an agent-based model for financial vulnerability

An agent-based model for financial vulnerabilityRichard Bookstaber, Mark Paddrik and Brian TivnanFebruary 2017

J Econ Interact Coord (2018) 13:433–466https://doi.org/10.1007/s11403-017-0188-1

REGULAR ARTICLE

An agent-based model for financial vulnerability

Richard Bookstaber1 · Mark Paddrik1 ·Brian Tivnan2

Received: 18 July 2014 / Accepted: 18 January 2017 / Published online: 13 February 2017© Springer-Verlag Berlin Heidelberg (outside the USA) 2017

Abstract This study addresses a critical regulatory shortfall by developing a platformto extend stress testing from a microprudential approach to a dynamic, macropruden-tial approach. This paper describes the ensuing agent-based model for analyzing thevulnerability of the financial system to asset- and funding-based fire sales. The modelcaptures the dynamic interactions of agents in the financial system extending fromthe suppliers of funding through the intermediation and transformation functions ofthe bank/dealers to the financial institutions that use the funds to trade in the assetmarkets. The model replicates the key finding that it is the reaction to initial losses,rather than the losses themselves, that determine the extent of a crisis. By building ona detailed mapping of the transformations and dynamics of the financial system, the

We wish to thank additional members of the MITRE team: Zoe Henscheid, David Slater, MatthewKoehler, Tony Bigbee, Matt McMahon, and Christine Harvey. We also would like to thank Nathan Palmerfor his work on the conceptual model. Finally, we would like to thank Charlie Brummitt, PaulGlasserman, Benjamin Kay, Blake LeBaron, Eric Schaanning, Larry Wall, and participants of MIT’sConsortium for Systemic Risk Analysis 2013, the INET Conference Toronto 2014, and Atlanta FederalReserve’s Conference on Nonbank Financial Firms and Financial Stability for their valuable comments.

Electronic supplementary material The online version of this article (doi:10.1007/s11403-017-0188-1)contains supplementary material, which is available to authorized users.

B Mark [email protected]

Richard [email protected]

Brian [email protected]

1 Office of Financial Research, United States Department of Treasury, Washington, DC, USA

2 MITRE Corporation, McLean, VA, USA

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434 R. Bookstaber et al.

agent-based model provides an avenue toward risk management that can illuminatethe pathways for the propagation of key crisis dynamics such as fire sales and fundingruns.

Keywords Agent-based models · Financial intermediation · Financial networks ·Contagion · Macroprudential · Stress testing

JEL Classification G01 · G14

1 Introduction

Stress testing gainedmomentumamongfinancial regulators after the 2007–2009finan-cial crisis because risk measures based on historical relationships failed to provideadequate insight. Although stress testing provides a better picture of a firm’s expo-sure in the face of proposed scenarios, the tests remain microprudential exercises thatdo not examine the impact of firms on one another beyond the initial stress event.Without incorporating the dynamics, feedback, and related complexities of financialintermediations, it is difficult to understand the impact that stress scenarios will haveon lending, borrowing, and asset markets, and it is impossible to assess the stabilityrisk to the aggregate financial system.

Agent-based models (ABMs) are well suited for incorporating these transforma-tions to explore crisis dynamics. ABMs follow the agents period by period, assessingtheir reaction to events and updating themacro system variables compiled frommicro-level agent decisions. This paper develops an ABM to provide a macroprudential viewof the transformations and dynamic interactions of agents in the financial system. Themodel extends from the suppliers of funding, such as money market funds, throughthe channels of bank/dealers to the financial institutions that use the funds, and thecollateral that moves in the opposite direction. The ABM integrates various chan-nels for crisis dynamics from specific failures in the transformations provided by theintermediaries.

This paper builds on the literature about financial shocks and their impact on col-lateralization. We examine the impact of bank/dealers in financing and collateralizing,as done by Cifuentes et al. (2005), and additionally consider the bank/dealers’ multi-faceted roles as prime brokers in trading and market making, in capital-raising, and asa counterparty. We integrate the funding side of the market by incorporating moneymarket funds and pension funds (Copeland et al. 2010) and hedge funds, which aremajor borrowers of cash and sources of leverage (Thurner et al. 2012). Our contribu-tion integrates these related literatures into a multi-agent framework that incorporatesthe major participants typically seen in U.S. funding markets.

Our model captures several interconnected, network-like relationships that cancause reverberations across a stressed financial system. The integration of marketasset-price correlations through overlapping portfolios with funding networks pro-vides the mechanisms for feedback cycles to occur. This is integral to explaining howthe credit crises in 2008 unfolded for several financial institutions (Brunnermeier andPedersen 2009; Tasca and Battiston 2016). This more detailed and integrated repre-

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An agent-based model for financial vulnerability 435

sentation of the financial system depicts the various paths of fire sale cascades andcontagion, including those that occur from leverage and funding-based fire sales, andfrom credit and redemption stress.

Through the integration and use of ABMs, this paper moves various facets of theanalysis of cascades and contagion closer to implementation. ABM can provide themacroprudential community, regulators and financial institution risk managers, with aflexible tool for enhanced stress and scenario analysis to discover vulnerabilities andforecast the potential implications of financial tail events.

To demonstrate the value of themodel,we test the effectiveness of critical regulatoryrisk measures during periods of market dislocation and crisis. We examine the abilityof these risk measures to capture the propagation of asset and funding risks duringsystemic shocks such as sudden price declines, funding restrictions, erosions of credit,or investor redemptions.

The remainder of the paper is organized as follows: Sect. 2 reviews the role ofmarketparticipants in the dynamics of fire sales, and the function of the asset, funding, andcredit channels. Section 3 presents an ABM to address these dynamics. Section 4demonstrates the model’s response to market shocks, and includes tests of modelvalidation and robustness. Section 5 provides additional details of the impact thatleverage, liquidity and crowding play in market dynamics. Section 6 presents theperformance of regulatory risk measures within the ABM during periods of shocksand their subsequent dynamics. The paper concludes with a summary in Sect. 7 ofthe current application of this analytical approach within the U.S. Treasury’s Office ofFinancial Research (2012) as an essential element of the Financial Stability OversightCouncil’s macroprudential toolkit.

2 Literature review

Wesummarize the prevailing literature relevant to our study andbeginwith anoverviewof current approaches to stress testing and their limitations.We followwith a review ofthe literature of the funding and asset markets, highlighting the dearth of literature thatfocuses on the interdependency of these two markets. We conclude with an overviewof agent-based modeling and its potential to represent the dynamic stressors of actualmarkets.

2.1 Stress testing

Prior to the 2007–2009 crisis, financial regulators placed a lower priority on assess-ing systemwide characteristics of the financial system (Haldane and May 2011).Since the crisis, bank supervisors have honed their financial stability monitoringtools and significantly expanded the use of stress testing with the largest financialinstitutions.

Though useful, stress testing such as the Federal Reserve’s Comprehensive Cap-ital Analysis and Review (CCAR) remains essentially a microprudential exercise. Itfocuses on the resilience of individual banks to specific shocks, rather than the broaderandmore complexmacroprudential question of how stress might be transmitted across

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Fig. 1 Funding cycle and fire sale. Source: Office of financial research annual report (2012), pp. 56–57

firms, financial markets, and into the real economy. Addressing that macroprudentialquestion requires a broad representation of the network of relationships among finan-cial market participants so transmissions of risk can be observed.

As shown in Fig. 1, a run often begins with concerns about counterparty credit-worthiness and a drying up of liquidity, which boost funding costs and place strainson vulnerable firms. The rise in funding costs promotes further concerns about coun-terparty risk and ever-wider funding spreads. In contrast, a fire sale often begins witha news development that prompts repricing of assets, combined with a concentrationof leveraged funds that are forced to sell assets to meet margin requirements. Theforced sales push prices lower and margin calls act as feedback to magnify the effects,triggering more selling.

As identified by the Office of Financial Research (2012, 2013, 2014), an importantchallenge is to increase the macroprudential value of supervisory stress testing byincorporating feedback and enhancing the models to allow for runs and fire sales.Figure 1a depicts an illustrative trajectory for a run, which often begins with a fundingshock. Figure 1b depicts an illustrative trajectory of a fire sale, which often beginswith a pricing shock. Ultimately, a macroprudential stress test should ask whether thefinancial system as a whole has the capital and liquidity to support lending and to beresilient to shocks.

2.2 Modeling direct and indirect stress: funding markets and asset markets

Many of the linkages needed to represent the macroprudential risks in a model mustconsider both fundingmarkets and asset markets. Their relationship is important in theU.S. financial system through interactions with leverage, pricing, and price discoveryof assets (Copeland et al. 2010). Understanding the dynamics requires looking at thefunding and asset markets in tandem. Yet their interdependence is not fully specifiedin the extant literature.

Traditional modeling discussions have focused on two main channels of risk con-tagion in the financial system: (1) direct inter-counterparty linkages between financialinstitutions, such as interbank lending networks, and (2) contagion due to changes insecurities values. The former, which has been given extensive empirical and theoreticalstudy (Wells 2002; Furfine 2003; Upper and Worms 2004) focuses on the dynamics

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of loss propagation via the direct counterparty exposures following an initial default.More directly to funding markets, Gorton and Metrick (2012) study an interdealer,bilateral repo market and show that haircuts increased dramatically during periods ofstress, similar to haircut spirals previously modeled (see Brunnermeier and Pedersen2009; Adrian and Shin 2010).

The linkage of assets to financial institutions through portfolios can impose furtherdownward pressure on asset values in the market. Damage can spread to institutionalinvestors, and the result is a cascading of risk propagation throughout the system(Cifuentes et al. 2005; Tsatskis 2012). These, as well as the models such as those inCaccioli et al. (2012) and Chen et al. (2014), have been able to show the importanceof diversification and bank leverage on the sensitivity of the system to shocks. Tascaet al. (2014) identify a critical level of diversification, mapping to regimes wherediversification can and cannot dampen higher systemic risks from leverage.

Though many models exist in these distinct research programs, few studies linkthe two channels in a single model that allows the dynamics to be transmitted acrossthe financial system. This is important to understand because during a financial crisisthese flows and agent objectives can be strained from the joint interaction of declinesin prices and funding, and forced liquidations in the face of reduced funding. Theseevents lead to the dynamics associated with cascade and contagion among finan-cial entities. The paths for these dynamics are variously characterized as fire sales(Shleifer and Vishny 2011), balance sheet constrictions (Danielsson et al. 2012), liq-uidity or margin spirals (Brunnermeier and Pedersen 2009), leverage cycles (Adrianand Shin 2014; Fostel and Geanakoplos 2008; Sato and Tasca 2015), and panics (Gor-ton 2010).

2.3 Modeling dynamic stress: agent-based models

Evaluating the implications to systemic risk within institutions and markets broadlyrequires that we model the individual participants as they make decisions and reactto the decisions of other participants, both individually and in aggregate. ABMs arewell suited for incorporating these transformations to explore crisis dynamics (2013,2014). ABMs follow the agents period by period, assessing their reactions to eventsand updating the system variables accordingly.

Agent-based models that seek to explain how the behavior of individual firms oragents can affect outcomes in complex systems offer the opportunity to understandpotential vulnerabilities and paths through which risks can propagate across the finan-cial system. Additionally, such models offer the ability to depict the heterogeneity ofagents, as well as idiosyncratic rules for how financial institutions operate, which areimportant to replicate real market conditions.

In this paper, we develop a systemwide view of the transformations and dynamicinteractions of agents in the financial system. This integration captures the individualdifferences of agents from the perspective of a network of relationships and from theperspective of behavior decisions. AnABM can depict the interdependent relationshipbetween the funding and asset markets and incorporate dynamic, stress testing bytransmitting agent-level decisions through these interconnections.

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2.4 Stress testing constraints

The cornerstone of the current international regulatory agenda has been to establishmechanisms to prevent fire sale dynamics through higher requirements for capi-tal and liquidity ratios (Haldane and May 2011). The traditional rationale for suchrequirements is that they reduce idiosyncratic risks to the balance sheets of individ-ual institutions. An alternative interpretation is that they may strengthen the financialsystem as a whole by limiting the potential for network spillovers.

The principal regulatory capital constraints as dictated by Basel Committee forBanking Supervision (Basel Committee onBanking Supervision 2010) have taken twoforms. One is a leverage constraint. The second is a constraint on risk-weighted assets,commonly known as a Value-at-Risk (VaR) measure (Jorion 1997). An immediateapplication of the ABM is to produce a VaR-like view of the risk of the financialentities when the dynamics of the system are considered.

Typically, liquidity requirements are specified as aminimum ratio of a bank’s liquidassets to its short-term liabilities. This liquidity ratio can be seen as a means of short-circuiting the potential for systemic liquidity spillovers arising from fire sales on theasset side of the balance sheet (liquidity shocks) or liquidity hoarding on the liabilitiesside (liquidity-hoarding shocks). The primary risk measure for this is the liquiditycoverage ratio (LCR) described in Basel Committee on Banking Supervision (2013).We can evaluate the dynamic implications of shocks to short-term, funding liquidityusing the methods discussed above for stress testing in two ways: First, a given shockwill propagate through funding and credit channels to affect the funding liquidity andthe LCR, and this will be seen in the simulation. Second, we can impose a shock tothe funding liquidity, and see how that initial shock creates a cascade affecting furtherfunding liquidity and affects the markets and the agents.

When either of these issues occurs, a combined asset-based and funding-basedfire sale can occur due to a degrading credit quality (Drehmann and Nikolaou 2013).When the credit quality of an agent drops, funding is reduced, leading to further creditdegradation and feeding the asset-based and funding-based aspects of the fire sale(Diamond and Rajan 2001; Acharya et al. 2012; Bluhm et al. 2014). The decline incredit quality comes from a perception that the agentwill be unable to pay its liabilities.Because the first liabilities to fail tend to be those that are shorter term, a measure ofcredit degradation is a drop in the ratio of short-term assets to short-term liabilities.A drop in one agent’s credit quality can also affect its counterparties, such as in theinterbank market (see Georg 2013; Ladley 2013).

3 Model formulation

The objective of our ABM is to integrate the network of relationships and functionsof the various participants in the lending and borrowing transformations to learn moreabout risk in financial intermediation transformations. The model also helps explorehow the asset portfolio creates a second level of linkages between agents.

The model includes the three classes of agents described above: cash providers, thebank/dealers, and the hedge funds (collateral providers). TheC cash providers, to lend

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Fig. 2 Diagram of model formulation. Source: Authors’ analysis

cash for collateral to K bank/dealers. The K bank/dealers can then intermediate thatlending down to N hedge funds. In addition, the K bank/dealers and n hedge fundsthen can invest into M assets, which they will buy and sell as their capital grows anddecrease (see Fig. 2 for relationships). The following comprise the key variables ofthe model, which will later be referenced further into the model.

The set of K bank/dealers and N hedge funds hold a quantity of each of the Massets:

Qk,m(t), Qn,m (t) where k = 1, . . . , K ; n = 1, . . . , N ;m = 1, . . . , M

To buy assets the K bank/dealers and N hedge funds will use their capital, Cap:

Capk (t) ,Capn (t) where k = 1, . . . , K ; n = 1, . . . , N

The M assets have a price:

Pm (t) where m = 1, . . . , M

On a day-to-day basis, as asset values change and the K bank/dealers’ and N hedgefunds’ capital, Capk (t) ,Capn (t) , changes their demand for the m assets:

QDn,m (t) , QDk,m (t)

where k = 1, . . . , K ; n = 1, . . . , N ;m = 1, . . . , M

The funding of the set of K bank/dealers and N hedge funds will need to makeleveraged purchases will be:

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Fk (t) , Fn (t) =M∑

i=0

(Pi (t) Qk,n,i (t) − Capk,n (t)

)

where k = 1, . . . , K ; n = 1, . . . , N (1)

The C cash providers govern the funding using haircuts for each of the K bank/dealersand N hedge funds:

HCk,m (t) where k = 1, . . . , K ;m = 1, . . . , M

The haircut implicitly determines the amount of collateral of asset m that can bepurchased and rehypothecated as collateral, CA:

CAk (t) ,CAn (t) =K ,N∑

i=0

(Ai (t) − Capi (t)) /(1 − HCc,k (t)

)(2)

Each of the K bank/dealers and n hedge funds govern how much funding, Fk, Fn ,they use with a leverage target, LevTarget :

LevTargetk , LevTargetn , where k = 1, . . . , K ; n = 1, . . . , N

Additionally, the K bank/dealers, to maintain good standing with their lenders, theC cash providers, govern the amount of free capital they have disposable in case ofliquidity shocks using a liquidity ratio target:

LiqRatek , where k = 1, . . . , K

This provides the bank/dealers with an accounting buffer, similar to capital require-ments that they can draw on in cases of liquidation constraints.

With the preceding set of variables that will govern agent behaviors in place, wewill now discuss how each of the variables interact within the model market, such thatwe may see the dynamics created by them.

3.1 Asset market

Asset markets in this model represent any number of different markets: equities,futures, commodities, mortgage-backed securities, etc. There is extensive literatureapplying ABMs to market microstructure, beginning with Maslov (2000). The priceimpact literature of Stoll (1978) and Kyle (1985) demonstrates a price impact to belinear and we employ a model along the lines of Greenwood et al. (2015) for linearprice impact with βm determining the market impact, and PRR is N (0, βm) .

PRm (t) = βm

∑ (QDpi

n (t) + PRRm (t)

)(3)

Pm (t + 1) = max (0, Pm (t) (1 + PRm (t))) (4)

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To keep the model tractable, we assume the error terms for the prices are indepen-dent, so that any correlation structure occurs through the interaction of the agents,in particular due to contagion during the fire sales. The implication of this model isthat the agents are atomistic with respect to the market except during times of forcedliquidation, and absent such forced sales the day-to-day movement in prices takes ona simple random process. That is, the firms are assumed to execute their buying andselling by placing orders, QDpi

n (t), which typically do not affect the price of assets.This is what would be expected during normal times, because agents have the optionof spacing out their trades to minimize the market impact. What does matter and is thefocus of the model are the occasions when a shock leads a bank/dealer or hedge fundinto fire sale mode and is forced to liquidate without regard to the market implicationsof its actions. In those cases, we assume that the executed orders can have a priceimpact denoted by QDpi

n (t). Note that using the product of βm and the quantity offorced sales will lead to a larger impact as prices drop, because for a lower price therewill need to be a higher quantity sold to sell the same dollar amount. This means thatwe are assuming liquidity drops proportionately with price; that is, liquidity is basedon the total market value or float available to sell. An alternative is to base forcedselling on a dollar amount rather than a quantity. This, of course, will change the unitsand size of βm , or we can add a further term to the determination of Bm to allow it toincrease as the fire sale evolves. For example, the Bm can increase as a function of theforced selling that enters the market.

3.2 Cash provider

The cash provider, c, lends to the finance desk based on the dollar value of the collateralit receives and a haircut it sets for bank/dealer k, HCc,k . The haircut is based on theperceived creditworthiness of a borrower and is used to cut the value of the asset usedas collateral, CAk (t), by a percent of its current market value. The target amount thatwill be loaned based on the haircut, which can be modeled to vary based on the cashprovider’s decision rule, is:

LTargetc,k (t) = CAk (t)

(1 − HCc,k (t)

)(5)

The loan, L , is checked to ensure it does not go over a maximum dollar amount thecash provider is willing to lend independent of any collateral or haircut, LMax

c,k (t):

Lc,k (t) = min(LMaxc,k (t) , LTarget

c,k (t))

(6)

This last aspect of the firm’s decision rule reflects the fact that many cash providers areextremely risk averse and have limits on their collateral holdings. The cash providerswill not lend more than a given amount, no matter what the size and nature of thecollateral. If LMax

c,k (t) is hit, the CAk(t) is revised to reflect the amount of collateral itwill submit.

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3.3 Bank/dealers

The bank/dealer acts as an intermediary between buyers and sellers of securities andbetween lenders and borrowers of funding. As we previously outlined, it employs anumber of subagents to do the various tasks. Just as a hedge fund can be modeled torepresent a wider set of institutions, so the bank/dealer can be modeled to representagents that have only a subset of these functions. For example, there might be anintermediary that provides only the market-making function of the trading desk, orthat does not have a derivatives function. Thus, the bank/dealer category encompassesmore than the major bank/dealers that provide all these functions.

3.3.1 Prime broker

The prime broker is the agent that interacts with all the hedge funds that bank/dealerk does business with (a subset Nk of all n hedge funds). The prime broker’s job is togather the collateral of the hedge funds, CAPB

k , so that it can then look for funding,FPBk , from the cash providers for any loans that hedge funds need to cover their

leveraged positions. As stated earlier, we make the simplifying assumption that theprime broker passes the funding from the finance desk throughwith no further haircuts,so the collateral of the prime broker is equal to that of the sum of the hedge funds itservices.

3.3.2 Finance desk

The finance desk is responsible for the financing of all the bank/dealer’s activities,which include the trading desk and prime broker’s funding needs. As the prime brokerdoes for the hedge fund, the finance desk also gathers the collateral, CAT D

k (t), of thetrading desk it will need to obtain the funding, FT D

k (t), for the assets it holds abovethe value of its capital. The finance desk takes in the securities posted by the primebroker and by the trading desk, CAFD

k (t) and these form the basis for the collateralit gives the cash provider. It receives FFD

k (t) and distributes this back to the primebroker and the trading desk.

3.3.3 Trading desk

Following the adjustment mechanism of Adrian and Shin (2014) and Greenwoodet al. (2015), the trading desk use three leverage constraints: Leverage Maximum,LevMax

n (t), which is set by the prime broker of the bank/dealer, k, which the tradingdesk is using for financing and is the inverse of the haircut it receives. The tradingdesk governs its leverage using LevMax

n (t) to set a Leverage Buffer, LevBu f f ern (t),

which it rebalances to in the event that it exceeds LevMaxn (t) and a Leverage Tar-

get, LevTargetn (t), which it rebalances to when the changes are small (i.e., normalfluctuations).

LevMaxn (t) ≥ LevBu f f er

n (t) ≥ LevTargetn (t) , (7)

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Additionally, the trading desk has two other parameters. The first is an initial capital,Capn (0)which it uses to fund all of its initial activities. The other is an asset allocationvector, AAllocation

n (t) that determines how to allocate the trading desk’s capital amongthe set of M assets. Using these parameters as the initial conditions, the trading deskfollows a sequential updating function that allows it to manage its asset portfolioand governing leverage constraints for every period t in the future. The trading deskdetermines current capital, Capn (t) , based on its evaluation of all its assets, An (t),and subtracting any slippage in trading, Sn, after estimatingwhat it expects to purchaseor sell assets at and funding FHF

n (t − 1).

An (t) =M∑

i=1

Pi (t − 1) ∗ Qn,i (t − 1) (8)

Sn (t) =M∑

i=1

QDn,i (t) ∗ (Pi (t − 1) − Pi (t − 2)) (9)

Given its new capital level, the hedge fund computes its target asset level; the dollarassets that the trading desk would own at its target leverage.

ATargetk (t) = Capn (t) ∗LevTargetn (t) (10)

The hedge fund can then determine how well it has been able to meet the constraintand target leverage by calculating the current leverage it has after the previous pricemovements. This allows the hedge fund to determine how it should change its portfolio.

LevCurrentn (t) = Capn (t) ∗LevTargetn (t) (11)

If LevCurrentn (t) ≥ LevMax

n (t), the hedge fund receives a forced margin call, which

we term a forced sale, and therefore must reduce its assets by QDpin (t), liquidating

enough shares to get back to the LevBu f f ern (t). If LevCurrent

n (t) < LevMaxn (t), the

hedge fund buys or sells QDpin (t) to move its portfolio with ATarget

n (t) dictated byLevTargetn (t) . This updates the quantity of shares held by the hedge fund to Qn (t)based on the trading decisions made as a result of QDn (t). This then allows the hedgefund to determine how much funding, Fn (t), it will need to achieve Qn,m (t) basedon its current capital.

Qn,m (t) = Qn,m (t − 1) + QDn,m (t) (12)

This completes the sequential process that each trading desk uses to determine itsportfolio management. If at the end of any daily sequence, the hedge fund’s Capn (t)is less than zero, the hedge fund will default. Additionally, this will cause the hedgefund to sell all its assets during the next period, t + 1, and will no longer allow itto participate within the financial system throughout the rest of the model’s run. Thetrading desk also acts as amarket maker for customers looking for liquidity inmarkets.

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As a result, the trading desk can suffer from limited liquidity because it sources thistransformation process as part of its business. We introduce the maximum liquidationthreshold, QMax

k (t), which is the maximum dollar value of assets that can be sold inany period. It reflects limits on liquidity, or more generally, on a bank/dealer’s inabilityto remove obligations from its portfolio.

QMaxk (t) =

∑M

i=1QDk,i (t) + Pi (t) (13)

The bank/dealer uses its liquidity reserve when the trading desk faces a drop in fundinggreater than the amount of inventory it can immediately liquidate. This is discussedin the next section.

3.3.4 Derivatives desk

Derivatives desk activities are represented by the counterparty credit exposure eachbank/dealer i has to bank/dealer j . The total credit exposure, CETotal

k (t), is calcu-lated as a dollar quantity of exposure to all other counterparties, CEk, and individualcreditworthiness, CWk:

CETotalk (t) =

K∑

i=1,i �= j

CEi (t) (100 − CEi (t)) (14)

Each bank/dealer has a percent of its initial capital exposed to other agents (similar towriting a credit default swap on another agent). At the close of day, the credit ratingof each agent is calculated based on the liquidity ratio of the agent. If an agent towhom the bank/dealer is exposed drops in its creditworthiness, CWk, there is a mark-to-market effect represented by a drop in the value of the exposed capital, Capk. anagent is detailed in the next section. The sum of mark-to-market is the total creditexposure, CETotal

k (t).

Capk (t) = Ak (t) − Fk (t − 1) − Sk (t) − CETotalk (t) (15)

3.3.5 Treasury desk

The bank/dealer’s treasury department acts as a maintenance agent to ensure thatsubagents’ financing and credit risks do not hurt the bank/dealer as a whole. Thetreasury department achieves this by maintaining the bank/dealer’s liquidity reserveand creditworthiness.

3.3.5.1 Liquidity Reserve Because of banking regulations and risks that leveragedinstitutions face, bank/dealers typically are required to hold a liquidity reserve incase of transaction stresses. The liquidity reserve, LiqReserve

k is a proportion of abank/dealer’s capital not used to buy assets. The liquidity reserve is a buffer if thebank/dealer’s funding drops and it cannot reduce assets by an equal amount due tomarket illiquidity. The reserve amount is based on the liquidity reserve rate, LiqRate

k ,

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a parameter solved for as a result of the liquidity ratio target, LiqReserve Targetk , whichis discussed in the following section.

LiqReservek (t) = LiqRate

k (t) ∗ Capk (t) (16)

If the quantity of shares a bank/dealer is trying to sell is above a liquidation thresholdQMax

k , the rest of the shares it needs to continue to hold will have to be funded usingthe liquidity reserve by debiting LiqDebit

k (t) up to the limit of LiqReservek (t). The

treasury department tries to keep the LiqDebitk (t) at zero due to its impact on the

creditworthiness of the firm (see next section of detail). The result is the treasury willtry to sell the shares in the next period, assuming the following conditions are met:

QMaxk ≥

M∑

i=1

QDk,i (t) + Pi (t) + LiqDebitk (t) (17)

The liquidity reserve variables are also part of the difference in the capital calculationof the bank/dealer.

Capk (t) = Ak (t) − Fk (t − 1) − Sk (t) − CETotalk (t)

+LiqReservek (t − 1) − LiqDebit

k (t − 1) (18)

If the bank/dealer has LiqDebitk (t) ≥ LiqReserve

k (t) it will suffer a liquidity default.This differs from a default due to the bank/dealer’s equity dropping to zero, becauseit still may have Capk (t) ≥ 0, but it can no longer meet its short-term obligationsbecause of liquidity constraints. In this case, a bank/dealer’s assets go into receivershipand no longer enter the market as forced sales.

3.3.5.2 Creditworthiness Cash providers and bank/dealers that hold exposures toother banks assess the creditworthiness of their counterparties with a creditworthi-ness rating, CWk . For cash providers, the rating determines the haircut and how muchfunding is provided to the bank/dealers. For bank/dealers, the rating determines thevalue of capital exposure one bank has to another through mark-to-market based onthe CWk . Both leverage and the liquidity ratio are measures that can be used to reflectcreditworthiness. To reflect the functions of the funding map, the treasury depart-ment determines the leverage measures and the liquidity reserve. The key measure forcreditworthiness is the liquidity ratio, LiqRate

k (t), determined by:

LiqRatiok (t) =

(LiqReserve

k (t) − LiqDebitk (t)

)/FT D

k (t)) (19)

The LiqRatiok (t) is significant in representing the bank/dealer’s ability to meet obli-

gations. As seen in Eq. (17), the bank/dealer works to target a liquidity ratio,LiqRatioTargt

k (t), so it can continue to have a good credit rating in the future.

LiqRatioT argetk (t) =

(LiqReserve

k (t) /FT Dk (t)

)(20)

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If the liquidity ratio goes below a minimum liquidity ratio, LiqRatioMink (t), the

bank/dealer’s creditworthiness begins to decrease, and the haircuts placed by the cashprovider will increase. This will cause forced sales by the bank/dealer and potentiallycause forced sales by counterparties exposed to the bank/dealer. The adjustment ruleis similar to that used for the adjustment of leverage, as described in Greenwood et al.(2015).

CWk (t) = 100 − ϕCW(LiqRatioMin

k (t) − LiqRatiok (t)

)(21)

HCc,k (t + 1) = HCc,k (t) + ϕHC(LiqRatioMin

k (t) − LiqRatiok (t)

)(22)

Where ϕCW and ϕHC are two global parameters set at the beginning of the simulationto govern the functions of CWk (t + 1) and HCck (t + 1) for all K bank/dealers.

3.4 Hedge fund

We focus on hedge funds because leverage is the critical feature that creates asset-based, fire sales.1 A hedge fund uses its capital and borrowed cash from the primebroker of a bank/dealer to buy assets. The broader universe of asset managers can beconsidered unleveraged hedge funds in this model. It is important to consider thesefirms because they face redemption risks, which make them susceptible to forced saledynamics similar to those that leverage creates for the hedge funds.

Hedge fund behavior is similar to that of a trading desk, and most of the time hasthe same sets of constraints and objectives as discussed in the trading desk section.The only difference is that we do not bind the hedge fund by the quantity of assets itcan sell into the market.

3.5 Delimitations of the model

We identify two delimitations of our current model: absence of a central banker andabsence of new entrants during a financial crisis. Our ABM, as presented here, doesnot include the central bank as an autonomous agent. Adding the central bank wouldenhance themodel from the perspective of policy analysis.However,we can impose thepolicy levers of the central bank into the model exogenously. For example, the centralbank’s injection of liquidity into the asset market can be represented by an exogenousdrop in the price elasticity of demand for assetm, βm . An injection of funding liquiditycan be represented by an exogenous increase in the funding lines for the hedge fundand bank/dealer. Support for the bank/dealer, either overall or in the specific, can berepresented by increasing the value of the bonds that reflect counterparty exposure.Insofar as the central bank has discernible rules, these exogenous policy effects canbe replaced by including the central bank explicitly.

1 Though we term this agent as a hedge fund, it can more broadly represent other institutional financialfirm classes, which have varying degrees of leverage in their portfolio.

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Table 1 Design of experiments. Source: Authors’ analysis

Experiment Independent variables Values Replicates

Baseline – – 1000

Benchmark tests Price shock {0%, 10%, 15%, 20%} 4 × 1000 = 4000

Robustness tests Price shock {10%, 13%, 15%} 3

βm {0.5, 1.0. 2.0} ×3

HCc,k {0.1, 0.13, 0.16, 0.19} ×4

ϕCW {100, 200, 300, 400} ×4

ϕHC {0.1, 0.2, 0.3, 0.4} ×4

Ak (0) {10%, 20%, 40%, 80%} ×4

QMaxk (t) {$500 M, $1 B, $2 B} ×3

(LiqRatioMin1 , LiqRatioMin

2 ,

LiqRatioT arget1 ,

LiqRatioT arget2 )

{(0.2, 0.25, 0.15, 0.25),(0.25, 0.3, 0.2, 0.3),(0.3, 0.35, 0.25, 0.35)}

×3× 500= 1,036,800

We do not currently allow for the entry of new market participants. As describedin greater detail in the OFR’s (2012) Annual Report to Congress (p. 56), fire sales canbe sweeping, cascading events lasting for only a short duration (e.g., a month). Forcontext, no new market participants arose during the peak of the financial crisis in thefall of 2008, mergers notwithstanding.

4 Model dynamics validation

In this section, we present the results from several sets of experiments testing modelvalidity and robustness. Table 1 provides an overview of our experimental designand the set of independent replications.2 The first two experiments serve as validationtesting of our baselinemodel and the interaction effects of variables when under stress.This allows us to verify that the model’s dynamics adhere to those depicted in Fig. 1.Finally, we perform an extensive set of experiments as a robustness test and validatethe statistical significance of various model parameters.

4.1 The baseline model

To illustrate the model’s dynamics, we explore the influence of asset, funding, andcredit shocks through the system in a tractable network of three assets, twohedge funds,two bank/dealers, and a single cash provider that treats each bank/dealer separately.Consistent with Battiston et al. (2012), we depict the couplings between financialinstitutions as a network and subsequently extend the network depiction to reflectendogenous dynamics. At initialization, all hedge funds and bank/dealers have equal

2 Bigbee et al. (2015).

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Fig. 3 Schematic of the relationship between the agents and markets. This figure has two bank/dealers(BD), two hedge funds (HF); one cash provider (CP), and three asset markets (A). The directed edges showthe flow of assets, funding, and collateral. Source: Bigbee et al. (2015)

capital, and all assets have equal values of price and liquidity. Table 7 of the Appendixcontains a detailed description of the model parameters.

Figure 3 depicts the relationship of the various agents we use in the scenario.Bank/Dealer 1 (BD1) and Hedge Fund 1 (HF1) hold equal weights in Asset 1 (A1)and Asset 2 (A2). Bank/Dealer 2 (BD2) and Hedge Fund 2 (HF2) hold equal weightsin Asset 2 (A2) and Asset 3 (A3). Finally, the Cash Provider (CP1) supplies fundingto the bank/dealers, which in turn supply funding to the hedge funds. All hedge fundsuse the assets they own as collateral.

We ran 1000 replicates of the baseline model without observing a single margin callfor the hedge fund and, therefore, no hedge fund defaults. Without any margin calls,we verify that the price returns of each asset fit the normal distribution as expected.Hence, we confirm that the baseline model serves as a valid benchmark for stresstesting. We then proceed to validate the stress testing of our model.

4.2 Shocking the baseline model

We present results from three experiments with the baseline model, each a dynamicstress test with increasing levels of an exogenous shock to one asset. Taken together,these three sets of experiments with the baseline model represent a comprehensive setof validation testing.

Our initial experiment entails the exogenous introduction of a 10% price shock toAsset 1 at Time 20. We chose Time 20 to allow for the dampening of any transientdynamics from initialization, therefore ensuring that all observed events of interestresulted from the experimental treatment (i.e., exogenous shock). We ran the same1000 replicates of the baseline model as above (i.e., controlling for randomness atinitialization), introducing the 10% exogenous shock at Time 20. There are a total of13 possible events of interest, as listed in Table 2.

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Table 2 Model events. Source:Authors’ analysis (1) Price shock

(2) qDemand event for hedge fund 1 asset 1




(6) Default of hedge fund 1

(7) Default of hedge fund 2

(8) qDemand event for bank/dealer 1 asset 1




(12) Default of trading unit 1

(13) Default of trading unit 2

In addition to the occurrence of these events, we also confirmed the validity oftheir sequence. Figure 4 depicts the sequence of events for all 1000 replicates fromthis experiment. Note that in 742 of the 1000 instances of this experiment, the 10%price shock to Asset 1 did not produce any additional events of interest. Stated anotherway, in 74.2% of the model realizations, the financial system was robust to the 10%shock that was dampened throughout the system without significance. We see a fewrare, but interesting events in this dynamic stress testing. In two instances, the priceshock produced contagion between the two hedge funds. Also, there are several uniqueinstances of extensive fire sales that cascade throughout the financial system.

Figure 4 reflects many aspects of dynamic and macroprudential stress testing. Forexample, there were two instances where the price shock generated margin calls forHedge Fund 1. This subsequently produced contagion between the two hedge fundsthat resulted in margin calls for Hedge Fund 2, which did not even hold the toxicasset. This sequence of events directly reflects the stylized fire sale dynamics depictedin Fig. 1b. Figure 3 also shows several unique instances of extensive fire sales thatcascade throughout the entire financial system. Note that consistent with Fig. 1b, allsequences of interest begin with a margin call for Hedge Fund 1 in the toxic asset,then propagate across the financial system to varying degrees.

For comparison and completeness of the discussion, we present the Event Sequencediagrams for the 15 and 20% shock to Asset 1 in Figs. 5 and 6, respectively. Thesefigures confirm there is no Hedge Fund default without a qDemand event precedingit. As depicted in Fig. 6, the 20% price shock to Asset 1 generates a contagion of firesales that sweeps across the system to entities that do not hold the shocked asset (i.e.,Hedge Fund 2 and the Bank/dealer 2).

While Figs. 5 and 6 depict only the event sequences that resulted in a count of sixreplicates or more, for completeness, we analyzed all the event sequences for both the15 and the 20% shock experiments to identify the following:

For the 15% price shock

• In 4 realizations out of 1000, the price shock had no effect.

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Fig. 4 Event sequence for events of interest from a 10% price shock of baseline model. Source: Authors’analysis, Bigbee et al. (2015)

Fig. 5 Event sequence for events of interest from 15% price shock of the baseline model. Source: Authors’model, Bigbee et al. (2015)

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Fig. 6 Event sequence for the events of interest from 20% price shock of the baseline model. Source:Authors’ model, Bigbee et al. (2015)

• In 250 realizations where Hedge Fund 1 experienced qDemand events that default,neither Hedge Fund 2 nor the bank/dealers had any qDemand events.

• In 60 realizations, all four entities defaulted.

For the 20% price shock

• In all 1000 realizations, the price shock had a material effect; that is, there was atleast one qDemand event for Hedge Fund 1.

• In only 24 realizations, Hedge Fund 1 did not default.• In 512 realizations, Hedge Fund 1 defaulted but neither Hedge Fund 2 nor the twoBank / Dealers had any qDemand events (i.e., Hedge Fund 1 defaulted quicklybefore the spread of any contagion).

• In 166 realizations, all four entities defaulted.

Taken together, these three sets of experiments with the baseline model represent acomprehensive set of validation tests, which clearly demonstrates our novel approachto dynamic, stress testing.

4.3 Robustness of validation testing

While the validation tests of our novel approach to dynamic stress testing variedonly the size of the exogenous shock (i.e., 0, 10, 15, and 20%), we demonstratethe robustness of these results using a full-factorial design of experiments (Kleijnenet al. 2005). We identified 20,736 different parametric combinations and generated50 realizations of the model for each combination (i.e., 20,736 combinations × 50replicates per combination = 1,036,800 realizations of the model). See Sect. 5 of theSupplementary Information for complete details of these experiments.

Figure 6 depicts approximately 90% of the event sequences from this com-prehensive set of experiments (i.e., 935,104 of the 1,036,800 realizations). Note

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Fig. 7 Event sequence diagram for comprehensive set of experiments. Note that the figure is not exhaustive,with the first decile of event sequences excluded for space considerations. Source: Authors’ analysis, Bigbeeet al. (2015)

that approximately 26% of the time, the price shock did not trigger any events ofinterest.

We developed a filtering capability to allow for the presentation of our EventSequence Diagrams at finer resolutions. While Fig. 7 provides an overview of theEvent Sequences for this comprehensive set of experiments, Fig. 8 demonstrates ourability to filter those results by specific parameter values. For example, in Fig. 8, wefilter by level of exogenous shock for the full factorial sweep of the remaining param-eters. Note that the results are robust and consistent across the sets of experiments.The size of the shock and the dampening of its effects are strongly anti-correlated(i.e., the 10% shock was fully dampened in two or fewer margin calls for 83% ofthe realizations, whereas the 13% shock was fully dampened in two or fewer mar-gin calls for 24% of the realizations and in only 15% of the realizations for the 15%shock).3

3 See the Supplementary Material for additional results from our robustness testing.

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Fig. 8 Event sequence diagrams filtered by level of exogenous shock, for shocks of 10, 13, and 15%. Thefigure is truncated to depict only those event sequences that represent at least 1% of the model realizationsfor those parametric combinations. Source: Authors’ analysis, Bigbee et al. (2015)

4.4 Significance testing

The model contains numerous parameters, so it is important to perform significancetests to verify that the model’s results remain valid as we explore the model’s param-eter space. We examined the parameter space by measuring the contribution of eachparameter to the results for a specified shock. As we can see in Table 3, the major-ity of the variables are statistically significant. The direction of the coefficients forLeverage Target and Max-liquidation are negative—meaning the larger the value, thelarger the size of asset sales would have to be, causing larger decreases in price returnsand capital—and for the Liquidity Ratio it is positive—the higher the value the lessamount of assets would have to be sold. These results confirm that the expected inter-dependencies are robust to variations in shock and risk associated tolerance.

5 Model dynamics and critical constraints

This section addresses the influence of critical variables in stress dynamics, leverage,liquidity, and crowding. We do this by running an experimental panel, where we vary

123


Table3

Significanceof

modelparametersin

contributin

gto

behavior

dynamicswhenstrained

with

a15

percentp

rice

shockto

asset1

.So

urce:A

uthors’analysis

Leveragetarget

Liquidity

ratio

Max-liquidatio

nAllo

catio

nof

firms

Adj.R2

Fstat

Estim

ate

Pr(>

|t|)

Estim

ate

Pr(>

|t|)

Estim

ate

Pr(>

|t|)

Pr(>

|t|)

Price

chan

ge

Asset1

0.03

360.00

12−0

.086

8**

***

*0.01

0634

.4

−0.018

9−0

.001

9−0

.008

8

Asset2

−0.143

3**

*0.01

61**

*−0

.267

6**

***

*0.15

9317

8.93

−0.025

8−0

.002

4−0

.012

Asset3

−0.160

1**

*0.03

56**

*−0

.317

7**

***

*0.28

8824

0.77

−0.026

3−0

.002

7−0

.012

2

Cap

ital

chan

ge

HF1/B

D1

−0.092

*0.00

01−0

.007

6**

***

*0.00

0811

.89

−0.031

1−0

.003

70.00

15

HF2/B

D2

0.10

2*

0.00

24−0

.000

7**

*0.01

890.2

−0.040

4−0

.003

1−0

.000

1

#of

forced

sales

HF1/B

D1

0.18

45**

−0.010

8**

0.48

32**

***

*0.04

436

.63

−0.021

2−0

.003

9−0

.012

4

HF2/B

D2

0.19

75**

−0.010

8**

0.51

62**

*0.01

43.2

−0.021

1−0

.003

9−0

.012

5

The

italicized

values

areheadersthatdescribe

thevariables(normalfont

text)in

theleftmostcolum

n∗ 9

0%;∗

∗ 95%

;∗∗∗

99%

significance

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Fig. 9 The impact of leverage on hedge fund portfolio values and asset prices. Source: Authors’ modeland analysis

each of these parameters and examine how they cause interactions among marketparticipant portfolios and asset values.

5.1 Leverage

Studies of the role of leverage in fire sale dynamics have generally focused on assetleverage and the influence it exerts through forced liquidation. To illustrate the effect ofleverage in our model, we apply a price shock of 10% to Asset 1 to the base model andmeasure asset prices and the capital of the hedge funds as the permissible maximumleverage is varied. Figure 9a, b show the effect that varying leverage has on the capitalof the agents and on prices.

The first-order effect of leverage to A1, the asset that is being shocked, and to HF1,which holds the asset, are generally well understood and predicable. However, thereis a second-order effect because HF2 shares asset exposure with HF1: Both HF1 andHF2 have exposure to A2. Leverage combined with overlapping portfolios is a sourceof contagion, as one agent that is under pressure in one asset starts to liquidate otherholdings (Caccioli et al. 2012). Figure 8 shows the expected effect of contagion witha shock having a great impact on the agents as leverage is increased. Of note, though,is that Fig. 8 shows that the effect on contagion occurs at a threshold level, with thethreshold lower for the asset that is held in common across the agents, and a higherthreshold for Asset 3, which is not held by the agent that is initially under pressure.

As would be expected, an increase of leverage reduces the capital of both hedgefunds and does so at an increasing rate. The effect is more rapid and severe for HF1because it has exposure to the shocked asset, A1. Contagion from HF1 and HF2 isimmediately evident, because the two HFs share A2. The drop in capital acceleratesonce the leverage increases toward 10; on occasion there will be forced selling withlower leverage, but for the baseline parameters, it is at this point where forced sellingstarts to occur with higher frequency and severity.

Of course, if there is contagion between the two hedge funds, there will also becontagion between the assets. A1 drops immediately based on the shock, but it is onlywhen the leverage approaches 10 and there is an increasing frequency of forced sellingthat the other two assets are affected by the price shock. The contagion first affects

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Fig. 10 The cross-market impact of asset liquidity on hedge fund portfolio values and asset price. Source:Authors’ analysis

A2, because it is shared between the two hedge funds, and then as leverage increaseseven further, the contagionmoves to A3. This is a case of “collateral damage,” becauseA3 is not even in HF1’s portfolio. For very high leverage, it is A2, not A1, where wefind the greatest price drop. Even though A1 started the process off with a 10% priceshock, A2 is more widely held and becomes more embroiled in the forced selling. Ifa third hedge fund were exclusively in A2, it ultimately could face a greater impacton its capital than HF1. This resembles the path of contagion during the Long-TermCapital Management failure, when the company had little exposure to the source ofthe initial shock, Russian debt, but was highly leveraged to other assets held by thosewho did.4 These results align with previous studies which identified the existence ofa positive feedback loop between institutional levels of leverage and asset prices thatcan amplify the effects of exogenous shocks (e.g., Office of Financial Research 2012;Tasca and Battiston 2016; Sato and Tasca 2015).

5.2 Liquidity

Liquidity concerns come in two forms, asset and funding. During periods of stress, thecost of accessing markets without incurring steep price reductions or funding haircutincreases becomes difficult. Asset liquidity is critical to fire sale behavior because ofthe price impact caused by sizable selling during the event. We model asset liquiditythrough βm , which measures the market impact of forced sale events.

In Fig. 10, we show the effect of variations in asset liquidity on capital and prices inthe face of a 10% shock to A1. Lower liquidity has the same effect as higher leverage.The shock has a larger impact on the capital of both hedge funds, as well as on theaverage decline of prices, even showing that the impact of A2 is the greatest whenthere is large erosion in liquidity.

Funding liquidity is the ability of a bank/dealer’s finance desk or prime broker toreplace external funding with cash equivalents in the event of a drop in funding. Wemodel the funding liquidity by the liquidity ratio, which is the ratio of liquid (cash-equivalent) assets to short-term, nondurable funding. The liquidity ratio has a role

4 See Summers et al. (1999) and Bookstaber (2007) for a first-hand account of one path of the contagionover the course of the Russian default to the failure of Long-Term Capital Management.

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Fig. 11 The impact of market liquidity constraints and liquidity ratios on prices and bank/dealer portfolios.Source: Authors’ analysis

similar to the leverage ratio, in the sense that there is a threshold for the liquidity ratio,where a bank/dealer is forced to liquidate its inventory so as to address the firm’screditworthiness. Note that this is consistent with the stylized dynamics of cyclesdepicted in Fig. 1a.

As the bank/dealer’s liquidity ratio drops below a targeted value, it must use partof its liquidity reserves to finance assets that it cannot liquidate as a result of themax-liquidation constraint. The bank/dealers can then attempt to liquidate the restof its assets in the following periods. Figure 11 presents a simple case of the effectof variations in the liquidity ratio, where the bank/dealers have the same liquidityratio for illustrative purposes. The plot to the left shows the capital of BD1 and theplot to the right shows the price of A1 for various values of its liquidity ratio and itsmax-liquidation constraint.

5.3 Crowding

Crowding is a loose term for a wide set of agents heavily invested in the same assets.It is a way of thinking of the dialing up or down of the extent of overlap in portfolios.The effect of crowding was manifest in the “Quant Quake” of 2007, when a number ofleveraged hedge funds using the same quantitative strategy were forced to exit similarpositions at the same time (Khandani and Lo 2011). Crowding in a trade can increasethe number of forced sellers and also attract attention from strategic sellers that enterthe market in anticipation of the effect of crowding (Stein 2009; Brunnermeier andPedersen 2009). This was famously explained by a principal at Long-Term CapitalManagement shortly after its failure (Lowenstein 2000, pp. 156–157): “The hurricaneis not more or less likely to hit because more hurricane insurance has been written. Inthe financial markets this is not true. The more people write financial insurance, themore likely it is that a disaster will happen, because the people who know you havesold the insurance can make that disaster happen.”

As expected, the effects of shock on the agents’ capital and on price will be greater,the greater the amount of crowding. We can see in Fig. 12 the effect of crowdedtrades by varying the asset allocation from this benchmark for a given pricing shock,

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Fig. 12 The effect of crowded trades due to varying asset allocations during a pricing shock. Source:Authors’ model

and comparing it to the results shown above, which are based on an equal allocationbetween A1 and A2 for HF1. In Fig. 11 we have HF2 hold 50% of its allocation in A2and see the effect of varying the proportion of the remaining 50% allocation held inA1 versus A3. In this analysis, we have A2 suffer the shock. As would be expected,a high allocation in A1 versus A3 leads to a larger price effect for A1 and A2, anda larger drop in capital for HF1 and HF2. However, as HF2 transfers its allocationfrom A3 to A1, the sensitivity to the allocations of HF2 affects both firms, though bydifferent amounts.

The greater the overall concentration in the shocked asset, the greater the effect willbe to those who are holding that asset. A higher allocation in A1 versus A3 leads to alarger price effect for A1 and a larger drop in capital for HF1.

6 Evaluation of regulatory risk measures under crisis

There are a wide range of proposed measures to address market and financial systemrisk. Only a small subset has been implemented by regulators. In this section, we assesskey regulatory risk measures used to impose risk constraints on the largest U.S. banksin light of the dynamics revealed by the agent-based model during fire sale events.These measures are used for three types of risk constraints: capital, stress, and fundingliquidity.

6.1 Capital constraints

The principal regulatory capital constraints from the Basel Committee on BankingSupervision (2013) take two forms. One is a leverage constraint; the second is aconstraint on risk-weighted assets, which commonly is known as a Value-at-Risk(VaR) measure. An immediate application of the agent-based model is to produce aVaR-like view of the risk of the financial entities when the dynamics of the system areconsidered.

VaR is computed by calculating the returns of the current portfolio over a pastperiod and then computing the standard deviation of those returns. The VaR measureis typically then taken to be the two- or three-standard deviations of returns. Of course,the relevance of this risk measure depends on how well the distributions of futurereturns are reflected by the returns used in the sample.

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Fig. 13 Capital of hedge fund 1 (HF1) after a 15% shock to asset 1 for 1000 runs of the simulation. Thesolid red line is the mean of the paths, the dotted lines are the 5th and 95th percentiles. Source: Authors’model

Figure 13 presents the results of introducing a one-time shock to one of the assetsheld by a hedge fund, showing the continuing downward path for the fund’s capitaldue to the subsequent contagion and cascade. Figure 13a shows the envelope of pathsfor the hedge fund’s capital over 1000 simulations. The solid red line is the averageacross the paths, and the dotted lines are the 5th and 95th percentiles. The distributionmanifests a marked skew. Figure 13b shows the distribution of changes in capital overtime. The skew is also apparent in this figure, as is the fat tail for the drop in capital; thefirst percentile pulls away from the fifth percentile during the period of severe drop.

We can express the results with a VaR measure, but it differs from the traditionalVaR in several respects. First, the conventional VaR measure is symmetric but here,unsurprisingly, there is asymmetry. Second, the time period matters; the variability ofthe paths is high during the early periods, but over a longer time the effect dissipates.And, notably, these results suggest a two-tiered response to the shock. While mostof the realizations of the model depict a capital loss of 30% or less, a distinct groupshows much greater losses in post-shock capital. This highlights a complexity to thedistribution in the extreme tail.

6.2 Stress testing

The failure of regulators to anticipate the events of 2007–2009 led to supervisory riskassessments adding stress testing of the largest U.S. financial institutions. The FederalReserve’s Supervisory Capital Assessment Program (SCAP) in 2009 evolved intothe Comprehensive Capital Analysis and Review (CCAR) in 2011 (Federal ReserveBoard 2013a, b). CCAR has since been combined with the Dodd-Frank ConsumerProtection and Wall Street Reform Act stress testing (DFAST) requirements as a toolfor U.S. bank supervision. On the international front, the Basel Committee on BankingSupervision (2013) lays out principles for stress testing.

The stress tests estimate the effect of specified shocks on the loans and assets heldby the banks. But, as pointed out in Bookstaber et al. (2013), the tests do not considerpossible second-round impacts, such as idiosyncratic increases in bank funding costsrelated to deterioration in capital adequacy or declines in liquidity. Once our modelhas shown the credit and trading losses at a bank, we must now ask questions such

123


as: What happens next? What are second-round effects of one bank’s stress on otherbanks? What impacts does a stress event have on other parts of the financial system?How do those events, in turn, alter the behavior of a bank? These questions relate to thedynamics of the process, the interconnections, and the health of the transformationswithin the financial system.

Table 4 illustrates how our agent-based model can be used to provide insight intothese subsequent trajectories by showing the dynamic impact of the stress to the entiresystem, and thereby serve as a tool for extending stress tests to capture the overalleffect of the stress scenarios. The table shows the progression of one simulation runof the agent-based model. Each period in the progression is depicted by a networkshowing which agents (nodes) influence other agents. The networks are depicted asan output of the model, and the network structure changes period-by-period as theenvironment changes due to the agents’ actions and as the agents adapt accordingly.For this example, the shock does not have far to go before it embroils the system; itreverberates through the system, demonstrating the contagion and cascades typical ofthe fire sales that this model seeks to address before running its course in six periods.5

In Table 4, the dark outline for the nodes shows the agents’ initial size (in this casewe assume all of the agents have the same starting capital and all initial prices areidentical), The shrinking of the colored area within the nodes is proportional to thedecline in capital in the case of the hedge funds and bank/dealers, the reduction infunding for the cash provider, and the drop in prices in the case of the assets. If thecolor within a node disappears, that agent has defaulted.6

6.3 Funding liquidity coverage constraints

The third area of regulatory focus for establishing risk constraints is in short-termfunding liquidity, the ability of a bank to fund short-term liabilities with liquid assets.The primary risk measure for this is the liquidity coverage ratio (LCR) as describedin Basel Committee on Banking Supervision (2013). We can evaluate the dynamicimplications of shocks to short-term funding liquidity using the same methods as wehave above for stress testing in two ways. First, a given shock will propagate throughfunding and credit channels to affect the funding liquidity and the LCR, and this willbe seen in the simulation. Second, we can impose a shock to the funding liquidity,and see how that initial shock creates a cascade for funding liquidity, as well as howit moves out to affect the markets and the agents.

As depicted in Table 5, the agent-based modeling framework can posit a rangeof shock originations: price shocks, as have been employed above, but also shocksto funding through the cash provider, to redemptions through the hedge funds, and

5 The parameter values used in this section are the same as used in Sect. 4.6 Each edge in the network denotes the relational impact of one node, i, on another, j, based on therelationship that exists in the agent-based model, normalized by running the simulation a number of timeswith variations on each variable. The width of the edge shows the cumulative effect of the transmissionwith respect to t periods and the n runs of the simulation, and the color of the edge in the figure shows theintensity of the interaction in the current period; a darker color means greater intensity or change in thesystem relative to other runs and periods observed.

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Table 4 A multi-period illustration of interaction between agents starting with a shock to asset 1 (A1),then providing snapshots for periods 2, 4, and 6. Source: Authors model

Network graphof basemodel

Description of events

The market is in an equilibrium statewhere the BD1 and HF1 hold A1,they are directly affected by theshock. CP1 is also affected becausethe value of collateral declines. In astatic stress test, the analysis endsat this point

BD1 and HF1 decrease theirpositions in both A1 and A2, BD1and HF1 decrease their positions inboth A1 and A2, resulting in a dropin A2. This in turn affects otheragents with holdings in A2, inparticular, BD2 and HF2. CP1 isaffected because it holds collateralin A2 as well as in A1

The Propagation from the shockleads to a default od HF1 and BD1.Credit exposure that BD2 has toBD1 spreads problems through thecredit channel. The drop in A2affects HF2, and its forced salesspread the shock to A3. Note thatno entities holding the shockedasset also hold A3. CP1 markedlyreduces its funding due to the dropin the value of its collateral

The system finally settels down withfunding all but shut off, and bothhedge funds and BD1 in default.A2 ultimately has a greater pricedrop than A1, the shocked asset

The thickness of the edges indicates the cumulative effect on, and the coloring within the agent symbolsindicates the total value of price in the case of assets, capital in the case of the bank/dealers (BD) and hedgefunds (HF), and funding in the case of the cash provider (CP). The darker the coloring of the edge, the morerecent the effect

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Table 5 Examples of the initial effect of types of shocks. Source: Authors’ analysis

Price shocks initially affect hedge funds and bank/dealers holding the asset, as well as the cash providerholding the asset as collateral. A funding shock passes from the cash provider through the bank/dealers tothe final funding users; a credit shock pass from the agent under credit stress to counterparties and providersof funding; and a redemption shock affects the assets held by the agent facing redemptions

funding liquidity and related credit effects through the bank/dealers. For all of these,we can show the propagation of the initial shock.

7 Conclusion

This paper contributes an agent-basedmodel that gives a broad view of the transforma-tions and dynamic interactions in the financial system. Our model provides an avenuetoward highlighting and monitoring key crisis dynamics such as fire sales and fundingruns. Using a map of funding and collateral flows, the model links these flows to assetmarkets, providing a structure for examining the effect of the individual firms’ actionson each other.

This study integrates several related literatures into a multi-agent framework thatincorporates the major market participants in the U.S. asset and funding markets. Byextending the Cifuentes et al. (2005) model and integrating the behavior of marketparticipants on both the funding and borrowing sides, we demonstrate how stresseson these firms can cause pricing consequences and feedback effects which are notGaussian distributed. By comparing the results of our model to various traditional riskmeasures, we show the limitation of these measures in capturing the outcomes causedby vulnerabilities in the financial system.

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This study demonstrates the importance of capturing various interconnected,network-like relationships among themarket functions, whichwhen stressed can causereverberations across the financial system. Previous studies have shown that feedbackcycles are integral to explaining how the credit crises in 2008 unfolded (Brunnermeierand Pedersen 2009; Tasca and Battiston 2016). By integrating market asset price cor-relations through overlapping portfolios with funding networks, our model generatesthese same feedback cycles.

While the model presented in this paper is more stylized than required for centralbank stress testing, it provides an existence proof of a prototype for macropruden-tial, stress testing. As such, this study represents a critical step toward creating anoperational model for integrating stress testing across firms.

8 Appendix

See Tables 6 and 7.

Table 6 Model variable glossary

ϕCW Global parameter that governs the impact of bank/dealer k going belowLiqRatio Min on its CWk

ϕHC Global parameter that governs the impact of bank/dealer k going belowLiqRatio Min on its HCc,k

An Assets held by bank/dealer or hedge fund n

AAllocationn Vector of % of assets m allocation for bank/dealer k or hedge fund n

ATargetn Target quantity of assets held by bank/dealer k or hedge fund n

βm Price elasticity of demand for asset m

C The number of cash provider in the model

CAn Collateral of bank/dealer or hedge fund n

C AHFk Collateral of hedge funds using bank/dealer k

C APBk Collateral of prime broker of bank/dealer k

C AT Dk Collateral of trading desk of bank/dealer k

Capk ,Capn Capital of bank/dealer k or hedge fund n

CEK−1 Credit exposure of bank/dealer k to another bank/dealer K − 1

CETotalk Credit exposure of bank/dealer k to all the other bank/dealers

CWk Creditworthiness of bank/dealer k

EDSn Funding driven sales of bank/dealer k or hedge fund n

Fn Funding to bank/dealer k or hedge fund n

F PBk Funding to prime broker of bank/dealer k

FT Dk Funding to trading desk of bank/dealer k

HCc,k The haircut cash provider c give to bank/dealer k

K The number of bank/dealer in the model

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Table 6 continued

ϕCW Global parameter that governs the impact of bank/dealer k going belowLiqRatioMin on its CWk

Lc,k Loan cash provider c gives to bank/dealer k

LMaxc,k Loan maximum of cash provider c for bank/dealer k

LTargetc,k Loan target of cash provider c for bank/dealer k

LevBu f f ern Leverage buffer of bank/dealer k or hedge fund n

LevCurrentn Current leverage of bank/dealer k or hedge fund n

LevMaxn Leverage maximum of bank/dealer k or hedge fund n

LevTargetn Leverage target of bank/dealer k or hedge fund n

LevBu f f er Raten Percent of LevMaxn that bank/dealer or hedge fund n sets LevBu f f ern

LevTarget Raten Percent of LevBu f f ern that bank/dealer or hedge fund n sets LevTargetn

Liqdebit Liquidity reserve debit

LiqRk Liquidity reserve of bank/dealer k

LiqRatek Liquidity reserve rate of bank/dealer k

LiqRatiok Liquidity ratio of bank/dealer k

LiqRatioMink Liquidity ratio minimum of bank/dealer k

LiqRatioTargetk Liquidity target of bank/dealer k

M The number of assets in the model

N The number of hedge funds in the model

Nk The subset of N hedge funds that prime broker of bank/dealer k works

On,m (t) Sum of all normal orders for buying or selling assets by bank/dealer kor hedge fund n

Pm Price of asset m

PLm Previous day’s trading profit/loss accounting

PRm Price return for asset m

PRRm Random price movement which is N (0, σm )

Qn,m Quantity of asset m held by bank/dealer k or hedge fund n

QMaxk Quantity of assets that a bank/dealer k can sell in a single period

t The current period of the model

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Table 7 Model parameters

Variable Value Value derived by

LevTargetn 0.85Adrian and Shin (2014) that leverage targets tend tobe close to the maximum leverage level

HC (0) 0.13 Price return for asset m

LiqRk 0.3 The levels specified for the liquidity coverage ratioin Basel Committee on Banking Supervisions(2013)

βm 10 basis points per $10billion Greenwood et al. (2015) use a market impact of 10

basis points per $10 billion of liquidation

P (0) 100 Authors’ choice

Capk (0) ,Capn (0) 10 Million Authors’ choice

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www.federalreserve.gov/bankinforeg/ccar-2013-results-20130314.pdf

www.federalreserve.gov/bankinforeg/ccar-2013-results-20130314.pdf

www.federalreserve.gov/newsevents/press/bcreg/dfast2013results

www.federalreserve.gov/newsevents/press/bcreg/dfast2013results

http://arxiv.org/abs/1504.07152

an agent-based model for financial vulnerability

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