere tat te value investing fund performs a price stabilizing role, wic can be seen a service to financial markets. As it goes long in under-priced assets and sort (if it can) in over-priced ones, due to its demand it drives te price of te asset toward its 'true' value. Note furter tat te 'true' value is a subjective estimate of investors. It can be wrong. It not only depends on te objective status of te company under investigation, but also on te beaviour of te collective of te oter investors in te same asset. Tere are of course situations, wen te collective of investors does not price te asset correctly over a longer time scale. An example wen tis is not te case is wen a bubble is present. It can be tat a burst of te bubble will eventually bring te price of a firm to its 'true' value, but te time scale can be too long to make profits for te fund. Even by finding te correct (absolute) 'true' value te fund can make losses. It is essential to also get te reaction time (time-scale) of te oter market participants rigt. It is caracteristic for times of crisis (panic) tat tese time-scales become sort, and dynamics becomes syncronized. Tis price stabilizing role of value investing investors is a service to markets in general since it reduces volatility of asset prices. Value investors reduce volatility. However, reduced volatility is not necessarily a sign of efficient markets. Tere is danger to te view tat low levels of volatility indicate a functioning market its management and its regulation. Tis as been a fundamental misconception, wic was famously declared by A. Greenspan [11]. It is well known tat oter investment strategies, suc as trend following can ave a price destabilizing effect. Trend followers reinforce price movements in any direction, leading eventual price corrections to oversoot. Trend followers play a role in bubble formation in financial markets. Tey are self-reinforcing: detecting an upward trend generates demand in trend followers, leading to increase of price, wic reinforces te trend. Te same is true for downward trends. For institutions interested in 14

15 moderate levels of volatility (suc as regulators, governments, real sector) it would be natural to consider options to limit trend followers. A andle for tis would be to impose maximum leverage levels for trend following strategies. Finding standards for controlling trend following strategies of investors would be ard to obtain under today s circumstances. However, one could tink of disclosure procedures for banks wen tey act as leverage providers to trend followers. Quantification could appen by imposing disclosure of investment portfolios, and performance on a relatively sort timescale. Knowledge of stabilizing versus destabilizing forces would be essential information for regulators To estimate te contribution of price stabilizing forces from value investors versus te destabilizing ones from trend followers, it would be essential to know te fraction at a given point in time of te demand generated by te respective groups. However, te fraction of value investors to trend followers is ard - maybe impossible - to find out witout furter disclosure requirements for (large) investors or leverage providers. At tis point agent based models could become of great importance. Te fraction of demand of value investors to trend followers is not constant, but depends on te performance of te two groups. If trend following is a good strategy at a time, it will attract more trend followers, and vice versa. Tis affects not only te asset prices but te relative fractions. Price and group sizes are co-evolving quantities, influencing eac oter. Agent based models are ideal to treat problems of tis kind. In suc models, see e.g. [12], populations of value- and trend following investors can be simulated in computer 'experiments'. Investors can switc between value investing and trend following (and oter strategies). Tese agents submit demand functions at various time-scales, a mecanism of e.g. market clearing leads to a unique (simulated) asset price. Given te demand functions of funds and te price at all times allows to compute te performance of te funds. Funds of one type wo perform bad wit respect to te oter type will eventually switc strategies. In tis way one obtains fluctuating group sizes of valueand trend following agents. Teir fraction can be compared wit volatility patterns in te price. Te influence of exogenous interventions suc as limits of leverage to one group or bot can be systematically studied. Tis field of researc pioneered by J.D. Farmer et al., see e.g. [12] sould be significantly boosted and sponsored Te role of leverage in times of crisis We continue te following discussion for leverage-taking value investors in financial markets. Te same basic structure olds true for leverage in e.g. te ousing markets, or on national levels. Te situation for trend following investors is less interesting since tey are mostly price destabilizing. By looking at value investors one investigates 'te best of possible worlds'. A fraction of trend following investors will make te situation worse. Here one looks at an idealized 'optimistic' scenario to identify te fundamental dangers. Imagine a leverage-taking value investor. Te leverage provider (e.g. a bank) allows a range of credit, caped by a maximum level of credit, say a maximum leverage of 5. If te fund finds no under priced assets on te market, its demand is zero, it will not be long in any asset. As soon as it finds under priced assets it will buy and old tem, until tey approac teir 'true' value. Te actual demand in tat asset depends generally on two aspects: First, ow big is te mispricing. Te more te distance of te observed market price from te 'true value', te larger is te demand in te asset - given te fund is beaving rationally. Note tat tis is an optimistic assumption - certainly very often massively violated by numerous investors. Second, ow muc does te investor 'believe' te 'true value'. If it believes strongly tat e computed or assessed te mispricing correctly, it will increase its demand strongly wit every unit of mispricing. Tis is termed an 'aggressive' investor. If e is not so sure about te correctness of te mispricing, se will not increase er demand as rapidly, se is a 'moderate' investor. 15

16 Now assume tat an investor finds an investment opportunity e.g. se detects an under priced asset. Given er level of aggression, se first invests a fraction of er wealt in te opportunity. Assume te mispricing gets bigger (price falls). Te investor takes bigger and bigger positions in te asset (now making losses!). As all er funds are used up for tese positions, te investor approaces a bank for credit, and keeps leveraging, until te demand is satisfied, or until no more credit is available. Tis appens in tis example at a leverage level of 5. If te price goes up, te investor makes profit under leverage. If te price continues to go down, te bank will now ensure tat te maximum leverage level is not exceeded. It will start recalling its credit. Tis is called a margin call. At er maximally allowed leverage level, te investor is now forced to sell assets on te market to maintain te maximum level of leverage. Te investor reduces its demand, even toug its investment strategy would suggest to increase it. Selling assets tend to drive te price down. If te investor olds a significant fraction of te asset, tis price drop can turn out to be self-reinforcing (selling into a falling market), and it can be forced to sell off assets again to keep te maximally allowed leverage. A single fund is owever unrealistic altogeter. To come toward a systemic understanding of te danger of leverage, let us look at an example sligtly more complicated: assume 2 funds, one aggressive, one moderate. Bot ave detected a mispriced asset, and old long positions. Te aggressive fund as reaced its maximum leverage level 5. Te price drops sligtly due to some external effects. Te aggressive fund now as to sell, kicking te price down a little too. Tis drop in price lowers te value of te portfolio of te moderate fund, and - at same credit volume wit its bank - experiences its leverage going up. Assume te price cange to be so big, tat te moderate fund now reaces its credit limit. It terefore as to sell off assets into te falling market too. It is clear ow tis can lead to cascading effects to all leveraged investors wo are invested in te same asset. However, tese effects are of course not limited to te same asset. Tere are spill over effects. Assume te example as above. Te aggressive fund, if it is invested in more tan one asset, as te coice to sell oter assets in its portfolio. If it as to sell off at massive rates, te sold assets will drop. Funds invested in tese assets migt ten be forces to srink teir portfolios. Te same cascading effect takes place as above, but now involving many assets. Note, tat tese assets migt be oterwise completely unrelated. Even if te assets would usually tend to sow no correlations (even anticorrelations in returns), as a result of te cascading deleveraging and spill over effects, teir correlations in (negative) returns will become positive, possibly large. Cointegration emerges, a systemic scenario unfolds. Let us continue te example on a systemic. During te selling into falling markets, a sufficiently leveraged fund can face illiquidity quickly. Its default and bankruptcy can create losses for te bank since it will in general not get te extended credit back entirely. Note tat te collateral for te credit te banks often olds as te assets in its possession. Te credit tat can get paid back is e.g. generated troug te selling of tese assets. Te loss is usually taken by te bank. If tese losses become severe, eiter because te bank as extended leverage to a variety of funds (wo now all perform badly), or because te bank itself (as a 'edge fund of its own') as made investments in now falling assets, banks temselves can - and will - get under stress and default. Naturally banks are connected troug mutual credit relations. Tese asset and liability 'networks' are extremely dense, and can contribute a furter layer of systemic risk [13, 14]. Tese networks ave been empirically studied for an entire economy in [15, 17, 16, 18, 19, 20]. It was found tat tey typically display a scale-free organization and are consequently igly vulnerable against te sortfall of one of te 'ubs', i.e. te big players. Realistic networks of tese kinds ave been used to simulate contagion effects and contagion dynamics troug tese banking networks [17]. It was sown tat te most sensible measure wic correlated to te systemic danger of an individual bank witin te system, is te betweenness centrality of te bank [21]. Banks wit ig betweenness centrality can bring down te entire banking system - due to consecutive situations of illiquidity in te system - given tere are no interventions. We 16

17 refrain from commenting on effects of te real economy under collapse of te financial system. Due to te importance of te matter, academic researc as to be boosted in tis direction. Systemic effects of te network of leverage providers - and banks need to be better understood Insurance of risks to leveraged investments Te objective of te following two suggestions for mandatory leveraged investment edges is to force te default risk onto financial markets, and to avoid te present situation tat effectively te public sector as to serve as te ultimate risk taker for leveraged financial speculations. In teory potential losses of leverage providers can be edged (insured). Again sticking to te scenario of financial markets. Tere are two ways of strategies for edging. One is insuring te collateral, te oter insures te credit. Witin te current financial framework bot ways are possible in principle but suffer severe problems wic make tem questionable in practice. In particular tese safety measures are costly. Effectively investors would eiter ave to pay iger interest rates for leverage to te leverage provider - wo will roll over its edging costs to te investors, or will directly pay te premiums for credit default swaps. Tis implies tat leverage becomes effectively less attractive for bot sides: te provider as te burden of monitoring investor's performance, of designing adequate edges and as to bear its immediate costs. Te investors suffer less favourable interest rates for te used leverage, or tey ave to pay premiums for CDSs wic works against te investors performance. New role of regulator: enforce edges for leveraged investments Hedging collateral: If collateral for leveraged investments are e.g. stocks, tey can be edged by conventional derivatives. Premiums for te appropriate derivatives will be paid for by te older of te collateral, e.g. te banks. Te costs for tese edges will ave to be rolled over to te investor in form of iger interest rates. It is possible tat tese rates become unattractively expensive. Naturally, it must be avoided tat te leverage provider itself is e.g. engaged in derivative trading, tat it could for example write options. It is necessary tat te risk is transferred from te leverage provider institution to te financial market. Te implementation of mandatory collateral edging of te leverage provider could be implemented troug an extended disclosure requirements for banks. Tey would report te collateral and te size of te extended credit, togeter wit te type of edge to te regulator or Central Bank. Tese requirements impose indirect costs to te leverage providers and would make leverage more expensive. It would be imperative tat all providers to leverage ave te same disclosure requirements. Hedging te credit: Te actual risk of credit default from te side of te investor could be dealt wit e.g. credit default swaps (CDS). Tese CDSs would be sold on te financial market. Te premium for tese instruments would be paid by te investors. Markets would estimate, assess and monitor te default risk of individual investors and would determine te corresponding premiums. Tis could lead to an implicit 'rating' of investors. In particular te prices for CDSs for te different investment companies could be used as an indicator of te perceived (from te market) level of aggressiveness. Naturally, more aggressive funds will ave iger default rates. Te issue of implementation of disclosure rules for suc a strategy in today's financial markets could be more associated wit tecnical problems. Investors would ave to declare te actual credit Te difficulty wit CDSs is, as for te collateral edge, tat te portfolio and te debt structure cosen by te investor can cange on relatively sort timescales. To avoid costly permanent readjustments it could become necessary to use more complex instruments tan ordinary CDSs. Maybe even new instruments for exactly tis purpose could be introduced. 17

18 Counterparty risk: Realistic or not, tese measures for insurance will work only as long as te corresponding counterparties are liquid. Te associated risk of financial collapse of counterparties is called counterparty risk. In te case of te collateral edge te risk is tat te issuer of te option disappears, in case for te credit edge, tere is te default risk of te buyer of te CDS. Usually, counter party risk is ard to anticipate because a wide range of scenarios can lead to te disappearance of tem. It is consequently almost impossible to price counterparty risk and to edge for it. Insurance possibilities against systemic collapse do not exist witin te financial system itself. From a systemic point of view leverage-investment edges could be igly desirable for two reasons: First, tey would transfer te credit / collateral risk from te leverage provider to te financial market. It would reduce te probability for te public sector to cover losses in form of e.g. bank defaults. Second, tey would make leverage more expensive and less attractive and would reduce its use to relatively save investments. To estimate te influence of edge costs on leveraged investments, again, agent based models could be te metod of coice to incorporate systemic and non-linear effects, wic would be not trackable witin traditional game teoretic or equilibrium approaces. It is a priory not clear to wat extend te practice of leverage will get reduced depending on te edging costs. It is a researc question in its own rigt to tink about te consequences of making disclosed data of CDS premiums or leverage levels of institutions available to all market participants. 1.3 Scales of leverage in te financial industry Typically leverage becomes ig in boom times and lowers in bad times. Tis can imply tat in boom times asset prices are overpriced and too low during crisis. Tis is te leverage cycle [11]. Leverage dramatically increased in te United States from 1999 to A bank tat wanted to buy a AAA-rated mortgage security in 2006 could borrow 98.4% of te price, using te security as collateral and pay only 1.6% in cas, i.e. a leverage ratio of 60. Te average leverage in 2006 across all of te US$2.5 trillion of so-called toxic mortgage securities was about 16, meaning tat te buyers paid down only $150 billion and borrowed te oter $2.35 trillion. Home buyers could get a mortgage leveraged 20 to 1, a 5% down payment. Security and ouse prices soared. Today leverage as been drastically curtailed by nervous lenders wanting more collateral for every dollar loaned. Tose toxic mortgage securities are now leveraged on average only about 1.5 to 1. Home buyers can now only leverage temselves 5 to 1 if tey can get a government loan, and less if tey need a private loan. De-leveraging is te main reason te prices of bot securities and omes are still falling. Te leverage cycle is a recurring penomenon, [22] A typical scale of leverage in investment banks just before te crisis was almost about a factor of 30. Many of tese borrowed funds to invest in so-called mortgage-backed securities. Leverage levels of te 5 largest investment firms in te US rose from about 17 in 2003 to about 30 in Te risks of tese leverage levels are reflected in te fate of tese companies in late Leman Broters went bankrupt, Merrill Lync and Bear Stearns were bougt by oter banks, or canged to commercial bank olding companies, as Morgan Stanley and Goldman Sacs. A typical value of leverage obtainable for trading in foreign excange markets is a factor of 100, meaning tat wit one dollar owned, one can speculate wit 101 dollars. 1.4 Scales of leverage on national levels For most western economies present leverage levels range between 50 and 100 % of GDP, Japan being iger at around 200 %. To wic extend suc levels are sustainable under crisis - i.e. at times wen governments migt be confronted wit major bank bailouts, or wit te fact tat outstanding debt from oter governments as to be written off (in case of defaulting countries) - is under eavy debate. It is 18

19 noteworty to mention tat sovereign governments in control of teir monetary policy, ave tree ways of managing teir debt wen under severe stress. Tey ave tree possible mecanisms for de-leveraging: tey can (i) default, tey can (ii) inflate teir debt away (tis is e.g. not possible for countries in te Euro zone), or tey could finally - given a competent and efficient government and public support - (iii) decrease spending and increase production and efficiency. Wat are Agent Based Models? Agent based models allow to simulate te outcome of te aggregated results of social processes, resulting from te overlay of actions of many individual agents. Individual agents beave according to a set of local rules, or randomly. Usually tis means tat an agent - wo interacts wit oter agents - can take decisions, wic in general depend on (or are determined to some extend by) te interactions wit te oters. Random decisions of agents are decisions wic are uncorrelated to interactions wit oters or oter events. For an example tink of a simple opinion formation model. All agents can take one of two decisions: vote for A or vote for B. Every agent as several friends. All friendsips are recorded in te friendsip network. At every time step of te model a particular player X cecks ow many of is fiends say, tey would vote for A. If a big fraction of is friends vote for A, player X will also say in te next time step tat e would also vote for A, even toug maybe in te previous time step e as voted for B. His local neigbourood (better te state of is neigbourood) made im cange is mind. Tis is done simultaneously wit say 1 million agents, and for 1000 times. It can now be cecked, ow often a majority for A forms. Tis can ten be repeated under different initial conditions - i.e. ow many people voted for A in te first time step. It can furter be cecked ow te connectivity of te friendsip network influences te probability for a final outcome of A winning, etc. In tis example a random decision of an agent would be tat e votes for A or B depending on trowing a coin and not copying wat te majority of is friends are doing. Tis model is clearly not of practical interest ere, but serves to illustrate te structure of an agent based models: Compute te probability of an outcome (election of A or B) under te knowledge, tat people ave a given communication network, and tat tey tend to copy te beaviour of teir friends. Te basic idea is tat often one as a clear concept of ow most people act locally under given circumstances, but te outcome of te collective action is unknown. Suc models can be seen as 'in-silico experiments'. Often tey are te only source to understand a problem. In te above example it is impossible to let people vote a tousand times. In computer experiments tis is no problem and can nowadays be carried out on laptops. It is clear tat agent base models only work if te basic rules of interaction (ere adaptation of opinions) is to a large scale correct. If tis part is modelled erroneously, te outcome can only be of little or zero value Wat tey are Agent based models consist of a set of agents of a certain type. In tis work, te different agent types will be banks, informed investors, investors to funds, regulators, etc. Usually every type is populated wit many of tese agents. Tese agents are in mutual contact wit eac oter. A bank lends to an investor wic leads to a money flow from bank to investor. Te investor invests te money in te stock market wic leads to a money flow from te investor to anoter agent owing te stock, etc. Te agents all are equipped wit beavioural rules, e.g. tat if te bank recalls borrowed money at a particular time, te borrower will pay it back in te next time step. If tis rule is broken, tere are oter rules wic manage te breaking of rules. For example if a loan is not paid back because te borrower illiquid, te borrower is declared bankrupt. Bankruptcy implies e.g. tat all assets of te borrower will be sold at te stock market, te proceeds will be sared between te lenders, te difference to te outstanding loans will be written off by te banks. Te essence of agent based models is to model te interactions of te agents rigt. In a 19

20 typical agent based model tere can be several dozen of rules wic govern tese interactions. It migt seem tat te given set of rules implies a deterministic model. Tis is by no means so. For example irrationality of agents can be modelled by random decisions of agents, wic again mimics reality well. In tis way models can mimic ypotetical worlds of only rational agents and ow tey differ from situations were a given fraction of decisions are made by pure coice. Te latter can be used to model decisions wit incomplete knowledge, ignorance, etc Wat can tey do tat oter approaces can not? Agent based models allow for estimating probabilities of systemic events, i.e. large scale collective beaviour based on te individual outcomes of locally connected agents. Teir prime advantage is tat tey can generate data were tere is none or too little to understand te system. Te collective effects can be studied by running touands of simulations. A sense can be reaced about te effect tat canging te interaction of rules between agents as on te collective outcome of te system. For example in tis work, te likeliood of default of investment funds is sown to depend on regulation, e.g. if a maximum leverage is imposed. Likewise it can be sown ow banking regulations (suc as te Basel accords) influence default rates. Wit tese models it can be demonstrated tat suc regulators measures can - under certain circumstances - lead to adverse effects, wic would be ard to guess oterwise. Tey allow to understand te concrete unfolding of events leading to a systemic cange in te system. For example te origin, te development and te unfolding of a financial crisis can be related to all model parameters. In reality usually tere is never all te relevant data present, wic would allow you to identify te relevant parameters. Te relevant parameters can be identified as te relevant ones. Also it is possible to follow te establisment of a new regime after a sock appened, suc as recovery rates, typical wealt re-distributions, etc. Some typical concrete questions tat can be answered wit an agent based model of financial markets would be: Can it be judged if one set of regulations is more efficient tan anoter? How can systematic dangers be identified? Wat precursors ave to be monitored? How can empirical facts be understood, suc as fat tailed return distributions of asset prices, clustered volatility?, etc. In sort, agent based models are toy worlds wic allow one can understands crucial dynamical aspects of complex systems suc as te financial markets. As wit toys in general tey are useful to develop a concept and feeling of ow tings work, exactly in te sense of ow a cild acquires a clear concept of wat a car by playing wit toy cars. Tey are not meant to be used for te following Wat agent based models cannot do Agent based models simulate actions and interactions of autonomous agents; but tey cannot capture te full complexity of reality. Tey attempt to re-create and predict te appearance of complex penomena, but sould not be used to predict outcomes of future states of te world. Tey cannot explain inventions and innovations saping and canging a given environment. Nor can tey be scaled to realistic situations witout considerable effort. Reasonable mappings of agent based models to real world situations requires large and competent modelling teams, of wic tere are very few. Greater support for tese efforts is needed. Agent based models are models tey are not reality. Agent based models cannot be used to predict outcomes of future states of te world. Tey cannot explain inventions and innovations saping and canging a given environment. Tey cannot be scaled to realistic situations witout considerable effort. Reasonable mappings of agent based models to real world situations need large and competent modelling teams, of wic tere are very few. It would be wise to focus efforts in tese directions. 20

21 A clear danger is tat agent based models are often in transparent. Tis is due to te fact tat it can be very ard to judge if particular rules are implemented realistically. Wat becomes essential is to test tese models against real data. Tis means tat certain dynamic patterns tat are generated by te model - wic are also observable in te real world analogon - ave to be compared to real world data. For example in an agent based model of te financial market, time series of model-prices ave to sow te same statistical features as real time series. If clustered volatility or fat tailed return distributions are not found in a model of financial markets, for example, tis would indicate tat te model missed essential parts and is not to be trusted. 1.5 Failure of economics and te necessity of agent based models of financial markets Te recent crisis as made it clear tat tere is poor understanding of systemic risk. Te regulation sceme Basle II, wic cost millions to be establised worldwide, as spectacularly failed. Based on tis sceme financial institutions devoted considerable resources to modelling risk. However, tey typically did tis under te assumption tat teir own impact on te market is negligible, or tat tey were alone in contemplating a radical cange in policy. In oter words tey did not take feedback loops and syncronization effects into account and terefore teir basic assumption of independence and uncorrelatedness turned out to be a poor approximation. Tis was especially relevant in te time of crisis wen risk models were most needed. A single institution, suc as e.g. Leman Broters, ad an enormous impact, and a group of institutions unknowingly acting in tandem ad an even larger impact. Wen tese impacts are taken into account te result can be dramatically different from tat predicted by standard VaR (value at risk) models based on statistical extrapolations of returns from past beaviour. Systemic risk is a classic complex systems problem: It is an emergent penomenon tat arises from te interactions of individual actors, generating collective beaviour at a system wide level wose properties are not obvious from te decision rules of eac of te individual actors. Typically te individual agents believe tey are acting prudently. Systemic risks occur because financial agents do not understand (or care) ow teir beaviour will affect everyone else and ow syncronization of beaviour builds up. Suc mutual influence leads to nonlinear feedbacks tat are not properly taken into account in classical models of risk management. Very muc against te usual intuition, it becomes possible tat tese nonlinear effects may lead to situations were te very policies designed to reduce individual risks may temselves lead to te creation of extreme risks, eventually able to bring te entire system down. Recent events in financial markets provide a good illustration on many different levels. Te current crisis is very likely te result of a combination of several components, suc as te boom of use of supposedly risk-reducing derivatives, excessive use of credit and a ig degree of lack of transparency troug te use of complicated and often overlapping agreements, and finally a significant portion of criminal beaviour. Practically it is not possible for te individual agent to collect and monitor te relevant data wic would be necessary to model te potential systemic risks. Furter, a point wic is severely underrepresented in current risk models is te possibility of disappearing counterparties. Especially in good times counterparty risk is often not taken into account properly in derivative prices, or oter financial agreements. Practically none of te current risk models took defaults properly into account, because none of tem addressed te complicated couplings and feedbacks of te interconnected components of te financial system. Most models of risk are based on idealized assumptions wic migt be reasonable to a certain degree during times of little or no cange. Under tese circumstances assumptions suc as te efficient market ypotesis, or te general equilibrium scenario migt provide a useful framework to develop tools and metods for managing risk - for times of no cange. Altoug tese concepts migt work under equilibrium conditions tey may become completely useless - sometimes even counterproductive - at times of cange, stress and crisis. In te aftermat of te current financial crisis reforms of te financial 21

22 system are actively being discussed. Many of tese reforms are indered, owever, by te simple fact tat wat policies would be optimal is unknown. To address tese questions te traditional tools associated to economics and financial economics suc as game teory, no-arbitrage concepts and teir tereof derived partial differential equations, as well as classical Gaussian statistics will not be sufficient. Risk management derived from tese concepts suc as VaR and eventually regulation scemes suc as Basle I or II, ignore systemic effects, suc as syncronization or feedback. Tese scemes ave clearly failed during te recent crisis. It is essential to look at feedback loops, syncronization mecanisms, successions of events, and statistics of igly correlated variables. A fantastic setup to understand tese issues is provided by so-called agent based models, were a set of different agents interact in a computer simulation. In tis report suc a model of financial agents is studied, wic was introduced recently in full detail. Rater tan continue pursuing te standard game teory approaces, te need for agent based models arises because of te nonlinearities in te decision rules, payoff functions, and te nonlinear actions between te agents, wic are simply too complex to treat wit game teory-based models. Tis is furter complicated by network effects. Te utility of agent-based models in tis context comes about because of te complicated interactions and dynamic feedback tat leverage introduces. Agent based models allow to study dynamic systems and te detailed unfolding of typical scenarios in te simulations. Furter tey allow to generate data were tere is none: simulations can be run tousands of times under various conditions, and assumptions, someting wic could never be done in te real world. From te study of te system under various conditions one can learn ow te systems beave under different forms of regulation for example. Te essence and te main difficulty for agent based simulations is to capture not only te properties of te agents properly, but also te interactions between tem. Agent based models furter ave to be built from variables and parameters, wic - at least in principle - ave to be accessible in te real world. Anoter difficulty is to calibrate agent models against real data, i.e. to set te scales of influences between agents to realistic values. Tese issues can be most relevant, since systemic properties of complex systems often depend on detailed balance of relations between parameters. Lastly, agent based models often make clear wic parameters are relevant and dominant and wic ones are marginal. As a consequence agent based models can be used to define wic data is needed to be collected from real markets, and to wic level of precision. 1.6 Secondary Impacts and te Global Context Financial markets are eavily intertwined wit te global economy, wic in its present form depends on tem. A crisis in te financial industry, te banking industry in particular, could ave impacts tat cascade into a broad range of different industries. Crisis in te financial industry can often lead to a tigtening of credit, and terefore ideas and creative potential can not be realized - wic can lead to a slowdown of economy. Related impacts are typically measured in bankruptcies, unemployment, loss of investment. Ultimately suc a situation could engender social unrest. Te fear of suc a succession of events is one of te reasons for many governments to bail out defaulted financial institutions lately. However, tese interventions - wic are effectively noting else but an active engagement (of te worst side) of governments in leveraged speculations on te financial markets- come at a tremendous price: Te danger of illiquidity of governments. A state of illiquidity leads to situation tat governments can no longer pursue teir core activities, suc as providing infra structure, finance education, ealt systems, pensions, etc. Suc a failure bears te risk of generating social unrest. Defaulting countries can kick oter countries into default troug cascading effects, wic migt be at te edge of appening at te present time wit Greece, Portugal, Spain and Italy. Wic of two scenarios - one 22

23 of no governmental bank bailouts or solidarity wit defaulting countries, te oter being te opposite - is more likely to lead to larger and more prolonged losses in te end is impossible to say wit te limited present knowledge of te dynamics of financial markets and teir links to economy. All te more it is imperative to cannel collective efforts to understand tese mecanisms. One of te starting points will be to extend agent based models of financial markets and try to link tem to agent based models of te economy. Current main stream economics as been incapable of making useful contributions in tis direction [23]. 2 A SIMPLE AGENT BASED MODEL OF FINANCIAL MARKETS For te new arcitecture of any future financial system it is essential to understand te dynamics of leveraged investments. In te following section discuss a model prased in terms of financial markets. Many results generated tere can be straigt forwardly applied to oter forms of leveraged investments - of course only after taking into account te necessary modifications, identifications etc., appropriate for te particular setup. Suc identifications for leveraged investments on national scales will be discussed in Section Overview Te following section reviews a recent agent based model of te financial market, tat allows to pinpoint at te systemic risk of te use of leverage [7]. Te model does not claim to be a realistic representation of financial markets. It is designed for making clear te basic mecanism under rater stylized conditions. Tis sort overview introduces te 'agents' of te model, tey will all be described in more detail in te following sub sections. Te model assumes a collection of interacting types of financial agents. Tese agent types are investors in financial assets, banks, investors to investment companies and regulators. Witin a type of agent, tere can be a variety or spectrum of eterogeneity, e.g. investors will in general differ in teir investment strategies, teir risk aversion, banks will differ in teir willingness to extend credit under given situations etc. In te model it is assumed tat all investors interact troug buying and selling on te financial market. Tis means tat te price formation is governed troug a simple market clearing mecanism. Tis means tat at every time step wen trading takes place, tere will be a unique price for a given financial asset. For simplicity, tink of financial assets as stocks or sares of a company, traded publicly. In te model no derivatives or more complex assets will be used. Trading appens suc tat all investors interested in a financial asset submit teir demand function, i.e. te quantity of sares tey want to buy, given tat te asset as a certain price, p. Te demand function is a function of te price, usually te iger te price of an asset te smaller te demand. Te combination of all demand functions of all market participants - under te assumption of market clearing - yields a unique price of te asset. Te set of investors is partitioned into tree types of investors, wic will be described in more detail below: informed investors, poorly uninformed investors, and investors in investment companies. Te first kind of investor can be tougt of as investment funds suc as edge funds, mutual funds or investment banks wo base teir investment decisions on researc. In particular tey try to find out weter te assets of a firm are over- or underpriced at a particular time. Researc may involve visiting firms, performing market analysis, etc. In case tey spot an underpriced asset te informed investor will 23

24 assume tat te market will sooner or later find out tat te asset is underpriced, and will eventually price it correctly, i.e. te price of te asset will tend to rise in te future. If te informed investor detects a supposedly overpriced asset, se migt want to 'go sort' in te asset, and wait for it to be priced correctly by te market, i.e. wait till it falls. Investors of tis kind basically try to estimate a 'true price' of assets. Te difference of te actual price (as quoted in te market) and te estimate of te 'true price' will be called te mispricing, or trading signal, trougout te paper. Investors will in general differ from eac oter in two ways. First, tey will differ in teir opinion ow 'correct' teir assessment of te true price of te asset is. Second, tey will differ in teir view of ow muc tat piece of information is wort, i.e. ow muc are tey willing to speculate on teir estimation of te 'true price'. An investment firm wic is willing to speculate muc on a given mispricing or trading signal will be called an 'aggressive' firm, a more risk averse, cautious firm will place moderate bets on teir assessed mispricing. Often investors of te informed kind will use teir know-ow for speculating for oters. For tis service tey generally take fees, often divided into a fixed fee and a performance fee. Typical fees before te financial turmoil were in te range of 1-4 % performance fee (of te capital invested) and 10-30% performance fee (from te actual profits made). At tis point it may be argued tat a model of tis kind is severely unrealistic because it ignores socalled trend followers. Te model pursued in [7] is in a certain way te best of all worlds. Te studied effects on price fluctuations, crases, etc., and te role of leverage played, can only be amplified by te presence of trend followers, wo are known to destabilize te system. Te situation in te real financial world will always be more unstable tan in te presented model. Te importance of te model lies partly in te fact tat even in a world of investors, wo contribute to price stability troug teir beaviour, leverage can be devastatingly dangerous. Te second type of investors are poorly- or uninformed investors. Tese are investors wo do not base teir investment decisions on solid researc or oter forms of estimating te 'true prices' of assets, but - as a collective - place basically random orders (demands). Tis is eiter because tey do not do any researc, or because teir researc is not providing a reasonable trading signal - te trading signal is 'noise', so tis kind of investor is ere called a noise trader. It is assumed tat tese random investments follow a very slow trend toward te 'true price' of te asset. More precisely te demand of poorly informed traders (noise traders) sall be modelled suc tat te price tends to approac a 'true price' of te asset in te long run. In tecnical terms te noise traders will be caracterized by a mean reverting random walk, wic will be specified in detail below. Tis mean reverting beaviour is again an optimistic assumption in terms of stability of asset prices. Tis mean reversion is introduced to mimic efficient markets, i.e. te wide spread textbook believe, or dogma, tat markets tend to price assets correctly [24]. Tat markets are not efficient is a well establised fact [25], wic as been famously expressed by A. Greenspan in 2008 [11]. Te tird kind of investors are investors, wo invest teir cas in investment firms, suc as mutual funds or edge funds. Tese fund will ten invest tis capital in financial markets. Investors wo invest in funds generally believe tat tese funds are informed investors, wic allows for iger profits, and justifies te fees. Investors to funds can be individuals, banks, insurance companies, governments, etc. In general tey will monitor te performance of te fund tey invested in, togeter wit te performance of alternatives for teir investments. If a fund performs well, investors will keep teir investment tere, maybe will increase teir stake. If te fund performs badly, investors will redeem teir investments. Te model below introduces tese investors wo constantly monitor te performance of teir investments. How te investment is monitored will be described in detail. Tere are no oter forms of investors in te model financial economy. Te fraction of investments made by informed vs. uninformed investors is constantly canging and depends on te performance of te informed investors. If tey perform well, tey will experience an influx 24

25 of capital from teir investors (of te tird kind) and will additionally leverage tis capital. Consequently te relative importance of informed investors will increase. In times wen informed investors are performing badly te fraction of uninformed investors will increase. In our model economy banks play te role of liquidity providers in te sense tat tey extend credit to investors to leverage teir speculative investments. Tey do tis in te form of sort term loans. Tese loans are often provided for one day only, but are automatically extended for anoter day under normal circumstances. Te size of tese loans (leverage) largely vary across industries, see above. Investors use te cas from tese credits to boost teir investments, i.e. tey will inflate teir demand by te leverage factor. Aggressive funds will use more leverage in general tan risk adverse ones. Banks earn interest on tese loans. In reality banks also play te role of direct investors to investment funds, or tey beave as if tey were edge funds temselves. Tis is often done to avoid regulation. In te following it is assumed tat only informed investors use leverage. Finally, tere are regulators. Tese are not modelled as agents per se, but as a framework of constraints under wic te above dynamics unfolds. Tese constraints regulate issues suc as te maximum of leverage tat can be allowed for speculative investments, te size of capital cusions for banks (e.g. for a Basle I like regulation sceme). Tese constraints are treated as variable parameters in te model, wic allows to directly study systemic risk under different regulation scemes. For example two fictitious worlds can be compared, one in a tigtly regulated setting, te oter witout regulation. In tese worlds one can measure te respective systemic financial stability, in terms of collapses of investment funds or banks, price crases, etc. Also one can directly compare te consequences of price caracteristics in te different worlds. 2.2 Te specific model economy In te following agent based model tere is a single financial asset wic does not pay a dividend. Tere are two types of agents wo buy and sell te asset, noise traders and value investors wic are ere referred to as 'edge funds'. Te noise traders buy and sell more or less at random, but wit a sligt bias tat makes te price weakly mean-reverting around a fundamental value. Te edge funds use a strategy tat exploits mispricings by taking a long-position (olding a net positive quantity of te asset) wen te price is below its perceived fundamental value. A pool of investors wo invest in edge funds contribute or witdraw money from edge funds depending on teir istorical performance relative to a bencmark return; successful edge funds attract more capital and unsuccessful ones lose capital. Te edge funds can leverage teir investments by borrowing money from a bank, but tey are required to maintain teir leverage below a fixed value tat is determined in te model. Prices are set using market clearing. Now te components of te model can be described in more detail. 25

26 2.3 Variables in te agent based model Table 1: Summary of parameters used in te model. Variable General parameters N H B I p m Informed investors D D nt V W C L Description Number of total assets in te particular stock Number of informed investors in te economy Number of banks in te economy Lending relationsip between banks and informed investors (matrix) Asset price Mispricing of te asset (perceived) Demand in asset of informed investor (in sares) Demand in asset from uninformed investors (in sares) Perceived fundamental value of te asset Wealt of informed investor Aggressivity of informed investor Cas position of informed investor Size of loan of informed investor W Minimum level of wealt before bankruptcy occurs min T w Time before defaulted fund gets replaced ait Actual leverage of informed investor Banks W Wealt of bank b b Maximum leverage Noise traders (uninformed investors) Parameter caracterizing mean reversion tendency of noise traders Volatility of noise trader demands Investors to Funds Time to compute variance for price volatility m r b Bencmark return for investors erf r p moving average parameter for performance monitoring b F performance based witdrawl/investment factor Volatility monitoring parameter performance based witdrawl/investment to informed investor AV r N Net asset value of fund (informed investor) slope of log-return distribution 26

27 2.4 Price formation We use a standard market clearing mecanism in wic prices are obtained by self-consistently solving te demand equation. Let D t ( p( t)) be te noise trader demand and D ( p( t)) be te edge fund n demand. N is te number of sares of te asset. Te asset price p (t) is found by solving D nt ( p( t)) D ( p( t)) = N, (1) were te sum extends over all edge funds in te system. 2.5 Uninformed investors (noise traders) We construct a noise trader process so tat in te absence of any oter investors te logaritm of te price of te asset is a weakly mean-reverting random walk. Te central value is cosen so tat te price reverts around fundamental value V. Te dollar value of te noise traders' oldings is defined by ( t), wic follows te equation log ( t 1) = log ( t) ( t) (1 )log( VN ). (2) nt nt Te noise traders' demand is nt D t ( t) t) =. p( t) n nt ( (3) Substituting into equation (1), and letting be normally distributed wit mean zero and standard deviation one, tis coice of te noise trader process guarantees tat wit < 1 te price will be a mean reverting random walk wit E[ log p] = logv. In te limit as 1 te log returns r( t) = log p( t 1) log p( t) are normally distributed wen tere are no edge funds. For te purposes of tis paper we fix V = 1, = and = Tus in te absence of te edge funds te log returns are close to being normally distributed, wit tails tat are sligtly truncated due to te mean reversion. 2.6 Informed investors (edge funds) At eac time step eac edge fund allocates its wealt between cas C (t) and its demand for te asset, D (t). To avoid dealing wit te complications of sort selling we require ( t) 0, i.e. te edge funds are long-only. Tis means tat wen te mispricing is zero te edge funds are out of te market. Tus, to study teir affect on prices we are only interested in situations were tere is a positive mispricing. Te edge fund's wealt is te value of te asset plus its cas, W ( t) = D ( t) p( t) C( t). (4) D 27

28 On any given step te edge fund may buy or sell sares of te asset and te cas C (t) canges according to D ( t) D ( t 1) p( ). C ( t) = C ( t 1) t (5) If te edge fund uses leverage te cas may become negative, and te edge fund is forced to take out a loan wose size is L ( t) = max[ C( t),0]. Te leverage of fund is te ratio of te value of te assets it olds to its wealt, i.e. D ( t) p( t) D ( t) p( t) ( t) = =. (6) W ( t) D ( t) p( t) C ( t) Te bank tries to limit te size of its risk by enforcing a maximum leverage. In purcasing sares a fund spends its cas first. If te mispricing is sufficiently strong, in order to purcase more sares it takes out a loan, wic can be as large as permitted by te maximum leverage. Suppose te fund is using te maximum leverage on time step t and te price decreases at t 1. If te fund takes no action its leverage at te next time step will exceed te maximum leverage. It is tus forced to sell sares and repay part of te loan in order to reduce te leverage to. Tis is called ``making a margin call". We require tat te edge funds attempt to stay below te maximum leverage at eac time step. We normally keep te maximum leverage constant, but we also investigate policies tat adjust te maximum leverage dynamically based on time dependent factors suc as price volatility, see Section 2.8. Our edge funds are value investors wo base teir demand on a mispricing signal m( t) = V p( t), (7) were as before V is te perceived fundamental value, wic is eld constant to keep tings simple. All edge funds perceive te same fundamental value V. Eac edge fund computes its demand D (t) based on te mispricing at time t. Te edge fund's demand function is sown in dollar terms in Fig. 1. As te mispricing increases te dollar value of te fund's position increases linearly until it reaces te maximum leverage, at wic point it is capped. It can be broken down into tree regions: 1. Te asset is over-priced. In tis case te fund olds only cas. M 2. Te asset is under-priced wit ( t) < AX. In tis case te dollar value of te asset is proportional to te mispricing and proportional to te wealt. M 3. Te asset is under-priced wit ( t) = AX are capped to remain under te maximum leverage.. In tis case its oldings of te asset Expressing all quantities at time t, te edge fund demand can be written: m < 0 : D = 0 28

29 0 < m < mc rit : D p = mw m m : D p = W. (8) crit We call > 0 te aggressiveness of te edge fund. It sets te slope of te demand function in te middle region, i.e. it relates te size of te position fund is willing to take for a given mispricing signal AX m. m c is defined as m rit = M c /. Tis is te critical mispricing beyond wic te fund cannot take rit on more leverage. For larger mispricings te leverage stays constant at. If te price decreases tis may require te fund to sell assets even toug te mispricing is ig. Tis is wat we mean by making a margin call 1. To compute te demand it is convenient to substitute equation (5) into equation (4), wic gives W( t) = C( t 1) D( t 1) p( t). (9) Tis is useful because it means tat in te expression for te demand everyting except te price is known, and te price can be found using market clearing, equation (1). 1 A more realistic margin policy would set a leverage band, wic as te effect of making margin calls larger but less frequent. For example, if te leverage band is (5,7), wen te edge fund reaced 7 it would need to make a margin call to reduce leverage to 5. To avoid introducing yet anoter free parameter, we simply ave te edge funds make continuous margin calls, so tat as long as te mispricing is sufficiently strong, tey constantly adjust teir leverage to maintain it at. Introducing a leverage band exaggerates te effects we observe ere. 29

30 Figure 1: (a) Demand function of a fund as a function of te mispricing signal m defined in equation (7). (b) Market clearing: price and demand are determined by te intersection of te demand functions of noise traders and funds. perf Investment firms typically generate teir income troug a performance fee and a management mgt fee,. Wile te management fee generally ranges from 1-5 % (for edge funds) of te 'assets under management' ( (t) ), te performance fee can be someting between 5-40% of te generated profits. Te W fund's payoff P is p P ( t) = ( erf NAV mgt (1 r ( t)) ) W ( t), (10) were r N AV is te rate of return of te NAV over a specified time interval; W is te average size of te fund over tat same interval. 2.7 Investors to investment fund A pool of edge fund investors (representative investor) contribute or witdraw money from eac fund based on a moving average of its recent performance. Tis kind of beaviour is well documented, 2 and guarantees a steady-state beaviour wit well-defined long term statistical averages of te wealt of te edge funds. Te performance of a fund is measured in terms of its Net Asset Value (NAV), wic can be tougt of as te value of a dollar initially invested in te fund. Letting F (t) be te flow of capital in 2 Some of te references tat document or discuss te flow of investors in and out of mutual funds include [34,35,36,37,38]. 30

31 or out of te fund at time t, and initializing NAV (0) = 1, te NAV is computed as Let W ( t 1) F ( t) NAV ( t 1) = NAV ( t). (11) W ( t) N r AV ( t) = NAV ( t) NAV ( t 1) NAV ( t) be te fractional cange in te NAV. Te investors make teir decisions about weter to invest in te p fund based on r erf ( t), an exponential moving average of te NAV, defined as r perf perf NAV ( t) = (1 a) r ( t 1) ar ( t). (12) Te flow of capital in or out of te fund, F (t), is perf bm F ( t) = b[ r ( t) r ] W ( t), (13) were b is a parameter controlling te fraction of capital witdrawn and is te bencmark return m of te investors. Te parameter r b plays te important role of determining te relative size of edge funds vs. noise traders. m r b Funds are initially given wealt W = (0). At te end of eac time step te wealt of te fund canges according to 0 W W ( t 1) = W ( t) [ p( t 1) p( t)] D ( t) F ( t). (14) bm In te simulations in tis paper, unless oterwise stated we set a = 0.1, b = 0.15, r = W = 2. 0, and 2.8 Setting maximum leverage In most of te work described ere we simply set te maximum leverage at a constant value. However, we explicitly test te effect of policies tat adapt leverage based on market conditions. A common policy for banks is to monitor volatility, increasing te allowable leverage wen volatility as recently been low and decreasing it wen it as recently been ig. We assume te bank computes a 2 moving average of te asset price volatility,, measured as te variance of p of over an observation period of time steps. Here we use = 10. Te bank adjusts te maximum allowable leverage according to te relationsip 31

32 t ) = max 1,. (15) 1 max ( 2 Tis policy lowers te maximum leverage as te volatility increases, wit a floor of one corresponding to no leverage at all. Te parameter sets te bank's responsiveness to canges in volatility. For most of te work presented te maximum leverage is constant, corresponding to = 0, in Fig. 4 of te main paper we compare te effects to = Banks extending leverage to funds Here we assume tat eac fund as only one bank wic extends leverage to it. Tere are no connections (influences) from one bank to anoter, oter tan te fact tat bot migt be invested in te same asset troug (different) funds Defaults If a fund's wealt falls below zero it defaults, i.e. it can not repay its loans. A fund can default because of redemptions or because of trading losses, or a combination of bot. Te fund is ten removed from te simulation. After a waiting period of T w ait a new fund is introduced wit wealt W 0 and wit te same parameters as te original fund. Furter, wenever a fund falls below a non-zero tresold, somewat arbitrarily set to t) < W /10, i.e. 10% of te initial endowment, it will be removed and reintroduced after T w ait W ( 0. Using tis tresold to reintroduce funds avoids te problem of ``zombie edge funds", i.e. funds wose wealt is very close to but not zero, wo take a very long time to recover Return to edge fund investors Since te investor pool actively invests and witdraws money from funds, te NAV does not properly capture te actual return to investors. For example, by witdrawing money troug time an investor may make a good return from a edge fund tat eventually defaults. To solve tis accounting problem we compute te effective return r i nv to investors from teir witdrawals by discounting te present value of te flows in and out of te fund. For any given period from t = 0 to t = T tis is done by solving te equation F(1) F(2) F( T ) F(0) = 0. (16) 2 T 1 r (1 r ) (1 r ) inv inv inv Based on te sequence F (t) of investments and witdrawals tis can be solved numerically for r i nv. Note tat if te simulation ends and te fund is still in business ten F (T ) is computed under te assumption tat all te oldings of te fund are liquidated at te current price. If te fund defaults ten F (T) = 0. During te course of a simulation a fund may default and be re-introduced several times, and it becomes necessary to compute an average performance for te full simulation. Suppose it defaults n times 32

33 at times T i over te total simulation period, i.e. it existed for n 1 time periods. For eac period were te fund remains in business witout defaulting we compute te corresponding return r inv[ T, T ], and ten average tem, weigted by te time over wic eac existed, according to n 1 r inv = ri nv[ T, T ]/( Ti Ti 1), (17) i=1 i 1 i i i 1 were T 0 = 1 is te first time step in te simulation, and by definition T n 1 is te ending time of te simulation Simulation procedure Te numerical implementation of te model on te t t time step proceeds as follows: Noise traders compute teir demand for te time period t 1 based on equation (3). Hedge funds compute teir demand for t 1 based on te mispricing signal m (t) according to equation (8). Note tat tis must be done in conjunction wit computing te new price p ( t 1) i.e. equations (1) and (8) are solved simultaneously. Tis includes computing te wealt ( t 1), wic involves te new cas oldings C ( t 1) and te leverage ( t 1). Investors monitor te NAV of eac fund and make capital contributions or witdrawals. If te maximum leverage is not being eld constant, banks compute te new maximum leverage. If a fund's wealt W (t) falls too low it gets replaced as described in Section Continue wit next time step Summary of parameters and teir default values W Parameters eld fixed: - number of assets: N = perceived fundamental value: V = 1 - initial wealt (cas) of funds: W 0 = C(0) = 2. L (0) = D (0) = 0 - noise trader parameters: = 0.99, = bankruptcy level: 10% of initial wealt W 0 - time to re-introducing defaulted fund T = 100 wait 33

34 - time to compute variance for price volatility, = 10 (see Section 2.8) bm - bencmark return for investors, r = moving average parameter for r p, a = 0.1 erf - investor witdrawal factor, b = 0.15 Parameters tat we vary: - number of funds and teir banks: 1 or 10 - aggressivity of funds, values range from = 5 to maximum leverage = 1 to 15 - volatility monitoring parameter = 0 or 100 (see Section 2.8) We are now ready to implement te model into a computer simulation and study te dynamics of te emergent system. Te most important results are collected in [7]. Here we give a more detailed overview. 34

35 2.14 An ``ecology'' of financial agents max Figure 2: Wealt time series W (t) for 10 funds wit 1, = 2,, = 10, and = 20 1 = b Fig. 2 sows te wealt time series for a system of ten competing funds wit te aggression parameter ranging from one to ten. Oterwise all teir parameters are te same, including teir initial wealt. To illustrate ow leverage can drive crases we ave also raised te maximum leverage to = 20. During te initial transient pase te most aggressive funds grow rapidly, wile te least aggressive srink. Tis is not surprising: Te aggressive funds exploit mispricings faster tan te non-aggressive funds, make better returns, and attract more investment capital. Tis situation continues until rougly t = 7,500, at wic point tere is a cras tat causes te funds wit = 8,9 and 10 to default. Te oters take severe losses, bringing tem below teir initial wealt, but tey do not default. Funds 8, 9, and 10 get reintroduced wit W 0, and once again tey grow rapidly. Tere is anoter cras at rougly t = 19,000, wen once again funds 8, 9 and 10 default. Fund 6, in contrast, was not fully leveraged wen te cras occurs and consequently it survives te cras taking only a small loss. Tis gives it a competitive advantage, and tere is a long period after te cras were it is te dominant fund. Wit time, owever, te oter more aggressive funds overtake it. Meanwile te oter funds ave all grown so small tat tey are effectively irrelevant. Altoug we do not sow tis ere, wit te passage of time fund 6 also dies out, leaving only te tree most aggressive funds wit any significant wealt. Tis simulation migt give te impression tat it is always better to be more aggressive. Tis is not te case. We ave also done simulations in wic one fund uses a significantly larger aggression level tan te oters, e.g. = 40. Tis also causes tis fund to use more leverage, leading to a muc iger default 10 rate. In tis case tis fund is not selected at te end. 35

36 2.15 Generic results Market impact and asymptotic wealt dynamics Figure 3: Time series of a fund for its demand, wealt, leverage, capital flow, loan size and bonds. Critical size ere is seen to be around a value of 100. Te fund as a = 10 and a = 5 ; it was introduced at a size W. (0) = 2 To demonstrate tat te model results in reasonable properties, in Fig. 3 we present results for a single fund in a parameter regime were te beaviour of te system is quite stable. We introduce te fund wit a fairly low wealt W (0) = 2. Te fund initially makes fairly steady profits and attracts more investment capital. Its wealt initially grows, bot due to te accumulation and reinvestment of profits, but also because of te flow of new funds into te market. Once te fund size grows sufficiently large, in tis case rougly W = 100, te growt trend ceases and te wealt fluctuates around an average value. As can be seen in te figure, te fluctuations are rater large: At some points te wealt dips down to almost zero, wile at oters it rises as large as W = 200. At time steps around t = 34,000 and 41,000 te fund defaults (indicated by triangles). Te central reason for tis beaviour is market impact. As te fund grows in size it becomes a significant part of te trading activity. Tis also limits te size of mispricings: As a mispricing starts to develop, te fund buys te asset, wic prevents te mispricing from growing large, and tere are correspondingly fewer profit opportunities. Tis effect increases wit wealt, so tat as te fund grows its returns decrease. Poorer performance causes an outflow of investment capital. Te result is a ceiling on te size to wic te fund can grow: If te fund gets too large, its returns go down and it loses capital. Noneteless, it is clear from te simulation tat te resulting steady state beaviour is only ``steady" in a statistical sense - te fluctuations about te average are substantial. b 36

37 Note tat te beaviour of te system is essentially independent of initial conditions. If we ad introduced te fund wit a wealt of W (0) = 1, for example, it would ave taken it longer to build up te to critical wealt of W 100, but after tat te beaviour would ave been essentially te same. target Te parameter r, te bencmark return for te investors, plays a key role in setting te critical target size of te funds. If r is small te funds will grow to be very large and tey will take over te market, so tat tere are never large mispricings. As a result te fundamental price V forms an effective lower bound on prices. At te opposite extreme, if r target is very large te funds will attract little capital and te beaviour will be essentially te same as it is wen tere are only noise traders. Te interesting regime is target in te middle zone were r is set so tat te demand of te funds is of te same order of magnitude as te demand of te noise traders. 37

38 Figure 4: (a) Mean AV r N of a fund conditioned on its wealt. (b) average mispricing m = log( V/ p) conditioned on te funds wealt. Simulation: single fund, = 10, = 10, (0) wait W = 2, = 10 T. 38

39 Volatility: eavy tails 39

40 Figure 5: (a) Distribution of log returns r. Sown is te density (r) p (conditioned on positive mispricing) in a semi-log scale. (b) cumulative density of negative returns conditioned on te mispricing being positive, = we fit a power P ( r > R m > 0), in log-log scale. Te red curve is te noise trader only case. For 10 law to te data across te indicated region and sow a line for comparison. T = 100, r = 0.005, max = 10 ; = 5,10,,50. (c) as a function of wait. max exp Te nonlinear feedback between leverage and prices demonstrated in te previous section dramatically alters te statistical properties of prices [7]. Tis is illustrated in Fig. 3, were we compare te distribution of logaritmic price returns wit only noise traders to tat wit edge funds but witout leverage ( = 1) to te case wen tere is substantial leverage ( = 10 ). Wit only noise traders te log returns are normally distributed. Wen edge funds are added witout leverage te volatility of prices drops sligtly, as it sould given tat te edge funds are damping mispricings. Noneteless, te log returns remain normally distributed. Wen we add leverage, owever, te distribution becomes peaked in te centre and develops eavy tails. We sow te case wen = 10, were we see tat te distribution of largest returns is rougly described by a straigt line in te double logaritmic plot over more tan an order of magnitude, suggesting tat te tails can be reasonably approximated as a power law, of te form P ( r > R) : R. Te transition from a normal distribution to a eavy tailed distribution as te leverage is increased occurs continuously: As we increase above one, te tails become eavier and eavier, and te centre 40

41 of te distribution becomes more and more sarply peaked. Increasing causes te tail exponent to decrease in a more or less continuous manner. For = 10, for example, we find 2.6, wic is sligtly more eavy tailed tan typical values measured for real price series [39,40]. For lower values of we get larger tail exponents, e.g. M AX : 7 gives : 3, a typical value tat is commonly measured in financial time series Clustered volatility Te origin of te penomenon of clustered volatility in markets is anoter important mystery. Under te random walk model te rate of price diffusion v is called te volatility. Clustered volatility refers to te property tat v as positive and persistent autocorrelations in time. In practice tis means tat tere are extended periods in wic volatility is ig and oters in wic it is low. Tis model also produces clustered volatility in prices, similar to tat observed in real markets. In Figure 6 we sow te log-returns as a function of time [7]. Te case wit = 1 is essentially indistinguisable from te pure noise trader case; tere are no large fluctuations and only mild temporal structure, corresponding to te mean reversion of te noise traders. Te case = 5, in contrast, sows large, temporally correlated fluctuations, wic become even stronger for =

42 42

43 Figure 6: Log-return time series (a) = 1; (b) = 5 ; (c) = 10 ; Triangles mark margin calls in te simulation, indicating a direct connection of pases of large price moves wit margin calls. (d) Autocorrelation function of te absolute values of log-returns for (a-c) obtained from a single run wit 100,000 time steps. Tis is plotted on log-log scale in order to illustrate te power law tails. (Te autocorrelation function is computed only wen te mispricing is positive.) Default parameter values. To provide a more quantitative measure of te clustered volatility, in Fig. 4(d) we present te autocorrelation function of te absolute value of te returns for te parameter values sown in Fig. 4(a-c). Since te beaviour for noise traders only is essentially equivalent to tat wit unleveraged edge funds, = 1, we sow only te latter case. In every case we observe positive autocorrelations out to lags of = Plotting te autocorrelation function on double logaritmic scale suggests power law beaviour at long lags, i.e. C ( ) :, as is commonly seen in real data [26]. For te pure noise trader case, or equivalently for unleveraged edge funds, te first autocorrelation is rougly 0.03, a factor of rougly 30 less tan one. Interestingly, after tis it drops as rougly a power law, wit an exponent 0.5. At = 5 te first autocorrelation is rougly 0.2, almost an order of magnitude iger, and for large values of it decreases somewat faster, wit = Finally, at = 10 te first autocorrelation is similar, but it decreases more slowly, wit Tus te beaviour as varies is not simple, but for large values of it is similar to tat observed in real data. 43

44 Anatomy of crases Figure 7: Anatomy of a cras: under te scenario of large levels of leverage a tiny - oterwise normal - fluctuation of te noise traders demand tips te system over te edge - it collapses. 10 funds, in te range of 5 :5: 50, = 5, r = We now sow ow crases are driven by nonlinear feedback between leverage and prices. In Fig. 5 we present a detailed view of wat appens during a cras. At about t = 41185, due to random fluctuations in te noise trader demand, te price dips below its fundamental value and a mispricing develops, as sown in te top two panels. To exploit tis te funds use leverage to take large positions. At t = te noise traders appen to sell, driving te price down. Tis causes te funds to take losses, driving te leverage up, and some of te funds begin sell in order to make teir margin calls. Tis drives te price down furter, generating more selling, generating ultimately a cras in te prices over a period of only about 2 3 time steps. We tus see ow crases are inerently nonlinear, involving te interaction between prices and leverage. It is immediately clear tat typically price fluctuations driven by random events of te noise traders will not cause big events. Many fluctuations ave been bigger tan te one wic triggers te cras. Te fluctuation only becomes systemically relevant, i.e. dangerous if it appens in a system wic is igly leveraged. Te leverage build-up from 0 10 during time steps t = is enoug to allow for a cras triggered by a normal size fluctuation. Tis is a typical scenario - a cras is triggered by a trivial seemingly irrelevant event. Te circumstances under wic it appens owever (te general leverage levels in te system) make it a trigger for system wide collapse. exp 44

45 3 EVOLUTIONARY PRESSURE FOR INCREASING LEVERAGE Free market advocates often argue tat markets are best left to operate in an unfettered manner. In tis section we demonstrate tat regulation of leverage is desirable from several different points of view. We first sow tat, under te parameter values investigated ere, increased leverage leads to increased returns 3. Tere is tus 'evolutionary pressure' driving leverage up, meaning tat witout exogenous regulation fund managers are under pressure to use iger leverage tan teir competitors. If tis process is left uncecked, leverage rises to levels tat are bad for everyone. Tis can lead to an increase in te number of defaults and lowers returns and profits for banks as well as for te funds temselves. To illustrate tis we do an experiment in wic we old te maximum leverage of all but one fund constant at = 3 wile we vary te maximum leverage of one fund from = 1 to = 10. We ten plot te returns to investors as a function of for te varying fund, as illustrated in Fig Wit very ig aggression parameters, e.g. > 40 tis effect can reverse itself, because leverage becomes so ig tat defaults also become very ig, lowering returns. 45

46 Figure 8: Results of a numerical experiment in wic te maximum leverage = 3 for nine funds wile varies from 1 10 for te remaining fund. (a) sows te returns to investors and (b) te number of defaults. We find tat te returns to investors go up and down wit leverage: All else being equal, wen te leverage of a given fund is below te average of te oter funds, te returns are below average, and wen te leverage is above average, te returns are above average. Tis result is not quite as obvious as it migt seem, since as sown in Fig. 6(b), as te leverage increases, te default rate also increases. Te NAV for a fund tat defaults is by definition 0, corresponding to te fact tat an investor wo put a dollar in at te beginning and never took anyting out loses everyting. However, our investors redeem and add to teir investments incrementally, so even in a situation were te fund defaults tey may still on average acieve a good return. Tis is te reason wy we ave computed returns using Eq. 16, wic properly takes tis effect into account. We see tat te enanced returns of increasing leverage dominates over defaults, so tat at least at tese parameter values, tere is a strong incentive for funds to increase leverage. 46

47 4 HOW LEVERAGE INCREASES VOLATILITY We now explain ow leverage increases volatility by stating te argument given in [7]. Let us begin wit te case of noise traders alone, and assume for a moment V = 1 for simplicity. Market clearing requires tat D nt = N, and equation 3 implies = Np. If we define x log / N = log p, te noise trader process can ten be written in terms of log prices as xt 1 = x t t. (18) Tus te price process is a simple AR(1) process. Defining te log return as r = x x and te volatility in terms of te squared log returns as E ], te volatility wit a pure noise trader process is E [ r t ] =. (19) 1 In te limit as 1 tis converges to [ r 2 t 1 2. Tus for < 1 t t 1 tere is a very mild amplification. Now assume te presence of a single edge fund wit aggressiveness and assume a positive mispricing 0 < m = 1, small enoug tat te price is sligtly below V and te edge fund is not at its maximum leverage, i.e. te edge fund demand is D p = mw. Te market clearing condition can ten we written as NV mw = Np. Wit te mispricing being m = V p togeter wit te definition W = C D p, tis gives te quadratic equation in m t t t t 2 D m [ N ( C DV )] m NV NV = 0. (20) t t t t t t At time t 1 we can make use of equation (9), W = t 1 Ct Dt pt 1, and write a similar equation for m t 1, wic is te same except for t t 1 : 2 Dt mt 1 [ N ( Ct DV t )] mt 1 NV t 1 NV = 0. (21) Solving tese two quadratic equations (denoting te coefficients of equations (20) and (21) by a = D t, b = N ( C t DtV ), c = NV NV t and c = NV t 1 NV ), te cange in price can be written t p t 1 p t 2 2 b 4ac b 4ac c c NV ( t 1 t ) = mt mt 1 = : =, (22) 2a b N ( C DV ) t t 47

48 2 2 assuming tat ac /b and a c/b to be small, wic is certainly true for large N. Comparing to te pure noise trader case, were p p V ( ), we see tat te volatility is reduced by a factor (1 ( Ct N DV )) t 1 t 1 t = t 1 t, wic is less tan 1 as soon as leverage is taken, i.e. > 1. At maximum leverage te market clearing condition is gives M NV AX W = pn. A similar calculation NV ( t 1 t ) pt 1 pt =. (23) N D t Bot D t and are positive. Comparing to te pure noise trader case, we see tat now te volatility is amplified [7]. 4.1 Te danger of pro-cyclicality troug prudence So far we ave assumed trougout tat te maximum leverage parameter is eld constant. However, risk levels vary, so it is natural to consider policies tat adjust leverage based on market conditions. We analyze two situations, one in wic leverage providers monitor te value of teir collateral. In case te volatility of te collateral gets ig, a prudent lender will reduce its outstanding loan, and recall some of it. Te oter situation is to study a 'toy' implementation of te Basle I regulation scenario. Leverage providers - ere banks - ave to keep a capital cusion, i.e. tey ave to keep a ratio of about 10% of cas (wit respect to teir assets) in teir vaults. Banks are forced to keep tat ratio by te Basle agreement (e.g. [27, 28]). Potential pro-cyclical effects ave been discussed in [29, 30]. Wit te agent based model described above we can now study te systemic effects of tese practices Volatility monitoring of collateral A common policy is for banks to monitor volatility, increasing te allowable leverage wen volatility as recently been low and decreasing it wen it as recently been ig. We assume te bank computes a 2 moving average of te asset price volatility,, measured as te variance of p of over an observation period of time steps. Here we use = 10 time steps. Te bank adjusts te maximum allowable leverage according to te relation t ) = max 1,. (24) 1 max ( 2 Tis policy lowers te maximum leverage as te volatility increases, wit a floor of one corresponding to no leverage at all. Te parameter sets te bank's responsiveness to canges in volatility. For most of te work presented above te maximum leverage was eld constant, corresponding to a = 0. In tis section we use =

49 49

50 50

51 Figure 9: Comparison between constant maximum leverage (red circles) vs. adjustable leverage based on recent istorical volatility, Eq. 24 (blue squares). 10 funds were considered wit values in te range of = 5,10,50. (a) Defaults in te system. (b) Losses to te banks due to defaults (sum of all losses of all banks). (c) Average return to investors as a function of, te average extends over te returns over all funds. (d) Total return to all (10) fund managers assuming a wit 2% management fee and a 20% performance fee. (e) Average variance of te price. (f) Distribution of te time-dependent max (t) in a simulation wit in Eq. (24) fixed to = 15 and = 100. All results are averages over 10 independent runs. In Fig. 9 we compare te active leverage management policy of Eq. 24 to a constant leverage policy. We sow a series of measures as a function of a ypotetical imposed (by an external regulator) on 51